tttjjj/elt_bench huggingface dataset:
Don't forget to give some consideration of why you want to know the average. It's often not a very useful summary of a distribution.
Please consider modifying your question, then, because the response you checked as your preferred answer does *not* answer the question as it stands (and the other one does).  Currently you are using the word "independent" in an idiosyncratic way.  Your example with Gamma shows this: one could simply reparameterize Gamma in terms of the mean (mu) and variance (sigma), because we can recover theta = sigma/mu and kappa = mu^2/sigma.  In other words, *functional* "independence" of the parameters is usually meaningless (except for single-parameter families).
To follow up on whuber's comment, if you can give more information about where these groups come from and/or what kinds of structure you expect to see in the data (and especially their relative sizes!) then it might be possible to work something out...
Symmetry is not meaningful for a *categorical* random variable, because there is no natural ordering. Even if there is ordering, usually we don't know anything about spacing, so it is unclear which category should correspond to which "symmetric" one. But there are discrete distributions where symmetry makes sense, eg binomial or poisson distributions.
Can I check -- "accuracy" with respect to what gold standard (benign/malignant status)?
If you come to edit your question at some point, you might like to fix the typo in the title.
Thank you for your reply. 
Indeed I am trying to find the relationship between different parameters and the price of machine parts. But I am also trying to find objects that are too expensive for their properties. I am not completely new to R but I am more familiar with Stata I know how to read tables in R and how to interpret coefficients :) But thank you very much for the time you spend to write your answer, I really appreciate it!

Regarding the correlations, I also wanted to look at the pairwise correlations between the price and the element properties.

Please see my updated post
@Glen_b Can you give some more details?
You use it because - when the data are normal - $Q = (n-1)\\frac{s^2}{\\sigma^2}\\sim \\chi^2_{n-1}$. (This makes $Q$ a pivotal quantity)
I'm trying to add formula-formatting to your equations using LaTeX, but I can't figure out the very last one... there must be a mistake in it, right? Could you review the formula you give after "...video the slope equals..."
Thanks. I'm doing a multilevel logistic analysis with 5 imputed datasets and I'm going to combine them manually in a pooled version
Comment: If $C\\neq\\alpha\\beta$, then $f_{X,Y}$ is not a density function (since it doesn't integrate 1). $C=\\alpha\\beta$ is the only value that makes $f_{X,Y}$ a density function. This is easy to prove by integrating with respect to $x$ and $y$. If you want to use a copula to modelling the dependence between the marginals, then the expression for the density will be different. Cheers.
Also, I should note that these appear to be intervals around four point estimates.  I'm hoping to be able to get a band around a fitted negative exponential curve.
Closely related: [What is the relationship between a chi square test and test of equal proportions?](http://stats.stackexchange.com/q/2391/930)
Because it might actually be a solution to the problem ;-)
_Which_ distributions do you say are equal?  $p(x\\mid w_1)$ equals $p(x\\mid w_2)$ exactly when $a = b = 1$ and $c = d = e = 0$.  There must be something else that you have been told but are not revealing to us.
This question seems too broad, but also, if you are attempting to identify possible factors then you are not doing a causal analysis. This sounds like an exploratory analysis.
Also, not all variables/distributions are averages.
When you're stuck it often helps to simplify the problem.  For instance, suppose $\\Pr(X=0)=1/3$ and $\\Pr(X=1)=2/3$ and you change four days to two days.  Because there are only four possible samples it is quick and easy to enumerate them, compute their probabilities, and identify those whose standard deviations are zero.  This ought to be enough to show you the point of Karl's approach (+1).
No. I have edited my response. Marginal effects have a "ceteris paribus" interpretation. They tell what happens if a given variable varies, while all the other variables remain unchanged.
This is not my research. The authors kept the first data point in and I was wondering why instead of doing a post-hoc analyses using a Tukey HSD they used some other technique that took into account the outlier.
@Harvey I don't know for a fact that the mean has a certain value, but I have a prediction (e.g. from theory or an earlier study).
Thanks for the info. Unfortunately the sensitivity is related to what it is measuring and it is measuring thousands of different things during the analysis. So essentially each row has a different LOD which is unknown. To make matters worse I'm not even sure all the zeros really = below the limit of detection. There is a chance they are really missing values because the machine does not sample fast enough to catch everything
A lot of essential detail is missing here--you would be better off explaining the real problem you're trying to solve. Apart from specifying the nature of the random quantities indicated here, what precisely does it mean to award points "equitably"? Are these shared by team members? What is the utility of these points and to whom? Are they used to determine which team "wins" in the competition?  Can individual members appear on more than one team?
(I also edited your answer with a math image from http://www.codecogs.com/latex/eqneditor.php while math is not rendering on the site!)
I find this discussion confusing, because [*by definition*](http://en.wikipedia.org/wiki/Histogram) a histogram represents probability by area.  Therefore it *must* be "normalized" in the sense that the total of the bar *areas* must equal unity.  (Anything else is just a frequency (or probability) bar chart.)  The answers to this set of questions follow easily from the definition.
(+1) Nicely written question. Welcome to the site. Just to clarify: In one instance you know the parameters and want to sample from the distribution, but in the other you have *data* and want to find the associated parameters?
I think the issue is getting from 'long' from 'short' form data.  There seems to be a restructuring 'wizard' for doing this, described here: https://www.youtube.com/watch?v=-GNRFeHAVy0
Proceeding with the dictionary abstraction, what I would like to do is to tell whether word usage differs significantly in the populations.
That is very clear. Thank you!
@Alexey After re-reading this paragraph, I think the author does not mention $2^N$ operations. He just points out that **"the belief state for $x$ is a mixture of $2^N$ Gaussians"**.
I suggest you give a better description of the data and analysis you are doing. One wonders if such over-weighting is really the right thing to do.
You might read about [modern portfolio theory](http://en.wikipedia.org/wiki/Modern_portfolio_theory) and the [capital asset pricing model](http://en.wikipedia.org/wiki/Capital_asset_pricing_model) both on Wikipedia and elsewhere, which takes covariance of returns into account.  The approach has its critics.
I think the most elementary way to approach it is simply to list all mutually exclusive events that satisfy the condition, and then add their probabilities. You need to be able to do it that way before you start trying to apply anything fancier.
Of course I am in R. Will look into `approx()` and get back to you.
Are the $\\sigma$'s known to the person doing the test or not?
I mean that you don't have any additional sources of information other than the tuples (user, timestamp). The threshold-based method (based on delta > N) is just an example of a method. Maybe something else is possible.
Are you using SciPy 0.9?  Also, could you post your data, or a subset thereof?
`var(fitted(lm_model))` do you need this one?
Beautifully said Peter.  I can't imagine a situation where dichotomization in analysis is a good idea.
What kind of exams are they (do they attempt to measure the same construct, related constructs, or independent constructs)? Do you have multiple exam scores per subject for each exam (ie, sub i took exam j k-times, w/ K>1)? Variance in what sense (internal reliability / measurement error, or between subjects / if the subs are more 'spread out' on 1 dimension than another)?
Not sure if that is a good idea or why you would want to that, but you can just fit the final model returned by train to the full data set.
I think what you're looking for is in my answer here: [How to generate correlated random numbers (given means, variances and degree of correlation)](http://stats.stackexchange.com/questions/38856//38867#38867), which both explains the basic idea conceptually, & includes some custom R code & discussion of the standard function. If there's anything you're asking about that isn't covered, edit your Q to clarify what you still need to understand.
What is the mean of a categorical variable?
I am trying to implement image recognition using LDA, my class1 matrix is of size 10*32256 of 10 sample images.like this i have 5 classes. I was confused how to take mean for this matrix. whether to add all the instance of row 1 and divide by 32256 or add the column and divide by 10.
Nicely explained!
Thank you Peter. As you had mentioned i will try with 2 options. 1) Excluding the bins with lower statistical power but this untimely i will end up with some data loss. 2)Try with the Classification trees.
You appear to be asking two fairly unrelated questions (and your title doesn't relate to the first one at all). You should probably ask two different questions. Please take care that they both fall inside the bounds for CV questions given in the faq.
Great suggestion, I will consider this a valid answer if there is no more answer about this, as this one still doesn't cover more advanced concepts such as ANOVA (or at least that what searching it suggests). Great for being free as well..
This cartoon...speaks to me.
I wrote `expand.grid(rep(x,k))` as an example of the function. My question is how would I perform the test after obtaining this matrix, i.e. how to calculate the p-value and check if it matches the one from `permTS()`
Then estimating local variograms is a logical thing to do. If they vary according to certain features including other predictors in the model is an option as well. IDW might be considered the simplest type of kriging model - so IDW should be no better than actually estimating the variogram from the data.
There are many puzzling things here.  Perhaps the most substantial is that the data document 110 observations whereas the total "expected frequency" is 135.827.  Because these disagree, the expected frequencies appear to have been mis-computed.  Regardless, the chi-squared test is inapplicable due to the many zero-frequency bins.  The remark about ordering is mysterious because it makes no sense in the context of a KS test.  For more about the subtleties of chi-square testing, please see http://stats.stackexchange.com/questions/16921/. The histogram exhibits moderate departures from normality.
That would then simply mean that you cannot assume normality of the posterior. If you assume the posterior is normal, then this would indeed be correct.
Thanks, most likely it will not be necessary to analyze all interactions, so the analysis will focus on just a couple of the possible interactions. Mostly the third factor and it's interactions with the other factors are interesting for the study. So I will just create new variables by averaging all the cases with the appropriate settings of the other factors and then analyze this interaction in isolation.
Thanks for the edits Michael, much better now.
I was thinking that i have to difference the time series twelve times and with the results out of the correlogram i get the parameters for my seasonal arima model.
Due the autocorrelation i thought the non-seasonal parameters are (1,1,0). 
(0,0,1) due that the seasonal lag spikes at 12,24,36 are falling the pacf is 0. Furthermore we have a identical lag spike at lag 12 in with acf correlogram, so the acf seasonal is 1.
The asymptotic distribution of the median is Normal with variance inversely proportional to the square of the pdf at the median.  Thus, experimental designs for median estimation have to make some specific assumptions about the nature of the distribution near its middle.  (See http://mathworld.wolfram.com/StatisticalMedian.html for example.)  Usually people study a range of parametric alternatives to compare the efficiency of the median to that of the mean; relative efficiencies translate via the usual square-root law into relative sample sizes.
Sure, you can ask about how to deal w/ this situation, & in the process of answering someone might be able to provide some helpful R code (or not...). It's just asking 'what package will do X' that's off-topic. If you want the question to stay here (& stay open), just edit your Q to make it on-topic. It may help you to read the [relevant section of the help page](http://stats.stackexchange.com/help/on-topic) & our [guide to asking questions](http://meta.stats.stackexchange.com/questions/1479/how-to-ask-a-good-question-on-crossvalidated) in reformulating your Q.
now i have one major problem, how do i know which of the 17 input features are the most significant ?
As far as I am aware (and has been demonstrated during the contest), the assumption from the given information is correct - no one person can have more than 6 entries, one of each "type."
thanks for detailed explanation. One thing I am curious about is that: the first-term in your three-termed expression is called "Missed Detection" and it is not usually assumed as a False-alarm terminology.
Thanks. "You should always aim to solve the problem at hand directly, rather than use a more general method and post-process the result" - I was wondering if you have any intuitive/mathematical justification of the statement? Will be especially helpful with the current context.
Hi, @Max, that seems to be the answer. Would you mind making it an official answer (& perhaps elaborating it a little) so we can upvote it & the OP can accept it?
Percentage change in sales between January and month $i$.
There are many possible models. You have to choose one. Are the test questions all indistinguishable? Are the performance of the students on each test question related?
+1 for a good question, as I'm also interested in this. Perhaps simulation? Irrespective of the solution, I'd like to see exactly how this was done. Or, perhaps a formula.
I don't think you are allowed to have one of the parameters in $h$. Otherwise you could say that $Var[Y_i]=\\frac{1}{1-p}E[Y_i]$, and the dispersion parameter is $\\frac{1}{1-p}$. If $r$ is fixed, however, then the argument goes through. And that is what `glm.nb` uses for fitting: rerunning `glm` with a fixed value of $r$ and dispersion parameter 1.
categorical or continuous moderator?
@arcade See the PMF in [binomial distribution](http://en.wikipedia.org/wiki/Binomial_distribution).  I don't see a typo, but that doesn't mean there isn't one.  As for an approximation that might be quicker for large $n$, see my update.
That really depends on you've done the clustering :) If you show a little example of the data you have i'm sure you'll get an answer
`mlogit()` assumes data in person-choice (long) format. For each person, you need a row for each Behavior-category, a logical variable (`TRUE` for the selected category), an id-variable naming the person, and a variable identifying the category for that row. `mlogit.data()` reshapes data that is in wide format. The model formula in `mlogit()` then has a different form, in your case maybe `Behavior ~ 0 | Age+Temp+...`. See `vignette("mlogit")` or [this toy example](http://www.uni-kiel.de/psychologie/rexrepos/Univariate/Regression/regressionMultinom.html) comparing `VGAM`, `nnet` and `mlogit`.
The thread at http://stats.stackexchange.com/questions/65866/good-methods-for-density-plots-of-non-negative-variables-in-r looks a like a duplicate in that it addresses all the substantive issues you raised.  Does that answer your questions?
One wonders why you wouldn't simply measure the proportion of evaporation by having matched (control) samples that simply lacked ants; the additional weight-loss for the ones with ants would give an estimate of the amount actually eaten.
Any answers here ?
@Ray: the meaning/definition of both terms  for regression (chemical calibration) is so different from their meaning in classification that this should IMHO more emphasized. I've never heard about your definition of specificity, though. Can you give a reference?
I realize I had not read the original question precisely enough. My answer is for the case of mixed models. I think NRH's answer below is for the case of classical models.
You seem to have a great predictor.  When plotted, there is a strong curvilinear relationship between percent dead and your predictor score.  Perhaps the Pearson Goodness-of-fit test is not the best way to index this fit.
I researched this problem intensely about 5 years ago. There are 3 aspects to a proper distance metrics; non-negativity, symmetry and the triangle inequality. The proper approach to metric learning is summarized here: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.91.4732&rep=rep1&type=pdf but if you're just build a recommender system from a classifier with side information, then the permutation suggestion is the simplest thing that could work. Try it and analyse the accuracy before discarding it. It works surprisingly well because you have all of the usual classifiers to experiment.
The question at http://stats.stackexchange.com/questions/46389/solving-the-kolmogorov-forward-equation-for-transition-probabilities seems to be the same as this (apart from the details of the process).  Does it provide an adequate answer?
As for a big hint: expand the quadratic. Separate out the terms into their own sums. Simplify the terms and cancel where appropriate.
Your question seems pretty broad (i.e., how do you analyse data?). This whole site is devoted to answering that question. Could you say a bit more about the goals of such an analysis?
+1: MacKay is awesome.
I like this question, and am disappointed that we won't be seeing any answers to it.
Another exception that proves the multi-edition rule is [Freedman, Pisani, & Purves](http://books.wwnorton.com/books/978-0-393-92972-0/) which is in its fourth edition. Somewhere (perhaps in the instructor's manual, maybe the preface) they note that the material is largely unchanged from the first, published some 35 years ago, but that instructors have asked for updated datasets. Thus it's possible to teach from any edition if you like--and old editions are widely available at hugely discounted prices. (I bought my first copy for 25 cents.)
Ok I think I understand what the author meant now.

The author pulls random samples from the distributions, to determine the best choice.
This post is being automatically flagged as low quality due to its shortness. It is somewhat more of a comment than an answer at present. Would you mind expanding this to make it more of an answer?
My bad, I mixed the variable names. Now it's fixed.
@xan, thank you for your response. Yes, I know that using only intercept and contin.var coefficient was not correct - therefore I have asked this question. Finally I managed it myself, see my answer bellow.
Perfect. But how would I calculate the end points? For instance, it wouldn't make sense to say that the number of mandates has increased with 11 on the first day (though that would be correct).
+1 This is your best account yet (on this site) of these ideas.
This answer is lacking precision by not differentiating the conditional and unconditional statistics. A lot of text on this topic has similar problems. It is spreading like a plague.
I am not sure what you're even asking with (1), but I think you misunderstand the purpose of sufficient statistics in your first sentence and that leads to your confusion. With (2) it means *any* other functions. Think of a function, and it won't help, the sufficient statistic has already eaten its lunch
If there is a trend then for any of the possible cases that I mentioned there will be a correlation between sucessive values which is also called auto-correlation
A bundle of questions to you. 1) Is your task this one: Somebody's done FA on large sample and published loadings. You have 18 more cases and wish to compute factor scores for these. 2) By "scores" in the last paragraph you mean 18x30 data values, don't you? 3) Do you have the original large sample at hand? 4) Did they publish (a) means and st. deviations of their 30-item data; (b) factor score regression coefficient matrix? 4) What does PCA tag do here? - PCA is not quite identical to principal axis factoring.
Fixed effects:
               Estimate     Std. Error  z value    Pr(>|z|)    
(Intercept)       -1.9205     0.6465    -2.970    0.002974 ** 
Food               3.1765     0.6703    4.739     2.15e-06 ***
Animal             2.1069     0.7329    2.875     0.004045 ** 
FoodxAnimal       -2.9430     0.7916    -3.718    0.000201 ***
HabitatxAnimal    -0.1393     0.3891    -0.358    0.720405
You may find that the process of writing down an explicit model for your regression answers most of these questions. It's always a good exercise, anyway, because it clarifies one's thinking about the analysis.
Since a probability distribution's mean and SD are just numbers, we have to understand this question in the sense of *a family* of distributions. In fact, even that sense is too general, because to get a positive answer you could just take a basket of various distributions and pick out a few for which the SD:mean ratios are the same.  I guess you are probably asking about a *naturally parameterized family* of distributions.
Divergence in physics is merely a 3D version of a derivative, while in stats it is a *total* difference between distributions. Apart from sort of similar premise (both are kind of differences between some things), I don't see any connection.
Are the "errors" you're concerned with the theoretical random variables in the model, or the residuals in a sample regression?
No worries. Your example is ok, include it in answer. I just bugged you, since the answer in its previous form could have been better. Since the goal of SE network is to make internet the better place, I thought I could bug you to improve the question. Welcome to the site!
Surely 'not so much different' is a question about effect size (how different are they?) not hypothesis test ("is our sample large enough to pick up even unimportant differences?"). Why a hypothesis test?
@MichaelLew Here's my understanding.  You seem to be telling him to set up a null hypothesis of the form $H_0: \\mu_x-\\mu_y=c$ vs $H_A: \\mu_x-\\mu_y>c$ where the constant $c$ is somehow supposed to correspond to a % difference in the two means.  I don't understand how the constant $c$ can simultaneously correspond to an absolute difference in the two means as well as a % difference.  If I reject this null hypothesis, it seems to me that the only thing I can conclude is that the absolute difference in means is greater than $c$.  Thanks for your help in clarifying my misunderstanding.
Thanks Glen. This is very informative. I am surprised there is not much information online and references considering it is useful everywhere in optimization and sampling. Please post here whenever you have other references.
If you sample size is small you can't hope to validate your modified questionnaire. Whatever the tools. There is no magic means.
(1) Why would Pearson correlation be a reasonable metric for quality of approximation? (2) Please ask a definite question. So far you have only statements.