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1195 | 1 | null | null | 7 | 431 | Many income surveys (especially older ones) truncate key variables, such as household income, at some arbitrary point, to protect confidentiality. This point changes over time. This reduces inequality measures associated with the variable. I am interested in fitting a Pareto tail to the truncated distribution, replacin... | How can I apply a Pareto tail to a truncated distribution? | CC BY-SA 2.5 | null | 2010-08-03T20:23:34.770 | 2011-05-12T17:12:20.963 | 2010-08-03T21:44:30.103 | null | 687 | [
"pareto-distribution",
"data-imputation"
] |
1196 | 2 | null | 1164 | 23 | null | Statistics is a tool for non-statistical-minded researchers, and they just don't care.
I once tried to help with a Medicine article my ex-wife was co-authoring. I wrote several pages describing the data, what it suggested, why certain observations had been excluded from the study... and the lead researcher, a doctor, t... | null | CC BY-SA 2.5 | null | 2010-08-03T20:26:23.000 | 2010-08-03T20:26:23.000 | null | null | 666 | null |
1197 | 2 | null | 1194 | 8 | null | I haven't read her work beyond the abstract of the linked paper, but my sense is that the distinction between "explanation" and "prediction" should be thrown away and replaced with the distinction between the aims of the practitioner, which are either "causal" or "predictive". In general, I think "explanation" is such ... | null | CC BY-SA 2.5 | null | 2010-08-03T20:39:20.213 | 2010-08-03T20:39:20.213 | null | null | 303 | null |
1198 | 2 | null | 652 | 7 | null | This book is dynamite:
[George E. P. Box, Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building](http://rads.stackoverflow.com/amzn/click/0471093157)
It starts from zero knowledge of Statistics but it doesn't insult the reader's intelligence. It's incredibly practical but with no lo... | null | CC BY-SA 2.5 | null | 2010-08-03T21:19:22.083 | 2010-08-03T21:39:31.527 | 2010-08-03T21:39:31.527 | 666 | 666 | null |
1199 | 2 | null | 1194 | 5 | null | as others have already said, the distinction is somewhat meaningless, except in so far as the aims of the researcher are concerned.
Brad Efron, one of the commentators on [The Two Cultures](http://www.stat.osu.edu/~bli/dmsl/papers/Breiman.pdf) paper, made the following observation (as discussed [in my earlier question... | null | CC BY-SA 2.5 | null | 2010-08-03T21:30:22.343 | 2010-08-03T21:46:47.963 | 2017-04-13T12:44:28.813 | -1 | 5 | null |
1200 | 2 | null | 1194 | 7 | null | I am still a bit unclear as to what the question is. Having said that, to my mind the fundamental difference between predictive and explanatory models is the difference in their focus.
Explanatory Models
By definition explanatory models have as their primary focus the goal of explaining something in the real world. In ... | null | CC BY-SA 2.5 | null | 2010-08-03T21:32:40.813 | 2010-08-03T21:32:40.813 | null | null | null | null |
1201 | 2 | null | 11 | 3 | null | Sorry, no quick answer. There are thick books dedicated to answering this question. Here's a 600-page long example: [Harrell's Regression Modeling Strategies](http://rads.stackoverflow.com/amzn/click/1441929185)
| null | CC BY-SA 2.5 | null | 2010-08-03T21:50:09.007 | 2010-08-03T21:50:09.007 | null | null | 666 | null |
1202 | 1 | null | null | 3 | 693 | The question in short: What methods can be used to quantify distributional relationships between data when the distribution is unknown?
Now the longer story: I have a list of distributions and would like to rank them based on their similarity to a given base-line distribution. Correlation jumps into my mind in such a c... | Ranking distributional data by similarity | CC BY-SA 3.0 | null | 2010-08-03T22:09:33.840 | 2018-02-11T00:49:37.040 | 2018-02-11T00:07:14.130 | 186294 | 608 | [
"distributions",
"relative-distribution"
] |
1203 | 2 | null | 608 | 6 | null | In an attempt to partially answer my own question, I read [Wikipedia's](https://stats.stackexchange.com/questions/577/is-there-any-reason-to-prefer-the-aic-or-bic-over-the-other) description of leave-one-out cross validation
>
involves using a single observation
from the original sample as the
validation data, an... | null | CC BY-SA 2.5 | null | 2010-08-03T22:23:31.813 | 2010-08-03T22:58:09.447 | 2017-04-13T12:44:35.347 | -1 | 196 | null |
1204 | 2 | null | 1202 | 0 | null | Edit: I misunderstood the question at first. Your observations are actually paired. Sample1 Bin1 to Baseline Bin1 etc.
What you could do is take the difference between sample and baseline for each bin, then use the Wilcoxon signed rank statistic on the differences.
[http://en.wikipedia.org/wiki/Wilcoxon_signed-rank_tes... | null | CC BY-SA 2.5 | null | 2010-08-03T22:25:14.913 | 2010-08-03T22:39:31.763 | 2010-08-03T22:39:31.763 | 287 | 287 | null |
1205 | 1 | null | null | 15 | 15603 | I want to perform a two-sample T-test to test for a difference between two independent samples which each sample abides by the assumptions of the T-test (each distribution can be assumed to be independent and identically distributed as Normal with equal variance). The only complication from the basic two-sample T-test ... | Two-sample T-test with weighted data | CC BY-SA 2.5 | null | 2010-08-03T22:56:48.420 | 2010-08-04T15:57:30.210 | null | null | 366 | [
"t-test"
] |
1206 | 2 | null | 1194 | 21 | null | One practical issue that arises here is variable selection in modelling. A variable can be an important explanatory variable (e.g., is statistically significant) but may not be useful for predictive purposes (i.e., its inclusion in the model leads to worse predictive accuracy). I see this mistake almost every day in pu... | null | CC BY-SA 3.0 | null | 2010-08-03T23:36:08.593 | 2015-07-28T11:25:31.220 | 2015-07-28T11:25:31.220 | 68299 | 159 | null |
1207 | 1 | null | null | 61 | 37684 | This post is the continuation of another post related to a [generic method for outlier detection in time series](https://stats.stackexchange.com/questions/1142/simple-algorithm-for-online-outlier-detection-of-a-generic-time-series).
Basically, at this point I'm interested in a robust way to discover the periodicity/sea... | Period detection of a generic time series | CC BY-SA 3.0 | null | 2010-08-04T00:32:13.360 | 2021-07-06T18:38:45.747 | 2017-04-13T12:44:56.303 | -1 | 667 | [
"time-series",
"algorithms",
"frequency",
"real-time"
] |
1210 | 2 | null | 1133 | 12 | null | You should look into "partitioning chi-squared". This is similar in logic to performing post-hoc tests in ANOVA. It will allow you to determine whether your significant overall test is primarily attributable to differences in particular categories or groups of categories.
A quick google turned up this presentation, w... | null | CC BY-SA 2.5 | null | 2010-08-04T01:26:59.243 | 2010-08-04T01:26:59.243 | null | null | 485 | null |
1211 | 2 | null | 770 | 4 | null | It's like that old joke. When asked for directions the philosopher said "Well, if I wanted to go there, I wouldn't start from here ..."
While I think each "culture" should be open to learning from the other, they have different ways of looking at the world.
I think the problem with learning statistics through studying ... | null | CC BY-SA 2.5 | null | 2010-08-04T02:05:29.027 | 2010-08-04T02:05:29.027 | null | null | 521 | null |
1212 | 2 | null | 1174 | 22 | null | First of all my advice is you must refrain from trying out a Poisson distribution just as it is to the data. I suggest you must first make a theory as to why should Poisson distribution fit a particular dataset or a phenomenon.
Once you have established this, the next question is whether the distribution is homogeneous... | null | CC BY-SA 3.0 | null | 2010-08-04T03:23:21.247 | 2015-09-03T22:17:07.177 | 2015-09-03T22:17:07.177 | 56216 | 25692 | null |
1213 | 2 | null | 138 | 57 | null |
## Some useful R links (find out the link that suits you):
Intro:
- for R basics http://cran.r-project.org/doc/contrib/usingR.pdf
- for data manipulation http://had.co.nz/plyr/plyr-intro-090510.pdf
- http://portal.stats.ox.ac.uk/userdata/ruth/APTS2012/APTS.html
- Interactive intro to R programming language https... | null | CC BY-SA 3.0 | null | 2010-08-04T03:28:53.913 | 2015-04-01T21:11:30.723 | 2020-06-11T14:32:37.003 | -1 | 25692 | null |
1214 | 2 | null | 1207 | 57 | null | If you really have no idea what the periodicity is, probably the best approach is to find the frequency corresponding to the maximum of the spectral density. However, the spectrum at low frequencies will be affected by trend, so you need to detrend the series first. The following R function should do the job for most s... | null | CC BY-SA 4.0 | null | 2010-08-04T05:41:02.697 | 2021-07-06T18:38:45.747 | 2021-07-06T18:38:45.747 | 11887 | 159 | null |
1215 | 1 | 1220 | null | 2 | 181 | I have a list of sold items by size. Shoes in this case
```
Size Qty
35 2
36 1
37 4
38 4
39 32
40 17
41 23
42 57
43 95
44 90
45 98
46 33
47 16
48 4
total: 476
```
I have to tell the owner how much of every size to buy. The problem is, I can't say him.
- You should buy 95 shoes size 43 for every one ... | Analysis of sells (what to buy) | CC BY-SA 4.0 | null | 2010-08-04T05:41:51.087 | 2020-01-01T17:57:19.090 | 2020-01-01T17:57:19.090 | 92235 | 698 | [
"data-visualization"
] |
1216 | 2 | null | 1215 | 3 | null | I suggest he stocks $y_s$ pairs of shoes of size $s$ where $y_s$ is chosen so that the probability of running out of stock before the next delivery is set to some acceptable level (e.g., 5%).
It seems reasonable to assume $y_s$ is Poisson with rate $\lambda_s$. You can estimate $\lambda_s$ as the average sales of that ... | null | CC BY-SA 2.5 | null | 2010-08-04T06:12:22.350 | 2010-08-04T08:13:41.147 | 2010-08-04T08:13:41.147 | 159 | 159 | null |
1217 | 2 | null | 1194 | 12 | null | Example: A classic example that I have seen is in the context of predicting human performance.
Self-efficacy (i.e., the degree to which a person thinks that they can perform a task well) is often a strong predictor of task performance. Thus, if you put self-efficacy into a multiple regression along with other variables... | null | CC BY-SA 2.5 | null | 2010-08-04T06:16:41.453 | 2010-08-04T06:16:41.453 | null | null | 183 | null |
1218 | 2 | null | 459 | 1 | null | I would strongly enjoin you to avoid red as an indicator: there are many sorts of colour-deficiency that make this choice problematic (see eg [http://en.wikipedia.org/wiki/Color_blindness#Design_implications_of_color_blindness](http://en.wikipedia.org/wiki/Color_blindness#Design_implications_of_color_blindness) ).
The ... | null | CC BY-SA 2.5 | null | 2010-08-04T06:51:12.213 | 2010-08-04T06:51:12.213 | null | null | null | null |
1219 | 2 | null | 764 | 10 | null | A very good article explaining the general approach of LMMs and their advantage over ANOVA is:
- Baayen, R. H., Davidson, D. J., & Bates, D. M. (2008). Mixed-effects modeling with crossed random effects for subjects and items. Journal of Memory and Language, 59, 390-412.
Linear mixed-effects models (LMMs) generalize... | null | CC BY-SA 2.5 | null | 2010-08-04T08:20:14.020 | 2010-08-04T08:20:14.020 | null | null | 702 | null |
1220 | 2 | null | 1215 | 4 | null | Following on from Rob's answer, you could estimate an acceptable level using a Normal approximation. So if an acceptable rate of running out is 5%, then use the following rule:
Qty + 1.644 * Qty^(0.5)
The value 1.644 comes from the [Normal](http://en.wikipedia.org/wiki/Normal_distribution) or Gaussian distribution.
Oth... | null | CC BY-SA 2.5 | null | 2010-08-04T08:20:54.580 | 2010-08-04T08:33:07.973 | 2010-08-04T08:33:07.973 | 8 | 8 | null |
1221 | 2 | null | 534 | 10 | null | In the twins example it is not just the correlation that suggests causality, but also the associated information or prior knowledge.
Suppose I add one further piece of information. Assume that the diligent twin spent 6 hours studying for a stats exam, but due to an unfortunate error the exam was in history. Would we s... | null | CC BY-SA 2.5 | null | 2010-08-04T08:39:21.830 | 2010-08-04T08:44:43.833 | 2010-08-04T08:44:43.833 | 521 | 521 | null |
1222 | 2 | null | 886 | 9 | null | I can't post a comment (the appropriate place for this comment) as I don't have enough reputation, but the answer accepted as the best answer by the question owner misses the point.
"If statistics is all about maximizing likelihood, then machine learning is all about minimizing loss."
The likelihood is a loss function.... | null | CC BY-SA 2.5 | null | 2010-08-04T09:07:44.193 | 2010-08-04T09:07:44.193 | null | null | 521 | null |
1223 | 1 | null | null | 18 | 9771 | I'm looking for some robust techniques to remove outliers and errors (whatever the cause) from financial time-series data (i.e. tickdata).
Tick-by-tick financial time-series data is very messy. It contains huge (time) gaps when the exchange is closed, and make huge jumps when the exchange opens again. When the exchang... | Robust outlier detection in financial timeseries | CC BY-SA 2.5 | null | 2010-08-04T10:02:35.090 | 2022-01-11T16:19:56.830 | 2017-04-13T12:44:46.083 | -1 | 127 | [
"time-series",
"outliers"
] |
1224 | 1 | null | null | 2 | 186 | With data from two centres I want to account for potential heterogeneity or confounders between two centers. So the analysis will initially be stratified by clinical center and a chi square test performed with one degree of freedom. Is this appropriate with just two centres? Or is there an alternative?
| Heterogeneity with two studies | CC BY-SA 2.5 | null | 2010-08-04T10:38:03.740 | 2010-10-19T06:44:37.850 | 2010-10-18T21:55:09.667 | 930 | null | [
"hypothesis-testing",
"clinical-trials"
] |
1225 | 2 | null | 652 | 3 | null | Statistics as Principled Argument by Abelson is a good side book to learning statistics, particularly if your substantive field is in the social sciences. It won't teach you how to do analysis, but it will teach you about statistical thinking.
I reviewed this book [here](http://www.statisticalanalysisconsulting.com/bo... | null | CC BY-SA 3.0 | null | 2010-08-04T10:55:31.480 | 2015-07-20T16:35:32.303 | 2015-07-20T16:35:32.303 | 686 | 686 | null |
1226 | 2 | null | 30 | 3 | null | There are two parts to testing a random number generator. If you're only concerned with testing a uniform generator, then yes, something like the DIEHARD test suite is a good idea.
But often you need to test a transformation of a uniform generator. For example, you might use a uniform generator to create exponentially... | null | CC BY-SA 2.5 | null | 2010-08-04T11:48:13.410 | 2010-08-04T11:48:13.410 | null | null | 319 | null |
1228 | 1 | 1273 | null | 10 | 1192 | I am using a control chart to try to work on some infection data, and will raise an alert if the infection is considered "out of control".
Problems arrive when I come to a set of data where most of the time points have zero infection, with only a few occasions of one to two infections, but these already exceed the cont... | How to interpret a control chart containing a majority of zero values? | CC BY-SA 4.0 | null | 2010-08-04T12:05:30.000 | 2022-11-27T06:33:46.727 | 2022-11-27T06:33:46.727 | 362671 | 588 | [
"data-visualization",
"quality-control"
] |
1229 | 2 | null | 652 | 3 | null | The best intro in my eyes is the following one:
>
David Howell - Statistical Methods for
Psychology
It is the BEST in making statistical concepts understandable for non mathematicians so that they get the math afterwards!
Unfortunately it is updated every year and, hence, pricey.
| null | CC BY-SA 2.5 | null | 2010-08-04T12:32:08.780 | 2010-08-04T12:32:08.780 | null | null | 442 | null |
1230 | 2 | null | 652 | 5 | null | I am a big fan of [Statistical Models - Theory and Practice](http://rads.stackoverflow.com/amzn/click/0521671051) by David Friedman. It succeeds remarkably well to introduce and motivate the different concepts of statistical modeling through concrete, and historically important problems (cholera in London, Yule on the... | null | CC BY-SA 2.5 | null | 2010-08-04T12:32:12.103 | 2010-08-04T12:32:12.103 | null | null | 358 | null |
1231 | 2 | null | 1228 | 2 | null | You are asking quite a tricky question!
This is outside my area of expertise, but I know that [Prof Farrington](https://web.archive.org/web/20100507070649/http://www.mcs.open.ac.uk/People/c.p.farrington) does some work on this problem. So I would look at a some of his papers and follow a few of his references. To get y... | null | CC BY-SA 4.0 | null | 2010-08-04T12:52:05.560 | 2022-11-27T06:33:10.127 | 2022-11-27T06:33:10.127 | 362671 | 8 | null |
1232 | 2 | null | 1228 | 1 | null | Would it make sense to plot the control chart based on an average of the weekly infections or another similar floating average? Would this then 'damp' out spikes due to daily high values whilst ensuring that changes in trends are picked up in a relatively timely manner.
| null | CC BY-SA 2.5 | null | 2010-08-04T12:56:10.217 | 2010-08-04T12:56:10.217 | null | null | 210 | null |
1233 | 2 | null | 1228 | 1 | null | Perhaps, you can build an edge case in your routine/software to deal with the situation. If you detect several zeros in the dataset then you set a separate control for that particular situation. This is obviously a hack and not a principled solution but may serve your present needs till you can come up with something b... | null | CC BY-SA 2.5 | null | 2010-08-04T13:18:12.700 | 2010-08-04T13:18:12.700 | null | null | null | null |
1234 | 2 | null | 1223 | 5 | null | I have (with some delay) changed my answer to reflect your concern about the lack of 'adaptability' of the unconditional mad/median.
You can address the problem of time varying volatility with the robust statistics framework. This is done by using a robust estimator of the conditional variance (instead of the robust es... | null | CC BY-SA 3.0 | null | 2010-08-04T13:32:07.383 | 2013-11-08T15:43:39.733 | 2013-11-08T15:43:39.733 | 603 | 603 | null |
1235 | 2 | null | 1223 | 8 | null | I'll add some paper references when I'm back at a computer, but here are some simple suggestions:
Definitely start by working with returns. This is critical to deal with the irregular spacing where you can naturally get big price gaps (especially around weekends). Then you can apply a simple filter to remove returns we... | null | CC BY-SA 2.5 | null | 2010-08-04T13:47:36.963 | 2010-08-04T15:45:37.880 | 2010-08-04T15:45:37.880 | 5 | 5 | null |
1236 | 2 | null | 652 | 1 | null | As a biologist, I found the Sokal and Rohlf text to be quite readable, despite its voluminous-ness. It's not so great as a quick reference, but does walk one through statistical theory.
R. R. Sokal and F. J. Rohlf, Biometry the principles and practice of statistics in biological research, Third. (New York: W.J. Freeman... | null | CC BY-SA 2.5 | null | 2010-08-04T14:02:21.880 | 2010-08-04T14:02:21.880 | null | null | 124 | null |
1237 | 2 | null | 109 | 1 | null | I usually like to run simulations to answer questions like this, but without confirmed details of the algorithm the question asker wants evaluated and no obvious implementation of the Holm-Sidak procedure available in R, that is not possible. For my answer I eyeballed the code provided [here](http://www.mathworks.com/... | null | CC BY-SA 2.5 | null | 2010-08-04T14:32:12.320 | 2010-08-04T14:38:22.450 | 2010-08-04T14:38:22.450 | 196 | 196 | null |
1238 | 2 | null | 30 | 2 | null | The [NIST publishes a list of statistical tests](http://csrc.nist.gov/groups/ST/toolkit/rng/stats_tests.html) with a reference implementation in C.
There is also [TestU01](http://www.iro.umontreal.ca/~simardr/testu01/tu01.html) by some smart folks, including respected PRNG researcher Pierre L'Ecuyer. Again, there is a ... | null | CC BY-SA 2.5 | null | 2010-08-04T14:47:27.783 | 2010-08-04T14:49:15.003 | 2010-08-04T14:49:15.003 | null | 729 | null |
1240 | 2 | null | 30 | 4 | null | Small correction to Colin's post: the CRAN package
[RDieHarder](http://cran.r-project.org/package=RDieHarder) is an interface to
[DieHarder](http://www.phy.duke.edu/~rgb/General/dieharder.php), the Diehard rewrite / extension / overhaul done by [Robert G. Brown](http://www.phy.duke.edu/~rgb/) (who kindly lists me as ... | null | CC BY-SA 2.5 | null | 2010-08-04T15:11:13.447 | 2010-08-04T15:11:13.447 | null | null | 334 | null |
1241 | 1 | 1264 | null | 12 | 5184 | When would you tend to use ROC curves over some other tests to determine the predictive ability of some measurement on an outcome?
When dealing with discrete outcomes (alive/dead, present/absent), what makes ROC curves more or less powerful than something like a chi-square?
| What do ROC curves tell you that traditional inference wouldn't? | CC BY-SA 2.5 | null | 2010-08-04T15:13:27.610 | 2011-10-25T18:17:27.177 | 2010-08-27T02:42:25.167 | 159 | 684 | [
"regression",
"roc"
] |
1243 | 2 | null | 1174 | 2 | null | I think the main point is the one sidmaestro raises...does the experimental setup or data generation mechanism support the premise that the data might arise from a Poisson distribution.
I'm not a big fan of testing for distributional assumptions, since those tests typically aren't very useful. What seems more useful t... | null | CC BY-SA 2.5 | null | 2010-08-04T15:24:11.960 | 2010-08-04T15:24:11.960 | null | null | 732 | null |
1244 | 2 | null | 1149 | 7 | null | The way I think about this really is in terms of information. Say each of $X_{1}$ and $X_{2}$ has some information about $Y$. The more correlated $X_{1}$ and $X_{2}$ are with each other, the more the information content about $Y$ from $X_{1}$ and $X_{2}$ are similar or overlapping, to the point that for perfectly corre... | null | CC BY-SA 3.0 | null | 2010-08-04T15:37:40.830 | 2012-08-20T18:30:25.480 | 2012-08-20T18:30:25.480 | 13091 | 732 | null |
1245 | 2 | null | 1223 | 14 | null | The problem is definitely hard.
Mechanical rules like the +/- N1 times standard deviations, or +/ N2 times MAD, or +/- N3 IQR or ... will fail because there are always some series that are different as for example:
- fixings like interbank rate may be constant for some time and then jump all of a sudden
- similarly... | null | CC BY-SA 2.5 | null | 2010-08-04T15:51:53.803 | 2010-08-04T15:51:53.803 | null | null | 334 | null |
1246 | 2 | null | 1241 | 6 | null | An ROC curve is used when the predictor is continuous and the outcome is discrete, so a chi-square test would not be applicable. In fact, ROC analysis is in some sense equivalent to the Mann-Whitney test: the area under the curve is P(X>Y) which is the quantity being tested by the M-W test. However Mann-Whitney analysi... | null | CC BY-SA 2.5 | null | 2010-08-04T15:55:11.867 | 2010-08-04T15:55:11.867 | null | null | 279 | null |
1247 | 2 | null | 1241 | 6 | null | The shortest answer is that traditional tests of signal detection only give you a single point on the ROC (receiver operating characteristic) while the curve allows you to see responses through a range of values. It's possible that the criteria and d' shift as one shifts throughout the curve. It's like the difference... | null | CC BY-SA 2.5 | null | 2010-08-04T15:56:39.150 | 2010-08-04T15:56:39.150 | null | null | 601 | null |
1248 | 2 | null | 1205 | 8 | null | Use regression methods. A simple linear regression with group coded as 0-1 (or 1-2, etc) is equivalent to a t-test, but regression software usually has the capability to incorporate weigths correctly.
| null | CC BY-SA 2.5 | null | 2010-08-04T15:57:30.210 | 2010-08-04T15:57:30.210 | null | null | 279 | null |
1249 | 1 | null | null | 7 | 575 | How to find a non-trivial upper bound on $E[\exp(Z^2)]$ when $Z \sim {\rm Bin}(n, n^{-\beta})$ with $\beta \in (0,1)$? A trivial bound is obtained for substituting $Z$ with $n$.
A background on this question. In the paper by Baraud, 2002 -- Non-asymptotic minimax rates of testing in signal detection, if one is to subst... | Non-trivial bound for $E[\exp(Z^2)]$ when $Z \sim {\rm Bin}(n, n^{-\beta})$ with $\beta \in (0,1)$ | CC BY-SA 2.5 | null | 2010-08-04T16:04:56.337 | 2011-04-29T00:25:44.380 | 2011-04-29T00:25:44.380 | 3911 | 168 | [
"probability",
"binomial-distribution",
"mathematical-statistics"
] |
1251 | 2 | null | 1173 | 16 | null | Thanks for all you answers. For completeness I thought I should include what I usually do. I tend to do a combination of the suggestions given: dots, boxplots (when n is large), and se (or sd) ranges.
(Removed by moderator because the site hosting the image no longer appears to work correctly.)
From the dot plot, it is... | null | CC BY-SA 3.0 | null | 2010-08-04T16:29:26.283 | 2015-01-20T22:43:57.627 | 2015-01-20T22:43:57.627 | 919 | 8 | null |
1252 | 1 | 1349 | null | 47 | 17341 | I was wondering if there is a statistical model "cheat sheet(s)" that lists any or more information:
- when to use the model
- when not to use the model
- required and optional inputs
- expected outputs
- has the model been tested in different fields (policy, bio, engineering, manufacturing, etc)?
- is it accepte... | Statistical models cheat sheet | CC BY-SA 3.0 | null | 2010-08-04T16:39:49.250 | 2021-05-10T18:32:04.360 | 2016-08-04T12:59:35.907 | 22468 | 59 | [
"references",
"modeling"
] |
1253 | 1 | null | null | 6 | 443 | I'm trying to compute item-item similarity using [Jaccard (specifically Tanimoto)](http://en.wikipedia.org/wiki/Jaccard_index#Tanimoto_coefficient_.28extended_Jaccard_coefficient.29) on a large list of data in the format
```
(userid, itemid)
```
An item is considered as rated if i have a userid-itemid pair. I have abo... | How to limit my input data for Jaccard item-item similarity calculation? | CC BY-SA 3.0 | null | 2010-08-04T16:53:04.197 | 2012-07-08T08:54:17.417 | 2012-07-08T08:54:17.417 | 930 | 738 | [
"distributions",
"data-mining"
] |
1254 | 2 | null | 1252 | 22 | null | Do you mean a statistical analysis decision tree? ([google search](http://www.google.com/images?um=1&hl=en&biw=1024&bih=628&tbs=isch:1&sa=1&q=statistical+analysis+decision+tree&aq=f&aqi=&aql=&oq=&gs_rfai=)), like this (only with extensions):
[](https://i.stack.imgur.com/VM7XX.gif)
(source: [processma.com](http://www.pr... | null | CC BY-SA 4.0 | null | 2010-08-04T17:30:03.033 | 2019-02-18T20:52:18.257 | 2019-02-18T20:52:18.257 | 253 | 253 | null |
1255 | 2 | null | 1253 | 2 | null | I'm confused: shouldn't you only need the 7900^2 item similarities, for which you use ratings from all users, which is still quite sparse?
UPDATE
I still think there's a more efficient way to do this, but maybe I'm just being dense. Specifically, consider item A and item B. For item A, generate a U-dimensional vector o... | null | CC BY-SA 2.5 | null | 2010-08-04T17:35:42.507 | 2010-08-04T18:26:22.043 | 2010-08-04T18:26:22.043 | 303 | 303 | null |
1256 | 1 | 1263 | null | 6 | 2895 | I have a colleague who calculates correlations in which one set of scores for a subject (e.g. 100 scores) is correlated with another set of scores for that same subject. The resulting correlation reflects the degree to which those sets of scores are associated for that subject. He needs to do this for N subjects. Cons... | How can one speed up this correlation calculation in R without multicore? | CC BY-SA 2.5 | null | 2010-08-04T17:39:29.803 | 2010-08-05T10:00:43.613 | null | null | 196 | [
"r",
"correlation",
"efficiency"
] |
1257 | 1 | 1312 | null | 2 | 200 | Consider the following sequential, adaptive data generating process for $Y_1$, $Y_2$, $Y_3$. (By sequential I mean that we generate $Y_1$, $Y_2$, $Y_3$ in sequence and by adaptive I mean that $Y_3$ is generated depending on the observed values of $Y_1$ and $Y_2$.):
$Y_1 = X_1\ \beta + \epsilon_1$
$Y_2 = X_2\ \beta + ... | What is the correct likelihood function for an sequential, adaptive data generation process? | CC BY-SA 2.5 | null | 2010-08-04T18:04:52.290 | 2017-04-26T18:56:56.637 | 2017-04-26T18:56:56.637 | 113090 | null | [
"time-series",
"likelihood"
] |
1258 | 2 | null | 1164 | 12 | null | As someone who has learned a little bit of statistics for my own research, I'll guess that the reasons are pedagogical and inertial.
I've observed within my own field that the order in which topics are taught reflects the history of the field. Those ideas which came first are taught first, and so on. For people who onl... | null | CC BY-SA 2.5 | null | 2010-08-04T18:12:23.507 | 2010-08-04T18:12:23.507 | null | null | 287 | null |
1259 | 2 | null | 726 | 11 | null | >
All information looks like noise until you break the code.
Hiro in Neal Stephenson's Snow Crash (1992)
| null | CC BY-SA 3.0 | null | 2010-08-04T18:29:57.000 | 2015-08-19T15:43:58.967 | 2015-08-19T15:43:58.967 | 49500 | 743 | null |
1260 | 2 | null | 1256 | 1 | null | It might be one of those cases where using a different [BLAS engine](http://cran.r-project.org/bin/windows/base/rw-FAQ.html#Can-I-use-a-fast-BLAS_003f) would help. But I am not sure of it - it needs testing (and depends on your machine)
| null | CC BY-SA 2.5 | null | 2010-08-04T18:35:22.340 | 2010-08-04T18:35:22.340 | null | null | 253 | null |
1261 | 1 | 1282 | null | 4 | 271 | Suppose I have a table of counts that look like this
```
A B C
Success 1261 230 3514
Failure 381 161 4012
```
I have a hypothesis that there is some probability $p$ such that $P(Success_A) = p^i$, $P(Success_B) = p^j$ and $P(Success_C) = p^k$.
Is there some way to produce estimates for $p$, $... | Fitting a fixed, exponential relationship between categories with categorical data | CC BY-SA 2.5 | null | 2010-08-04T19:06:50.250 | 2010-08-05T10:19:24.293 | null | null | 287 | [
"modeling",
"categorical-data",
"chi-squared-test"
] |
1262 | 2 | null | 1261 | 2 | null | You could write the likelihood function like so:
$L(p,i,j,k|-) \propto (p^i)^{1261} (1-p^i)^{381} (p^j)^{230} (1-p^j)^{161} (p^k)^{3514} (1-p^k)^{4012}$
Maximize the above likelihood function to estimate your parameters. Constructing confidence intervals and hypothesis testing should be straight forward once you have t... | null | CC BY-SA 2.5 | null | 2010-08-04T19:19:23.767 | 2010-08-05T10:19:24.293 | 2010-08-05T10:19:24.293 | null | null | null |
1263 | 2 | null | 1256 | 3 | null | While making a call to `diag` you throw out a lot of information, so you can save time by simply not calculating it. You code is equivalent to:
```
sapply(1:100,function(i) cor(x[i,],y[i,]))
```
Extended to reflect comments: This code will be slower for small matrices since it does not use the full "vectorization powe... | null | CC BY-SA 2.5 | null | 2010-08-04T19:30:30.623 | 2010-08-05T10:00:43.613 | 2010-08-05T10:00:43.613 | null | null | null |
1264 | 2 | null | 1241 | 13 | null | The ROC function (it is not necessarily a curve) allows you to assess the discrimination ability provided by a a specific statistical model (comprised of a predictor variable or a set of them).
A main consideration of ROCs is that model predictions do not only stem from the model's ability to discriminate/make predict... | null | CC BY-SA 2.5 | null | 2010-08-04T19:31:22.247 | 2010-08-04T19:31:22.247 | null | null | 447 | null |
1265 | 2 | null | 1256 | 2 | null | This really depends on the relative numbers of "scores" and "subjects". The method you use calculates lots of cross-correlations which are not required. However, if there are relatively few "subjects" relative to "scores", then this probably doesn't matter too much, and the method you suggest is probably as good as any... | null | CC BY-SA 2.5 | null | 2010-08-04T19:35:08.557 | 2010-08-04T19:35:08.557 | null | null | 643 | null |
1266 | 1 | 2338 | null | 16 | 17516 | The following question is one of those holy grails for me for some time now, I hope someone might be able to offer a good advice.
I wish to perform a non-parametric repeated measures multiway anova using R.
I have been doing some online searching and reading for some time, and so far was able to find solutions for only... | A non-parametric repeated-measures multi-way Anova in R? | CC BY-SA 4.0 | null | 2010-08-04T20:01:07.787 | 2019-05-28T08:27:05.370 | 2019-05-28T08:27:05.370 | 11887 | 253 | [
"r",
"anova",
"repeated-measures",
"nonparametric",
"manova"
] |
1268 | 1 | 1393 | null | 14 | 6317 | I am using Singular Value Decomposition as a dimensionality reduction technique.
Given `N` vectors of dimension `D`, the idea is to represent the features in a transformed space of uncorrelated dimensions, which condenses most of the information of the data in the eigenvectors of this space in a decreasing order of imp... | SVD dimensionality reduction for time series of different length | CC BY-SA 2.5 | null | 2010-08-04T20:51:04.053 | 2010-12-17T07:59:12.010 | 2010-12-17T07:59:12.010 | 223 | 170 | [
"time-series",
"machine-learning",
"pca",
"data-transformation",
"multivariate-analysis"
] |
1269 | 2 | null | 1268 | 2 | null | Filling with zero is bad. Try filling with resampling using observations from the past.
| null | CC BY-SA 2.5 | null | 2010-08-04T21:10:02.473 | 2010-08-04T21:10:02.473 | null | null | 223 | null |
1270 | 1 | 1276 | null | 5 | 834 | If I have a (financial) time series, and I sample it with two different periods, at 5 and at 60 minute intervals, can I create an exponential moving average on the 5 minute sampled data which is the same as an exponential moving average on the 60 minute sampled data?
Something like this:
e1 = EMA(a1) applied on sampled... | Exponential moving averages of a time series with varying sampling | CC BY-SA 2.5 | null | 2010-08-04T21:12:03.857 | 2010-08-04T23:36:10.817 | null | null | 749 | [
"time-series"
] |
1271 | 2 | null | 1266 | 6 | null | When in doubt, bootstrap! Really, I don't know of a canned procedure to handle such a scenario.
Bootstrapping is a generally applicable way of generating some error parameters from the data at hand. Rather than relying on the typical parametric assumptions, bootstrap procedures capitalize on the characteristics of th... | null | CC BY-SA 2.5 | null | 2010-08-04T21:16:37.787 | 2010-08-04T21:27:49.517 | 2010-08-04T21:27:49.517 | 485 | 485 | null |
1272 | 1 | null | null | 0 | 574 | I have a set of data which consists of many different types (measurable, categorical)
For example:
name measurable_attribute_1 categorical_attribute_1 measurable_attribute_2 categorical_attribute_2 ...
Number of attributes may grows quite quickly during my study: into my spreadsheet, I can as many new entries a... | How to organize a dataset with many attributes | CC BY-SA 2.5 | null | 2010-08-04T21:30:19.030 | 2010-08-05T16:33:34.410 | 2017-04-13T12:44:24.667 | -1 | null | [
"algorithms",
"categorical-data",
"classification"
] |
1273 | 2 | null | 1228 | 6 | null | Change the variable. Run a control chart for the "time between infections" variable. That way, instead of a discrete variable with a very small range of values, you have a continuous variable with an adequate range of values. If the interval between infections gets too small, the chart will give an "out of control" ind... | null | CC BY-SA 2.5 | null | 2010-08-04T21:54:28.677 | 2010-08-04T23:48:08.457 | 2010-08-04T23:48:08.457 | 666 | 666 | null |
1274 | 1 | 1277 | null | 6 | 768 | [0,1,0,2,4,1,0,1,5,1,4,2,1,3,1,1,1,1,0,1,1,0,2,0,2,0,0,1,0,1,2,2,1,2,4,1,4,1,0,0,4,1,0,1,0,1,1,2,1,1,0,0]
What is the best way to convince myself that these data are correlated? that no univariate discrete distribution would approximate them well? that a time series model is necessary to better estimate the future dist... | Best way to show these or similar count data are not independent? | CC BY-SA 2.5 | null | 2010-08-04T22:17:59.627 | 2010-08-07T08:40:07.287 | null | null | 273 | [
"correlation",
"count-data"
] |
1275 | 2 | null | 1268 | 1 | null | You could estimate univariate time series models for the 'short' series and extrapolate them into the future to 'align' all the series.
| null | CC BY-SA 2.5 | null | 2010-08-04T22:46:30.457 | 2010-08-04T22:46:30.457 | null | null | null | null |
1276 | 2 | null | 1270 | 5 | null | The trivial and non-helpful answer is "Yes, downsample your 5-minute data to 60-minute data."
More practically, without throwing out 90% of your data, the answer is generally "No, unless you get extremely lucky with sampling your five-minute data." You should get an answer that's close (and under most noise models I s... | null | CC BY-SA 2.5 | null | 2010-08-04T23:36:10.817 | 2010-08-04T23:36:10.817 | null | null | 61 | null |
1277 | 2 | null | 1274 | 5 | null | You could just plot the ACF and check if the first coefficient is inside the critical values. The critical values are ok for non-Gaussian time series (at least asymptotically).
Alternatively, fit a simple count time series model such as the INAR(1) and see if the coefficient is significantly different from zero.
| null | CC BY-SA 2.5 | null | 2010-08-04T23:46:28.333 | 2010-08-04T23:46:28.333 | null | null | 159 | null |
1278 | 1 | null | null | 4 | 7892 | (This is part-2 of my long question, you can have a look at part-1 [here](https://stats.stackexchange.com/questions/1099/how-to-handle-count-data-categorical-data-when-it-has-been-converted-to-a-rate))
I am going to do a quasi-experiment, with measuring the base line of a sample (actually not quite a sample, but a ward... | Is calculating Incidence Rate Difference/Ratio appropriate for single case experimental design? | CC BY-SA 4.0 | null | 2010-08-05T01:24:22.440 | 2020-03-06T12:38:57.593 | 2020-03-06T12:38:57.593 | 11887 | 588 | [
"epidemiology",
"incidence-rate-ratio"
] |
1279 | 2 | null | 485 | 3 | null | [UCCS mathematics video archive](http://www.uccs.edu/~math/vidarchive.html) has
archived videos from a range of courses in mathematics. Several subjects called Mathematical Statistics I and Mathematical Statistics II are available. The main site requires a free registration to access.
Slightly more accessible are the... | null | CC BY-SA 3.0 | null | 2010-08-05T02:36:08.460 | 2012-08-21T07:26:40.843 | 2012-08-21T07:26:40.843 | 183 | 183 | null |
1280 | 2 | null | 726 | 140 | null | >
87% of statistics are made up on the spot
-Unknown
[Dilbert.com](http://dilbert.com/strips/comic/2008-05-08/)
| null | CC BY-SA 2.5 | null | 2010-08-05T02:42:11.413 | 2011-01-14T09:38:16.137 | 2011-01-14T09:38:16.137 | 442 | 553 | null |
1281 | 2 | null | 1272 | 0 | null | You could look at Exploratory Factor Analysis. It will tell you which attributes are the most similar to each other.
| null | CC BY-SA 2.5 | null | 2010-08-05T02:57:24.893 | 2010-08-05T02:57:24.893 | null | null | 74 | null |
1282 | 2 | null | 1261 | 5 | null | Following up on my comment, this question would be very simple if i, j, and k were not restricted to be integers. The reason is as follows: pA, pB, and pC denote the observed probability of success in the three groups. Then let p=pA, i=1, j=log(pB)/log(pA), and k=log(pC)/log(pA). These will easily satisfy the required... | null | CC BY-SA 2.5 | null | 2010-08-05T03:06:36.013 | 2010-08-05T03:06:36.013 | null | null | 279 | null |
1285 | 2 | null | 1272 | 1 | null | One option would involve using optimal scaling principal components analysis.
The approach allows you to state your measurement assumptions about each variable (e.g., nominal, ordinal, numeric).
I've used it in SPSS: see the Categories Add-On module (i.e., Analyze - Dimension Reduction - Optimal Scaling).
I'm not sure,... | null | CC BY-SA 2.5 | null | 2010-08-05T03:55:13.897 | 2010-08-05T04:14:12.710 | 2010-08-05T04:14:12.710 | 183 | 183 | null |
1286 | 1 | 1335 | null | 8 | 9247 | Sometimes I want to do an exact test by examining all possible combinations of the data to build an empirical distribution against which I can test my observed differences between means. To find the possible combinations I'd typically use the combn function. The choose function can show me how many possible combinati... | How can I obtain some of all possible combinations in R? | CC BY-SA 2.5 | null | 2010-08-05T04:54:46.270 | 2016-08-25T20:43:28.717 | 2016-08-25T20:43:28.717 | 101426 | 196 | [
"r",
"nonparametric",
"combinatorics"
] |
1287 | 2 | null | 1252 | 9 | null | Reading "Using Multivariate Statistics (4th Edition) Barbara G. Tabachnick"
I found these decision trees based on major research question. I think they are quite useful. Following this link you'll find an extract of the book
[http://www.psychwiki.com/images/d/d8/TF2.pdf](http://www.psychwiki.com/images/d/d8/TF2.pdf)
s... | null | CC BY-SA 2.5 | null | 2010-08-05T04:58:19.663 | 2010-08-05T04:58:19.663 | null | null | 10229 | null |
1288 | 2 | null | 165 | 13 | null | So there are plenty of answers here paraphrased from statistics/probability textbooks, Wikipedia, etc. I believe we have "laypersons" where I work; I think they are in the marketing department. If I ever have to explain anything technical to them, I apply the rule "show don't tell." With that rule in mind, I would prob... | null | CC BY-SA 3.0 | null | 2010-08-05T09:15:34.467 | 2013-10-02T12:17:36.667 | 2013-10-02T12:17:36.667 | 24521 | 438 | null |
1289 | 1 | null | null | 9 | 3209 | I am having difficulties to select the right way to visualize data. Let's say we have bookstores that sells books, and every book has at least one category.
For a bookstore, if we count all the categories of books, we acquire a histogram that shows the number of books that falls into a specific category for that bookst... | Visualizing multiple "histograms" (bar-charts) | CC BY-SA 4.0 | null | 2010-08-05T10:03:34.997 | 2019-08-25T23:50:08.677 | 2020-06-11T14:32:37.003 | -1 | 760 | [
"pca",
"data-visualization",
"histogram",
"barplot"
] |
1290 | 2 | null | 1286 | 1 | null | Generating combinations is pretty easy, see for instance [this](http://compprog.wordpress.com/2007/10/17/generating-combinations-1/); write this code in R and then process each combination at a time it appears.
| null | CC BY-SA 2.5 | null | 2010-08-05T10:17:47.227 | 2010-08-05T10:17:47.227 | null | null | null | null |
1291 | 2 | null | 1289 | 12 | null | As you have found out there are no easy answers to your question!
I presume that you interested in finding strange or different book stores? If this is the case then you could try things like [PCA](https://stats.stackexchange.com/questions/53/pca-on-correlation-or-covariance/78#78) (see the wikipedia [cluster analysis]... | null | CC BY-SA 2.5 | null | 2010-08-05T10:31:54.357 | 2010-08-05T12:58:40.073 | 2017-04-13T12:44:33.237 | -1 | 8 | null |
1292 | 1 | 1297 | null | 36 | 47849 | [Decision trees](http://en.wikipedia.org/wiki/Decision_tree) seems to be a very understandable machine learning method.
Once created it can be easily inspected by a human which is a great advantage in some applications.
What are the practical weak sides of Decision Trees?
| What is the weak side of decision trees? | CC BY-SA 2.5 | null | 2010-08-05T10:42:44.327 | 2019-04-05T16:31:35.390 | null | null | 217 | [
"machine-learning",
"nonparametric",
"cart"
] |
1293 | 1 | null | null | 2 | 740 | My girlfriend (B.B.A) is really interested in Actuarial science. She's looking at self teaching her self. She's good with basic math (Calculus 1 and 2) and stats
What are some of the essential sources she needs to read in order to learn and excel in the field?
| Essential reads for people interested in Actuary and Actuarial science | CC BY-SA 2.5 | null | 2010-08-05T10:47:12.837 | 2010-08-07T17:47:26.950 | 2010-08-07T17:47:26.950 | null | 59 | [
"references"
] |
1294 | 2 | null | 1289 | 3 | null | I would suggest something that hasn't got a defined name (probably "parallel plot") and looks like this:
Basically you plot all counts for all bookstores as points over categories listed on x axis and connect the results from each bookstore with a line. Still this may ... | null | CC BY-SA 4.0 | null | 2010-08-05T11:22:54.870 | 2019-01-07T11:15:31.740 | 2019-01-07T11:15:31.740 | 79696 | null | null |
1295 | 2 | null | 1292 | 27 | null | One disadvantage is that all terms are assumed to interact. That is, you can't have two explanatory variables that behave independently. Every variable in the tree is forced to interact with every variable further up the tree. This is extremely inefficient if there are variables that have no or weak interactions.
| null | CC BY-SA 2.5 | null | 2010-08-05T11:58:42.133 | 2010-08-05T11:58:42.133 | null | null | 159 | null |
1296 | 1 | 1353 | null | 10 | 4144 | I have detector which will detect an event with some probability p. If the detector says that an event occured, then that is always the case, so there are not false-positives. After I run it for some time, I get k events detected. I would like to calculate what the total number of events that occured was, detected or o... | How to find a confidence interval for the total number of events | CC BY-SA 2.5 | null | 2010-08-05T11:59:25.080 | 2011-04-29T00:26:13.040 | 2011-04-29T00:26:13.040 | 3911 | 762 | [
"probability",
"confidence-interval"
] |
1297 | 2 | null | 1292 | 40 | null | Here are a couple I can think of:
- They can be extremely sensitive to small perturbations in the data: a slight change can result in a drastically different tree.
- They can easily overfit. This can be negated by validation methods and pruning, but this is a grey area.
- They can have problems out-of-sample predict... | null | CC BY-SA 2.5 | null | 2010-08-05T12:08:23.523 | 2010-08-05T13:31:59.900 | 2010-08-05T13:31:59.900 | 495 | 495 | null |
1298 | 2 | null | 1296 | 2 | null | I think you misunderstood the purpose of confidence intervals. Confidence intervals allow you to assess where the true value of the parameter is located. So, in your case, you can construct a confidence interval for $p$. It does not make sense to construct an interval for the data.
Having said that, once you have an es... | null | CC BY-SA 2.5 | null | 2010-08-05T12:12:16.980 | 2010-08-05T12:12:16.980 | null | null | null | null |
1299 | 2 | null | 1292 | 12 | null | My answer is directed to CART (the C 4.5/C 5 implementations) though i don't think are limited to it. My guess is that this is what the OP has in mind--it's usually what someone means when they say "Decision Tree."
Limitations of Decision Trees:
---
Low-Performance
By 'performance' i don't mean resolution, but execut... | null | CC BY-SA 2.5 | null | 2010-08-05T12:47:45.727 | 2010-08-05T12:47:45.727 | null | null | 438 | null |
1302 | 2 | null | 1293 | 2 | null | sources:
1) statistics upto ANOVA
2) probability upto Central limit theorem
3) Basic programming and data analysis skills
4) Familiarity with business economics
5) Financial mathematics
This is a verrry basic smattering of things that she needs to read.
| null | CC BY-SA 2.5 | null | 2010-08-05T13:04:27.533 | 2010-08-05T13:04:27.533 | null | null | null | null |
1304 | 2 | null | 1296 | 3 | null | If you measure $k$ events and know your detection efficiency is $p$ you can automatically correct your measured result up to the "true" count $k_\mathrm{true} = k/p$.
Your question is then about finding the range of $k_\mathrm{true}$ where 95%
of the observations will fall. You can use the [Feldman-Cousins method](http... | null | CC BY-SA 2.5 | null | 2010-08-05T13:35:59.983 | 2010-08-05T16:34:40.807 | 2010-08-05T16:34:40.807 | 56 | 56 | null |
1306 | 2 | null | 856 | 10 | null | I have a suspicion that 'exact' tests and sampling weights are essentially incompatible concepts. I checked in Stata, which has good facilities for sample surveys and reasonable ones for exact tests, and its 8 possible test statistics for a crosstab with sample weights don't include any 'exact' tests such as Fisher's. ... | null | CC BY-SA 2.5 | null | 2010-08-05T13:56:03.143 | 2010-08-05T13:56:03.143 | null | null | 449 | null |
1307 | 2 | null | 1293 | 4 | null | And for bedtime reading, "Against the Gods: The Remarkable Story of Risk" by Peter L. Bernstein. She'll find out how Lloyds of London started, among many other interesting bits. Highly recommended and only $13 from amazon.
| null | CC BY-SA 2.5 | null | 2010-08-05T13:57:42.843 | 2010-08-05T13:57:42.843 | null | null | 247 | null |
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