idx int64 1 56k | question stringlengths 15 155 | answer stringlengths 2 29.2k ⌀ | question_cut stringlengths 15 100 | answer_cut stringlengths 2 200 ⌀ | conversation stringlengths 47 29.3k | conversation_cut stringlengths 47 301 |
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44,201 | Regressions. Why a and b explains more than a+b? | The reason is flexibility.
Option 1: When you regress $X_1, X_2$ on $Y$ you are allowing the coefficients to be different. In other words, your regression equation is $Y = \alpha + \beta_1X_1 + \beta_2X_2 + \epsilon$. Notice $\beta_1$ may equal $\beta_2$ if it wants to - if that's what the data suggests.
Option 2: When... | Regressions. Why a and b explains more than a+b? | The reason is flexibility.
Option 1: When you regress $X_1, X_2$ on $Y$ you are allowing the coefficients to be different. In other words, your regression equation is $Y = \alpha + \beta_1X_1 + \beta_ | Regressions. Why a and b explains more than a+b?
The reason is flexibility.
Option 1: When you regress $X_1, X_2$ on $Y$ you are allowing the coefficients to be different. In other words, your regression equation is $Y = \alpha + \beta_1X_1 + \beta_2X_2 + \epsilon$. Notice $\beta_1$ may equal $\beta_2$ if it wants to -... | Regressions. Why a and b explains more than a+b?
The reason is flexibility.
Option 1: When you regress $X_1, X_2$ on $Y$ you are allowing the coefficients to be different. In other words, your regression equation is $Y = \alpha + \beta_1X_1 + \beta_ |
44,202 | How to organise the variable names in R without messing up? [closed] | Use simple phrases
Rather than relying on very generic variable names like x1, x2, y1, y2, develop a very simple variable name system:
Common variables: Some variables are so ubiquitous that it's just easier to name them as is: male, edu, income or inc, age, etc. As time goes on the naming of this variable will be near... | How to organise the variable names in R without messing up? [closed] | Use simple phrases
Rather than relying on very generic variable names like x1, x2, y1, y2, develop a very simple variable name system:
Common variables: Some variables are so ubiquitous that it's just | How to organise the variable names in R without messing up? [closed]
Use simple phrases
Rather than relying on very generic variable names like x1, x2, y1, y2, develop a very simple variable name system:
Common variables: Some variables are so ubiquitous that it's just easier to name them as is: male, edu, income or in... | How to organise the variable names in R without messing up? [closed]
Use simple phrases
Rather than relying on very generic variable names like x1, x2, y1, y2, develop a very simple variable name system:
Common variables: Some variables are so ubiquitous that it's just |
44,203 | How to organise the variable names in R without messing up? [closed] | Got too long for a comment, so I guess it's an answer now.
Use long, descriptive names for anything you need to keep around for more than a couple of minutes.
The long names might be a nuisance to type, but if you're working by running scripts, writing functions and so on, the actual typing is not that much anyway.
B... | How to organise the variable names in R without messing up? [closed] | Got too long for a comment, so I guess it's an answer now.
Use long, descriptive names for anything you need to keep around for more than a couple of minutes.
The long names might be a nuisance to ty | How to organise the variable names in R without messing up? [closed]
Got too long for a comment, so I guess it's an answer now.
Use long, descriptive names for anything you need to keep around for more than a couple of minutes.
The long names might be a nuisance to type, but if you're working by running scripts, writi... | How to organise the variable names in R without messing up? [closed]
Got too long for a comment, so I guess it's an answer now.
Use long, descriptive names for anything you need to keep around for more than a couple of minutes.
The long names might be a nuisance to ty |
44,204 | How to organise the variable names in R without messing up? [closed] | Just don't use obscure names like x1. It's important, both from a computing as well as an analyzing perspective, to take a minute to look at your variable and understand what it represents. It's name should naturally be what it represents. For example, if it's a vector of ages, why not name it age? If it's a large name... | How to organise the variable names in R without messing up? [closed] | Just don't use obscure names like x1. It's important, both from a computing as well as an analyzing perspective, to take a minute to look at your variable and understand what it represents. It's name | How to organise the variable names in R without messing up? [closed]
Just don't use obscure names like x1. It's important, both from a computing as well as an analyzing perspective, to take a minute to look at your variable and understand what it represents. It's name should naturally be what it represents. For example... | How to organise the variable names in R without messing up? [closed]
Just don't use obscure names like x1. It's important, both from a computing as well as an analyzing perspective, to take a minute to look at your variable and understand what it represents. It's name |
44,205 | In R, what does a probability density function compute? [duplicate] | Densities are not probabilities.
$$
$$
The density $f$ of a continuous random variable $X$ is defined as
$$
f(x) = F'(x) = \lim_{\epsilon \rightarrow 0} \frac{\Pr\big(x < X \leq x + \epsilon\big)}{\epsilon}
$$
with $F$ the cumulative distribution function (cdf).
$$
$$
$f(x)$ can be larger than $1$.
The area under th... | In R, what does a probability density function compute? [duplicate] | Densities are not probabilities.
$$
$$
The density $f$ of a continuous random variable $X$ is defined as
$$
f(x) = F'(x) = \lim_{\epsilon \rightarrow 0} \frac{\Pr\big(x < X \leq x + \epsilon\big)}{\e | In R, what does a probability density function compute? [duplicate]
Densities are not probabilities.
$$
$$
The density $f$ of a continuous random variable $X$ is defined as
$$
f(x) = F'(x) = \lim_{\epsilon \rightarrow 0} \frac{\Pr\big(x < X \leq x + \epsilon\big)}{\epsilon}
$$
with $F$ the cumulative distribution func... | In R, what does a probability density function compute? [duplicate]
Densities are not probabilities.
$$
$$
The density $f$ of a continuous random variable $X$ is defined as
$$
f(x) = F'(x) = \lim_{\epsilon \rightarrow 0} \frac{\Pr\big(x < X \leq x + \epsilon\big)}{\e |
44,206 | In R, what does a probability density function compute? [duplicate] | You shoud distinguish between probability mass and probability density.
A discrete random variable is like a set of little stones: each stone has its own mass (weight). For example, if you toss a regular coin the mass of the "head-stone" is 1/2. The probability "density" function of a discrete variable is actually a pr... | In R, what does a probability density function compute? [duplicate] | You shoud distinguish between probability mass and probability density.
A discrete random variable is like a set of little stones: each stone has its own mass (weight). For example, if you toss a regu | In R, what does a probability density function compute? [duplicate]
You shoud distinguish between probability mass and probability density.
A discrete random variable is like a set of little stones: each stone has its own mass (weight). For example, if you toss a regular coin the mass of the "head-stone" is 1/2. The pr... | In R, what does a probability density function compute? [duplicate]
You shoud distinguish between probability mass and probability density.
A discrete random variable is like a set of little stones: each stone has its own mass (weight). For example, if you toss a regu |
44,207 | Is it possible to convert a Rayleigh distribution into a Gaussian distribution? | If you know the Rayleigh parameter, then the conversion to a standard normal is readily achieved by the probability integral transform followed by an inverse normal cdf. If $X\sim\text{Rayleigh}(\sigma)$, with cdf $F_\sigma(x)$, then $F_\sigma (X)$ is uniform, and $\Phi^{-1}(F_\sigma (X))$ is standard normal (where $\P... | Is it possible to convert a Rayleigh distribution into a Gaussian distribution? | If you know the Rayleigh parameter, then the conversion to a standard normal is readily achieved by the probability integral transform followed by an inverse normal cdf. If $X\sim\text{Rayleigh}(\sigm | Is it possible to convert a Rayleigh distribution into a Gaussian distribution?
If you know the Rayleigh parameter, then the conversion to a standard normal is readily achieved by the probability integral transform followed by an inverse normal cdf. If $X\sim\text{Rayleigh}(\sigma)$, with cdf $F_\sigma(x)$, then $F_\si... | Is it possible to convert a Rayleigh distribution into a Gaussian distribution?
If you know the Rayleigh parameter, then the conversion to a standard normal is readily achieved by the probability integral transform followed by an inverse normal cdf. If $X\sim\text{Rayleigh}(\sigm |
44,208 | Is it possible to convert a Rayleigh distribution into a Gaussian distribution? | If $R$ is a Rayleigh random variable and $\Theta \sim U[0,2\pi)$ is independent of $R$, then $X=R\cos \Theta$ and $Y=R \sin \Theta$ are independent
zero-mean normal random variables with identical variance
$\sigma^2 = \frac{1}{2}E[R^2]$. Thus, if you transform your data set as
$$\{r_1, r_2, \ldots, r_n\} \longrightar... | Is it possible to convert a Rayleigh distribution into a Gaussian distribution? | If $R$ is a Rayleigh random variable and $\Theta \sim U[0,2\pi)$ is independent of $R$, then $X=R\cos \Theta$ and $Y=R \sin \Theta$ are independent
zero-mean normal random variables with identical var | Is it possible to convert a Rayleigh distribution into a Gaussian distribution?
If $R$ is a Rayleigh random variable and $\Theta \sim U[0,2\pi)$ is independent of $R$, then $X=R\cos \Theta$ and $Y=R \sin \Theta$ are independent
zero-mean normal random variables with identical variance
$\sigma^2 = \frac{1}{2}E[R^2]$. ... | Is it possible to convert a Rayleigh distribution into a Gaussian distribution?
If $R$ is a Rayleigh random variable and $\Theta \sim U[0,2\pi)$ is independent of $R$, then $X=R\cos \Theta$ and $Y=R \sin \Theta$ are independent
zero-mean normal random variables with identical var |
44,209 | Can you do statistics with 4 data points? | I have a friend who used to work for the US defense department (long time ago, cold war era) and was once asked to answer a question using a single data point. When he insisted that he needed more data he was told that the person who had provided the single data point had been caught and executed for espionage shortly... | Can you do statistics with 4 data points? | I have a friend who used to work for the US defense department (long time ago, cold war era) and was once asked to answer a question using a single data point. When he insisted that he needed more da | Can you do statistics with 4 data points?
I have a friend who used to work for the US defense department (long time ago, cold war era) and was once asked to answer a question using a single data point. When he insisted that he needed more data he was told that the person who had provided the single data point had been... | Can you do statistics with 4 data points?
I have a friend who used to work for the US defense department (long time ago, cold war era) and was once asked to answer a question using a single data point. When he insisted that he needed more da |
44,210 | Can you do statistics with 4 data points? | Short answer: yes, but your results will usually be useless.
Long answer: Statistics often involves forming some kind of inference about underlying parameters based on data, with constraints on the probability of a False-Positive and/or of a False-Negative. In a typical test, i.e. testing if a sample came from a given ... | Can you do statistics with 4 data points? | Short answer: yes, but your results will usually be useless.
Long answer: Statistics often involves forming some kind of inference about underlying parameters based on data, with constraints on the pr | Can you do statistics with 4 data points?
Short answer: yes, but your results will usually be useless.
Long answer: Statistics often involves forming some kind of inference about underlying parameters based on data, with constraints on the probability of a False-Positive and/or of a False-Negative. In a typical test, i... | Can you do statistics with 4 data points?
Short answer: yes, but your results will usually be useless.
Long answer: Statistics often involves forming some kind of inference about underlying parameters based on data, with constraints on the pr |
44,211 | Can you do statistics with 4 data points? | I helped with a geological project where the researchers had a single data point, accompanied by a very reliable uncertainty bound. They were interested in testing a geological model (a set of differential equations describing the evolution of tectonic plates) which made a very specific prediction for the value of that... | Can you do statistics with 4 data points? | I helped with a geological project where the researchers had a single data point, accompanied by a very reliable uncertainty bound. They were interested in testing a geological model (a set of differe | Can you do statistics with 4 data points?
I helped with a geological project where the researchers had a single data point, accompanied by a very reliable uncertainty bound. They were interested in testing a geological model (a set of differential equations describing the evolution of tectonic plates) which made a very... | Can you do statistics with 4 data points?
I helped with a geological project where the researchers had a single data point, accompanied by a very reliable uncertainty bound. They were interested in testing a geological model (a set of differe |
44,212 | Why is Hedonic Regression used instead of Linear Regression | There is no package for hedonic regression because it isn't a specific kind of regression but a specific application of regression. Linear regression is used. It's called hedonic regression to highlight the method of price estimating and interpretation.
You might want to check this question out for more details on mo... | Why is Hedonic Regression used instead of Linear Regression | There is no package for hedonic regression because it isn't a specific kind of regression but a specific application of regression. Linear regression is used. It's called hedonic regression to highl | Why is Hedonic Regression used instead of Linear Regression
There is no package for hedonic regression because it isn't a specific kind of regression but a specific application of regression. Linear regression is used. It's called hedonic regression to highlight the method of price estimating and interpretation.
You ... | Why is Hedonic Regression used instead of Linear Regression
There is no package for hedonic regression because it isn't a specific kind of regression but a specific application of regression. Linear regression is used. It's called hedonic regression to highl |
44,213 | Why is Hedonic Regression used instead of Linear Regression | It's called hedonic regression because it is used to remove 'hedonic' product features (i.e. features that consumers 'like' or get pleasure from), from the price. The hypothesis is that remaining price changes from period to period are caused by inflation.
It is useful where the same product is not sold from period to... | Why is Hedonic Regression used instead of Linear Regression | It's called hedonic regression because it is used to remove 'hedonic' product features (i.e. features that consumers 'like' or get pleasure from), from the price. The hypothesis is that remaining pri | Why is Hedonic Regression used instead of Linear Regression
It's called hedonic regression because it is used to remove 'hedonic' product features (i.e. features that consumers 'like' or get pleasure from), from the price. The hypothesis is that remaining price changes from period to period are caused by inflation.
It... | Why is Hedonic Regression used instead of Linear Regression
It's called hedonic regression because it is used to remove 'hedonic' product features (i.e. features that consumers 'like' or get pleasure from), from the price. The hypothesis is that remaining pri |
44,214 | What have to be normally distributed: groups or whole sample? | Generally it is the residuals that need to be normally distributed. This implies that each group is normally distributed, but you can do the diagnostics on the residuals (values minus group mean) as a whole rather than group by group. It is possible (and even common) that the data will be approximately normal within ... | What have to be normally distributed: groups or whole sample? | Generally it is the residuals that need to be normally distributed. This implies that each group is normally distributed, but you can do the diagnostics on the residuals (values minus group mean) as | What have to be normally distributed: groups or whole sample?
Generally it is the residuals that need to be normally distributed. This implies that each group is normally distributed, but you can do the diagnostics on the residuals (values minus group mean) as a whole rather than group by group. It is possible (and e... | What have to be normally distributed: groups or whole sample?
Generally it is the residuals that need to be normally distributed. This implies that each group is normally distributed, but you can do the diagnostics on the residuals (values minus group mean) as |
44,215 | What have to be normally distributed: groups or whole sample? | @GregSnow has a really excellent answer here, which I would like to augment with 2 small points. First, while he's right that only the normality of the residuals matters, we can ask the question 'matters for what'? Everyone who has taken stats 101 leaves with the impression that normality is crucial, and that's not r... | What have to be normally distributed: groups or whole sample? | @GregSnow has a really excellent answer here, which I would like to augment with 2 small points. First, while he's right that only the normality of the residuals matters, we can ask the question 'mat | What have to be normally distributed: groups or whole sample?
@GregSnow has a really excellent answer here, which I would like to augment with 2 small points. First, while he's right that only the normality of the residuals matters, we can ask the question 'matters for what'? Everyone who has taken stats 101 leaves w... | What have to be normally distributed: groups or whole sample?
@GregSnow has a really excellent answer here, which I would like to augment with 2 small points. First, while he's right that only the normality of the residuals matters, we can ask the question 'mat |
44,216 | A mathematical formula for K-fold cross-validation prediction error? | There are formulae for computing the leave-one-out cross-validation error in closed form for many models, including least-squares regression, but as far as I am aware there isn't a general formula for k-fold cross-validation (or at least it may be possible but the computational advantage is too small to be worthwhile).... | A mathematical formula for K-fold cross-validation prediction error? | There are formulae for computing the leave-one-out cross-validation error in closed form for many models, including least-squares regression, but as far as I am aware there isn't a general formula for | A mathematical formula for K-fold cross-validation prediction error?
There are formulae for computing the leave-one-out cross-validation error in closed form for many models, including least-squares regression, but as far as I am aware there isn't a general formula for k-fold cross-validation (or at least it may be pos... | A mathematical formula for K-fold cross-validation prediction error?
There are formulae for computing the leave-one-out cross-validation error in closed form for many models, including least-squares regression, but as far as I am aware there isn't a general formula for |
44,217 | A mathematical formula for K-fold cross-validation prediction error? | The reason people do cross-validation is that there is no mathematical formula to accurately get at the same thing except under very restrictive conditions. And note that k-fold cross-validation does not have adequate precision in most cases, so you have to repeat k-fold cross-validation often 50-100 times (and averag... | A mathematical formula for K-fold cross-validation prediction error? | The reason people do cross-validation is that there is no mathematical formula to accurately get at the same thing except under very restrictive conditions. And note that k-fold cross-validation does | A mathematical formula for K-fold cross-validation prediction error?
The reason people do cross-validation is that there is no mathematical formula to accurately get at the same thing except under very restrictive conditions. And note that k-fold cross-validation does not have adequate precision in most cases, so you ... | A mathematical formula for K-fold cross-validation prediction error?
The reason people do cross-validation is that there is no mathematical formula to accurately get at the same thing except under very restrictive conditions. And note that k-fold cross-validation does |
44,218 | A mathematical formula for K-fold cross-validation prediction error? | The truth is that cross-validation is simply a heuristic for model selection.
If what you are really looking for is obtaining a theoretically-backed estimate of your generalization prediction, cross-validation can only give a good estimate of it but there are no guarantees. A better fit for that would be learning theo... | A mathematical formula for K-fold cross-validation prediction error? | The truth is that cross-validation is simply a heuristic for model selection.
If what you are really looking for is obtaining a theoretically-backed estimate of your generalization prediction, cross- | A mathematical formula for K-fold cross-validation prediction error?
The truth is that cross-validation is simply a heuristic for model selection.
If what you are really looking for is obtaining a theoretically-backed estimate of your generalization prediction, cross-validation can only give a good estimate of it but ... | A mathematical formula for K-fold cross-validation prediction error?
The truth is that cross-validation is simply a heuristic for model selection.
If what you are really looking for is obtaining a theoretically-backed estimate of your generalization prediction, cross- |
44,219 | A mathematical formula for K-fold cross-validation prediction error? | I think that what the person was asking the question needs is simply a formula that is more explanatory, or a full-blown explanation of the formula. I'm posting this for the sake of others looking for an answer.
Here is how I understand it.
Start with a less abstract loss function, say the MSE.
Once you divide your d... | A mathematical formula for K-fold cross-validation prediction error? | I think that what the person was asking the question needs is simply a formula that is more explanatory, or a full-blown explanation of the formula. I'm posting this for the sake of others looking for | A mathematical formula for K-fold cross-validation prediction error?
I think that what the person was asking the question needs is simply a formula that is more explanatory, or a full-blown explanation of the formula. I'm posting this for the sake of others looking for an answer.
Here is how I understand it.
Start wit... | A mathematical formula for K-fold cross-validation prediction error?
I think that what the person was asking the question needs is simply a formula that is more explanatory, or a full-blown explanation of the formula. I'm posting this for the sake of others looking for |
44,220 | What is the distribution of the Binomial distribution parameter $p$ given a sample k and n? | From a bayesian point of view the distribution of p with k empirical successes and n trials is the Beta-Distribution, in detail $p\sim Beta(\alpha,\beta)$ with $\alpha=k+1$ and $\beta=n-k+1$. It represents the unnormalized density $prob(p|data)$, i.e. the unormalized probability that the unknown parameter is $p$ given ... | What is the distribution of the Binomial distribution parameter $p$ given a sample k and n? | From a bayesian point of view the distribution of p with k empirical successes and n trials is the Beta-Distribution, in detail $p\sim Beta(\alpha,\beta)$ with $\alpha=k+1$ and $\beta=n-k+1$. It repre | What is the distribution of the Binomial distribution parameter $p$ given a sample k and n?
From a bayesian point of view the distribution of p with k empirical successes and n trials is the Beta-Distribution, in detail $p\sim Beta(\alpha,\beta)$ with $\alpha=k+1$ and $\beta=n-k+1$. It represents the unnormalized densi... | What is the distribution of the Binomial distribution parameter $p$ given a sample k and n?
From a bayesian point of view the distribution of p with k empirical successes and n trials is the Beta-Distribution, in detail $p\sim Beta(\alpha,\beta)$ with $\alpha=k+1$ and $\beta=n-k+1$. It repre |
44,221 | What is the distribution of the Binomial distribution parameter $p$ given a sample k and n? | The sample proportion $\hat{p}=k/n$ has a scaled Binomial distribution. That is $k\sim\text{Binomial}(n,p)$ which is scaled by the sample size $n$. I don't think it has any other name. | What is the distribution of the Binomial distribution parameter $p$ given a sample k and n? | The sample proportion $\hat{p}=k/n$ has a scaled Binomial distribution. That is $k\sim\text{Binomial}(n,p)$ which is scaled by the sample size $n$. I don't think it has any other name. | What is the distribution of the Binomial distribution parameter $p$ given a sample k and n?
The sample proportion $\hat{p}=k/n$ has a scaled Binomial distribution. That is $k\sim\text{Binomial}(n,p)$ which is scaled by the sample size $n$. I don't think it has any other name. | What is the distribution of the Binomial distribution parameter $p$ given a sample k and n?
The sample proportion $\hat{p}=k/n$ has a scaled Binomial distribution. That is $k\sim\text{Binomial}(n,p)$ which is scaled by the sample size $n$. I don't think it has any other name. |
44,222 | Uncorrected pairwise p-values for one-way ANOVA? | For the multcomp package, see the help page for glht; you want to use the "Tukey" option; this does not actually use the Tukey correction, it just sets up all pairwise comparisons. In the example section there's an example that does exactly what you want.
This calculates the estimates and se's for each comparison but ... | Uncorrected pairwise p-values for one-way ANOVA? | For the multcomp package, see the help page for glht; you want to use the "Tukey" option; this does not actually use the Tukey correction, it just sets up all pairwise comparisons. In the example sec | Uncorrected pairwise p-values for one-way ANOVA?
For the multcomp package, see the help page for glht; you want to use the "Tukey" option; this does not actually use the Tukey correction, it just sets up all pairwise comparisons. In the example section there's an example that does exactly what you want.
This calculate... | Uncorrected pairwise p-values for one-way ANOVA?
For the multcomp package, see the help page for glht; you want to use the "Tukey" option; this does not actually use the Tukey correction, it just sets up all pairwise comparisons. In the example sec |
44,223 | Uncorrected pairwise p-values for one-way ANOVA? | You can use pairwise.t.test() with one of the available options for multiple comparison correction in the p.adjust.method= argument; see help(p.adjust) for more information on the available option for single-step and step-down methods (e.g., BH for FDR or bonf for Bonferroni). Of note, you can directly give p.adjust() ... | Uncorrected pairwise p-values for one-way ANOVA? | You can use pairwise.t.test() with one of the available options for multiple comparison correction in the p.adjust.method= argument; see help(p.adjust) for more information on the available option for | Uncorrected pairwise p-values for one-way ANOVA?
You can use pairwise.t.test() with one of the available options for multiple comparison correction in the p.adjust.method= argument; see help(p.adjust) for more information on the available option for single-step and step-down methods (e.g., BH for FDR or bonf for Bonfer... | Uncorrected pairwise p-values for one-way ANOVA?
You can use pairwise.t.test() with one of the available options for multiple comparison correction in the p.adjust.method= argument; see help(p.adjust) for more information on the available option for |
44,224 | Uncorrected pairwise p-values for one-way ANOVA? | library(multcomp)
df = mtcars
df$am = as.factor(df$am)
m1 <- aov(mpg ~ am, data= df)
ht = glht(m1, linfct = mcp(am = "Tukey"))
summary(ht, test = adjusted("none"))
# Linear Hypotheses:
# Estimate Std. Error t value Pr(>|t|)
# 1 - 0 == 0 7.245 1.764 4.106 0.000285 *** | Uncorrected pairwise p-values for one-way ANOVA? | library(multcomp)
df = mtcars
df$am = as.factor(df$am)
m1 <- aov(mpg ~ am, data= df)
ht = glht(m1, linfct = mcp(am = "Tukey"))
summary(ht, test = adjusted("none"))
# Linear Hypotheses:
# Es | Uncorrected pairwise p-values for one-way ANOVA?
library(multcomp)
df = mtcars
df$am = as.factor(df$am)
m1 <- aov(mpg ~ am, data= df)
ht = glht(m1, linfct = mcp(am = "Tukey"))
summary(ht, test = adjusted("none"))
# Linear Hypotheses:
# Estimate Std. Error t value Pr(>|t|)
# 1 - 0 == 0 7.245 1.764 4... | Uncorrected pairwise p-values for one-way ANOVA?
library(multcomp)
df = mtcars
df$am = as.factor(df$am)
m1 <- aov(mpg ~ am, data= df)
ht = glht(m1, linfct = mcp(am = "Tukey"))
summary(ht, test = adjusted("none"))
# Linear Hypotheses:
# Es |
44,225 | Identify probability distributions | The short answer is that you can't.
The longer answer is that you really need to think about what you are trying to accomplish and what question(s) you are trying to answer.
Tests on distributions are not designed to prove a particular distribution, but to disprove (they are not perfect for that, you still have type I ... | Identify probability distributions | The short answer is that you can't.
The longer answer is that you really need to think about what you are trying to accomplish and what question(s) you are trying to answer.
Tests on distributions are | Identify probability distributions
The short answer is that you can't.
The longer answer is that you really need to think about what you are trying to accomplish and what question(s) you are trying to answer.
Tests on distributions are not designed to prove a particular distribution, but to disprove (they are not perfe... | Identify probability distributions
The short answer is that you can't.
The longer answer is that you really need to think about what you are trying to accomplish and what question(s) you are trying to answer.
Tests on distributions are |
44,226 | Identify probability distributions | The only way to "prove" that data comes from a certain distribution (without an infinite number of samples) is to know precisely how that data is generated. For example, if you know that the data came from the magnitude of a circular bivariate normal random variable, it has a Rician distribution. Or if the data came ... | Identify probability distributions | The only way to "prove" that data comes from a certain distribution (without an infinite number of samples) is to know precisely how that data is generated. For example, if you know that the data cam | Identify probability distributions
The only way to "prove" that data comes from a certain distribution (without an infinite number of samples) is to know precisely how that data is generated. For example, if you know that the data came from the magnitude of a circular bivariate normal random variable, it has a Rician ... | Identify probability distributions
The only way to "prove" that data comes from a certain distribution (without an infinite number of samples) is to know precisely how that data is generated. For example, if you know that the data cam |
44,227 | Identify probability distributions | If you're trying to do Exploratory Data Analysis, you could use some graphical techniques.
I can suggest chapter 1.3.4 in the NIST handbook. In particular, any of the probability graphs might be insightful (e.g. Probability Plot Correlation Coefficient Plot, Quantile-Quantile Plot, etc.).
For a number of common distrib... | Identify probability distributions | If you're trying to do Exploratory Data Analysis, you could use some graphical techniques.
I can suggest chapter 1.3.4 in the NIST handbook. In particular, any of the probability graphs might be insig | Identify probability distributions
If you're trying to do Exploratory Data Analysis, you could use some graphical techniques.
I can suggest chapter 1.3.4 in the NIST handbook. In particular, any of the probability graphs might be insightful (e.g. Probability Plot Correlation Coefficient Plot, Quantile-Quantile Plot, et... | Identify probability distributions
If you're trying to do Exploratory Data Analysis, you could use some graphical techniques.
I can suggest chapter 1.3.4 in the NIST handbook. In particular, any of the probability graphs might be insig |
44,228 | How to fit a model to self-reported number of friend interactions over a 20 day period? | First off, your response variable is discrete. The Cauchy distribution is continuous. Second, your response variable is non-negative. The Cauchy distribution with the parameters you specified puts about 1/5 of its mass on negative values. Whatever you have been reading about the QQ norm plot is false. Points falling cl... | How to fit a model to self-reported number of friend interactions over a 20 day period? | First off, your response variable is discrete. The Cauchy distribution is continuous. Second, your response variable is non-negative. The Cauchy distribution with the parameters you specified puts abo | How to fit a model to self-reported number of friend interactions over a 20 day period?
First off, your response variable is discrete. The Cauchy distribution is continuous. Second, your response variable is non-negative. The Cauchy distribution with the parameters you specified puts about 1/5 of its mass on negative v... | How to fit a model to self-reported number of friend interactions over a 20 day period?
First off, your response variable is discrete. The Cauchy distribution is continuous. Second, your response variable is non-negative. The Cauchy distribution with the parameters you specified puts abo |
44,229 | How to fit a model to self-reported number of friend interactions over a 20 day period? | Agree with HairyBeast (+1) that Cauchy is not appropriate here (it's symmetric for one thing) and that negative binomial might well be better.
Disagree about QQ-plot though. You can do a QQ-plot for any distribution, not just normal. What you say about interpretation of a QQ-plot is correct, but note that 2 of your poi... | How to fit a model to self-reported number of friend interactions over a 20 day period? | Agree with HairyBeast (+1) that Cauchy is not appropriate here (it's symmetric for one thing) and that negative binomial might well be better.
Disagree about QQ-plot though. You can do a QQ-plot for a | How to fit a model to self-reported number of friend interactions over a 20 day period?
Agree with HairyBeast (+1) that Cauchy is not appropriate here (it's symmetric for one thing) and that negative binomial might well be better.
Disagree about QQ-plot though. You can do a QQ-plot for any distribution, not just normal... | How to fit a model to self-reported number of friend interactions over a 20 day period?
Agree with HairyBeast (+1) that Cauchy is not appropriate here (it's symmetric for one thing) and that negative binomial might well be better.
Disagree about QQ-plot though. You can do a QQ-plot for a |
44,230 | Is it possible to use machine learning as a method for learning stats, rather than vice-versa? | I really wouldn't suggest using machine learning in order to learn statistics. The mathematics employed in machine learning is often different because there's a real emphasis on the computational algorithm. Even treatment of the same concept will be different.
A simple example of this would be to compare the treatm... | Is it possible to use machine learning as a method for learning stats, rather than vice-versa? | I really wouldn't suggest using machine learning in order to learn statistics. The mathematics employed in machine learning is often different because there's a real emphasis on the computational alg | Is it possible to use machine learning as a method for learning stats, rather than vice-versa?
I really wouldn't suggest using machine learning in order to learn statistics. The mathematics employed in machine learning is often different because there's a real emphasis on the computational algorithm. Even treatment o... | Is it possible to use machine learning as a method for learning stats, rather than vice-versa?
I really wouldn't suggest using machine learning in order to learn statistics. The mathematics employed in machine learning is often different because there's a real emphasis on the computational alg |
44,231 | Is it possible to use machine learning as a method for learning stats, rather than vice-versa? | It's like that old joke. When asked for directions the philosopher said "Well, if I wanted to go there, I wouldn't start from here ..."
While I think each "culture" should be open to learning from the other, they have different ways of looking at the world.
I think the problem with learning statistics through studying ... | Is it possible to use machine learning as a method for learning stats, rather than vice-versa? | It's like that old joke. When asked for directions the philosopher said "Well, if I wanted to go there, I wouldn't start from here ..."
While I think each "culture" should be open to learning from the | Is it possible to use machine learning as a method for learning stats, rather than vice-versa?
It's like that old joke. When asked for directions the philosopher said "Well, if I wanted to go there, I wouldn't start from here ..."
While I think each "culture" should be open to learning from the other, they have differe... | Is it possible to use machine learning as a method for learning stats, rather than vice-versa?
It's like that old joke. When asked for directions the philosopher said "Well, if I wanted to go there, I wouldn't start from here ..."
While I think each "culture" should be open to learning from the |
44,232 | Is it possible to use machine learning as a method for learning stats, rather than vice-versa? | Depends on what you mean.
I've gotten deeper and deeper into statistics based on exposure to machine learning. (I had previously been more of a general AI guy, and hadn't had good experience with statistics, but gained greater understanding and appreciation of statistics as time has gone on.) So it's certainly a useful... | Is it possible to use machine learning as a method for learning stats, rather than vice-versa? | Depends on what you mean.
I've gotten deeper and deeper into statistics based on exposure to machine learning. (I had previously been more of a general AI guy, and hadn't had good experience with stat | Is it possible to use machine learning as a method for learning stats, rather than vice-versa?
Depends on what you mean.
I've gotten deeper and deeper into statistics based on exposure to machine learning. (I had previously been more of a general AI guy, and hadn't had good experience with statistics, but gained greate... | Is it possible to use machine learning as a method for learning stats, rather than vice-versa?
Depends on what you mean.
I've gotten deeper and deeper into statistics based on exposure to machine learning. (I had previously been more of a general AI guy, and hadn't had good experience with stat |
44,233 | Is it possible to use machine learning as a method for learning stats, rather than vice-versa? | I really dont think so, as there are fundamental aspects in statistics that are simply overlooked in machine learning. For instance, in statistics, when fitting a model to data, the discrpeancy function that is used (e.g., G^2, RMSEA) is essential because they have different statistical properties. In machine learning,... | Is it possible to use machine learning as a method for learning stats, rather than vice-versa? | I really dont think so, as there are fundamental aspects in statistics that are simply overlooked in machine learning. For instance, in statistics, when fitting a model to data, the discrpeancy functi | Is it possible to use machine learning as a method for learning stats, rather than vice-versa?
I really dont think so, as there are fundamental aspects in statistics that are simply overlooked in machine learning. For instance, in statistics, when fitting a model to data, the discrpeancy function that is used (e.g., G^... | Is it possible to use machine learning as a method for learning stats, rather than vice-versa?
I really dont think so, as there are fundamental aspects in statistics that are simply overlooked in machine learning. For instance, in statistics, when fitting a model to data, the discrpeancy functi |
44,234 | Is it possible to use machine learning as a method for learning stats, rather than vice-versa? | I think that learning machine learning requires only an elementary subset of statistics; too much may be dangerous, since some intuitions are in conflict. Still, the answer to the question can it be reversed is no. | Is it possible to use machine learning as a method for learning stats, rather than vice-versa? | I think that learning machine learning requires only an elementary subset of statistics; too much may be dangerous, since some intuitions are in conflict. Still, the answer to the question can it be r | Is it possible to use machine learning as a method for learning stats, rather than vice-versa?
I think that learning machine learning requires only an elementary subset of statistics; too much may be dangerous, since some intuitions are in conflict. Still, the answer to the question can it be reversed is no. | Is it possible to use machine learning as a method for learning stats, rather than vice-versa?
I think that learning machine learning requires only an elementary subset of statistics; too much may be dangerous, since some intuitions are in conflict. Still, the answer to the question can it be r |
44,235 | Relationship between z-score and the normal distribution [duplicate] | The ``normal distribution'' is an entire family of different distributions. We use the notation $\textbf{Normal}(\mu,\sigma^2)$ to indicate what type of normal we get. If you pick a certain choice for $\mu$ and you pick another choice (positive) for $\sigma$, then you get a different type of Normal. Here are some pictu... | Relationship between z-score and the normal distribution [duplicate] | The ``normal distribution'' is an entire family of different distributions. We use the notation $\textbf{Normal}(\mu,\sigma^2)$ to indicate what type of normal we get. If you pick a certain choice for | Relationship between z-score and the normal distribution [duplicate]
The ``normal distribution'' is an entire family of different distributions. We use the notation $\textbf{Normal}(\mu,\sigma^2)$ to indicate what type of normal we get. If you pick a certain choice for $\mu$ and you pick another choice (positive) for $... | Relationship between z-score and the normal distribution [duplicate]
The ``normal distribution'' is an entire family of different distributions. We use the notation $\textbf{Normal}(\mu,\sigma^2)$ to indicate what type of normal we get. If you pick a certain choice for |
44,236 | Relationship between z-score and the normal distribution [duplicate] | There is no relationship. The (sample) z-score is defined as
$$
z_i = \frac{ x_i - \bar x } {s}
$$
where $i$ indexes observations $\{x\}$, $\bar x$ is the sample mean, and $s$ is the sample standard deviation.
There is nothing in this definition which states that the data has to be normally distributed, or that you can... | Relationship between z-score and the normal distribution [duplicate] | There is no relationship. The (sample) z-score is defined as
$$
z_i = \frac{ x_i - \bar x } {s}
$$
where $i$ indexes observations $\{x\}$, $\bar x$ is the sample mean, and $s$ is the sample standard d | Relationship between z-score and the normal distribution [duplicate]
There is no relationship. The (sample) z-score is defined as
$$
z_i = \frac{ x_i - \bar x } {s}
$$
where $i$ indexes observations $\{x\}$, $\bar x$ is the sample mean, and $s$ is the sample standard deviation.
There is nothing in this definition which... | Relationship between z-score and the normal distribution [duplicate]
There is no relationship. The (sample) z-score is defined as
$$
z_i = \frac{ x_i - \bar x } {s}
$$
where $i$ indexes observations $\{x\}$, $\bar x$ is the sample mean, and $s$ is the sample standard d |
44,237 | Relationship between z-score and the normal distribution [duplicate] | I think there is some confusion here due to the word "normalization". In this context, normalization means that the data are transformed to have zero mean and unit standard deviation. The transformed data will also be dimensionless, i.e. lacking physical units.
Z-score normalization does not mean that the data become n... | Relationship between z-score and the normal distribution [duplicate] | I think there is some confusion here due to the word "normalization". In this context, normalization means that the data are transformed to have zero mean and unit standard deviation. The transformed | Relationship between z-score and the normal distribution [duplicate]
I think there is some confusion here due to the word "normalization". In this context, normalization means that the data are transformed to have zero mean and unit standard deviation. The transformed data will also be dimensionless, i.e. lacking physi... | Relationship between z-score and the normal distribution [duplicate]
I think there is some confusion here due to the word "normalization". In this context, normalization means that the data are transformed to have zero mean and unit standard deviation. The transformed |
44,238 | Weak law vs strong law of large numbers - intuition | After a second read of your question, I found that there are two pairs of concepts need clarification. First, the an event happens vs. the probability of an event happens. Second, convergence in probability vs convergence almost surely.
1. $A$ happens vs. The probability of $A$ happens
Let $(\Omega, \mathscr{F}, P)$ b... | Weak law vs strong law of large numbers - intuition | After a second read of your question, I found that there are two pairs of concepts need clarification. First, the an event happens vs. the probability of an event happens. Second, convergence in prob | Weak law vs strong law of large numbers - intuition
After a second read of your question, I found that there are two pairs of concepts need clarification. First, the an event happens vs. the probability of an event happens. Second, convergence in probability vs convergence almost surely.
1. $A$ happens vs. The probabi... | Weak law vs strong law of large numbers - intuition
After a second read of your question, I found that there are two pairs of concepts need clarification. First, the an event happens vs. the probability of an event happens. Second, convergence in prob |
44,239 | Weak law vs strong law of large numbers - intuition | This is a supplement to Zhanxiong's comprehensive answer that addressed OP's concerns.
It would be apt to have a brief recollection of the concepts of almost everywhere convergence and convergence in probability.
$\bullet$ Consider a measure space $(X, \boldsymbol{\mathfrak A}, \mu).$ A sequence of $\boldsymbol{\mathfr... | Weak law vs strong law of large numbers - intuition | This is a supplement to Zhanxiong's comprehensive answer that addressed OP's concerns.
It would be apt to have a brief recollection of the concepts of almost everywhere convergence and convergence in | Weak law vs strong law of large numbers - intuition
This is a supplement to Zhanxiong's comprehensive answer that addressed OP's concerns.
It would be apt to have a brief recollection of the concepts of almost everywhere convergence and convergence in probability.
$\bullet$ Consider a measure space $(X, \boldsymbol{\ma... | Weak law vs strong law of large numbers - intuition
This is a supplement to Zhanxiong's comprehensive answer that addressed OP's concerns.
It would be apt to have a brief recollection of the concepts of almost everywhere convergence and convergence in |
44,240 | Deduce the logistic regression formula | There are several ways how the logistic function can occur as an expression that is derived based on some underlying principle.
Derivation as Bayes factor
The formula occurs when one computes the Bayes factor where one assumes that the groups are normal distributed.
$$\frac{P(Y = 1 | X=x)}{P(Y = 0 | X=x)} = \frac{P(Y =... | Deduce the logistic regression formula | There are several ways how the logistic function can occur as an expression that is derived based on some underlying principle.
Derivation as Bayes factor
The formula occurs when one computes the Baye | Deduce the logistic regression formula
There are several ways how the logistic function can occur as an expression that is derived based on some underlying principle.
Derivation as Bayes factor
The formula occurs when one computes the Bayes factor where one assumes that the groups are normal distributed.
$$\frac{P(Y = ... | Deduce the logistic regression formula
There are several ways how the logistic function can occur as an expression that is derived based on some underlying principle.
Derivation as Bayes factor
The formula occurs when one computes the Baye |
44,241 | Deduce the logistic regression formula | The logistic regression model is an instance of Generalized Linear Models (GLM) and arises as follows. Suppose we observe random variables that can take only zeros and ones, that is, let us have a sample $Y_1,\ldots, Y_n$, where $Y_i$ is a binary random variable. Suppose also that the elements of the sample are indepen... | Deduce the logistic regression formula | The logistic regression model is an instance of Generalized Linear Models (GLM) and arises as follows. Suppose we observe random variables that can take only zeros and ones, that is, let us have a sam | Deduce the logistic regression formula
The logistic regression model is an instance of Generalized Linear Models (GLM) and arises as follows. Suppose we observe random variables that can take only zeros and ones, that is, let us have a sample $Y_1,\ldots, Y_n$, where $Y_i$ is a binary random variable. Suppose also that... | Deduce the logistic regression formula
The logistic regression model is an instance of Generalized Linear Models (GLM) and arises as follows. Suppose we observe random variables that can take only zeros and ones, that is, let us have a sam |
44,242 | Why do we need Multi-Level Models when we have Logistic Regression? | As another comment notes - multilevel regression and logistic regression are not mutually exclusive, and in fact it might be that the best model to run is a mutlilevel logit model! In this case, the key issue is whether or not you want to consider "school" as a variable or a unit of analysis.
But first, this question c... | Why do we need Multi-Level Models when we have Logistic Regression? | As another comment notes - multilevel regression and logistic regression are not mutually exclusive, and in fact it might be that the best model to run is a mutlilevel logit model! In this case, the k | Why do we need Multi-Level Models when we have Logistic Regression?
As another comment notes - multilevel regression and logistic regression are not mutually exclusive, and in fact it might be that the best model to run is a mutlilevel logit model! In this case, the key issue is whether or not you want to consider "sch... | Why do we need Multi-Level Models when we have Logistic Regression?
As another comment notes - multilevel regression and logistic regression are not mutually exclusive, and in fact it might be that the best model to run is a mutlilevel logit model! In this case, the k |
44,243 | Why do we need Multi-Level Models when we have Logistic Regression? | To address you final comment - "If this is really the case - why not just fit a whole bunch of individual models for each combination of factors within the independent variables? I understand that this might result in fitting a large number of models, but aren't modern computers strong enough to do this?"
It is possibl... | Why do we need Multi-Level Models when we have Logistic Regression? | To address you final comment - "If this is really the case - why not just fit a whole bunch of individual models for each combination of factors within the independent variables? I understand that thi | Why do we need Multi-Level Models when we have Logistic Regression?
To address you final comment - "If this is really the case - why not just fit a whole bunch of individual models for each combination of factors within the independent variables? I understand that this might result in fitting a large number of models, ... | Why do we need Multi-Level Models when we have Logistic Regression?
To address you final comment - "If this is really the case - why not just fit a whole bunch of individual models for each combination of factors within the independent variables? I understand that thi |
44,244 | Why do we need Multi-Level Models when we have Logistic Regression? | Multilevel models provide us with a few benefits when compared to logistic regression. Here are a few I can think of without being too pedantic:
Shrinkage
Suppose you are estimating the mean/proportion of a response variable per group. Sometimes a group can have a mean too high just because that group has fewer samples... | Why do we need Multi-Level Models when we have Logistic Regression? | Multilevel models provide us with a few benefits when compared to logistic regression. Here are a few I can think of without being too pedantic:
Shrinkage
Suppose you are estimating the mean/proportio | Why do we need Multi-Level Models when we have Logistic Regression?
Multilevel models provide us with a few benefits when compared to logistic regression. Here are a few I can think of without being too pedantic:
Shrinkage
Suppose you are estimating the mean/proportion of a response variable per group. Sometimes a grou... | Why do we need Multi-Level Models when we have Logistic Regression?
Multilevel models provide us with a few benefits when compared to logistic regression. Here are a few I can think of without being too pedantic:
Shrinkage
Suppose you are estimating the mean/proportio |
44,245 | Why do we need Multi-Level Models when we have Logistic Regression? | A couple quick points to add on to what has already been said:
Multilevel models are not mutually exclusive from logistic regression. You can have a multilevel logistic regression model.
If you have the ability to do some basic simulation, I would highly recommend it to understand multilevel models (or statistical m... | Why do we need Multi-Level Models when we have Logistic Regression? | A couple quick points to add on to what has already been said:
Multilevel models are not mutually exclusive from logistic regression. You can have a multilevel logistic regression model.
If you hav | Why do we need Multi-Level Models when we have Logistic Regression?
A couple quick points to add on to what has already been said:
Multilevel models are not mutually exclusive from logistic regression. You can have a multilevel logistic regression model.
If you have the ability to do some basic simulation, I would h... | Why do we need Multi-Level Models when we have Logistic Regression?
A couple quick points to add on to what has already been said:
Multilevel models are not mutually exclusive from logistic regression. You can have a multilevel logistic regression model.
If you hav |
44,246 | Why do we need Multi-Level Models when we have Logistic Regression? | Standard Logistic Regression
All of the answers here are great. I just wanted to add a visual example because it is often illustrative of why this can matter. Using R as an example in case you may want to look at this yourself, we can load these libraries and data:
#### Load Libraries ####
library(datarium)
library(tid... | Why do we need Multi-Level Models when we have Logistic Regression? | Standard Logistic Regression
All of the answers here are great. I just wanted to add a visual example because it is often illustrative of why this can matter. Using R as an example in case you may wan | Why do we need Multi-Level Models when we have Logistic Regression?
Standard Logistic Regression
All of the answers here are great. I just wanted to add a visual example because it is often illustrative of why this can matter. Using R as an example in case you may want to look at this yourself, we can load these librar... | Why do we need Multi-Level Models when we have Logistic Regression?
Standard Logistic Regression
All of the answers here are great. I just wanted to add a visual example because it is often illustrative of why this can matter. Using R as an example in case you may wan |
44,247 | Alternatives to scatterplot for visualisation for large samples | I would begin by considering a transformation - very like to be useful for the Y-axis, and possibly for the X too. Log if there are no zeroes or negative values, square root if there are no negative values (zeroes are fine), cube root if there are negative values.
After transformation, you could consider:
A scatterplo... | Alternatives to scatterplot for visualisation for large samples | I would begin by considering a transformation - very like to be useful for the Y-axis, and possibly for the X too. Log if there are no zeroes or negative values, square root if there are no negative v | Alternatives to scatterplot for visualisation for large samples
I would begin by considering a transformation - very like to be useful for the Y-axis, and possibly for the X too. Log if there are no zeroes or negative values, square root if there are no negative values (zeroes are fine), cube root if there are negative... | Alternatives to scatterplot for visualisation for large samples
I would begin by considering a transformation - very like to be useful for the Y-axis, and possibly for the X too. Log if there are no zeroes or negative values, square root if there are no negative v |
44,248 | Alternatives to scatterplot for visualisation for large samples | You could indicate the density of points with a heatmap, e.g., using a black-body radiation palette. Alternatively, use a grayscale. Or use a hexbinplot, which pretty much does the grayscaling for you. | Alternatives to scatterplot for visualisation for large samples | You could indicate the density of points with a heatmap, e.g., using a black-body radiation palette. Alternatively, use a grayscale. Or use a hexbinplot, which pretty much does the grayscaling for you | Alternatives to scatterplot for visualisation for large samples
You could indicate the density of points with a heatmap, e.g., using a black-body radiation palette. Alternatively, use a grayscale. Or use a hexbinplot, which pretty much does the grayscaling for you. | Alternatives to scatterplot for visualisation for large samples
You could indicate the density of points with a heatmap, e.g., using a black-body radiation palette. Alternatively, use a grayscale. Or use a hexbinplot, which pretty much does the grayscaling for you |
44,249 | Alternatives to scatterplot for visualisation for large samples | Binscatter plots are designed for this exact problem: visualizing two-way relationships in huge datasets. See here for R and python packages and some references.
The name says it all. Binscatter procedure partitions the data domain into bins and plots only sample averages in the bins rather than all data points.
Main a... | Alternatives to scatterplot for visualisation for large samples | Binscatter plots are designed for this exact problem: visualizing two-way relationships in huge datasets. See here for R and python packages and some references.
The name says it all. Binscatter proce | Alternatives to scatterplot for visualisation for large samples
Binscatter plots are designed for this exact problem: visualizing two-way relationships in huge datasets. See here for R and python packages and some references.
The name says it all. Binscatter procedure partitions the data domain into bins and plots only... | Alternatives to scatterplot for visualisation for large samples
Binscatter plots are designed for this exact problem: visualizing two-way relationships in huge datasets. See here for R and python packages and some references.
The name says it all. Binscatter proce |
44,250 | Practical implication of failing to reject a null hypothesis | Here is a discussion of a simple version of your question. Suppose the dosages are normal. And you have looked
at 30 bottles sampled before the faulty filling may have begun and 30 after.
Then 'Before' dosages may have been
$X_i \sim\mathsf{Norm}(50, 2),$ $i=1,2,\dots,30,$
and 'After' dosages
$Y_i \sim\mathsf{Norm}(52,... | Practical implication of failing to reject a null hypothesis | Here is a discussion of a simple version of your question. Suppose the dosages are normal. And you have looked
at 30 bottles sampled before the faulty filling may have begun and 30 after.
Then 'Before | Practical implication of failing to reject a null hypothesis
Here is a discussion of a simple version of your question. Suppose the dosages are normal. And you have looked
at 30 bottles sampled before the faulty filling may have begun and 30 after.
Then 'Before' dosages may have been
$X_i \sim\mathsf{Norm}(50, 2),$ $i=... | Practical implication of failing to reject a null hypothesis
Here is a discussion of a simple version of your question. Suppose the dosages are normal. And you have looked
at 30 bottles sampled before the faulty filling may have begun and 30 after.
Then 'Before |
44,251 | Practical implication of failing to reject a null hypothesis | If you approached the problem this way, then you made a mistake!
What you could have done is calculate the sample size required to detect a difference of practical importance. You know it isn’t filling exactly 50, and you have some tolerable difference from that value. Maybe you can tolerate 49-51, or maybe you can tol... | Practical implication of failing to reject a null hypothesis | If you approached the problem this way, then you made a mistake!
What you could have done is calculate the sample size required to detect a difference of practical importance. You know it isn’t fillin | Practical implication of failing to reject a null hypothesis
If you approached the problem this way, then you made a mistake!
What you could have done is calculate the sample size required to detect a difference of practical importance. You know it isn’t filling exactly 50, and you have some tolerable difference from t... | Practical implication of failing to reject a null hypothesis
If you approached the problem this way, then you made a mistake!
What you could have done is calculate the sample size required to detect a difference of practical importance. You know it isn’t fillin |
44,252 | Practical implication of failing to reject a null hypothesis | This seems like a situation where the failure is relying on null hypothesis significance testing in the first place.
Monitoring that a machine is well calibrated is an estimation problem. You might do better to control accuracy (margin of error) of the dosage rather than the type I error. A type I error in this situati... | Practical implication of failing to reject a null hypothesis | This seems like a situation where the failure is relying on null hypothesis significance testing in the first place.
Monitoring that a machine is well calibrated is an estimation problem. You might do | Practical implication of failing to reject a null hypothesis
This seems like a situation where the failure is relying on null hypothesis significance testing in the first place.
Monitoring that a machine is well calibrated is an estimation problem. You might do better to control accuracy (margin of error) of the dosage... | Practical implication of failing to reject a null hypothesis
This seems like a situation where the failure is relying on null hypothesis significance testing in the first place.
Monitoring that a machine is well calibrated is an estimation problem. You might do |
44,253 | Testing difference between coefficients of nonlinear regression models | Define a new variable red in your data set that is 1 for the red points and 0 for the blue points. Then include an extra coefficient c in your regression equation:
1−1/(1+(x/(a+c*red))^b)
c denotes the offset between the curve fitted to the blue and the red points. If c is significantly different from zero then you can... | Testing difference between coefficients of nonlinear regression models | Define a new variable red in your data set that is 1 for the red points and 0 for the blue points. Then include an extra coefficient c in your regression equation:
1−1/(1+(x/(a+c*red))^b)
c denotes th | Testing difference between coefficients of nonlinear regression models
Define a new variable red in your data set that is 1 for the red points and 0 for the blue points. Then include an extra coefficient c in your regression equation:
1−1/(1+(x/(a+c*red))^b)
c denotes the offset between the curve fitted to the blue and... | Testing difference between coefficients of nonlinear regression models
Define a new variable red in your data set that is 1 for the red points and 0 for the blue points. Then include an extra coefficient c in your regression equation:
1−1/(1+(x/(a+c*red))^b)
c denotes th |
44,254 | Testing difference between coefficients of nonlinear regression models | I would probably approach this using an approximate permutation test.
The general idea is that, if the data comes from the same function (note that this would assume $a_1 = a_2$ AND $b_1 = b_2$ for the two sets of data, as well as that the random components of the two datasets were drawn from the same distribution), wh... | Testing difference between coefficients of nonlinear regression models | I would probably approach this using an approximate permutation test.
The general idea is that, if the data comes from the same function (note that this would assume $a_1 = a_2$ AND $b_1 = b_2$ for th | Testing difference between coefficients of nonlinear regression models
I would probably approach this using an approximate permutation test.
The general idea is that, if the data comes from the same function (note that this would assume $a_1 = a_2$ AND $b_1 = b_2$ for the two sets of data, as well as that the random co... | Testing difference between coefficients of nonlinear regression models
I would probably approach this using an approximate permutation test.
The general idea is that, if the data comes from the same function (note that this would assume $a_1 = a_2$ AND $b_1 = b_2$ for th |
44,255 | Testing difference between coefficients of nonlinear regression models | I feel a bit bad by overcrowding the comments section with pedantic notes and questions about the origins of the noise in the data. So I will make up for it with an answer that is a bit more decent than those little comments.
This answer will demonstrate how a straightforward (non linear) least squares fit (which can a... | Testing difference between coefficients of nonlinear regression models | I feel a bit bad by overcrowding the comments section with pedantic notes and questions about the origins of the noise in the data. So I will make up for it with an answer that is a bit more decent th | Testing difference between coefficients of nonlinear regression models
I feel a bit bad by overcrowding the comments section with pedantic notes and questions about the origins of the noise in the data. So I will make up for it with an answer that is a bit more decent than those little comments.
This answer will demons... | Testing difference between coefficients of nonlinear regression models
I feel a bit bad by overcrowding the comments section with pedantic notes and questions about the origins of the noise in the data. So I will make up for it with an answer that is a bit more decent th |
44,256 | Correct use of Chi-square and hypothesis testing? | All of your concerns are valid and well articulated. The chi-square test simply provides the weight of the evidence that there is an association between attendance and purchase. Any causality is an extra layer of interpretation that in many instances is unverifiable. There are many causal hypotheses one could entert... | Correct use of Chi-square and hypothesis testing? | All of your concerns are valid and well articulated. The chi-square test simply provides the weight of the evidence that there is an association between attendance and purchase. Any causality is an | Correct use of Chi-square and hypothesis testing?
All of your concerns are valid and well articulated. The chi-square test simply provides the weight of the evidence that there is an association between attendance and purchase. Any causality is an extra layer of interpretation that in many instances is unverifiable. ... | Correct use of Chi-square and hypothesis testing?
All of your concerns are valid and well articulated. The chi-square test simply provides the weight of the evidence that there is an association between attendance and purchase. Any causality is an |
44,257 | Correct use of Chi-square and hypothesis testing? | An alternative version of your chi-squared test is the prop.test procedure in R. You have the proportion of purchasers $\hat p_a = 190/1540 = 0.1234$ among those who attended
events and the proportion $\hat p_n = 893/16571 = 0.0593$ among those
who did not attend events. The question is whether $\hat p_a$ and $\hat p_n... | Correct use of Chi-square and hypothesis testing? | An alternative version of your chi-squared test is the prop.test procedure in R. You have the proportion of purchasers $\hat p_a = 190/1540 = 0.1234$ among those who attended
events and the proportion | Correct use of Chi-square and hypothesis testing?
An alternative version of your chi-squared test is the prop.test procedure in R. You have the proportion of purchasers $\hat p_a = 190/1540 = 0.1234$ among those who attended
events and the proportion $\hat p_n = 893/16571 = 0.0593$ among those
who did not attend events... | Correct use of Chi-square and hypothesis testing?
An alternative version of your chi-squared test is the prop.test procedure in R. You have the proportion of purchasers $\hat p_a = 190/1540 = 0.1234$ among those who attended
events and the proportion |
44,258 | Correct use of Chi-square and hypothesis testing? | On a shallow level, the analysis you've done is good enough to show to the business. This is what matters.
Now, "is our marketing good and people buy products because of these events?" is quite different from "should we keep conducting these events, and if so, how many?". Maybe events make people feel better about the ... | Correct use of Chi-square and hypothesis testing? | On a shallow level, the analysis you've done is good enough to show to the business. This is what matters.
Now, "is our marketing good and people buy products because of these events?" is quite differ | Correct use of Chi-square and hypothesis testing?
On a shallow level, the analysis you've done is good enough to show to the business. This is what matters.
Now, "is our marketing good and people buy products because of these events?" is quite different from "should we keep conducting these events, and if so, how many?... | Correct use of Chi-square and hypothesis testing?
On a shallow level, the analysis you've done is good enough to show to the business. This is what matters.
Now, "is our marketing good and people buy products because of these events?" is quite differ |
44,259 | Can I find MLE of probability of X greater than x [duplicate] | As I'm sure you know (or can easily derive), the maximum likelihood estimator of $(\mu,\sigma^2)$ is
$$(\hat\mu,\hat\sigma^2) = \left(\bar X, \operatorname{Var}(X)\right)$$
where, as usual, $n\bar X = X_1+X_2+\cdots + X_n$ and, a little unusually,
$$\operatorname{Var}(X) = \frac{1}{n}\left((X_1-\bar X)^2 + (X_2-\bar X)... | Can I find MLE of probability of X greater than x [duplicate] | As I'm sure you know (or can easily derive), the maximum likelihood estimator of $(\mu,\sigma^2)$ is
$$(\hat\mu,\hat\sigma^2) = \left(\bar X, \operatorname{Var}(X)\right)$$
where, as usual, $n\bar X = | Can I find MLE of probability of X greater than x [duplicate]
As I'm sure you know (or can easily derive), the maximum likelihood estimator of $(\mu,\sigma^2)$ is
$$(\hat\mu,\hat\sigma^2) = \left(\bar X, \operatorname{Var}(X)\right)$$
where, as usual, $n\bar X = X_1+X_2+\cdots + X_n$ and, a little unusually,
$$\operato... | Can I find MLE of probability of X greater than x [duplicate]
As I'm sure you know (or can easily derive), the maximum likelihood estimator of $(\mu,\sigma^2)$ is
$$(\hat\mu,\hat\sigma^2) = \left(\bar X, \operatorname{Var}(X)\right)$$
where, as usual, $n\bar X = |
44,260 | Meaning of vertical bar | in loss function? | It denotes that the function is parameterized by $\theta$ and the $x_i$ are the inputs to the function. For example $f(x|\theta)=x\cdot \theta$ is the dot product of the input, $x$, and the parameters $\theta$. User @Underminer adds a note about reading: if you wanted to read the symbols $f(x|\theta)$ aloud, you might ... | Meaning of vertical bar | in loss function? | It denotes that the function is parameterized by $\theta$ and the $x_i$ are the inputs to the function. For example $f(x|\theta)=x\cdot \theta$ is the dot product of the input, $x$, and the parameters | Meaning of vertical bar | in loss function?
It denotes that the function is parameterized by $\theta$ and the $x_i$ are the inputs to the function. For example $f(x|\theta)=x\cdot \theta$ is the dot product of the input, $x$, and the parameters $\theta$. User @Underminer adds a note about reading: if you wanted to read... | Meaning of vertical bar | in loss function?
It denotes that the function is parameterized by $\theta$ and the $x_i$ are the inputs to the function. For example $f(x|\theta)=x\cdot \theta$ is the dot product of the input, $x$, and the parameters |
44,261 | Using cross-entropy for regression problems | In a regression problem you have pairs $(x_i, y_i)$. And some true model $q$ that characterizes $q(y|x)$. Let's say you assume that your density
$$f_\theta(y|x)= \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left\{-\frac{1}{2\sigma^2}(y_i-\mu_\theta(x_i))^2\right\}$$
and you fix $\sigma^2$ to some value
The mean $\mu(x_i)$ is the... | Using cross-entropy for regression problems | In a regression problem you have pairs $(x_i, y_i)$. And some true model $q$ that characterizes $q(y|x)$. Let's say you assume that your density
$$f_\theta(y|x)= \frac{1}{\sqrt{2\pi\sigma^2}} \exp\lef | Using cross-entropy for regression problems
In a regression problem you have pairs $(x_i, y_i)$. And some true model $q$ that characterizes $q(y|x)$. Let's say you assume that your density
$$f_\theta(y|x)= \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left\{-\frac{1}{2\sigma^2}(y_i-\mu_\theta(x_i))^2\right\}$$
and you fix $\sigma... | Using cross-entropy for regression problems
In a regression problem you have pairs $(x_i, y_i)$. And some true model $q$ that characterizes $q(y|x)$. Let's say you assume that your density
$$f_\theta(y|x)= \frac{1}{\sqrt{2\pi\sigma^2}} \exp\lef |
44,262 | Using cross-entropy for regression problems | The mean squared error is the cross-entropy between the data distribution $p^*(x)$ and your Gaussian model distribution $p_{\theta}$. Note that the standard MLE procedure is:
$$
\begin{align}
\max_{\theta} E_{x \sim p^*}[\log p_{\theta}(x)] &= \min_{\theta} \left(- E_{x \sim p^*}[\log p_{\theta}(x)]\right)\\
&= \min_{... | Using cross-entropy for regression problems | The mean squared error is the cross-entropy between the data distribution $p^*(x)$ and your Gaussian model distribution $p_{\theta}$. Note that the standard MLE procedure is:
$$
\begin{align}
\max_{\t | Using cross-entropy for regression problems
The mean squared error is the cross-entropy between the data distribution $p^*(x)$ and your Gaussian model distribution $p_{\theta}$. Note that the standard MLE procedure is:
$$
\begin{align}
\max_{\theta} E_{x \sim p^*}[\log p_{\theta}(x)] &= \min_{\theta} \left(- E_{x \sim ... | Using cross-entropy for regression problems
The mean squared error is the cross-entropy between the data distribution $p^*(x)$ and your Gaussian model distribution $p_{\theta}$. Note that the standard MLE procedure is:
$$
\begin{align}
\max_{\t |
44,263 | Why does ordinary least squares have to be linear in the parameters? | What is it about the process of solving OLS that requires the parameters to be linear?
Because equations which are nonlinear in their parameters can't be written as $y=X\beta$. OLS estimates $\beta$ in the equation
$$
y = X\beta +\epsilon.
$$
This is a linear relationship, so when we say that $\hat{\beta} = (X^\top X... | Why does ordinary least squares have to be linear in the parameters? | What is it about the process of solving OLS that requires the parameters to be linear?
Because equations which are nonlinear in their parameters can't be written as $y=X\beta$. OLS estimates $\beta$ | Why does ordinary least squares have to be linear in the parameters?
What is it about the process of solving OLS that requires the parameters to be linear?
Because equations which are nonlinear in their parameters can't be written as $y=X\beta$. OLS estimates $\beta$ in the equation
$$
y = X\beta +\epsilon.
$$
This i... | Why does ordinary least squares have to be linear in the parameters?
What is it about the process of solving OLS that requires the parameters to be linear?
Because equations which are nonlinear in their parameters can't be written as $y=X\beta$. OLS estimates $\beta$ |
44,264 | Why prices are usually not stationary, but returns are more likely to be stationary? | The return $Y_t$ represents the increase in the value of the stock as a percentage of its previous value. This return fluctuates a great deal in an economy, but in a properly functioning economy, it does tend to fluctuate around a small positive value. Consequently, the total stock price for a company tends to grow r... | Why prices are usually not stationary, but returns are more likely to be stationary? | The return $Y_t$ represents the increase in the value of the stock as a percentage of its previous value. This return fluctuates a great deal in an economy, but in a properly functioning economy, it | Why prices are usually not stationary, but returns are more likely to be stationary?
The return $Y_t$ represents the increase in the value of the stock as a percentage of its previous value. This return fluctuates a great deal in an economy, but in a properly functioning economy, it does tend to fluctuate around a sma... | Why prices are usually not stationary, but returns are more likely to be stationary?
The return $Y_t$ represents the increase in the value of the stock as a percentage of its previous value. This return fluctuates a great deal in an economy, but in a properly functioning economy, it |
44,265 | Why prices are usually not stationary, but returns are more likely to be stationary? | Stock prices can be thought of being a cumulative sum of mean-independent increments due to economic (and other types of) shocks. This is per definition a process with a unit root:
$$
X_t=X_{t-1}+\varepsilon_t=(X_{t-2}+\varepsilon_{t-1})+\varepsilon_t=\dots=\sum_{\tau=0}^t\varepsilon_\tau.
$$
(After the first equality... | Why prices are usually not stationary, but returns are more likely to be stationary? | Stock prices can be thought of being a cumulative sum of mean-independent increments due to economic (and other types of) shocks. This is per definition a process with a unit root:
$$
X_t=X_{t-1}+\va | Why prices are usually not stationary, but returns are more likely to be stationary?
Stock prices can be thought of being a cumulative sum of mean-independent increments due to economic (and other types of) shocks. This is per definition a process with a unit root:
$$
X_t=X_{t-1}+\varepsilon_t=(X_{t-2}+\varepsilon_{t-... | Why prices are usually not stationary, but returns are more likely to be stationary?
Stock prices can be thought of being a cumulative sum of mean-independent increments due to economic (and other types of) shocks. This is per definition a process with a unit root:
$$
X_t=X_{t-1}+\va |
44,266 | Why prices are usually not stationary, but returns are more likely to be stationary? | This is a generalization but I think it is useful to think of the price of the stock as
$$X_t = E_t P_t$$
where $E_t$ is a company's earnings and $P_t$ the multiple of earnings investors are willing to pay for the stock (also known as a P/E ratio).
$E_t$ is non-stationary since earnings tend to grow over time due to ec... | Why prices are usually not stationary, but returns are more likely to be stationary? | This is a generalization but I think it is useful to think of the price of the stock as
$$X_t = E_t P_t$$
where $E_t$ is a company's earnings and $P_t$ the multiple of earnings investors are willing t | Why prices are usually not stationary, but returns are more likely to be stationary?
This is a generalization but I think it is useful to think of the price of the stock as
$$X_t = E_t P_t$$
where $E_t$ is a company's earnings and $P_t$ the multiple of earnings investors are willing to pay for the stock (also known as ... | Why prices are usually not stationary, but returns are more likely to be stationary?
This is a generalization but I think it is useful to think of the price of the stock as
$$X_t = E_t P_t$$
where $E_t$ is a company's earnings and $P_t$ the multiple of earnings investors are willing t |
44,267 | Shrinkage priors | I don't quite know your specific use case though I can provide you with a bunch of resources so that you can make your own decision:
Michael Betancourt's case study on sparse regressions: https://betanalpha.github.io/assets/case_studies/bayes_sparse_regression.html
That is a particularly great resource because he shows... | Shrinkage priors | I don't quite know your specific use case though I can provide you with a bunch of resources so that you can make your own decision:
Michael Betancourt's case study on sparse regressions: https://beta | Shrinkage priors
I don't quite know your specific use case though I can provide you with a bunch of resources so that you can make your own decision:
Michael Betancourt's case study on sparse regressions: https://betanalpha.github.io/assets/case_studies/bayes_sparse_regression.html
That is a particularly great resource... | Shrinkage priors
I don't quite know your specific use case though I can provide you with a bunch of resources so that you can make your own decision:
Michael Betancourt's case study on sparse regressions: https://beta |
44,268 | Shrinkage priors | You can check the following paper by Sarah Van Erp et al (2019), who discuss different shrinkage priors (below you can see table from their paper).
Those priors vary greatly in their shapes, and so amount of shrinkage they provide.
Besides discussing pros and cons of those priors, the authors describe a simulation st... | Shrinkage priors | You can check the following paper by Sarah Van Erp et al (2019), who discuss different shrinkage priors (below you can see table from their paper).
Those priors vary greatly in their shapes, and so a | Shrinkage priors
You can check the following paper by Sarah Van Erp et al (2019), who discuss different shrinkage priors (below you can see table from their paper).
Those priors vary greatly in their shapes, and so amount of shrinkage they provide.
Besides discussing pros and cons of those priors, the authors describ... | Shrinkage priors
You can check the following paper by Sarah Van Erp et al (2019), who discuss different shrinkage priors (below you can see table from their paper).
Those priors vary greatly in their shapes, and so a |
44,269 | Expectation of truncated normal | Let $\phi(\cdot)$ and $\Phi(\cdot)$ be the PDF and CDF of standard normal distribution, as usual.
Suppose $\Sigma=\left[\begin{matrix}\sigma_1^2&\rho \sigma_1\sigma_2\\\rho\sigma_1\sigma_2&\sigma_2^2\end{matrix}\right]$ is the dispersion matrix of $(X,Y)$.
By definition,
\begin{align}
E\left[X\mid a<Y<b\right]&=\frac{E... | Expectation of truncated normal | Let $\phi(\cdot)$ and $\Phi(\cdot)$ be the PDF and CDF of standard normal distribution, as usual.
Suppose $\Sigma=\left[\begin{matrix}\sigma_1^2&\rho \sigma_1\sigma_2\\\rho\sigma_1\sigma_2&\sigma_2^2\ | Expectation of truncated normal
Let $\phi(\cdot)$ and $\Phi(\cdot)$ be the PDF and CDF of standard normal distribution, as usual.
Suppose $\Sigma=\left[\begin{matrix}\sigma_1^2&\rho \sigma_1\sigma_2\\\rho\sigma_1\sigma_2&\sigma_2^2\end{matrix}\right]$ is the dispersion matrix of $(X,Y)$.
By definition,
\begin{align}
E\... | Expectation of truncated normal
Let $\phi(\cdot)$ and $\Phi(\cdot)$ be the PDF and CDF of standard normal distribution, as usual.
Suppose $\Sigma=\left[\begin{matrix}\sigma_1^2&\rho \sigma_1\sigma_2\\\rho\sigma_1\sigma_2&\sigma_2^2\ |
44,270 | Expectation of truncated normal | Let's generalize the question so that a key idea can be revealed.
Let $Y$ and $R$ be independent random variables. Define $$X=f(Y)+R$$ for a specified (measurable) function $f$. For any $a\lt b,$ what is the conditional expectation $$E[X\mid a\le Y \le b]?$$
(This states that $f$ is the regression of $X$ on $Y$ wit... | Expectation of truncated normal | Let's generalize the question so that a key idea can be revealed.
Let $Y$ and $R$ be independent random variables. Define $$X=f(Y)+R$$ for a specified (measurable) function $f$. For any $a\lt b,$ w | Expectation of truncated normal
Let's generalize the question so that a key idea can be revealed.
Let $Y$ and $R$ be independent random variables. Define $$X=f(Y)+R$$ for a specified (measurable) function $f$. For any $a\lt b,$ what is the conditional expectation $$E[X\mid a\le Y \le b]?$$
(This states that $f$ is ... | Expectation of truncated normal
Let's generalize the question so that a key idea can be revealed.
Let $Y$ and $R$ be independent random variables. Define $$X=f(Y)+R$$ for a specified (measurable) function $f$. For any $a\lt b,$ w |
44,271 | Expectation of truncated normal | I am posting the complete answer for reference:
\begin{eqnarray*}
E(X|a<Y<b) &= &\int_a^bE(X|Y=y)f_{Y|a<Y<b}(y)dy \\
& = &\rho_{xy}\frac{\sigma_x}{\sigma_y}\int_a^bf_{Y|a<Y<b}(y)dy \\
& = &\rho_{xy}\frac{\sigma_x}{\sigma_y}\sigma_y\frac{\phi(a)-\phi(b)}{\Phi(b)-\Phi(a)} \\
& = &\frac{\sigma_{xy}}{\sigma_y}\frac{\phi(a)... | Expectation of truncated normal | I am posting the complete answer for reference:
\begin{eqnarray*}
E(X|a<Y<b) &= &\int_a^bE(X|Y=y)f_{Y|a<Y<b}(y)dy \\
& = &\rho_{xy}\frac{\sigma_x}{\sigma_y}\int_a^bf_{Y|a<Y<b}(y)dy \\
& = &\rho_{xy}\f | Expectation of truncated normal
I am posting the complete answer for reference:
\begin{eqnarray*}
E(X|a<Y<b) &= &\int_a^bE(X|Y=y)f_{Y|a<Y<b}(y)dy \\
& = &\rho_{xy}\frac{\sigma_x}{\sigma_y}\int_a^bf_{Y|a<Y<b}(y)dy \\
& = &\rho_{xy}\frac{\sigma_x}{\sigma_y}\sigma_y\frac{\phi(a)-\phi(b)}{\Phi(b)-\Phi(a)} \\
& = &\frac{\si... | Expectation of truncated normal
I am posting the complete answer for reference:
\begin{eqnarray*}
E(X|a<Y<b) &= &\int_a^bE(X|Y=y)f_{Y|a<Y<b}(y)dy \\
& = &\rho_{xy}\frac{\sigma_x}{\sigma_y}\int_a^bf_{Y|a<Y<b}(y)dy \\
& = &\rho_{xy}\f |
44,272 | Does this graph support the assumption of homoscedasticity? | I don't think a graph can necessarily "show" homoscedasticity, but it can indicate to deviations from it. Your plot shows a very obvious trend in residuals vs. predicted. Anytime you see a some sort of a structure in these plots it's a source of concern. Ideally you should see a shapeless cloud of dots without any kind... | Does this graph support the assumption of homoscedasticity? | I don't think a graph can necessarily "show" homoscedasticity, but it can indicate to deviations from it. Your plot shows a very obvious trend in residuals vs. predicted. Anytime you see a some sort o | Does this graph support the assumption of homoscedasticity?
I don't think a graph can necessarily "show" homoscedasticity, but it can indicate to deviations from it. Your plot shows a very obvious trend in residuals vs. predicted. Anytime you see a some sort of a structure in these plots it's a source of concern. Ideal... | Does this graph support the assumption of homoscedasticity?
I don't think a graph can necessarily "show" homoscedasticity, but it can indicate to deviations from it. Your plot shows a very obvious trend in residuals vs. predicted. Anytime you see a some sort o |
44,273 | Does this graph support the assumption of homoscedasticity? | I must confess that I've never seen a plot where the fitted values are standardized - usually, we standardize the residuals but not the fitted values.
The first thing you should do is draw an imaginary horizontal line through zero in this plot. This line will anchor the plot and really help you understand what is goi... | Does this graph support the assumption of homoscedasticity? | I must confess that I've never seen a plot where the fitted values are standardized - usually, we standardize the residuals but not the fitted values.
The first thing you should do is draw an imagina | Does this graph support the assumption of homoscedasticity?
I must confess that I've never seen a plot where the fitted values are standardized - usually, we standardize the residuals but not the fitted values.
The first thing you should do is draw an imaginary horizontal line through zero in this plot. This line wil... | Does this graph support the assumption of homoscedasticity?
I must confess that I've never seen a plot where the fitted values are standardized - usually, we standardize the residuals but not the fitted values.
The first thing you should do is draw an imagina |
44,274 | Does this graph support the assumption of homoscedasticity? | The graph does show that there are issues with this model, but the question is whether or not it shows deviations from the hypothesis of homoscedasticity, and that is not quite clear from the graph. I think it does show some deviation: sigma seems smaller at low values of the predicted variable, but it is hard to tell.... | Does this graph support the assumption of homoscedasticity? | The graph does show that there are issues with this model, but the question is whether or not it shows deviations from the hypothesis of homoscedasticity, and that is not quite clear from the graph. I | Does this graph support the assumption of homoscedasticity?
The graph does show that there are issues with this model, but the question is whether or not it shows deviations from the hypothesis of homoscedasticity, and that is not quite clear from the graph. I think it does show some deviation: sigma seems smaller at l... | Does this graph support the assumption of homoscedasticity?
The graph does show that there are issues with this model, but the question is whether or not it shows deviations from the hypothesis of homoscedasticity, and that is not quite clear from the graph. I |
44,275 | Does this graph support the assumption of homoscedasticity? | Note: I use capital letters to refer to an entire vector, and lower case letters to refer to a specific observation of a vector. Hopefully this isn't confusing.
I think the answer ought to be no. I offer an intuitive explanation, as well as some quick and dirty (emphasis on dirty) R code that supports my assertion.
The... | Does this graph support the assumption of homoscedasticity? | Note: I use capital letters to refer to an entire vector, and lower case letters to refer to a specific observation of a vector. Hopefully this isn't confusing.
I think the answer ought to be no. I of | Does this graph support the assumption of homoscedasticity?
Note: I use capital letters to refer to an entire vector, and lower case letters to refer to a specific observation of a vector. Hopefully this isn't confusing.
I think the answer ought to be no. I offer an intuitive explanation, as well as some quick and dirt... | Does this graph support the assumption of homoscedasticity?
Note: I use capital letters to refer to an entire vector, and lower case letters to refer to a specific observation of a vector. Hopefully this isn't confusing.
I think the answer ought to be no. I of |
44,276 | What does it mean if I have high Cronbach alpha, but poor results in Exploratory Factor Analysis | Not necessarily. You might have high Cronbach's alpha while items are weakly correlated. The reason is that Cronbach's alpha does not only depend on the correlation/covariance of the items, but also the number of items. If the number of items are high enough (which is probably the case in your analysis), you will have ... | What does it mean if I have high Cronbach alpha, but poor results in Exploratory Factor Analysis | Not necessarily. You might have high Cronbach's alpha while items are weakly correlated. The reason is that Cronbach's alpha does not only depend on the correlation/covariance of the items, but also t | What does it mean if I have high Cronbach alpha, but poor results in Exploratory Factor Analysis
Not necessarily. You might have high Cronbach's alpha while items are weakly correlated. The reason is that Cronbach's alpha does not only depend on the correlation/covariance of the items, but also the number of items. If ... | What does it mean if I have high Cronbach alpha, but poor results in Exploratory Factor Analysis
Not necessarily. You might have high Cronbach's alpha while items are weakly correlated. The reason is that Cronbach's alpha does not only depend on the correlation/covariance of the items, but also t |
44,277 | What does it mean if I have high Cronbach alpha, but poor results in Exploratory Factor Analysis | You should not consider Cronbach's alpha along with exploratory Factor analysis. Alpha is from the domain of reliability, specifically, one of measures of item-item homogeneity (interchangeability) within a scale; or more specifically about alpha - how much sufficient a scale is supplied with items. Factor analysis is ... | What does it mean if I have high Cronbach alpha, but poor results in Exploratory Factor Analysis | You should not consider Cronbach's alpha along with exploratory Factor analysis. Alpha is from the domain of reliability, specifically, one of measures of item-item homogeneity (interchangeability) wi | What does it mean if I have high Cronbach alpha, but poor results in Exploratory Factor Analysis
You should not consider Cronbach's alpha along with exploratory Factor analysis. Alpha is from the domain of reliability, specifically, one of measures of item-item homogeneity (interchangeability) within a scale; or more s... | What does it mean if I have high Cronbach alpha, but poor results in Exploratory Factor Analysis
You should not consider Cronbach's alpha along with exploratory Factor analysis. Alpha is from the domain of reliability, specifically, one of measures of item-item homogeneity (interchangeability) wi |
44,278 | What does it mean if I have high Cronbach alpha, but poor results in Exploratory Factor Analysis | So for example, let’s say I do a survey on healthy eating. My questions are:
1) Do you like cherry tomatoes?
2) Do you like yellow tomatoes?
Whenever the respondent answers these questions, he sneezes and slightly shifts his pen on the paper. If I add the answers to the 2 questions, I average out this error. As I add... | What does it mean if I have high Cronbach alpha, but poor results in Exploratory Factor Analysis | So for example, let’s say I do a survey on healthy eating. My questions are:
1) Do you like cherry tomatoes?
2) Do you like yellow tomatoes?
Whenever the respondent answers these questions, he sneez | What does it mean if I have high Cronbach alpha, but poor results in Exploratory Factor Analysis
So for example, let’s say I do a survey on healthy eating. My questions are:
1) Do you like cherry tomatoes?
2) Do you like yellow tomatoes?
Whenever the respondent answers these questions, he sneezes and slightly shifts ... | What does it mean if I have high Cronbach alpha, but poor results in Exploratory Factor Analysis
So for example, let’s say I do a survey on healthy eating. My questions are:
1) Do you like cherry tomatoes?
2) Do you like yellow tomatoes?
Whenever the respondent answers these questions, he sneez |
44,279 | How can I apply Akaike Information Criterion and calculate it for Linear Regression? | A simple formula for the calculation of the AIC in the OLS framework (since you say linear regression) can be found in Gordon (2015, p. 201):
$$\text{AIC} = n *\ln\Big(\frac{SSE}{n}\Big)+2k $$
Where SSE means Sum of Squared Errors ($\sum(Y_i-\hat Y_i)^2$), $n$ is the sample size, and $k$ is the number of predictors in ... | How can I apply Akaike Information Criterion and calculate it for Linear Regression? | A simple formula for the calculation of the AIC in the OLS framework (since you say linear regression) can be found in Gordon (2015, p. 201):
$$\text{AIC} = n *\ln\Big(\frac{SSE}{n}\Big)+2k $$
Where S | How can I apply Akaike Information Criterion and calculate it for Linear Regression?
A simple formula for the calculation of the AIC in the OLS framework (since you say linear regression) can be found in Gordon (2015, p. 201):
$$\text{AIC} = n *\ln\Big(\frac{SSE}{n}\Big)+2k $$
Where SSE means Sum of Squared Errors ($\s... | How can I apply Akaike Information Criterion and calculate it for Linear Regression?
A simple formula for the calculation of the AIC in the OLS framework (since you say linear regression) can be found in Gordon (2015, p. 201):
$$\text{AIC} = n *\ln\Big(\frac{SSE}{n}\Big)+2k $$
Where S |
44,280 | Distribution of 2X - Y when X and Y are known | For two variables $X$ and $Y$, the variance of $aX-bY$ is:
$$Var(aX-bY)=a^2Var(X)+b^2Var(Y)-2ab·Cov(X,Y)$$
Where $Cov(X,Y)$ is the covariance between $X$ and $Y$. (source)
Usually the problem statement should say whether the variables $X$ and $Y$ are dependent or independent. Since this one apparently doesn't, I'm taki... | Distribution of 2X - Y when X and Y are known | For two variables $X$ and $Y$, the variance of $aX-bY$ is:
$$Var(aX-bY)=a^2Var(X)+b^2Var(Y)-2ab·Cov(X,Y)$$
Where $Cov(X,Y)$ is the covariance between $X$ and $Y$. (source)
Usually the problem statemen | Distribution of 2X - Y when X and Y are known
For two variables $X$ and $Y$, the variance of $aX-bY$ is:
$$Var(aX-bY)=a^2Var(X)+b^2Var(Y)-2ab·Cov(X,Y)$$
Where $Cov(X,Y)$ is the covariance between $X$ and $Y$. (source)
Usually the problem statement should say whether the variables $X$ and $Y$ are dependent or independen... | Distribution of 2X - Y when X and Y are known
For two variables $X$ and $Y$, the variance of $aX-bY$ is:
$$Var(aX-bY)=a^2Var(X)+b^2Var(Y)-2ab·Cov(X,Y)$$
Where $Cov(X,Y)$ is the covariance between $X$ and $Y$. (source)
Usually the problem statemen |
44,281 | Distribution of 2X - Y when X and Y are known | There are a few important results that are required here. Based on the provided answer, I think an assumption of the question was that $X$ and $Y$ are independent random variables.
Let's try and start by proving an important result
$$X\sim N(\mu,\sigma^{2})\Leftrightarrow aX\sim N(a\mu,a^{2}\sigma^{2})$$
We can do this... | Distribution of 2X - Y when X and Y are known | There are a few important results that are required here. Based on the provided answer, I think an assumption of the question was that $X$ and $Y$ are independent random variables.
Let's try and start | Distribution of 2X - Y when X and Y are known
There are a few important results that are required here. Based on the provided answer, I think an assumption of the question was that $X$ and $Y$ are independent random variables.
Let's try and start by proving an important result
$$X\sim N(\mu,\sigma^{2})\Leftrightarrow a... | Distribution of 2X - Y when X and Y are known
There are a few important results that are required here. Based on the provided answer, I think an assumption of the question was that $X$ and $Y$ are independent random variables.
Let's try and start |
44,282 | stratification in cox model | In a Cox model, stratification allows for as many different hazard functions as there are strata. Beta coefficients (hazard ratios) optimized for all strata are then fitted.
In your example, the model coxph(Surv(futime, fustat) ~ age + strata(rx) will output a hazard ratio for age in the presence of two (or more) hazar... | stratification in cox model | In a Cox model, stratification allows for as many different hazard functions as there are strata. Beta coefficients (hazard ratios) optimized for all strata are then fitted.
In your example, the model | stratification in cox model
In a Cox model, stratification allows for as many different hazard functions as there are strata. Beta coefficients (hazard ratios) optimized for all strata are then fitted.
In your example, the model coxph(Surv(futime, fustat) ~ age + strata(rx) will output a hazard ratio for age in the pre... | stratification in cox model
In a Cox model, stratification allows for as many different hazard functions as there are strata. Beta coefficients (hazard ratios) optimized for all strata are then fitted.
In your example, the model |
44,283 | stratification in cox model | In addition to the good answer from Todd B, here's a process you can follow, and some more of the background math.
A reasonable process to follow
Fit a Cox model (fit.unstrat) with all of the covariates and no stratifying variables.
Use cox.zph(fit.unstrat) to check for violations of the proportional hazards assumpti... | stratification in cox model | In addition to the good answer from Todd B, here's a process you can follow, and some more of the background math.
A reasonable process to follow
Fit a Cox model (fit.unstrat) with all of the covaria | stratification in cox model
In addition to the good answer from Todd B, here's a process you can follow, and some more of the background math.
A reasonable process to follow
Fit a Cox model (fit.unstrat) with all of the covariates and no stratifying variables.
Use cox.zph(fit.unstrat) to check for violations of the p... | stratification in cox model
In addition to the good answer from Todd B, here's a process you can follow, and some more of the background math.
A reasonable process to follow
Fit a Cox model (fit.unstrat) with all of the covaria |
44,284 | How to interpret $\mathbb{P}(B|A)$ when $\mathbb{P}(A) = 0$ | No. First, probability is bounded in $[0, 1]$, so it cannot be infinite.
Notice that there are two cases where probability can be equal to zero:
When you are dealing with empty set $\Pr(\varnothing)=0$. So for example, if you ask "what is the probability that person being -31 years old dies in a car crash?", then it i... | How to interpret $\mathbb{P}(B|A)$ when $\mathbb{P}(A) = 0$ | No. First, probability is bounded in $[0, 1]$, so it cannot be infinite.
Notice that there are two cases where probability can be equal to zero:
When you are dealing with empty set $\Pr(\varnothing)= | How to interpret $\mathbb{P}(B|A)$ when $\mathbb{P}(A) = 0$
No. First, probability is bounded in $[0, 1]$, so it cannot be infinite.
Notice that there are two cases where probability can be equal to zero:
When you are dealing with empty set $\Pr(\varnothing)=0$. So for example, if you ask "what is the probability that... | How to interpret $\mathbb{P}(B|A)$ when $\mathbb{P}(A) = 0$
No. First, probability is bounded in $[0, 1]$, so it cannot be infinite.
Notice that there are two cases where probability can be equal to zero:
When you are dealing with empty set $\Pr(\varnothing)= |
44,285 | How to interpret $\mathbb{P}(B|A)$ when $\mathbb{P}(A) = 0$ | No (no such thing as infinite probability: a certain event has probability 1).
In fact if it is a probability of an event (and not of a continuous random variable) then it is an ill-posed problem: if $P(A)=0$, what is the probability of $B$ given that $A$ happened?
$P(B|A)=\frac{P(B\cap A)}{P(A)}=\frac{0}{0}$
So I th... | How to interpret $\mathbb{P}(B|A)$ when $\mathbb{P}(A) = 0$ | No (no such thing as infinite probability: a certain event has probability 1).
In fact if it is a probability of an event (and not of a continuous random variable) then it is an ill-posed problem: if | How to interpret $\mathbb{P}(B|A)$ when $\mathbb{P}(A) = 0$
No (no such thing as infinite probability: a certain event has probability 1).
In fact if it is a probability of an event (and not of a continuous random variable) then it is an ill-posed problem: if $P(A)=0$, what is the probability of $B$ given that $A$ hap... | How to interpret $\mathbb{P}(B|A)$ when $\mathbb{P}(A) = 0$
No (no such thing as infinite probability: a certain event has probability 1).
In fact if it is a probability of an event (and not of a continuous random variable) then it is an ill-posed problem: if |
44,286 | How to interpret $\mathbb{P}(B|A)$ when $\mathbb{P}(A) = 0$ | As far as I know (Stochastic lecture and quick wikipedia lookup), the conditional probability P(B|A) is not defined for P(A)=0. (rule of Bayes)
where A and B are events and P(B) ≠ 0.
So I would say no, not valid.
But if P(A) = 0 then the variables are independent anyways.. | How to interpret $\mathbb{P}(B|A)$ when $\mathbb{P}(A) = 0$ | As far as I know (Stochastic lecture and quick wikipedia lookup), the conditional probability P(B|A) is not defined for P(A)=0. (rule of Bayes)
where A and B are events and P(B) ≠ 0.
So I would say | How to interpret $\mathbb{P}(B|A)$ when $\mathbb{P}(A) = 0$
As far as I know (Stochastic lecture and quick wikipedia lookup), the conditional probability P(B|A) is not defined for P(A)=0. (rule of Bayes)
where A and B are events and P(B) ≠ 0.
So I would say no, not valid.
But if P(A) = 0 then the variables are indepe... | How to interpret $\mathbb{P}(B|A)$ when $\mathbb{P}(A) = 0$
As far as I know (Stochastic lecture and quick wikipedia lookup), the conditional probability P(B|A) is not defined for P(A)=0. (rule of Bayes)
where A and B are events and P(B) ≠ 0.
So I would say |
44,287 | general solution sum of two uniform random variables aY+bX=Z? | If we have a variable $X\sim U(0,1)$ and multiply it by $a$, then $aX\sim U(0,a)$.
Assume that we're dealing with independent continuous uniform on $(0,a)$ and $(0,b)$ respectively (with $a<b$)
(This assumption is not restrictive since we can obtain the general case from this easily.)
Then the joint density is $\frac{1... | general solution sum of two uniform random variables aY+bX=Z? | If we have a variable $X\sim U(0,1)$ and multiply it by $a$, then $aX\sim U(0,a)$.
Assume that we're dealing with independent continuous uniform on $(0,a)$ and $(0,b)$ respectively (with $a<b$)
(This | general solution sum of two uniform random variables aY+bX=Z?
If we have a variable $X\sim U(0,1)$ and multiply it by $a$, then $aX\sim U(0,a)$.
Assume that we're dealing with independent continuous uniform on $(0,a)$ and $(0,b)$ respectively (with $a<b$)
(This assumption is not restrictive since we can obtain the gene... | general solution sum of two uniform random variables aY+bX=Z?
If we have a variable $X\sim U(0,1)$ and multiply it by $a$, then $aX\sim U(0,a)$.
Assume that we're dealing with independent continuous uniform on $(0,a)$ and $(0,b)$ respectively (with $a<b$)
(This |
44,288 | general solution sum of two uniform random variables aY+bX=Z? | Define $Y'=aY$ and $X'=bX$, find their distribution, and you are back to the problem you know how to solve: $Y'+X'=Z$ (convolution). | general solution sum of two uniform random variables aY+bX=Z? | Define $Y'=aY$ and $X'=bX$, find their distribution, and you are back to the problem you know how to solve: $Y'+X'=Z$ (convolution). | general solution sum of two uniform random variables aY+bX=Z?
Define $Y'=aY$ and $X'=bX$, find their distribution, and you are back to the problem you know how to solve: $Y'+X'=Z$ (convolution). | general solution sum of two uniform random variables aY+bX=Z?
Define $Y'=aY$ and $X'=bX$, find their distribution, and you are back to the problem you know how to solve: $Y'+X'=Z$ (convolution). |
44,289 | Linear Transformation of Gaussian Random Variable | I deprecate your dreadful notation, it will not help you achieve
any kind of rigor at all.
If $\mathbf X$ is a (jointly continuous) vector random variable with density function $f_{\mathbf X}({\mathbf x})$, then with $\mathbf G$
being an invertible matrix, the density function of
$\mathbf{Y} =\mathbf{XG}$ is
$$f_{\ma... | Linear Transformation of Gaussian Random Variable | I deprecate your dreadful notation, it will not help you achieve
any kind of rigor at all.
If $\mathbf X$ is a (jointly continuous) vector random variable with density function $f_{\mathbf X}({\mathbf | Linear Transformation of Gaussian Random Variable
I deprecate your dreadful notation, it will not help you achieve
any kind of rigor at all.
If $\mathbf X$ is a (jointly continuous) vector random variable with density function $f_{\mathbf X}({\mathbf x})$, then with $\mathbf G$
being an invertible matrix, the density ... | Linear Transformation of Gaussian Random Variable
I deprecate your dreadful notation, it will not help you achieve
any kind of rigor at all.
If $\mathbf X$ is a (jointly continuous) vector random variable with density function $f_{\mathbf X}({\mathbf |
44,290 | Linear Transformation of Gaussian Random Variable | This can be shown very succinctly by using the characteristic function of distributions. Let $\phi_X(t) = E[ \exp(i t ^ \mathsf T X) ]$ be the characteristic function of a random variable $X \in \mathbb R^n$.
If $x$ is normally distributed $x \sim \mathcal N(\mu, \Sigma)$, then we have $\phi_x(t) = \exp \Big (i t ^ \ma... | Linear Transformation of Gaussian Random Variable | This can be shown very succinctly by using the characteristic function of distributions. Let $\phi_X(t) = E[ \exp(i t ^ \mathsf T X) ]$ be the characteristic function of a random variable $X \in \math | Linear Transformation of Gaussian Random Variable
This can be shown very succinctly by using the characteristic function of distributions. Let $\phi_X(t) = E[ \exp(i t ^ \mathsf T X) ]$ be the characteristic function of a random variable $X \in \mathbb R^n$.
If $x$ is normally distributed $x \sim \mathcal N(\mu, \Sigma... | Linear Transformation of Gaussian Random Variable
This can be shown very succinctly by using the characteristic function of distributions. Let $\phi_X(t) = E[ \exp(i t ^ \mathsf T X) ]$ be the characteristic function of a random variable $X \in \math |
44,291 | Intuition for this observation//how restrictive is this assumption? | The OP really seeks the distributions with log-concave density functions -the quotient he differentiates is the reciprocal of the one we examine in order to determine the log-concavity or log-convexity of a function.
Specifically:
For $f$ to be log-concave, it means that $\ln f$ is concave, for which we require that
... | Intuition for this observation//how restrictive is this assumption? | The OP really seeks the distributions with log-concave density functions -the quotient he differentiates is the reciprocal of the one we examine in order to determine the log-concavity or log-convexit | Intuition for this observation//how restrictive is this assumption?
The OP really seeks the distributions with log-concave density functions -the quotient he differentiates is the reciprocal of the one we examine in order to determine the log-concavity or log-convexity of a function.
Specifically:
For $f$ to be log-co... | Intuition for this observation//how restrictive is this assumption?
The OP really seeks the distributions with log-concave density functions -the quotient he differentiates is the reciprocal of the one we examine in order to determine the log-concavity or log-convexit |
44,292 | Intuition for this observation//how restrictive is this assumption? | Let $f(x)=F^\prime(x)$. Since
$$\frac{d}{dx} \frac{F^\prime(x)}{F^{\prime\prime}(x)} = \frac{d}{dx}\left(1/\frac{f^\prime(x)}{f(x)}\right)= \frac{d}{dx}\left(1/\frac{d}{dx}\log(f(x))\right)=-\frac{\frac{d^2}{dx^2}\log(x)}{{(\cdots)}^2},$$ the positivity of the left hand side assures the negativity of the second deriva... | Intuition for this observation//how restrictive is this assumption? | Let $f(x)=F^\prime(x)$. Since
$$\frac{d}{dx} \frac{F^\prime(x)}{F^{\prime\prime}(x)} = \frac{d}{dx}\left(1/\frac{f^\prime(x)}{f(x)}\right)= \frac{d}{dx}\left(1/\frac{d}{dx}\log(f(x))\right)=-\frac{\f | Intuition for this observation//how restrictive is this assumption?
Let $f(x)=F^\prime(x)$. Since
$$\frac{d}{dx} \frac{F^\prime(x)}{F^{\prime\prime}(x)} = \frac{d}{dx}\left(1/\frac{f^\prime(x)}{f(x)}\right)= \frac{d}{dx}\left(1/\frac{d}{dx}\log(f(x))\right)=-\frac{\frac{d^2}{dx^2}\log(x)}{{(\cdots)}^2},$$ the positivi... | Intuition for this observation//how restrictive is this assumption?
Let $f(x)=F^\prime(x)$. Since
$$\frac{d}{dx} \frac{F^\prime(x)}{F^{\prime\prime}(x)} = \frac{d}{dx}\left(1/\frac{f^\prime(x)}{f(x)}\right)= \frac{d}{dx}\left(1/\frac{d}{dx}\log(f(x))\right)=-\frac{\f |
44,293 | Fitted Confidence Intervals Forecast Function R | Presumably you mean prediction intervals rather than confidence intervals.
The fitted values are in-sample one-step forecasts. Assuming normally distributed errors, 95% prediction intervals are given by
$$\hat{y}_t \pm 1.96 \hat{\sigma}$$
where $\hat\sigma^2$ is the estimated variance of the residuals.
Here is an examp... | Fitted Confidence Intervals Forecast Function R | Presumably you mean prediction intervals rather than confidence intervals.
The fitted values are in-sample one-step forecasts. Assuming normally distributed errors, 95% prediction intervals are given | Fitted Confidence Intervals Forecast Function R
Presumably you mean prediction intervals rather than confidence intervals.
The fitted values are in-sample one-step forecasts. Assuming normally distributed errors, 95% prediction intervals are given by
$$\hat{y}_t \pm 1.96 \hat{\sigma}$$
where $\hat\sigma^2$ is the estim... | Fitted Confidence Intervals Forecast Function R
Presumably you mean prediction intervals rather than confidence intervals.
The fitted values are in-sample one-step forecasts. Assuming normally distributed errors, 95% prediction intervals are given |
44,294 | How to explain degrees of freedom term to a layman? [duplicate] | You get the next 7 days off work, but you use the first day planning the rest of the days, so you have 6 days free.
"the number of observations minus the number of necessary relations among these observations." -Walker
Degrees of freedom can be very complex though and contrary to popular belief, see this answer for mor... | How to explain degrees of freedom term to a layman? [duplicate] | You get the next 7 days off work, but you use the first day planning the rest of the days, so you have 6 days free.
"the number of observations minus the number of necessary relations among these obse | How to explain degrees of freedom term to a layman? [duplicate]
You get the next 7 days off work, but you use the first day planning the rest of the days, so you have 6 days free.
"the number of observations minus the number of necessary relations among these observations." -Walker
Degrees of freedom can be very comple... | How to explain degrees of freedom term to a layman? [duplicate]
You get the next 7 days off work, but you use the first day planning the rest of the days, so you have 6 days free.
"the number of observations minus the number of necessary relations among these obse |
44,295 | How to explain degrees of freedom term to a layman? [duplicate] | Definition for Layman
Degrees of freedom is the number of values that are free to vary when the value of some statistic, like $\bar{X}$ or $\hat{\sigma}^2$, is known. In other words, it is the number of values that need to be known in order to know all of the values.
I will provide two examples. The first example is a... | How to explain degrees of freedom term to a layman? [duplicate] | Definition for Layman
Degrees of freedom is the number of values that are free to vary when the value of some statistic, like $\bar{X}$ or $\hat{\sigma}^2$, is known. In other words, it is the number | How to explain degrees of freedom term to a layman? [duplicate]
Definition for Layman
Degrees of freedom is the number of values that are free to vary when the value of some statistic, like $\bar{X}$ or $\hat{\sigma}^2$, is known. In other words, it is the number of values that need to be known in order to know all of... | How to explain degrees of freedom term to a layman? [duplicate]
Definition for Layman
Degrees of freedom is the number of values that are free to vary when the value of some statistic, like $\bar{X}$ or $\hat{\sigma}^2$, is known. In other words, it is the number |
44,296 | How to explain degrees of freedom term to a layman? [duplicate] | The Wikipedia article Degrees of freedom (statistics) is pretty good at explaining it (see the first few paragraphs).
Imagine you have some system or a black box which behavior is defined by a number of values (or parameters). The concrete values of the parameters set one particular mode of operation. Another set of pa... | How to explain degrees of freedom term to a layman? [duplicate] | The Wikipedia article Degrees of freedom (statistics) is pretty good at explaining it (see the first few paragraphs).
Imagine you have some system or a black box which behavior is defined by a number | How to explain degrees of freedom term to a layman? [duplicate]
The Wikipedia article Degrees of freedom (statistics) is pretty good at explaining it (see the first few paragraphs).
Imagine you have some system or a black box which behavior is defined by a number of values (or parameters). The concrete values of the pa... | How to explain degrees of freedom term to a layman? [duplicate]
The Wikipedia article Degrees of freedom (statistics) is pretty good at explaining it (see the first few paragraphs).
Imagine you have some system or a black box which behavior is defined by a number |
44,297 | Using years when calculating linear regression? | In principle, it doesn't matter - only the intercept term will be affected. Say that you want to estimate the regression Y = a + bX + e. Remember that the slope coefficient can be calculated as b = Cov(Y, X) / Var(X), and a = Ym - bXm, where Ym and Xm are the sample means of the respective variables. Now, let's add a c... | Using years when calculating linear regression? | In principle, it doesn't matter - only the intercept term will be affected. Say that you want to estimate the regression Y = a + bX + e. Remember that the slope coefficient can be calculated as b = Co | Using years when calculating linear regression?
In principle, it doesn't matter - only the intercept term will be affected. Say that you want to estimate the regression Y = a + bX + e. Remember that the slope coefficient can be calculated as b = Cov(Y, X) / Var(X), and a = Ym - bXm, where Ym and Xm are the sample means... | Using years when calculating linear regression?
In principle, it doesn't matter - only the intercept term will be affected. Say that you want to estimate the regression Y = a + bX + e. Remember that the slope coefficient can be calculated as b = Co |
44,298 | Using years when calculating linear regression? | The second series can be written as the first one minus 2008.
Ought it to make a difference to how we think unemployment changes over time when we start counting years from—the birth of Christ or the start of the data series?
Look at the least-squares equations & try to work out the effect of subtracting a constant fr... | Using years when calculating linear regression? | The second series can be written as the first one minus 2008.
Ought it to make a difference to how we think unemployment changes over time when we start counting years from—the birth of Christ or the | Using years when calculating linear regression?
The second series can be written as the first one minus 2008.
Ought it to make a difference to how we think unemployment changes over time when we start counting years from—the birth of Christ or the start of the data series?
Look at the least-squares equations & try to ... | Using years when calculating linear regression?
The second series can be written as the first one minus 2008.
Ought it to make a difference to how we think unemployment changes over time when we start counting years from—the birth of Christ or the |
44,299 | Using years when calculating linear regression? | a) If you're sure (or in this case specifically told) you only need to use year as linear variable (no interactions, no quadratic terms, no other terms), and you only have one timeseries, then in this case it doesn't make any difference (just results in a constant offset). So might as well use year as is.
b) In general... | Using years when calculating linear regression? | a) If you're sure (or in this case specifically told) you only need to use year as linear variable (no interactions, no quadratic terms, no other terms), and you only have one timeseries, then in this | Using years when calculating linear regression?
a) If you're sure (or in this case specifically told) you only need to use year as linear variable (no interactions, no quadratic terms, no other terms), and you only have one timeseries, then in this case it doesn't make any difference (just results in a constant offset)... | Using years when calculating linear regression?
a) If you're sure (or in this case specifically told) you only need to use year as linear variable (no interactions, no quadratic terms, no other terms), and you only have one timeseries, then in this |
44,300 | Using years when calculating linear regression? | Yes, you can use years as the predictor variable in linear regression. The basic code would be Outcome = Year. The beta coefficient from such a model would allow you to predict the outcome for an unobserved year.
It is important to remember that the p-value for the beta coefficient is testing whether there is a linear... | Using years when calculating linear regression? | Yes, you can use years as the predictor variable in linear regression. The basic code would be Outcome = Year. The beta coefficient from such a model would allow you to predict the outcome for an unob | Using years when calculating linear regression?
Yes, you can use years as the predictor variable in linear regression. The basic code would be Outcome = Year. The beta coefficient from such a model would allow you to predict the outcome for an unobserved year.
It is important to remember that the p-value for the beta ... | Using years when calculating linear regression?
Yes, you can use years as the predictor variable in linear regression. The basic code would be Outcome = Year. The beta coefficient from such a model would allow you to predict the outcome for an unob |
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