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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5,001 | Cumming (2008) claims that distribution of p-values obtained in replications depends only on the original p-value. How can it be true? | Thanks for all the interesting discussions! When writing that 2008 article, it took me a while to convince myself that the distribution of replication p (the p value given by an exact replication of a study, meaning a study that is exactly the same, but with a new sample) is dependent only on p given by the original st... | Cumming (2008) claims that distribution of p-values obtained in replications depends only on the ori | Thanks for all the interesting discussions! When writing that 2008 article, it took me a while to convince myself that the distribution of replication p (the p value given by an exact replication of a | Cumming (2008) claims that distribution of p-values obtained in replications depends only on the original p-value. How can it be true?
Thanks for all the interesting discussions! When writing that 2008 article, it took me a while to convince myself that the distribution of replication p (the p value given by an exact r... | Cumming (2008) claims that distribution of p-values obtained in replications depends only on the ori
Thanks for all the interesting discussions! When writing that 2008 article, it took me a while to convince myself that the distribution of replication p (the p value given by an exact replication of a |
5,002 | Cumming (2008) claims that distribution of p-values obtained in replications depends only on the original p-value. How can it be true? | The issue has been clarified by @GeoMatt22, and I've been delighted to see @GeoffCumming coming here to participate in the discussion. I am posting this answer as a further commentary.
As it turns out, this discussion goes back at least to Goodman (1992) A comment on replication, P‐values and evidence and a later repl... | Cumming (2008) claims that distribution of p-values obtained in replications depends only on the ori | The issue has been clarified by @GeoMatt22, and I've been delighted to see @GeoffCumming coming here to participate in the discussion. I am posting this answer as a further commentary.
As it turns ou | Cumming (2008) claims that distribution of p-values obtained in replications depends only on the original p-value. How can it be true?
The issue has been clarified by @GeoMatt22, and I've been delighted to see @GeoffCumming coming here to participate in the discussion. I am posting this answer as a further commentary.
... | Cumming (2008) claims that distribution of p-values obtained in replications depends only on the ori
The issue has been clarified by @GeoMatt22, and I've been delighted to see @GeoffCumming coming here to participate in the discussion. I am posting this answer as a further commentary.
As it turns ou |
5,003 | Cumming (2008) claims that distribution of p-values obtained in replications depends only on the original p-value. How can it be true? | Thanks everyone for further interesting discussion. Rather than making my comments, point by point, I’ll offer some general reflections.
Bayes. I have nothing at all against Bayesian approaches. From the beginning I’ve expected that a Bayesian analysis, assuming a flat or diffuse prior, would give the same or very simi... | Cumming (2008) claims that distribution of p-values obtained in replications depends only on the ori | Thanks everyone for further interesting discussion. Rather than making my comments, point by point, I’ll offer some general reflections.
Bayes. I have nothing at all against Bayesian approaches. From | Cumming (2008) claims that distribution of p-values obtained in replications depends only on the original p-value. How can it be true?
Thanks everyone for further interesting discussion. Rather than making my comments, point by point, I’ll offer some general reflections.
Bayes. I have nothing at all against Bayesian ap... | Cumming (2008) claims that distribution of p-values obtained in replications depends only on the ori
Thanks everyone for further interesting discussion. Rather than making my comments, point by point, I’ll offer some general reflections.
Bayes. I have nothing at all against Bayesian approaches. From |
5,004 | Derive Variance of regression coefficient in simple linear regression | At the start of your derivation you multiply out the brackets $\sum_i (x_i - \bar{x})(y_i - \bar{y})$, in the process expanding both $y_i$ and $\bar{y}$. The former depends on the sum variable $i$, whereas the latter doesn't. If you leave $\bar{y}$ as is, the derivation is a lot simpler, because
\begin{align}
\sum_i (x... | Derive Variance of regression coefficient in simple linear regression | At the start of your derivation you multiply out the brackets $\sum_i (x_i - \bar{x})(y_i - \bar{y})$, in the process expanding both $y_i$ and $\bar{y}$. The former depends on the sum variable $i$, wh | Derive Variance of regression coefficient in simple linear regression
At the start of your derivation you multiply out the brackets $\sum_i (x_i - \bar{x})(y_i - \bar{y})$, in the process expanding both $y_i$ and $\bar{y}$. The former depends on the sum variable $i$, whereas the latter doesn't. If you leave $\bar{y}$ a... | Derive Variance of regression coefficient in simple linear regression
At the start of your derivation you multiply out the brackets $\sum_i (x_i - \bar{x})(y_i - \bar{y})$, in the process expanding both $y_i$ and $\bar{y}$. The former depends on the sum variable $i$, wh |
5,005 | Derive Variance of regression coefficient in simple linear regression | I believe the problem in your proof is the step where you take the expected value of the square of $\sum_i (x_i - \bar{x} )\left( u_i -\sum_j \frac{u_j}{n} \right)$. This is of the form $E \left[\left(\sum_i a_i b_i \right)^2 \right]$, where $a_i = x_i -\bar{x}; b_i = u_i -\sum_j \frac{u_j}{n}$. So, upon squaring, we g... | Derive Variance of regression coefficient in simple linear regression | I believe the problem in your proof is the step where you take the expected value of the square of $\sum_i (x_i - \bar{x} )\left( u_i -\sum_j \frac{u_j}{n} \right)$. This is of the form $E \left[\left | Derive Variance of regression coefficient in simple linear regression
I believe the problem in your proof is the step where you take the expected value of the square of $\sum_i (x_i - \bar{x} )\left( u_i -\sum_j \frac{u_j}{n} \right)$. This is of the form $E \left[\left(\sum_i a_i b_i \right)^2 \right]$, where $a_i = x... | Derive Variance of regression coefficient in simple linear regression
I believe the problem in your proof is the step where you take the expected value of the square of $\sum_i (x_i - \bar{x} )\left( u_i -\sum_j \frac{u_j}{n} \right)$. This is of the form $E \left[\left |
5,006 | Derive Variance of regression coefficient in simple linear regression | Begin from "The derivation is as follow:"
The 7th "=" is wrong.
Because
$\sum_i (x_i - \bar{x})(u_i - \bar{u})$
$ = \sum_i (x_i - \bar{x})u_i - \sum_i (x_i - \bar{x}) \bar{u}$
$ = \sum_i (x_i - \bar{x})u_i - \bar{u} \sum_i (x_i - \bar{x})$
$ = \sum_i (x_i - \bar{x})u_i - \bar{u} (\sum_i{x_i} -n \bar{x})$
$ = \sum_i (... | Derive Variance of regression coefficient in simple linear regression | Begin from "The derivation is as follow:"
The 7th "=" is wrong.
Because
$\sum_i (x_i - \bar{x})(u_i - \bar{u})$
$ = \sum_i (x_i - \bar{x})u_i - \sum_i (x_i - \bar{x}) \bar{u}$
$ = \sum_i (x_i - \bar | Derive Variance of regression coefficient in simple linear regression
Begin from "The derivation is as follow:"
The 7th "=" is wrong.
Because
$\sum_i (x_i - \bar{x})(u_i - \bar{u})$
$ = \sum_i (x_i - \bar{x})u_i - \sum_i (x_i - \bar{x}) \bar{u}$
$ = \sum_i (x_i - \bar{x})u_i - \bar{u} \sum_i (x_i - \bar{x})$
$ = \sum... | Derive Variance of regression coefficient in simple linear regression
Begin from "The derivation is as follow:"
The 7th "=" is wrong.
Because
$\sum_i (x_i - \bar{x})(u_i - \bar{u})$
$ = \sum_i (x_i - \bar{x})u_i - \sum_i (x_i - \bar{x}) \bar{u}$
$ = \sum_i (x_i - \bar |
5,007 | Determining sample size necessary for bootstrap method / Proposed Method | I took interest in this question because I saw the word bootstrap and I have written books on the bootstrap. Also people often ask "How many bootstrap samples do I need to get a good Monte Carlo approximation to the bootstrap result?" My suggested answer to that question is to keep increasing the size until you get c... | Determining sample size necessary for bootstrap method / Proposed Method | I took interest in this question because I saw the word bootstrap and I have written books on the bootstrap. Also people often ask "How many bootstrap samples do I need to get a good Monte Carlo appr | Determining sample size necessary for bootstrap method / Proposed Method
I took interest in this question because I saw the word bootstrap and I have written books on the bootstrap. Also people often ask "How many bootstrap samples do I need to get a good Monte Carlo approximation to the bootstrap result?" My suggest... | Determining sample size necessary for bootstrap method / Proposed Method
I took interest in this question because I saw the word bootstrap and I have written books on the bootstrap. Also people often ask "How many bootstrap samples do I need to get a good Monte Carlo appr |
5,008 | Determining sample size necessary for bootstrap method / Proposed Method | The resampling process creates many possible samples that a study could have drawn. The various combinations of values in the simulated samples collectively provide an estimate of the variability between random samples drawn from the same population. The range of these potential samples allows the procedure to construc... | Determining sample size necessary for bootstrap method / Proposed Method | The resampling process creates many possible samples that a study could have drawn. The various combinations of values in the simulated samples collectively provide an estimate of the variability betw | Determining sample size necessary for bootstrap method / Proposed Method
The resampling process creates many possible samples that a study could have drawn. The various combinations of values in the simulated samples collectively provide an estimate of the variability between random samples drawn from the same populati... | Determining sample size necessary for bootstrap method / Proposed Method
The resampling process creates many possible samples that a study could have drawn. The various combinations of values in the simulated samples collectively provide an estimate of the variability betw |
5,009 | What algorithm is used in linear regression? | Regarding the question in the title, about what is the algorithm that is used:
In a linear algebra perspective, the linear regression algorithm is the way to solve a linear system $\mathbf{A}x=b$ with more equations than unknowns. In most of the cases there is no solution to this problem. And this is because the vector... | What algorithm is used in linear regression? | Regarding the question in the title, about what is the algorithm that is used:
In a linear algebra perspective, the linear regression algorithm is the way to solve a linear system $\mathbf{A}x=b$ with | What algorithm is used in linear regression?
Regarding the question in the title, about what is the algorithm that is used:
In a linear algebra perspective, the linear regression algorithm is the way to solve a linear system $\mathbf{A}x=b$ with more equations than unknowns. In most of the cases there is no solution to... | What algorithm is used in linear regression?
Regarding the question in the title, about what is the algorithm that is used:
In a linear algebra perspective, the linear regression algorithm is the way to solve a linear system $\mathbf{A}x=b$ with |
5,010 | What algorithm is used in linear regression? | To answer the letter of the question, "ordinary least squares" is not an algorithm; rather it is a type of problem in computational linear algebra, of which linear regression is one example. Usually one has data $\{(x_1,y_1),\dots,(x_m,y_m)\}$ and a tentative function ("model") to fit the data against, of the form $f(x... | What algorithm is used in linear regression? | To answer the letter of the question, "ordinary least squares" is not an algorithm; rather it is a type of problem in computational linear algebra, of which linear regression is one example. Usually o | What algorithm is used in linear regression?
To answer the letter of the question, "ordinary least squares" is not an algorithm; rather it is a type of problem in computational linear algebra, of which linear regression is one example. Usually one has data $\{(x_1,y_1),\dots,(x_m,y_m)\}$ and a tentative function ("mode... | What algorithm is used in linear regression?
To answer the letter of the question, "ordinary least squares" is not an algorithm; rather it is a type of problem in computational linear algebra, of which linear regression is one example. Usually o |
5,011 | What algorithm is used in linear regression? | The wiki link: Estimation Methods for Linear Regression gives a fairly comprehensive list of estimation methods including OLS and the contexts in which alternative estimation methods are used. | What algorithm is used in linear regression? | The wiki link: Estimation Methods for Linear Regression gives a fairly comprehensive list of estimation methods including OLS and the contexts in which alternative estimation methods are used. | What algorithm is used in linear regression?
The wiki link: Estimation Methods for Linear Regression gives a fairly comprehensive list of estimation methods including OLS and the contexts in which alternative estimation methods are used. | What algorithm is used in linear regression?
The wiki link: Estimation Methods for Linear Regression gives a fairly comprehensive list of estimation methods including OLS and the contexts in which alternative estimation methods are used. |
5,012 | What algorithm is used in linear regression? | It is easy to get confused between definitions and terminology. Both terms are used, sometimes interchangeably. A quick lookup on Wikipedia should help:
ordinary least squares
lnear regression
Ordinary Least Squares (OLS) is a method used to fit linear regression models. Because of the demonstrable consistency and ... | What algorithm is used in linear regression? | It is easy to get confused between definitions and terminology. Both terms are used, sometimes interchangeably. A quick lookup on Wikipedia should help:
ordinary least squares
lnear regression
Ord | What algorithm is used in linear regression?
It is easy to get confused between definitions and terminology. Both terms are used, sometimes interchangeably. A quick lookup on Wikipedia should help:
ordinary least squares
lnear regression
Ordinary Least Squares (OLS) is a method used to fit linear regression models.... | What algorithm is used in linear regression?
It is easy to get confused between definitions and terminology. Both terms are used, sometimes interchangeably. A quick lookup on Wikipedia should help:
ordinary least squares
lnear regression
Ord |
5,013 | What algorithm is used in linear regression? | I tend to think of 'least squares' as a criterion for defining the best fitting regression line (i.e., that which makes the sum of 'squared' residuals 'least') and the 'algorithm' in this context as the set of steps used to determine the regression coefficients that satisfy that criterion. This distinction suggests tha... | What algorithm is used in linear regression? | I tend to think of 'least squares' as a criterion for defining the best fitting regression line (i.e., that which makes the sum of 'squared' residuals 'least') and the 'algorithm' in this context as t | What algorithm is used in linear regression?
I tend to think of 'least squares' as a criterion for defining the best fitting regression line (i.e., that which makes the sum of 'squared' residuals 'least') and the 'algorithm' in this context as the set of steps used to determine the regression coefficients that satisfy ... | What algorithm is used in linear regression?
I tend to think of 'least squares' as a criterion for defining the best fitting regression line (i.e., that which makes the sum of 'squared' residuals 'least') and the 'algorithm' in this context as t |
5,014 | What algorithm is used in linear regression? | An old book, yet one I find myself repeatedly turning to, is
Lawson, C.L. and Hanson, R.J. Solving Least Squares Problems, Prentice-Hall, 1974.
It contains a detailed and very readable discussion of some of the algorithms that previous answers have mentioned. You might want to look at it. | What algorithm is used in linear regression? | An old book, yet one I find myself repeatedly turning to, is
Lawson, C.L. and Hanson, R.J. Solving Least Squares Problems, Prentice-Hall, 1974.
It contains a detailed and very readable discussion of | What algorithm is used in linear regression?
An old book, yet one I find myself repeatedly turning to, is
Lawson, C.L. and Hanson, R.J. Solving Least Squares Problems, Prentice-Hall, 1974.
It contains a detailed and very readable discussion of some of the algorithms that previous answers have mentioned. You might want... | What algorithm is used in linear regression?
An old book, yet one I find myself repeatedly turning to, is
Lawson, C.L. and Hanson, R.J. Solving Least Squares Problems, Prentice-Hall, 1974.
It contains a detailed and very readable discussion of |
5,015 | What references should be cited to support using 30 as a large enough sample size? | The choice of n = 30 for a boundary between small and large samples is a rule of thumb, only. There is a large number of books that quote (around) this value, for example, Hogg and Tanis' Probability and Statistical Inference (7e) says "greater than 25 or 30".
That said, the story told to me was that the only reason 30... | What references should be cited to support using 30 as a large enough sample size? | The choice of n = 30 for a boundary between small and large samples is a rule of thumb, only. There is a large number of books that quote (around) this value, for example, Hogg and Tanis' Probability | What references should be cited to support using 30 as a large enough sample size?
The choice of n = 30 for a boundary between small and large samples is a rule of thumb, only. There is a large number of books that quote (around) this value, for example, Hogg and Tanis' Probability and Statistical Inference (7e) says "... | What references should be cited to support using 30 as a large enough sample size?
The choice of n = 30 for a boundary between small and large samples is a rule of thumb, only. There is a large number of books that quote (around) this value, for example, Hogg and Tanis' Probability |
5,016 | What references should be cited to support using 30 as a large enough sample size? | Actually, the "magic number" 30 is a fallacy. See Jacob's Cohen's delightful paper, Things I Have Learned (So Far) (Am. Psych. December 1990 45 #12, pp 1304-1312). This myth is his first example of how "some things you learn aren't so".
[O]ne of my fellow doctoral candidates undertook a dissertation [with] a sample of... | What references should be cited to support using 30 as a large enough sample size? | Actually, the "magic number" 30 is a fallacy. See Jacob's Cohen's delightful paper, Things I Have Learned (So Far) (Am. Psych. December 1990 45 #12, pp 1304-1312). This myth is his first example of ho | What references should be cited to support using 30 as a large enough sample size?
Actually, the "magic number" 30 is a fallacy. See Jacob's Cohen's delightful paper, Things I Have Learned (So Far) (Am. Psych. December 1990 45 #12, pp 1304-1312). This myth is his first example of how "some things you learn aren't so".
... | What references should be cited to support using 30 as a large enough sample size?
Actually, the "magic number" 30 is a fallacy. See Jacob's Cohen's delightful paper, Things I Have Learned (So Far) (Am. Psych. December 1990 45 #12, pp 1304-1312). This myth is his first example of ho |
5,017 | What references should be cited to support using 30 as a large enough sample size? | Mostly arbitrary rule of thumb. This statement depends on a number of factor to be true. For example on the distribution of the data. If the data comes from a Cauchy for example, even 30^30 observations are not enough to estimate the mean (in that case even an infinite number of observations would not be enough to caus... | What references should be cited to support using 30 as a large enough sample size? | Mostly arbitrary rule of thumb. This statement depends on a number of factor to be true. For example on the distribution of the data. If the data comes from a Cauchy for example, even 30^30 observatio | What references should be cited to support using 30 as a large enough sample size?
Mostly arbitrary rule of thumb. This statement depends on a number of factor to be true. For example on the distribution of the data. If the data comes from a Cauchy for example, even 30^30 observations are not enough to estimate the mea... | What references should be cited to support using 30 as a large enough sample size?
Mostly arbitrary rule of thumb. This statement depends on a number of factor to be true. For example on the distribution of the data. If the data comes from a Cauchy for example, even 30^30 observatio |
5,018 | What references should be cited to support using 30 as a large enough sample size? | IMO, it all depends on what you want to use your sample for. Two "silly" examples to illustrate what I mean: If you need to estimate a mean, 30 observations is more than enough. If you need to estimate a linear regression with 100 predictors, 30 observations will not be close to enough. | What references should be cited to support using 30 as a large enough sample size? | IMO, it all depends on what you want to use your sample for. Two "silly" examples to illustrate what I mean: If you need to estimate a mean, 30 observations is more than enough. If you need to estim | What references should be cited to support using 30 as a large enough sample size?
IMO, it all depends on what you want to use your sample for. Two "silly" examples to illustrate what I mean: If you need to estimate a mean, 30 observations is more than enough. If you need to estimate a linear regression with 100 pred... | What references should be cited to support using 30 as a large enough sample size?
IMO, it all depends on what you want to use your sample for. Two "silly" examples to illustrate what I mean: If you need to estimate a mean, 30 observations is more than enough. If you need to estim |
5,019 | What references should be cited to support using 30 as a large enough sample size? | This is meant to supplement user1108's answer stating that:
That said, the story told to me was that the only reason 30 was regarded as a good boundary was because it made for pretty Student's t tables in the back of textbooks to fit nicely on one page. That, and the critical values (between Student's t and Normal) ar... | What references should be cited to support using 30 as a large enough sample size? | This is meant to supplement user1108's answer stating that:
That said, the story told to me was that the only reason 30 was regarded as a good boundary was because it made for pretty Student's t tabl | What references should be cited to support using 30 as a large enough sample size?
This is meant to supplement user1108's answer stating that:
That said, the story told to me was that the only reason 30 was regarded as a good boundary was because it made for pretty Student's t tables in the back of textbooks to fit ni... | What references should be cited to support using 30 as a large enough sample size?
This is meant to supplement user1108's answer stating that:
That said, the story told to me was that the only reason 30 was regarded as a good boundary was because it made for pretty Student's t tabl |
5,020 | Bootstrap vs. permutation hypothesis testing | Both are popular and useful, but primarily for different uses. The permutation test is best for testing hypotheses and bootstrapping is best for estimating confidence intervals.
Permutation tests test a specific null hypothesis of exchangeability, i.e. that only the random sampling/randomization explains the differenc... | Bootstrap vs. permutation hypothesis testing | Both are popular and useful, but primarily for different uses. The permutation test is best for testing hypotheses and bootstrapping is best for estimating confidence intervals.
Permutation tests tes | Bootstrap vs. permutation hypothesis testing
Both are popular and useful, but primarily for different uses. The permutation test is best for testing hypotheses and bootstrapping is best for estimating confidence intervals.
Permutation tests test a specific null hypothesis of exchangeability, i.e. that only the random ... | Bootstrap vs. permutation hypothesis testing
Both are popular and useful, but primarily for different uses. The permutation test is best for testing hypotheses and bootstrapping is best for estimating confidence intervals.
Permutation tests tes |
5,021 | Bootstrap vs. permutation hypothesis testing | If you are using R, then they are all easy to implement. See, for instance, http://www.burns-stat.com/pages/Tutor/bootstrap_resampling.html
I would say there is a third major technique: cross validation. This is used to test the predictive power of models. | Bootstrap vs. permutation hypothesis testing | If you are using R, then they are all easy to implement. See, for instance, http://www.burns-stat.com/pages/Tutor/bootstrap_resampling.html
I would say there is a third major technique: cross valida | Bootstrap vs. permutation hypothesis testing
If you are using R, then they are all easy to implement. See, for instance, http://www.burns-stat.com/pages/Tutor/bootstrap_resampling.html
I would say there is a third major technique: cross validation. This is used to test the predictive power of models. | Bootstrap vs. permutation hypothesis testing
If you are using R, then they are all easy to implement. See, for instance, http://www.burns-stat.com/pages/Tutor/bootstrap_resampling.html
I would say there is a third major technique: cross valida |
5,022 | Bootstrap vs. permutation hypothesis testing | My question is which resampling technique has gained the more popularity
Bootstrapping or permutation tests?
Bootstrapping is mostly about generating large sample standard errors or confidence intervals; permutation tests as the name suggests are mostly about testing. (Each can be adapted to be used for the other t... | Bootstrap vs. permutation hypothesis testing | My question is which resampling technique has gained the more popularity
Bootstrapping or permutation tests?
Bootstrapping is mostly about generating large sample standard errors or confidence int | Bootstrap vs. permutation hypothesis testing
My question is which resampling technique has gained the more popularity
Bootstrapping or permutation tests?
Bootstrapping is mostly about generating large sample standard errors or confidence intervals; permutation tests as the name suggests are mostly about testing. (E... | Bootstrap vs. permutation hypothesis testing
My question is which resampling technique has gained the more popularity
Bootstrapping or permutation tests?
Bootstrapping is mostly about generating large sample standard errors or confidence int |
5,023 | Why is "statistically significant" not enough? | Hypothesis testing versus parameter estimation
Typically, hypotheses are framed in a binary way. I'll put directional hypotheses to one side, as they don't change the issue much. It is common, at least in psychology, to talk about hypotheses such as: the difference between group means is or is not zero; the correlation... | Why is "statistically significant" not enough? | Hypothesis testing versus parameter estimation
Typically, hypotheses are framed in a binary way. I'll put directional hypotheses to one side, as they don't change the issue much. It is common, at leas | Why is "statistically significant" not enough?
Hypothesis testing versus parameter estimation
Typically, hypotheses are framed in a binary way. I'll put directional hypotheses to one side, as they don't change the issue much. It is common, at least in psychology, to talk about hypotheses such as: the difference between... | Why is "statistically significant" not enough?
Hypothesis testing versus parameter estimation
Typically, hypotheses are framed in a binary way. I'll put directional hypotheses to one side, as they don't change the issue much. It is common, at leas |
5,024 | Why is "statistically significant" not enough? | Just to add to the existing answers (which are great, by the way). It is important to be aware that statistical significance is a function of sample size.
When you get more and more data, you can find statistically significant differences wherever you look. When the amount of data is huge, even the tiniest effects can... | Why is "statistically significant" not enough? | Just to add to the existing answers (which are great, by the way). It is important to be aware that statistical significance is a function of sample size.
When you get more and more data, you can fin | Why is "statistically significant" not enough?
Just to add to the existing answers (which are great, by the way). It is important to be aware that statistical significance is a function of sample size.
When you get more and more data, you can find statistically significant differences wherever you look. When the amoun... | Why is "statistically significant" not enough?
Just to add to the existing answers (which are great, by the way). It is important to be aware that statistical significance is a function of sample size.
When you get more and more data, you can fin |
5,025 | Why is "statistically significant" not enough? | If there was a reasonable basis for suspecting your hypothesis might be true before you ran your study; and you ran a good study (e.g., you didn't induce any confounds); and your results were consistent with your hypothesis and statistically significant; then I think you are fine, as far as that goes.
However, you sh... | Why is "statistically significant" not enough? | If there was a reasonable basis for suspecting your hypothesis might be true before you ran your study; and you ran a good study (e.g., you didn't induce any confounds); and your results were consiste | Why is "statistically significant" not enough?
If there was a reasonable basis for suspecting your hypothesis might be true before you ran your study; and you ran a good study (e.g., you didn't induce any confounds); and your results were consistent with your hypothesis and statistically significant; then I think you a... | Why is "statistically significant" not enough?
If there was a reasonable basis for suspecting your hypothesis might be true before you ran your study; and you ran a good study (e.g., you didn't induce any confounds); and your results were consiste |
5,026 | Why is "statistically significant" not enough? | Before reporting this and this and this and this, start by formulating what do you want to learn from you experimental data.
The main problem with usual hypothesis tests (these tests we learn at school...) is not the binarity: the main problem is that these are tests for hypotheses which are not hypotheses of interest... | Why is "statistically significant" not enough? | Before reporting this and this and this and this, start by formulating what do you want to learn from you experimental data.
The main problem with usual hypothesis tests (these tests we learn at scho | Why is "statistically significant" not enough?
Before reporting this and this and this and this, start by formulating what do you want to learn from you experimental data.
The main problem with usual hypothesis tests (these tests we learn at school...) is not the binarity: the main problem is that these are tests for ... | Why is "statistically significant" not enough?
Before reporting this and this and this and this, start by formulating what do you want to learn from you experimental data.
The main problem with usual hypothesis tests (these tests we learn at scho |
5,027 | Why is "statistically significant" not enough? | I'm far from an expert on statistics, but one thing that has been emphasised in the stats courses I have done to date is the issue of "practical significance". I believe this alludes to what what Jeromy and gung are talking about when referring to "effect size".
We had an example in class of a 12 week diet that had st... | Why is "statistically significant" not enough? | I'm far from an expert on statistics, but one thing that has been emphasised in the stats courses I have done to date is the issue of "practical significance". I believe this alludes to what what Jero | Why is "statistically significant" not enough?
I'm far from an expert on statistics, but one thing that has been emphasised in the stats courses I have done to date is the issue of "practical significance". I believe this alludes to what what Jeromy and gung are talking about when referring to "effect size".
We had an... | Why is "statistically significant" not enough?
I'm far from an expert on statistics, but one thing that has been emphasised in the stats courses I have done to date is the issue of "practical significance". I believe this alludes to what what Jero |
5,028 | Why is "statistically significant" not enough? | This is impossible to answer accurately without knowing more details of your study and the person's criticism. But here's one possibility: if you've run multiple tests, and you choose to focus on the one that came out at p<0.05 and ignore others, then that "significance" has been diluted by the fact of your selective a... | Why is "statistically significant" not enough? | This is impossible to answer accurately without knowing more details of your study and the person's criticism. But here's one possibility: if you've run multiple tests, and you choose to focus on the | Why is "statistically significant" not enough?
This is impossible to answer accurately without knowing more details of your study and the person's criticism. But here's one possibility: if you've run multiple tests, and you choose to focus on the one that came out at p<0.05 and ignore others, then that "significance" h... | Why is "statistically significant" not enough?
This is impossible to answer accurately without knowing more details of your study and the person's criticism. But here's one possibility: if you've run multiple tests, and you choose to focus on the |
5,029 | Why is "statistically significant" not enough? | I suggest you read the following:
Anderson, D.R., Burnham, K.P., Thompson, W.L., 2000. Null hypothesis testing: Problems, prevalence, and an alternative. J. Wildl. Manage. 64, 912-923.
Gigerenzer, G., 2004. Mindless statistics. Journal of Socio-Economics 33, 587-606.
Johnson, D.H., 1999. The Insignificance of Statisti... | Why is "statistically significant" not enough? | I suggest you read the following:
Anderson, D.R., Burnham, K.P., Thompson, W.L., 2000. Null hypothesis testing: Problems, prevalence, and an alternative. J. Wildl. Manage. 64, 912-923.
Gigerenzer, G. | Why is "statistically significant" not enough?
I suggest you read the following:
Anderson, D.R., Burnham, K.P., Thompson, W.L., 2000. Null hypothesis testing: Problems, prevalence, and an alternative. J. Wildl. Manage. 64, 912-923.
Gigerenzer, G., 2004. Mindless statistics. Journal of Socio-Economics 33, 587-606.
John... | Why is "statistically significant" not enough?
I suggest you read the following:
Anderson, D.R., Burnham, K.P., Thompson, W.L., 2000. Null hypothesis testing: Problems, prevalence, and an alternative. J. Wildl. Manage. 64, 912-923.
Gigerenzer, G. |
5,030 | Statistical tests when sample size is 1 | Unfortunately, your student has a problem.
The idea of any (inferential) statistical analysis is to understand whether a pattern of observations can be simply due to natural variation or chance, or whether there is something systematic there. If the natural variation is large, then the observed difference may be simply... | Statistical tests when sample size is 1 | Unfortunately, your student has a problem.
The idea of any (inferential) statistical analysis is to understand whether a pattern of observations can be simply due to natural variation or chance, or wh | Statistical tests when sample size is 1
Unfortunately, your student has a problem.
The idea of any (inferential) statistical analysis is to understand whether a pattern of observations can be simply due to natural variation or chance, or whether there is something systematic there. If the natural variation is large, th... | Statistical tests when sample size is 1
Unfortunately, your student has a problem.
The idea of any (inferential) statistical analysis is to understand whether a pattern of observations can be simply due to natural variation or chance, or wh |
5,031 | Statistical tests when sample size is 1 | Two-way ANOVA with One Observation per Cell
After you finish your important 'lecture' about consulting a statistician before starting to take data, you can tell your student that there is barely enough data
here to support a legitimate experimental design.
If the subjects were chosen at random from some relevant
popul... | Statistical tests when sample size is 1 | Two-way ANOVA with One Observation per Cell
After you finish your important 'lecture' about consulting a statistician before starting to take data, you can tell your student that there is barely enou | Statistical tests when sample size is 1
Two-way ANOVA with One Observation per Cell
After you finish your important 'lecture' about consulting a statistician before starting to take data, you can tell your student that there is barely enough data
here to support a legitimate experimental design.
If the subjects were c... | Statistical tests when sample size is 1
Two-way ANOVA with One Observation per Cell
After you finish your important 'lecture' about consulting a statistician before starting to take data, you can tell your student that there is barely enou |
5,032 | Statistical tests when sample size is 1 | BruceET has described the proper analysis (Two-way ANOVA without interaction), so I'll put a more positive spin on the experiment.
I'm assuming that the design was three pairs, where there is variability between pairs. One of each pair was given insulin and the other without, hopefully randomized. Then each sample (p... | Statistical tests when sample size is 1 | BruceET has described the proper analysis (Two-way ANOVA without interaction), so I'll put a more positive spin on the experiment.
I'm assuming that the design was three pairs, where there is variabil | Statistical tests when sample size is 1
BruceET has described the proper analysis (Two-way ANOVA without interaction), so I'll put a more positive spin on the experiment.
I'm assuming that the design was three pairs, where there is variability between pairs. One of each pair was given insulin and the other without, ho... | Statistical tests when sample size is 1
BruceET has described the proper analysis (Two-way ANOVA without interaction), so I'll put a more positive spin on the experiment.
I'm assuming that the design was three pairs, where there is variabil |
5,033 | Statistical tests when sample size is 1 | Delightful question and one with historical precedent. As much as we might fault our budding high school junior scientist for his experimental design, it has a nearly perfect historical precedent.
What some consider the first controlled scientific medical experiment did the same thing. This high school student tested... | Statistical tests when sample size is 1 | Delightful question and one with historical precedent. As much as we might fault our budding high school junior scientist for his experimental design, it has a nearly perfect historical precedent.
Wh | Statistical tests when sample size is 1
Delightful question and one with historical precedent. As much as we might fault our budding high school junior scientist for his experimental design, it has a nearly perfect historical precedent.
What some consider the first controlled scientific medical experiment did the same... | Statistical tests when sample size is 1
Delightful question and one with historical precedent. As much as we might fault our budding high school junior scientist for his experimental design, it has a nearly perfect historical precedent.
Wh |
5,034 | Statistical tests when sample size is 1 | If the student were willing to make a rather deep dive, you might redirect their interest from sampling variation to uncertainty, and from a hypothesis test to an expanded uncertainty interval. Sampling variation is only one component of uncertainty. While the student is not in position to assess sampling variability, ... | Statistical tests when sample size is 1 | If the student were willing to make a rather deep dive, you might redirect their interest from sampling variation to uncertainty, and from a hypothesis test to an expanded uncertainty interval. Sampli | Statistical tests when sample size is 1
If the student were willing to make a rather deep dive, you might redirect their interest from sampling variation to uncertainty, and from a hypothesis test to an expanded uncertainty interval. Sampling variation is only one component of uncertainty. While the student is not in p... | Statistical tests when sample size is 1
If the student were willing to make a rather deep dive, you might redirect their interest from sampling variation to uncertainty, and from a hypothesis test to an expanded uncertainty interval. Sampli |
5,035 | Statistical tests when sample size is 1 | A major problem is the small sample size reducing the degrees of freedom in model selection along with the model's required/sensitivity to normality of error assumption. Preserving degrees of freedom and being robust in methodology appears likely to be the best path. I would even advise generating random errors from po... | Statistical tests when sample size is 1 | A major problem is the small sample size reducing the degrees of freedom in model selection along with the model's required/sensitivity to normality of error assumption. Preserving degrees of freedom | Statistical tests when sample size is 1
A major problem is the small sample size reducing the degrees of freedom in model selection along with the model's required/sensitivity to normality of error assumption. Preserving degrees of freedom and being robust in methodology appears likely to be the best path. I would even... | Statistical tests when sample size is 1
A major problem is the small sample size reducing the degrees of freedom in model selection along with the model's required/sensitivity to normality of error assumption. Preserving degrees of freedom |
5,036 | Statistical tests when sample size is 1 | While the student does not have type A repeatability measurements, the student may/should be able to estimate the type B error contribution caused by equipment supplied from elsewhere ("For an estimate xi of an input quantity Xi that has not been obtained from repeated observations").
This is detailed in the SI/bipm G... | Statistical tests when sample size is 1 | While the student does not have type A repeatability measurements, the student may/should be able to estimate the type B error contribution caused by equipment supplied from elsewhere ("For an estimat | Statistical tests when sample size is 1
While the student does not have type A repeatability measurements, the student may/should be able to estimate the type B error contribution caused by equipment supplied from elsewhere ("For an estimate xi of an input quantity Xi that has not been obtained from repeated observatio... | Statistical tests when sample size is 1
While the student does not have type A repeatability measurements, the student may/should be able to estimate the type B error contribution caused by equipment supplied from elsewhere ("For an estimat |
5,037 | Statistical tests when sample size is 1 | What a good example of the old question of bias and random errors in observational errors.
If the biased estimation of the standard deviation is, as you mention:
$ \sigma = \sqrt{\frac{\sum{(x_i-\bar{x})^2}}{n}} = \frac {0}{1}=0$,
the unbiased estimation is
$ \sigma = \sqrt{\frac{\sum{(x_i-\bar{x})^2}}{n-1}} = \frac {... | Statistical tests when sample size is 1 | What a good example of the old question of bias and random errors in observational errors.
If the biased estimation of the standard deviation is, as you mention:
$ \sigma = \sqrt{\frac{\sum{(x_i-\bar{ | Statistical tests when sample size is 1
What a good example of the old question of bias and random errors in observational errors.
If the biased estimation of the standard deviation is, as you mention:
$ \sigma = \sqrt{\frac{\sum{(x_i-\bar{x})^2}}{n}} = \frac {0}{1}=0$,
the unbiased estimation is
$ \sigma = \sqrt{\fra... | Statistical tests when sample size is 1
What a good example of the old question of bias and random errors in observational errors.
If the biased estimation of the standard deviation is, as you mention:
$ \sigma = \sqrt{\frac{\sum{(x_i-\bar{ |
5,038 | Bayesian vs frequentist Interpretations of Probability | In the frequentist approach, it is asserted that the only sense in which probabilities have meaning is as the limiting value of the number of successes in a sequence of trials, i.e. as
$$p = \lim_{n\to\infty} \frac{k}{n}$$
where $k$ is the number of successes and $n$ is the number of trials. In particular, it doesn't m... | Bayesian vs frequentist Interpretations of Probability | In the frequentist approach, it is asserted that the only sense in which probabilities have meaning is as the limiting value of the number of successes in a sequence of trials, i.e. as
$$p = \lim_{n\t | Bayesian vs frequentist Interpretations of Probability
In the frequentist approach, it is asserted that the only sense in which probabilities have meaning is as the limiting value of the number of successes in a sequence of trials, i.e. as
$$p = \lim_{n\to\infty} \frac{k}{n}$$
where $k$ is the number of successes and $... | Bayesian vs frequentist Interpretations of Probability
In the frequentist approach, it is asserted that the only sense in which probabilities have meaning is as the limiting value of the number of successes in a sequence of trials, i.e. as
$$p = \lim_{n\t |
5,039 | Bayesian vs frequentist Interpretations of Probability | You're right about your interpretation of Frequentist probability: randomness in this setup is merely due to incomplete sampling. From the Bayesian viewpoint probabilities are "subjective", in that they reflect an agent's uncertainty about the world. It's not quite right to say that the parameters of the distributions ... | Bayesian vs frequentist Interpretations of Probability | You're right about your interpretation of Frequentist probability: randomness in this setup is merely due to incomplete sampling. From the Bayesian viewpoint probabilities are "subjective", in that th | Bayesian vs frequentist Interpretations of Probability
You're right about your interpretation of Frequentist probability: randomness in this setup is merely due to incomplete sampling. From the Bayesian viewpoint probabilities are "subjective", in that they reflect an agent's uncertainty about the world. It's not quite... | Bayesian vs frequentist Interpretations of Probability
You're right about your interpretation of Frequentist probability: randomness in this setup is merely due to incomplete sampling. From the Bayesian viewpoint probabilities are "subjective", in that th |
5,040 | Bayesian vs frequentist Interpretations of Probability | The Bayesian interpretation of probability is a degree-of-belief interpretation.
A Bayesian may say that the probability that there was life on Mars a billion years ago is $1/2$.
A frequentist will refuse to assign a probability to that proposition. It is not something that could be said to be true in half of all case... | Bayesian vs frequentist Interpretations of Probability | The Bayesian interpretation of probability is a degree-of-belief interpretation.
A Bayesian may say that the probability that there was life on Mars a billion years ago is $1/2$.
A frequentist will re | Bayesian vs frequentist Interpretations of Probability
The Bayesian interpretation of probability is a degree-of-belief interpretation.
A Bayesian may say that the probability that there was life on Mars a billion years ago is $1/2$.
A frequentist will refuse to assign a probability to that proposition. It is not some... | Bayesian vs frequentist Interpretations of Probability
The Bayesian interpretation of probability is a degree-of-belief interpretation.
A Bayesian may say that the probability that there was life on Mars a billion years ago is $1/2$.
A frequentist will re |
5,041 | Bayesian vs frequentist Interpretations of Probability | Chris gives a nice simplistic explanation that properly differentiates the two approaches to probability. But frequentist theory of probability is more than just looking at the long range proportion of successes. We also consider data sampled at random from a distribution and estimate parameters of the distribution s... | Bayesian vs frequentist Interpretations of Probability | Chris gives a nice simplistic explanation that properly differentiates the two approaches to probability. But frequentist theory of probability is more than just looking at the long range proportion | Bayesian vs frequentist Interpretations of Probability
Chris gives a nice simplistic explanation that properly differentiates the two approaches to probability. But frequentist theory of probability is more than just looking at the long range proportion of successes. We also consider data sampled at random from a dis... | Bayesian vs frequentist Interpretations of Probability
Chris gives a nice simplistic explanation that properly differentiates the two approaches to probability. But frequentist theory of probability is more than just looking at the long range proportion |
5,042 | Bayesian vs frequentist Interpretations of Probability | From a "real world" point of view, I find one major difference between a frequentist and a classical or Bayesian "solution" that applies to at least three major scenarios. The difference in selecting a methodology depends on whether you need a solution that is impacted by the population probability, or one that is impa... | Bayesian vs frequentist Interpretations of Probability | From a "real world" point of view, I find one major difference between a frequentist and a classical or Bayesian "solution" that applies to at least three major scenarios. The difference in selecting | Bayesian vs frequentist Interpretations of Probability
From a "real world" point of view, I find one major difference between a frequentist and a classical or Bayesian "solution" that applies to at least three major scenarios. The difference in selecting a methodology depends on whether you need a solution that is impa... | Bayesian vs frequentist Interpretations of Probability
From a "real world" point of view, I find one major difference between a frequentist and a classical or Bayesian "solution" that applies to at least three major scenarios. The difference in selecting |
5,043 | Bayesian vs frequentist Interpretations of Probability | The following is taken from my manuscript on p-value functions - Johnson, Geoffrey S. "Decision Making in Drug Development via Inference on Power" Researchgate.net (2021).
In any quantitative field it is not enough to simply apply a set of mathematical operations. One must also provide an interpretation. The field of... | Bayesian vs frequentist Interpretations of Probability | The following is taken from my manuscript on p-value functions - Johnson, Geoffrey S. "Decision Making in Drug Development via Inference on Power" Researchgate.net (2021).
In any quantitative field it | Bayesian vs frequentist Interpretations of Probability
The following is taken from my manuscript on p-value functions - Johnson, Geoffrey S. "Decision Making in Drug Development via Inference on Power" Researchgate.net (2021).
In any quantitative field it is not enough to simply apply a set of mathematical operations. ... | Bayesian vs frequentist Interpretations of Probability
The following is taken from my manuscript on p-value functions - Johnson, Geoffrey S. "Decision Making in Drug Development via Inference on Power" Researchgate.net (2021).
In any quantitative field it |
5,044 | Bayesian vs frequentist Interpretations of Probability | “I was just wondering whether anyone could give me a quick summary of their interpretation of Bayesian vs. Frequentist approach including Bayesian statistical equivalents of the Frequentist p-value and confidence interval. In addition, specific examples of where one method would be preferable to the other are appreciat... | Bayesian vs frequentist Interpretations of Probability | “I was just wondering whether anyone could give me a quick summary of their interpretation of Bayesian vs. Frequentist approach including Bayesian statistical equivalents of the Frequentist p-value an | Bayesian vs frequentist Interpretations of Probability
“I was just wondering whether anyone could give me a quick summary of their interpretation of Bayesian vs. Frequentist approach including Bayesian statistical equivalents of the Frequentist p-value and confidence interval. In addition, specific examples of where on... | Bayesian vs frequentist Interpretations of Probability
“I was just wondering whether anyone could give me a quick summary of their interpretation of Bayesian vs. Frequentist approach including Bayesian statistical equivalents of the Frequentist p-value an |
5,045 | Bayesian vs frequentist Interpretations of Probability | The choice of interpretation depends on the question. If you wish to know the odds in a game of chance, classical interpretation will solve your problem, but statistical data is useless since fair dice have no memory.
If you wish to predict a future event based on past experience, the frequentist interpretation is cor... | Bayesian vs frequentist Interpretations of Probability | The choice of interpretation depends on the question. If you wish to know the odds in a game of chance, classical interpretation will solve your problem, but statistical data is useless since fair dic | Bayesian vs frequentist Interpretations of Probability
The choice of interpretation depends on the question. If you wish to know the odds in a game of chance, classical interpretation will solve your problem, but statistical data is useless since fair dice have no memory.
If you wish to predict a future event based on... | Bayesian vs frequentist Interpretations of Probability
The choice of interpretation depends on the question. If you wish to know the odds in a game of chance, classical interpretation will solve your problem, but statistical data is useless since fair dic |
5,046 | Bayesian vs frequentist Interpretations of Probability | The other answers do a good job explaining this topic, but I think there's room for a more motivated explanation.
What is the probability of rolling a 1 with a fair six-sided die? The traditional answer would be 1 in 6, because no one side is favored over another. This concept can be extended to the principle of indiff... | Bayesian vs frequentist Interpretations of Probability | The other answers do a good job explaining this topic, but I think there's room for a more motivated explanation.
What is the probability of rolling a 1 with a fair six-sided die? The traditional answ | Bayesian vs frequentist Interpretations of Probability
The other answers do a good job explaining this topic, but I think there's room for a more motivated explanation.
What is the probability of rolling a 1 with a fair six-sided die? The traditional answer would be 1 in 6, because no one side is favored over another. ... | Bayesian vs frequentist Interpretations of Probability
The other answers do a good job explaining this topic, but I think there's room for a more motivated explanation.
What is the probability of rolling a 1 with a fair six-sided die? The traditional answ |
5,047 | Approximate order statistics for normal random variables | The classic reference is Royston (1982)[1] which has algorithms going beyond explicit formulas. It also quotes a well-known formula by Blom (1958):
$E(r:n) \approx \mu + \Phi^{-1}(\frac{r-\alpha}{n-2\alpha+1})\sigma$ with $\alpha=0.375$. This formula gives a multiplier of -2.73 for $n=200, r=1$.
[1]: Algorithm AS 177: ... | Approximate order statistics for normal random variables | The classic reference is Royston (1982)[1] which has algorithms going beyond explicit formulas. It also quotes a well-known formula by Blom (1958):
$E(r:n) \approx \mu + \Phi^{-1}(\frac{r-\alpha}{n-2\ | Approximate order statistics for normal random variables
The classic reference is Royston (1982)[1] which has algorithms going beyond explicit formulas. It also quotes a well-known formula by Blom (1958):
$E(r:n) \approx \mu + \Phi^{-1}(\frac{r-\alpha}{n-2\alpha+1})\sigma$ with $\alpha=0.375$. This formula gives a mult... | Approximate order statistics for normal random variables
The classic reference is Royston (1982)[1] which has algorithms going beyond explicit formulas. It also quotes a well-known formula by Blom (1958):
$E(r:n) \approx \mu + \Phi^{-1}(\frac{r-\alpha}{n-2\ |
5,048 | Approximate order statistics for normal random variables | $$\newcommand{\Pr}{\mathrm{Pr}}\newcommand{\Beta}{\mathrm{Beta}}\newcommand{\Var}{\mathrm{Var}}$$The distribution of the ith order statistic of any continuous random variable with a PDF is given by the "beta-F" compound distribution. The intuitive way to think about this distribution, is to consider the ith order stat... | Approximate order statistics for normal random variables | $$\newcommand{\Pr}{\mathrm{Pr}}\newcommand{\Beta}{\mathrm{Beta}}\newcommand{\Var}{\mathrm{Var}}$$The distribution of the ith order statistic of any continuous random variable with a PDF is given by th | Approximate order statistics for normal random variables
$$\newcommand{\Pr}{\mathrm{Pr}}\newcommand{\Beta}{\mathrm{Beta}}\newcommand{\Var}{\mathrm{Var}}$$The distribution of the ith order statistic of any continuous random variable with a PDF is given by the "beta-F" compound distribution. The intuitive way to think a... | Approximate order statistics for normal random variables
$$\newcommand{\Pr}{\mathrm{Pr}}\newcommand{\Beta}{\mathrm{Beta}}\newcommand{\Var}{\mathrm{Var}}$$The distribution of the ith order statistic of any continuous random variable with a PDF is given by th |
5,049 | Approximate order statistics for normal random variables | Aniko's answer relies on Blom's well known formula that involves a choice of $\alpha = 3/8$. It turns out that this formula is itself a mere approximation of an exact answer due to G. Elfving (1947), The asymptotical distribution of range in samples from a normal population, Biometrika, Vol. 34, pp. 111-119. Elfving's ... | Approximate order statistics for normal random variables | Aniko's answer relies on Blom's well known formula that involves a choice of $\alpha = 3/8$. It turns out that this formula is itself a mere approximation of an exact answer due to G. Elfving (1947), | Approximate order statistics for normal random variables
Aniko's answer relies on Blom's well known formula that involves a choice of $\alpha = 3/8$. It turns out that this formula is itself a mere approximation of an exact answer due to G. Elfving (1947), The asymptotical distribution of range in samples from a normal... | Approximate order statistics for normal random variables
Aniko's answer relies on Blom's well known formula that involves a choice of $\alpha = 3/8$. It turns out that this formula is itself a mere approximation of an exact answer due to G. Elfving (1947), |
5,050 | Approximate order statistics for normal random variables | Depending on what you want to do, this answer may or may not help - I got the following exact formula from Maple's Statistics package.
with(Statistics):
X := OrderStatistic(Normal(0, 1), 1, n):
m := Mean(X):
m;
$$\int _{-\infty }^{\infty }\!1/2\,{\frac {{\it \_t0}\,n!\,\sqrt {2}{
{\rm e}^{-1/2\,{{\it \_t0}}^{2}}} \le... | Approximate order statistics for normal random variables | Depending on what you want to do, this answer may or may not help - I got the following exact formula from Maple's Statistics package.
with(Statistics):
X := OrderStatistic(Normal(0, 1), 1, n):
m := M | Approximate order statistics for normal random variables
Depending on what you want to do, this answer may or may not help - I got the following exact formula from Maple's Statistics package.
with(Statistics):
X := OrderStatistic(Normal(0, 1), 1, n):
m := Mean(X):
m;
$$\int _{-\infty }^{\infty }\!1/2\,{\frac {{\it \_t... | Approximate order statistics for normal random variables
Depending on what you want to do, this answer may or may not help - I got the following exact formula from Maple's Statistics package.
with(Statistics):
X := OrderStatistic(Normal(0, 1), 1, n):
m := M |
5,051 | What is a good resource on table design? | Ed Tufte has a few pages on this in his classic "The Visual Display of Quantitative Information".
For a much more detailed treatment, there is Jane Miller's Chicago Guide to Writing about Numbers. I've never seen anything else like it. It has a whole chapter on "Creating Effective Tables". | What is a good resource on table design? | Ed Tufte has a few pages on this in his classic "The Visual Display of Quantitative Information".
For a much more detailed treatment, there is Jane Miller's Chicago Guide to Writing about Numbers. I'v | What is a good resource on table design?
Ed Tufte has a few pages on this in his classic "The Visual Display of Quantitative Information".
For a much more detailed treatment, there is Jane Miller's Chicago Guide to Writing about Numbers. I've never seen anything else like it. It has a whole chapter on "Creating Effecti... | What is a good resource on table design?
Ed Tufte has a few pages on this in his classic "The Visual Display of Quantitative Information".
For a much more detailed treatment, there is Jane Miller's Chicago Guide to Writing about Numbers. I'v |
5,052 | What is a good resource on table design? | Stephen Few's book Show Me the Numbers: Designing Tables and Graphs to Enlighten has a couple of chapters devoted to tabular display of information. It's good and recommended, but it's not quite Grammar of Graphics if that's what you're after.
Update This sounds interesting, but I haven't read it: Handbook of tabular ... | What is a good resource on table design? | Stephen Few's book Show Me the Numbers: Designing Tables and Graphs to Enlighten has a couple of chapters devoted to tabular display of information. It's good and recommended, but it's not quite Gram | What is a good resource on table design?
Stephen Few's book Show Me the Numbers: Designing Tables and Graphs to Enlighten has a couple of chapters devoted to tabular display of information. It's good and recommended, but it's not quite Grammar of Graphics if that's what you're after.
Update This sounds interesting, bu... | What is a good resource on table design?
Stephen Few's book Show Me the Numbers: Designing Tables and Graphs to Enlighten has a couple of chapters devoted to tabular display of information. It's good and recommended, but it's not quite Gram |
5,053 | What is a good resource on table design? | If you are interested in table design, I would definitely recommend two papers on the subject by Andrew Gelman:
A necessary preface to the paper on table design is Gelman et al, 2002 Let's practice what we preach: Turning Tables Into Graphs
Gelman argues that graphs are better than tables in the above paper. Then his ... | What is a good resource on table design? | If you are interested in table design, I would definitely recommend two papers on the subject by Andrew Gelman:
A necessary preface to the paper on table design is Gelman et al, 2002 Let's practice wh | What is a good resource on table design?
If you are interested in table design, I would definitely recommend two papers on the subject by Andrew Gelman:
A necessary preface to the paper on table design is Gelman et al, 2002 Let's practice what we preach: Turning Tables Into Graphs
Gelman argues that graphs are better ... | What is a good resource on table design?
If you are interested in table design, I would definitely recommend two papers on the subject by Andrew Gelman:
A necessary preface to the paper on table design is Gelman et al, 2002 Let's practice wh |
5,054 | What is a good resource on table design? | You might check out the documentation for the LaTeX package booktabs; it gives general guidance and implements its design suggestions in LaTeX tables. | What is a good resource on table design? | You might check out the documentation for the LaTeX package booktabs; it gives general guidance and implements its design suggestions in LaTeX tables. | What is a good resource on table design?
You might check out the documentation for the LaTeX package booktabs; it gives general guidance and implements its design suggestions in LaTeX tables. | What is a good resource on table design?
You might check out the documentation for the LaTeX package booktabs; it gives general guidance and implements its design suggestions in LaTeX tables. |
5,055 | What is a good resource on table design? | I hope this answer is not too off topic, but a couple of days ago I have seen this link on visualizing tables at StackExchange:
Visual Representation of Tabular Information – How to Fix the Uncommunicative Table | What is a good resource on table design? | I hope this answer is not too off topic, but a couple of days ago I have seen this link on visualizing tables at StackExchange:
Visual Representation of Tabular Information – How to Fix the Uncommunic | What is a good resource on table design?
I hope this answer is not too off topic, but a couple of days ago I have seen this link on visualizing tables at StackExchange:
Visual Representation of Tabular Information – How to Fix the Uncommunicative Table | What is a good resource on table design?
I hope this answer is not too off topic, but a couple of days ago I have seen this link on visualizing tables at StackExchange:
Visual Representation of Tabular Information – How to Fix the Uncommunic |
5,056 | What is a good resource on table design? | I cover table design in the seminars I offer. My sources are primarily Chapter 8 of Few’s Show Me the Numbers and a paper by Martin Koschat:
Koschat, Martin. 2005. “A Case for Simple Tables,” The American Statistician 59:1, 31-40. https://doi.org/10.1198/000313005X21429
Also, Howard Wainer discusses table design in Vi... | What is a good resource on table design? | I cover table design in the seminars I offer. My sources are primarily Chapter 8 of Few’s Show Me the Numbers and a paper by Martin Koschat:
Koschat, Martin. 2005. “A Case for Simple Tables,” The Amer | What is a good resource on table design?
I cover table design in the seminars I offer. My sources are primarily Chapter 8 of Few’s Show Me the Numbers and a paper by Martin Koschat:
Koschat, Martin. 2005. “A Case for Simple Tables,” The American Statistician 59:1, 31-40. https://doi.org/10.1198/000313005X21429
Also, H... | What is a good resource on table design?
I cover table design in the seminars I offer. My sources are primarily Chapter 8 of Few’s Show Me the Numbers and a paper by Martin Koschat:
Koschat, Martin. 2005. “A Case for Simple Tables,” The Amer |
5,057 | What is a good resource on table design? | This CV blog post by @AndyW is a really excellent. It gathers a number of best practices, useful examples, and a helpful literature review with links to papers and other resources. | What is a good resource on table design? | This CV blog post by @AndyW is a really excellent. It gathers a number of best practices, useful examples, and a helpful literature review with links to papers and other resources. | What is a good resource on table design?
This CV blog post by @AndyW is a really excellent. It gathers a number of best practices, useful examples, and a helpful literature review with links to papers and other resources. | What is a good resource on table design?
This CV blog post by @AndyW is a really excellent. It gathers a number of best practices, useful examples, and a helpful literature review with links to papers and other resources. |
5,058 | What is a good resource on table design? | The UN Document "Making Data Meaningful" provides a nice overview, with rules and examples, of table design in Section 3 (p 12-17).
This is in part 2 of a set of guidelines on 'using text and visualizations to bring statistics to life' https://www.unece.org/stats/documents/writing/ | What is a good resource on table design? | The UN Document "Making Data Meaningful" provides a nice overview, with rules and examples, of table design in Section 3 (p 12-17).
This is in part 2 of a set of guidelines on 'using text and visualiz | What is a good resource on table design?
The UN Document "Making Data Meaningful" provides a nice overview, with rules and examples, of table design in Section 3 (p 12-17).
This is in part 2 of a set of guidelines on 'using text and visualizations to bring statistics to life' https://www.unece.org/stats/documents/writi... | What is a good resource on table design?
The UN Document "Making Data Meaningful" provides a nice overview, with rules and examples, of table design in Section 3 (p 12-17).
This is in part 2 of a set of guidelines on 'using text and visualiz |
5,059 | Logistic regression vs. LDA as two-class classifiers | It sounds to me that you are correct. Logistic regression indeed does not assume any specific shapes of densities in the space of predictor variables, but LDA does. Here are some differences between the two analyses, briefly.
Binary Logistic regression (BLR) vs Linear Discriminant analysis (with 2 groups: also known as... | Logistic regression vs. LDA as two-class classifiers | It sounds to me that you are correct. Logistic regression indeed does not assume any specific shapes of densities in the space of predictor variables, but LDA does. Here are some differences between t | Logistic regression vs. LDA as two-class classifiers
It sounds to me that you are correct. Logistic regression indeed does not assume any specific shapes of densities in the space of predictor variables, but LDA does. Here are some differences between the two analyses, briefly.
Binary Logistic regression (BLR) vs Linea... | Logistic regression vs. LDA as two-class classifiers
It sounds to me that you are correct. Logistic regression indeed does not assume any specific shapes of densities in the space of predictor variables, but LDA does. Here are some differences between t |
5,060 | Logistic regression vs. LDA as two-class classifiers | Let me add some points to @ttnphns nice list:
The Bayes prediction of the LDA's posterior class membership probability follows a logistic curve as well.
[Efron, B. The efficiency of logistic regression compared to normal discriminant analysis, J Am Stat Assoc, 70, 892-898 (1975).]
While that paper shows that the rela... | Logistic regression vs. LDA as two-class classifiers | Let me add some points to @ttnphns nice list:
The Bayes prediction of the LDA's posterior class membership probability follows a logistic curve as well.
[Efron, B. The efficiency of logistic regressi | Logistic regression vs. LDA as two-class classifiers
Let me add some points to @ttnphns nice list:
The Bayes prediction of the LDA's posterior class membership probability follows a logistic curve as well.
[Efron, B. The efficiency of logistic regression compared to normal discriminant analysis, J Am Stat Assoc, 70, 8... | Logistic regression vs. LDA as two-class classifiers
Let me add some points to @ttnphns nice list:
The Bayes prediction of the LDA's posterior class membership probability follows a logistic curve as well.
[Efron, B. The efficiency of logistic regressi |
5,061 | Logistic regression vs. LDA as two-class classifiers | I just wanted to add one more point.
LDA works when all the independent/predictor variables are
continuous (not categorical) and follow a Normal distribution.
Whereas in Logistic Regression this is not the case
and categorical variables can be used as independent variables while making predictions. | Logistic regression vs. LDA as two-class classifiers | I just wanted to add one more point.
LDA works when all the independent/predictor variables are
continuous (not categorical) and follow a Normal distribution.
Whereas in Logistic Regression this is | Logistic regression vs. LDA as two-class classifiers
I just wanted to add one more point.
LDA works when all the independent/predictor variables are
continuous (not categorical) and follow a Normal distribution.
Whereas in Logistic Regression this is not the case
and categorical variables can be used as independent ... | Logistic regression vs. LDA as two-class classifiers
I just wanted to add one more point.
LDA works when all the independent/predictor variables are
continuous (not categorical) and follow a Normal distribution.
Whereas in Logistic Regression this is |
5,062 | Can a deep neural network approximate multiplication function? | NN with relu activation function can approximate multiplication when range of inputs is limited. Recall that relu(x) = max(x, 0).
It is enough if NN approximates a square function g(z) = z^2, because x*y = ((x-y)^2 - x^2 - y^2)/(-2). Right-hand side has just linear combinations and squares.
NN can approximate z^2 with ... | Can a deep neural network approximate multiplication function? | NN with relu activation function can approximate multiplication when range of inputs is limited. Recall that relu(x) = max(x, 0).
It is enough if NN approximates a square function g(z) = z^2, because | Can a deep neural network approximate multiplication function?
NN with relu activation function can approximate multiplication when range of inputs is limited. Recall that relu(x) = max(x, 0).
It is enough if NN approximates a square function g(z) = z^2, because x*y = ((x-y)^2 - x^2 - y^2)/(-2). Right-hand side has jus... | Can a deep neural network approximate multiplication function?
NN with relu activation function can approximate multiplication when range of inputs is limited. Recall that relu(x) = max(x, 0).
It is enough if NN approximates a square function g(z) = z^2, because |
5,063 | Can a deep neural network approximate multiplication function? | A big multiplication function gradient forces the net probably almost immediately into some horrifying state where all its hidden nodes have a zero gradient (because of neural network implementation details and limitations). We can use two approaches:
Divide by a constant. We are just dividing everything before the le... | Can a deep neural network approximate multiplication function? | A big multiplication function gradient forces the net probably almost immediately into some horrifying state where all its hidden nodes have a zero gradient (because of neural network implementation d | Can a deep neural network approximate multiplication function?
A big multiplication function gradient forces the net probably almost immediately into some horrifying state where all its hidden nodes have a zero gradient (because of neural network implementation details and limitations). We can use two approaches:
Divi... | Can a deep neural network approximate multiplication function?
A big multiplication function gradient forces the net probably almost immediately into some horrifying state where all its hidden nodes have a zero gradient (because of neural network implementation d |
5,064 | Can a deep neural network approximate multiplication function? | I'm unable to comment due to being a newly active user on StackExchange. But I think this is an important question because it's so friggin simple to understand yet difficult to explain. With respect, I don't think the accepted answer is sufficient. If you think about the core operations of a standard feed-forward NN,... | Can a deep neural network approximate multiplication function? | I'm unable to comment due to being a newly active user on StackExchange. But I think this is an important question because it's so friggin simple to understand yet difficult to explain. With respect | Can a deep neural network approximate multiplication function?
I'm unable to comment due to being a newly active user on StackExchange. But I think this is an important question because it's so friggin simple to understand yet difficult to explain. With respect, I don't think the accepted answer is sufficient. If you... | Can a deep neural network approximate multiplication function?
I'm unable to comment due to being a newly active user on StackExchange. But I think this is an important question because it's so friggin simple to understand yet difficult to explain. With respect |
5,065 | Can a deep neural network approximate multiplication function? | Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
A similar question struck me earlier today, and I was ... | Can a deep neural network approximate multiplication function? | Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
| Can a deep neural network approximate multiplication function?
Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
... | Can a deep neural network approximate multiplication function?
Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
|
5,066 | Can a deep neural network approximate multiplication function? | Traditional neural network consists of linear maps and Lipschitiz activation function. As a composition of Lischitz continuous functions, neural network is also Lipschitz continuous, but multiplication is not Lipschitz continuous. This means that neural network cannot approximate multiplication when one of the x or y g... | Can a deep neural network approximate multiplication function? | Traditional neural network consists of linear maps and Lipschitiz activation function. As a composition of Lischitz continuous functions, neural network is also Lipschitz continuous, but multiplicatio | Can a deep neural network approximate multiplication function?
Traditional neural network consists of linear maps and Lipschitiz activation function. As a composition of Lischitz continuous functions, neural network is also Lipschitz continuous, but multiplication is not Lipschitz continuous. This means that neural net... | Can a deep neural network approximate multiplication function?
Traditional neural network consists of linear maps and Lipschitiz activation function. As a composition of Lischitz continuous functions, neural network is also Lipschitz continuous, but multiplicatio |
5,067 | Can a deep neural network approximate multiplication function? | "one hidden layer" does not limit the number of neurons and kinds of activate function used, it still has a large representation space. One simple method to validate the existence of this problem: Train this regress problem with a real neuron network, record each weights and bias, use these parameters plot the predict ... | Can a deep neural network approximate multiplication function? | "one hidden layer" does not limit the number of neurons and kinds of activate function used, it still has a large representation space. One simple method to validate the existence of this problem: Tra | Can a deep neural network approximate multiplication function?
"one hidden layer" does not limit the number of neurons and kinds of activate function used, it still has a large representation space. One simple method to validate the existence of this problem: Train this regress problem with a real neuron network, recor... | Can a deep neural network approximate multiplication function?
"one hidden layer" does not limit the number of neurons and kinds of activate function used, it still has a large representation space. One simple method to validate the existence of this problem: Tra |
5,068 | Can a deep neural network approximate multiplication function? | Feed-forward REUL, will never be able to learn multiplication exactly or approximately with absolute bounded error, if numbers are unbounded. | Can a deep neural network approximate multiplication function? | Feed-forward REUL, will never be able to learn multiplication exactly or approximately with absolute bounded error, if numbers are unbounded. | Can a deep neural network approximate multiplication function?
Feed-forward REUL, will never be able to learn multiplication exactly or approximately with absolute bounded error, if numbers are unbounded. | Can a deep neural network approximate multiplication function?
Feed-forward REUL, will never be able to learn multiplication exactly or approximately with absolute bounded error, if numbers are unbounded. |
5,069 | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach? | Two reasons one may go with a Bayesian approach even if you're using highly non-informative priors:
Convergence problems. There are some distributions (binomial, negative binomial and generalized gamma are the ones I'm most familiar with) that have convergence issues a non-trivial amount of the time. You can use a "Ba... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas | Two reasons one may go with a Bayesian approach even if you're using highly non-informative priors:
Convergence problems. There are some distributions (binomial, negative binomial and generalized gam | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach?
Two reasons one may go with a Bayesian approach even if you're using highly non-informative priors:
Convergence problems. There are some distributions (binomial, negative binomial and generalized gamma a... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas
Two reasons one may go with a Bayesian approach even if you're using highly non-informative priors:
Convergence problems. There are some distributions (binomial, negative binomial and generalized gam |
5,070 | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach? | Although the results are going to be very similar, their interpretations differ.
Confidence intervals imply the notion of repeating an experiment many times and being able to capture the true parameter 95% of times. But you cannot say you have a 95% chance of capturing it.
Credible intervals (Bayesian), on the other h... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas | Although the results are going to be very similar, their interpretations differ.
Confidence intervals imply the notion of repeating an experiment many times and being able to capture the true paramet | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach?
Although the results are going to be very similar, their interpretations differ.
Confidence intervals imply the notion of repeating an experiment many times and being able to capture the true parameter 9... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas
Although the results are going to be very similar, their interpretations differ.
Confidence intervals imply the notion of repeating an experiment many times and being able to capture the true paramet |
5,071 | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach? | I believe one reason to do so is that a Bayesian analysis provides you with a full posterior distribution. This can result in more detailed intervals than the typical frequentist $\pm 2 \sigma$. An applicable quote, from Reis and Stedinger 2005, is:
Providing a full posterior distribution of the parameters is an
adv... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas | I believe one reason to do so is that a Bayesian analysis provides you with a full posterior distribution. This can result in more detailed intervals than the typical frequentist $\pm 2 \sigma$. An ap | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach?
I believe one reason to do so is that a Bayesian analysis provides you with a full posterior distribution. This can result in more detailed intervals than the typical frequentist $\pm 2 \sigma$. An applic... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas
I believe one reason to do so is that a Bayesian analysis provides you with a full posterior distribution. This can result in more detailed intervals than the typical frequentist $\pm 2 \sigma$. An ap |
5,072 | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach? | Sir Harold Jeffreys was a strong proponent of the Bayesian approach. He showed that if you use diffuse improper priors the resulting Bayesian inference would be the same as the frequentist inferential approach (that is, Bayesian credible regions are the same as frequentist confidence intervals). Most Bayesians advocate... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas | Sir Harold Jeffreys was a strong proponent of the Bayesian approach. He showed that if you use diffuse improper priors the resulting Bayesian inference would be the same as the frequentist inferential | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach?
Sir Harold Jeffreys was a strong proponent of the Bayesian approach. He showed that if you use diffuse improper priors the resulting Bayesian inference would be the same as the frequentist inferential app... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas
Sir Harold Jeffreys was a strong proponent of the Bayesian approach. He showed that if you use diffuse improper priors the resulting Bayesian inference would be the same as the frequentist inferential |
5,073 | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach? | We could argue forever about foundations of inference to defend both approaches, but let me propose something different. A $\textit{practical}$ reason to favor a Bayesian analysis over a classical one is shown clearly by how both approaches deal with prediction. Suppose that we have the usual conditionally i.i.d. case.... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas | We could argue forever about foundations of inference to defend both approaches, but let me propose something different. A $\textit{practical}$ reason to favor a Bayesian analysis over a classical one | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach?
We could argue forever about foundations of inference to defend both approaches, but let me propose something different. A $\textit{practical}$ reason to favor a Bayesian analysis over a classical one is ... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas
We could argue forever about foundations of inference to defend both approaches, but let me propose something different. A $\textit{practical}$ reason to favor a Bayesian analysis over a classical one |
5,074 | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach? | The Bayesian approach has practical advantages. It helps with estimation, often being mandatory. And it enables novel model families, and helps in construction of more complicated (hierarchical, multilevel) models.
For example, with mixed models (including random effects with variance parameters) one gets better estima... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas | The Bayesian approach has practical advantages. It helps with estimation, often being mandatory. And it enables novel model families, and helps in construction of more complicated (hierarchical, multi | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach?
The Bayesian approach has practical advantages. It helps with estimation, often being mandatory. And it enables novel model families, and helps in construction of more complicated (hierarchical, multileve... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas
The Bayesian approach has practical advantages. It helps with estimation, often being mandatory. And it enables novel model families, and helps in construction of more complicated (hierarchical, multi |
5,075 | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach? | There are several reasons:
In many situations constructing test statistics or confidence intervals is quite difficult, because normal approximations – even after using an appropriate link function – to work with $\pm \text{SE}$ are often not working too well for sparse data situations. By using Bayesian inference wit... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas | There are several reasons:
In many situations constructing test statistics or confidence intervals is quite difficult, because normal approximations – even after using an appropriate link function – | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach?
There are several reasons:
In many situations constructing test statistics or confidence intervals is quite difficult, because normal approximations – even after using an appropriate link function – to ... | Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the clas
There are several reasons:
In many situations constructing test statistics or confidence intervals is quite difficult, because normal approximations – even after using an appropriate link function – |
5,076 | Why does inversion of a covariance matrix yield partial correlations between random variables? | When a multivariate random variable $(X_1,X_2,\ldots,X_n)$ has a nondegenerate covariance matrix $\mathbb{C} = (\gamma_{ij}) = (\text{Cov}(X_i,X_j))$, the set of all real linear combinations of the $X_i$ forms an $n$-dimensional real vector space with basis $E=(X_1,X_2,\ldots, X_n)$ and a non-degenerate inner product g... | Why does inversion of a covariance matrix yield partial correlations between random variables? | When a multivariate random variable $(X_1,X_2,\ldots,X_n)$ has a nondegenerate covariance matrix $\mathbb{C} = (\gamma_{ij}) = (\text{Cov}(X_i,X_j))$, the set of all real linear combinations of the $X | Why does inversion of a covariance matrix yield partial correlations between random variables?
When a multivariate random variable $(X_1,X_2,\ldots,X_n)$ has a nondegenerate covariance matrix $\mathbb{C} = (\gamma_{ij}) = (\text{Cov}(X_i,X_j))$, the set of all real linear combinations of the $X_i$ forms an $n$-dimensio... | Why does inversion of a covariance matrix yield partial correlations between random variables?
When a multivariate random variable $(X_1,X_2,\ldots,X_n)$ has a nondegenerate covariance matrix $\mathbb{C} = (\gamma_{ij}) = (\text{Cov}(X_i,X_j))$, the set of all real linear combinations of the $X |
5,077 | Why does inversion of a covariance matrix yield partial correlations between random variables? | Here is a proof with just matrix calculations.
I appreciate the answer by whuber. It is very insightful on the math behind the scene. However, it is still not so trivial how to use his answer to obtain the minus sign in the formula stated in the wikipediaPartial_correlation#Using_matrix_inversion.
$$
\rho_{X_iX_j\cdot ... | Why does inversion of a covariance matrix yield partial correlations between random variables? | Here is a proof with just matrix calculations.
I appreciate the answer by whuber. It is very insightful on the math behind the scene. However, it is still not so trivial how to use his answer to obtai | Why does inversion of a covariance matrix yield partial correlations between random variables?
Here is a proof with just matrix calculations.
I appreciate the answer by whuber. It is very insightful on the math behind the scene. However, it is still not so trivial how to use his answer to obtain the minus sign in the f... | Why does inversion of a covariance matrix yield partial correlations between random variables?
Here is a proof with just matrix calculations.
I appreciate the answer by whuber. It is very insightful on the math behind the scene. However, it is still not so trivial how to use his answer to obtai |
5,078 | Why does inversion of a covariance matrix yield partial correlations between random variables? | Note that the sign of the answer actually depends on how you define partial correlation. There is a difference between regressing $X_i$ and $X_j$ on the other $n - 1$ variables separately vs. regressing $X_i$ and $X_j$ on the other $n - 2$ variables together. Under the second definition, let the correlation between res... | Why does inversion of a covariance matrix yield partial correlations between random variables? | Note that the sign of the answer actually depends on how you define partial correlation. There is a difference between regressing $X_i$ and $X_j$ on the other $n - 1$ variables separately vs. regressi | Why does inversion of a covariance matrix yield partial correlations between random variables?
Note that the sign of the answer actually depends on how you define partial correlation. There is a difference between regressing $X_i$ and $X_j$ on the other $n - 1$ variables separately vs. regressing $X_i$ and $X_j$ on the... | Why does inversion of a covariance matrix yield partial correlations between random variables?
Note that the sign of the answer actually depends on how you define partial correlation. There is a difference between regressing $X_i$ and $X_j$ on the other $n - 1$ variables separately vs. regressi |
5,079 | Why does inversion of a covariance matrix yield partial correlations between random variables? | For another perspective, this will examine the left inverse of a finite data matrix $A$. We can consider the data to be a sample rather than a theoretical distribution. While any distribution -- even continuous -- will have a covariance matrix, you can't generally talk about a data matrix unless you get into infinite... | Why does inversion of a covariance matrix yield partial correlations between random variables? | For another perspective, this will examine the left inverse of a finite data matrix $A$. We can consider the data to be a sample rather than a theoretical distribution. While any distribution -- eve | Why does inversion of a covariance matrix yield partial correlations between random variables?
For another perspective, this will examine the left inverse of a finite data matrix $A$. We can consider the data to be a sample rather than a theoretical distribution. While any distribution -- even continuous -- will have... | Why does inversion of a covariance matrix yield partial correlations between random variables?
For another perspective, this will examine the left inverse of a finite data matrix $A$. We can consider the data to be a sample rather than a theoretical distribution. While any distribution -- eve |
5,080 | Maximum Likelihood Estimators - Multivariate Gaussian | Deriving the Maximum Likelihood Estimators
Assume that we have $m$ random vectors, each of size $p$: $\mathbf{X^{(1)}, X^{(2)}, \dotsc, X^{(m)}}$ where each random vectors can be interpreted as an observation (data point) across $p$ variables. If each $\mathbf{X}^{(i)}$ are i.i.d. as multivariate Gaussian vectors:
$$ \... | Maximum Likelihood Estimators - Multivariate Gaussian | Deriving the Maximum Likelihood Estimators
Assume that we have $m$ random vectors, each of size $p$: $\mathbf{X^{(1)}, X^{(2)}, \dotsc, X^{(m)}}$ where each random vectors can be interpreted as an obs | Maximum Likelihood Estimators - Multivariate Gaussian
Deriving the Maximum Likelihood Estimators
Assume that we have $m$ random vectors, each of size $p$: $\mathbf{X^{(1)}, X^{(2)}, \dotsc, X^{(m)}}$ where each random vectors can be interpreted as an observation (data point) across $p$ variables. If each $\mathbf{X}^{(... | Maximum Likelihood Estimators - Multivariate Gaussian
Deriving the Maximum Likelihood Estimators
Assume that we have $m$ random vectors, each of size $p$: $\mathbf{X^{(1)}, X^{(2)}, \dotsc, X^{(m)}}$ where each random vectors can be interpreted as an obs |
5,081 | Maximum Likelihood Estimators - Multivariate Gaussian | An alternate proof for $\widehat{\Sigma}$ that takes the derivative with respect to $\Sigma$ directly:
Picking up with the log-likelihood as above:
\begin{eqnarray}
\ell(\mu, \Sigma) &=& C - \frac{m}{2}\log|\Sigma|-\frac{1}{2} \sum_{i=1}^m \text{tr}\left[(\mathbf{x}^{(i)}-\mu)^T \Sigma^{-1} (\mathbf{x}^{(i)}-\mu)\r... | Maximum Likelihood Estimators - Multivariate Gaussian | An alternate proof for $\widehat{\Sigma}$ that takes the derivative with respect to $\Sigma$ directly:
Picking up with the log-likelihood as above:
\begin{eqnarray}
\ell(\mu, \Sigma) &=& C - \frac | Maximum Likelihood Estimators - Multivariate Gaussian
An alternate proof for $\widehat{\Sigma}$ that takes the derivative with respect to $\Sigma$ directly:
Picking up with the log-likelihood as above:
\begin{eqnarray}
\ell(\mu, \Sigma) &=& C - \frac{m}{2}\log|\Sigma|-\frac{1}{2} \sum_{i=1}^m \text{tr}\left[(\mathb... | Maximum Likelihood Estimators - Multivariate Gaussian
An alternate proof for $\widehat{\Sigma}$ that takes the derivative with respect to $\Sigma$ directly:
Picking up with the log-likelihood as above:
\begin{eqnarray}
\ell(\mu, \Sigma) &=& C - \frac |
5,082 | Maximum Likelihood Estimators - Multivariate Gaussian | While previous answers are correct, mentioning the trace is unnecessary (from a personal point of view).
The following derivation might be more succinct: | Maximum Likelihood Estimators - Multivariate Gaussian | While previous answers are correct, mentioning the trace is unnecessary (from a personal point of view).
The following derivation might be more succinct: | Maximum Likelihood Estimators - Multivariate Gaussian
While previous answers are correct, mentioning the trace is unnecessary (from a personal point of view).
The following derivation might be more succinct: | Maximum Likelihood Estimators - Multivariate Gaussian
While previous answers are correct, mentioning the trace is unnecessary (from a personal point of view).
The following derivation might be more succinct: |
5,083 | Regression: Transforming Variables | One transforms the dependent variable to achieve approximate symmetry and homoscedasticity of the residuals. Transformations of the independent variables have a different purpose: after all, in this regression all the independent values are taken as fixed, not random, so "normality" is inapplicable. The main objectiv... | Regression: Transforming Variables | One transforms the dependent variable to achieve approximate symmetry and homoscedasticity of the residuals. Transformations of the independent variables have a different purpose: after all, in this | Regression: Transforming Variables
One transforms the dependent variable to achieve approximate symmetry and homoscedasticity of the residuals. Transformations of the independent variables have a different purpose: after all, in this regression all the independent values are taken as fixed, not random, so "normality" ... | Regression: Transforming Variables
One transforms the dependent variable to achieve approximate symmetry and homoscedasticity of the residuals. Transformations of the independent variables have a different purpose: after all, in this |
5,084 | PP-plots vs. QQ-plots | As @vector07 notes, probability plot is the more abstract category of which pp-plots and qq-plots are members. Thus, I will discuss the distinction between the latter two. The best way to understand the differences is to think about how they are constructed, and to understand that you need to recognize the difference... | PP-plots vs. QQ-plots | As @vector07 notes, probability plot is the more abstract category of which pp-plots and qq-plots are members. Thus, I will discuss the distinction between the latter two. The best way to understand | PP-plots vs. QQ-plots
As @vector07 notes, probability plot is the more abstract category of which pp-plots and qq-plots are members. Thus, I will discuss the distinction between the latter two. The best way to understand the differences is to think about how they are constructed, and to understand that you need to re... | PP-plots vs. QQ-plots
As @vector07 notes, probability plot is the more abstract category of which pp-plots and qq-plots are members. Thus, I will discuss the distinction between the latter two. The best way to understand |
5,085 | PP-plots vs. QQ-plots | Here is a definition from v8doc.sas.com:
A P-P plot compares the empirical cumulative distribution function of a data set with a specified theoretical cumulative distribution function F(·). A Q-Q plot compares the quantiles of a data distribution with the quantiles of a standardized theoretical distribution from a spe... | PP-plots vs. QQ-plots | Here is a definition from v8doc.sas.com:
A P-P plot compares the empirical cumulative distribution function of a data set with a specified theoretical cumulative distribution function F(·). A Q-Q plo | PP-plots vs. QQ-plots
Here is a definition from v8doc.sas.com:
A P-P plot compares the empirical cumulative distribution function of a data set with a specified theoretical cumulative distribution function F(·). A Q-Q plot compares the quantiles of a data distribution with the quantiles of a standardized theoretical d... | PP-plots vs. QQ-plots
Here is a definition from v8doc.sas.com:
A P-P plot compares the empirical cumulative distribution function of a data set with a specified theoretical cumulative distribution function F(·). A Q-Q plo |
5,086 | How to interpret error measures? | Let's denote the true value of interest as $\theta$ and the value estimated using some algorithm as $\hat{\theta}$.
Correlation tells you how much $\theta$ and $\hat{\theta}$ are related. It gives values between $-1$ and $1$, where $0$ is no relation, $1$ is very strong, linear relation and $-1$ is an inverse linear r... | How to interpret error measures? | Let's denote the true value of interest as $\theta$ and the value estimated using some algorithm as $\hat{\theta}$.
Correlation tells you how much $\theta$ and $\hat{\theta}$ are related. It gives va | How to interpret error measures?
Let's denote the true value of interest as $\theta$ and the value estimated using some algorithm as $\hat{\theta}$.
Correlation tells you how much $\theta$ and $\hat{\theta}$ are related. It gives values between $-1$ and $1$, where $0$ is no relation, $1$ is very strong, linear relatio... | How to interpret error measures?
Let's denote the true value of interest as $\theta$ and the value estimated using some algorithm as $\hat{\theta}$.
Correlation tells you how much $\theta$ and $\hat{\theta}$ are related. It gives va |
5,087 | How are we defining 'reproducible research'? | "Reproducible research" as reproducible analysis
Reproducible research is a term used in some research domains to refer specifically to conducting analyses such that
code transforms raw data and meta-data into processed data,
code runs analyses on the data, and
code incorporates analyses into a report.
When such dat... | How are we defining 'reproducible research'? | "Reproducible research" as reproducible analysis
Reproducible research is a term used in some research domains to refer specifically to conducting analyses such that
code transforms raw data and met | How are we defining 'reproducible research'?
"Reproducible research" as reproducible analysis
Reproducible research is a term used in some research domains to refer specifically to conducting analyses such that
code transforms raw data and meta-data into processed data,
code runs analyses on the data, and
code incorp... | How are we defining 'reproducible research'?
"Reproducible research" as reproducible analysis
Reproducible research is a term used in some research domains to refer specifically to conducting analyses such that
code transforms raw data and met |
5,088 | How are we defining 'reproducible research'? | Having access to the data and code for the analysis in an easy-to-execute form is a sine qua non of reproducible research. Once you verify that the analysis works, you can substitute your own code/data where you are skeptical of the original author's. I'd say that the majority of statistics-containing papers I read h... | How are we defining 'reproducible research'? | Having access to the data and code for the analysis in an easy-to-execute form is a sine qua non of reproducible research. Once you verify that the analysis works, you can substitute your own code/da | How are we defining 'reproducible research'?
Having access to the data and code for the analysis in an easy-to-execute form is a sine qua non of reproducible research. Once you verify that the analysis works, you can substitute your own code/data where you are skeptical of the original author's. I'd say that the majo... | How are we defining 'reproducible research'?
Having access to the data and code for the analysis in an easy-to-execute form is a sine qua non of reproducible research. Once you verify that the analysis works, you can substitute your own code/da |
5,089 | How are we defining 'reproducible research'? | Being able to re-run everything is a starting point for the reproducible research. It permits to show that you are actually using the same procedure. After that -and only after that- you can pursue the research of your peer. In other words, the strict reproducibility is not to be perceived as a time at which the resear... | How are we defining 'reproducible research'? | Being able to re-run everything is a starting point for the reproducible research. It permits to show that you are actually using the same procedure. After that -and only after that- you can pursue th | How are we defining 'reproducible research'?
Being able to re-run everything is a starting point for the reproducible research. It permits to show that you are actually using the same procedure. After that -and only after that- you can pursue the research of your peer. In other words, the strict reproducibility is not ... | How are we defining 'reproducible research'?
Being able to re-run everything is a starting point for the reproducible research. It permits to show that you are actually using the same procedure. After that -and only after that- you can pursue th |
5,090 | how to weight KLD loss vs reconstruction loss in variational auto-encoder | For anyone stumbling on this post also looking for an answer, this twitter thread has added a lot of very useful insight.
Namely:
beta-VAE: Learning Basic Visual Concepts with a Constrained Variational Framework
discusses my exact question with a few experiments. Interestingly, it seems their $\beta_{norm}$ (which is s... | how to weight KLD loss vs reconstruction loss in variational auto-encoder | For anyone stumbling on this post also looking for an answer, this twitter thread has added a lot of very useful insight.
Namely:
beta-VAE: Learning Basic Visual Concepts with a Constrained Variationa | how to weight KLD loss vs reconstruction loss in variational auto-encoder
For anyone stumbling on this post also looking for an answer, this twitter thread has added a lot of very useful insight.
Namely:
beta-VAE: Learning Basic Visual Concepts with a Constrained Variational Framework
discusses my exact question with a... | how to weight KLD loss vs reconstruction loss in variational auto-encoder
For anyone stumbling on this post also looking for an answer, this twitter thread has added a lot of very useful insight.
Namely:
beta-VAE: Learning Basic Visual Concepts with a Constrained Variationa |
5,091 | how to weight KLD loss vs reconstruction loss in variational auto-encoder | I would like to add one more paper relating to this issue (I cannot comment due to my low reputation at the moment).
In subsection 3.1 of the paper, the authors specified that they failed to train a straight implementation of VAE that equally weighted the likelihood and the KL divergence. In their case, the KL loss was... | how to weight KLD loss vs reconstruction loss in variational auto-encoder | I would like to add one more paper relating to this issue (I cannot comment due to my low reputation at the moment).
In subsection 3.1 of the paper, the authors specified that they failed to train a s | how to weight KLD loss vs reconstruction loss in variational auto-encoder
I would like to add one more paper relating to this issue (I cannot comment due to my low reputation at the moment).
In subsection 3.1 of the paper, the authors specified that they failed to train a straight implementation of VAE that equally wei... | how to weight KLD loss vs reconstruction loss in variational auto-encoder
I would like to add one more paper relating to this issue (I cannot comment due to my low reputation at the moment).
In subsection 3.1 of the paper, the authors specified that they failed to train a s |
5,092 | how to weight KLD loss vs reconstruction loss in variational auto-encoder | I faced the same problem of not knowing how to weigh the reconstruction and KL terms, and here I want to add an answer with some concrete values of $\beta$ (the weight of KL term) for future reference.
In the $\beta$-VAE paper, they seem to use the values of $\beta_{norm}$ ranging between $0.001$ and $10$ (Fig. 6 from ... | how to weight KLD loss vs reconstruction loss in variational auto-encoder | I faced the same problem of not knowing how to weigh the reconstruction and KL terms, and here I want to add an answer with some concrete values of $\beta$ (the weight of KL term) for future reference | how to weight KLD loss vs reconstruction loss in variational auto-encoder
I faced the same problem of not knowing how to weigh the reconstruction and KL terms, and here I want to add an answer with some concrete values of $\beta$ (the weight of KL term) for future reference.
In the $\beta$-VAE paper, they seem to use t... | how to weight KLD loss vs reconstruction loss in variational auto-encoder
I faced the same problem of not knowing how to weigh the reconstruction and KL terms, and here I want to add an answer with some concrete values of $\beta$ (the weight of KL term) for future reference |
5,093 | how to weight KLD loss vs reconstruction loss in variational auto-encoder | Update on Dec. 6th 2020: I made a blog post to explain this in details.
I finally manage to figure out the reason of weighting KL divergence in VAE. It is more about the normalized constant of the distribution modeled the target variable. Here, I am going to present some output distributions we often use. Most of the ... | how to weight KLD loss vs reconstruction loss in variational auto-encoder | Update on Dec. 6th 2020: I made a blog post to explain this in details.
I finally manage to figure out the reason of weighting KL divergence in VAE. It is more about the normalized constant of the di | how to weight KLD loss vs reconstruction loss in variational auto-encoder
Update on Dec. 6th 2020: I made a blog post to explain this in details.
I finally manage to figure out the reason of weighting KL divergence in VAE. It is more about the normalized constant of the distribution modeled the target variable. Here, ... | how to weight KLD loss vs reconstruction loss in variational auto-encoder
Update on Dec. 6th 2020: I made a blog post to explain this in details.
I finally manage to figure out the reason of weighting KL divergence in VAE. It is more about the normalized constant of the di |
5,094 | How to make a time series stationary? | De-trending is fundamental. This includes regressing against covariates other than time.
Seasonal adjustment is a version of taking differences but could be construed as a separate technique.
Transformation of the data implicitly converts a difference operator into something else; e.g., differences of the logarithms a... | How to make a time series stationary? | De-trending is fundamental. This includes regressing against covariates other than time.
Seasonal adjustment is a version of taking differences but could be construed as a separate technique.
Transfo | How to make a time series stationary?
De-trending is fundamental. This includes regressing against covariates other than time.
Seasonal adjustment is a version of taking differences but could be construed as a separate technique.
Transformation of the data implicitly converts a difference operator into something else;... | How to make a time series stationary?
De-trending is fundamental. This includes regressing against covariates other than time.
Seasonal adjustment is a version of taking differences but could be construed as a separate technique.
Transfo |
5,095 | How to make a time series stationary? | I still think using the % change from one period to the next is the best way to render a non-stationary variable stationary as you first suggest. A transformation such as a log works reasonably well (it flattens the non-stationary quality; but does not eliminate it entirely).
The third way is to deseasonalize and de... | How to make a time series stationary? | I still think using the % change from one period to the next is the best way to render a non-stationary variable stationary as you first suggest. A transformation such as a log works reasonably well | How to make a time series stationary?
I still think using the % change from one period to the next is the best way to render a non-stationary variable stationary as you first suggest. A transformation such as a log works reasonably well (it flattens the non-stationary quality; but does not eliminate it entirely).
Th... | How to make a time series stationary?
I still think using the % change from one period to the next is the best way to render a non-stationary variable stationary as you first suggest. A transformation such as a log works reasonably well |
5,096 | How to make a time series stationary? | Logs and reciprocals and other power transformations often yield unexpected results.
As for detrending residuals(ie Tukey), this may have some application in some cases but could be dangerous. On the other hand, detecting level shifts and trend changes are systematically available to researchers employing interventi... | How to make a time series stationary? | Logs and reciprocals and other power transformations often yield unexpected results.
As for detrending residuals(ie Tukey), this may have some application in some cases but could be dangerous. On t | How to make a time series stationary?
Logs and reciprocals and other power transformations often yield unexpected results.
As for detrending residuals(ie Tukey), this may have some application in some cases but could be dangerous. On the other hand, detecting level shifts and trend changes are systematically availab... | How to make a time series stationary?
Logs and reciprocals and other power transformations often yield unexpected results.
As for detrending residuals(ie Tukey), this may have some application in some cases but could be dangerous. On t |
5,097 | How to make a time series stationary? | Could you fit a loess/spline through the data and use the residuals? Would the residuals be stationary?
Seems fraught with issues to consider, and perhaps there would not be as clear an indication of an overly-flexible curve as there is for over-differencing. | How to make a time series stationary? | Could you fit a loess/spline through the data and use the residuals? Would the residuals be stationary?
Seems fraught with issues to consider, and perhaps there would not be as clear an indication of | How to make a time series stationary?
Could you fit a loess/spline through the data and use the residuals? Would the residuals be stationary?
Seems fraught with issues to consider, and perhaps there would not be as clear an indication of an overly-flexible curve as there is for over-differencing. | How to make a time series stationary?
Could you fit a loess/spline through the data and use the residuals? Would the residuals be stationary?
Seems fraught with issues to consider, and perhaps there would not be as clear an indication of |
5,098 | How to make a time series stationary? | Difference with another series. i.e. Brent oil prices are not stationary,
but the spread brent-light sweet crude might be. A more risky proposition for
forecasting is to bet on the existence of a co integration relationship
with another time series. | How to make a time series stationary? | Difference with another series. i.e. Brent oil prices are not stationary,
but the spread brent-light sweet crude might be. A more risky proposition for
forecasting is to bet on the existence of a co i | How to make a time series stationary?
Difference with another series. i.e. Brent oil prices are not stationary,
but the spread brent-light sweet crude might be. A more risky proposition for
forecasting is to bet on the existence of a co integration relationship
with another time series. | How to make a time series stationary?
Difference with another series. i.e. Brent oil prices are not stationary,
but the spread brent-light sweet crude might be. A more risky proposition for
forecasting is to bet on the existence of a co i |
5,099 | Is it possible to give variable sized images as input to a convolutional neural network? | There are a number of ways to do it. Most of these have already been covered in a number of posts over StackOverflow, Quora and other content websites.
To summarize, most of the techniques listed can be grouped into two classes of solutions, namely,
Transformations
Inherent Network Property
In transformations, one ca... | Is it possible to give variable sized images as input to a convolutional neural network? | There are a number of ways to do it. Most of these have already been covered in a number of posts over StackOverflow, Quora and other content websites.
To summarize, most of the techniques listed can | Is it possible to give variable sized images as input to a convolutional neural network?
There are a number of ways to do it. Most of these have already been covered in a number of posts over StackOverflow, Quora and other content websites.
To summarize, most of the techniques listed can be grouped into two classes of ... | Is it possible to give variable sized images as input to a convolutional neural network?
There are a number of ways to do it. Most of these have already been covered in a number of posts over StackOverflow, Quora and other content websites.
To summarize, most of the techniques listed can |
5,100 | Is it possible to give variable sized images as input to a convolutional neural network? | The convolutional layers and pooling layers themselves are independent of the input dimensions. However, the output of the convolutional layers will have different spatial sizes for differently sized images, and this will cause an issue if we have a fully connected layer afterwards (since our fully connected layer requ... | Is it possible to give variable sized images as input to a convolutional neural network? | The convolutional layers and pooling layers themselves are independent of the input dimensions. However, the output of the convolutional layers will have different spatial sizes for differently sized | Is it possible to give variable sized images as input to a convolutional neural network?
The convolutional layers and pooling layers themselves are independent of the input dimensions. However, the output of the convolutional layers will have different spatial sizes for differently sized images, and this will cause an ... | Is it possible to give variable sized images as input to a convolutional neural network?
The convolutional layers and pooling layers themselves are independent of the input dimensions. However, the output of the convolutional layers will have different spatial sizes for differently sized |
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