tttjjj/elt_bench huggingface dataset:
On the contrary, the distribution of $t$ is *different* for every $n$ because the degrees of freedom matter!
Quite so. The only small issue is whether the OP, or anyone else  reading the question, is familiar with the term "ordination". As you say, that is ecologists' way of describing what is being done with PCA. Your answer is positive as pointing to correspondence analysis as likely to be a better method.
Because I believe, with great regret, that every one of your statements is incorrect or misleading, I would like to invite you to support them with authoritative citations if you can.
Good point. I measured it four times, but am only interested at time 1, for now.
+1, this is good info. To quibble: in this context, an additional benefit of centering is that you avoid the multicollinearity that would otherwise occur in creating the interaction terms.
@image_doctor
Don't want to make it too complicated for now, I think my last question was miss-leading. Lets consider only a single unbalanced dataset:
How can accuracy be useful here? Predicting only the majority class might give a higher accuracy than a better classifier that predicts some of the minority instances right.
Perhaps, fit a finite mixture model with one component for the signal and one for the noise.  I think this is a standard problem (and solution) in astrophysics.
It sounds like your data are too sparse for the high-dimension representations.
Are you planning on running regression models with this data?
Another useful stats exchange article: http://stats.stackexchange.com/questions/2149/what-is-the-difference-between-a-particle-filter-sequential-monte-carlo-and-a
This is the perfect answer, I realize now that I was getting lost with my own proof. :)
I added some to the original post to provide more details. A question, though: what do you mean when you say the plan is sound? The plan just to train it on a set which excludes the set I intend to demonstrate on? Or the idea to make a validation set?
I seen examples where a model has ten data points and nine parameters. On pointing out that the model has too many parameters, I was told that the R^2 was 0.999 so the model must be correct!
As part of figuring out what could go wrong, I'd look at output from PROC UNIVARIATE - mean, sd, and percentiles. I'd also plot the distribution
@HongOoi Sorry, I've wrongly added C++. Sorry about that! I've changed it to PHP.
@Yaroslaw: I don't really understand your issue with cross-validation, but to be honest I am no expert there. However, I would really like to make a point for measuring model mimicry. Therefore, see my updated answer.
No that does not influence the data source. By influence, I mean, if the output of the algorithm (lets say an ad) affects the data source (say user behavior such as click/no click) which is then fed back to update the algorithm.
(+1) I've tried a few experiments in Stata on their standard auto.dta example dataset and they agree with this. The two $t$-statistic, and hence the two $p$-values, are always identical. So i think the questioner's intuition is right and he is indeed doing something wrong with the software.
It would be nice to have a clearer idea of what you mean by "inflated." This suggests a comparison, so one is moved to ask, inflated *compared to what*?  If you mean the correlation coefficient of the distribution of *all* temperatures and humidities at that moment, we should pause for a moment to consider what relationship the *sample* correlation would have to it and why--if at all--it could be expected to be "inflated," and how--if at all--that would be related to "spatial correlation." I believe that if you clarify these points, you might be able to answer this question yourself.
I cannot find a definition of $w_i$ other than the one I uploaded here. It looks familiar to the inverse-variance weighting.
Adding non-significant variables can increase your R^2 superficially. So don't do it. They say with enough variables you fit an elephant. You need new and more informative variables. If not this is pretty much what you will have.
I'm pleased this was helpful.
*more negative* would be better than *lower* as some readers might instead think *smaller*
Do you have sources for analyses which get just outside 1% or was that figure off-the-cuff?
just to add to this, in some cases, you can use figures without explicit permission. Most publishers will have a permissions page that outlines what they allow: e.g., this APA one: http://www.apa.org/about/contact/copyright/index.aspx
@kanzen_master: different test sets allow you to judge the variance due to finite test sample size. Different training sets show the variance of (true) accuracy around the average accuracy for training sample size $n$.
Could you elaborate a bit on the "paradoxes in data" a bit more? I didn't quite follow and would like to understand what you are explaining.
@means-to-meaning, yes it would be interesting to compare, pls do share if you have some benchmark results.
@Bill
Dependent variable - _Tmax daily_ (Surface Observatory Data).
For independent variables, we have daily data for four time periods(5:30, 11:30, 17:30, 23:30). So for _Tmax daily_ we are using reanalysed data from nearest grid point(only) at time 11:30 & 17:30 as independent variables - _Tmax, RH, solar radiation (Long & Short Wave), wind (N-S & E-W)_. So total of 11(=1+2+4+4) independent variables. 
Do let me know if that is only you meant to confirm.
Closely related: http://stats.stackexchange.com/questions/9358.  The classic text in the theory of valuing multiple attributes is [Keeney & Raiffa](http://www.amazon.com/Decisions-Multiple-Objectives-Preferences-Tradeoffs/dp/0521438837).
Could you elaborate on the nature of your concern?  There does not seem to be any problem here.  The formulas make sense for all $n\\ge 1$ and can be computed no matter what values the $x_i$ take on.
(partially in jest; not an answer) Don't reject the null.  Nulls have feelings too.
A very clear, comprehensive treatment of this question is jthetzel's answer at http://stats.stackexchange.com/questions/18480/interpretation-of-log-transformed-predictor
Would you like to elaborate on why the argument on "objective measurement" sensu Rasch (or more correctly the possibility of specific objective comparisons) is "strikingly uncompelling"?
Never knew about this - thank you...
All these questions are answered near the top of the Wikipedia article on the [multivariate normal distribution](http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Definition).
@Max  i'd love to see a reference for your point about the change at 3-4  years old.
thanks - I can get my head around this.
This is not a statistical question. Which variables and interactions to include can be answered by specialists in your (sub-(sub-(sub-)))discipline but not statisticians.
@user23774 if you want, you can accept it as the answer
(+1). Nice overview. Could you explain what "multiway splits" and "surrogate splits" are? Are multiway splits if the splits are not dichotomous?
Clear and helpful answer, thanks.
+1, excellent answer, clear & well-documented.
Thank you for the answer!  I'm a little confused, though, as another member posted that it wouldn't be wise to use the Wilcoxon signed rank test to test pair differences with ordinal data: @Scortchi (http://stats.stackexchange.com/a/47230/49494).  Anyway, there's no statistician I can speak to in person about my (unnecessarily complicated) study that is familiar with non-parametric statistics in SPSS, so I appreciate the feedback!
If you have flat priors and no shrinkage, smoothing, random effects or other regularization on the relevant variable, the one with a single observation could be a concern.
Do you want to check if two categorical variables $A$ and $B$, each with two levels ($A = 1$ and $A \\ne 1$ plus $B = 1$ and $B \\ne 1$) are independent?
Each person contributes two alleles to the gene population. AA individuals contribute 2 instances of A each, Aa individuals contribute 1, and aa individuals contribute none. So the proportion of A present must be $(2\\cdot P^2 + 1\\cdot 2PQ)/2$.
More specific information about the data, the purpose of the "tests," and the kinds of tests is needed. Would you be able to edit this question to provide that necessary amplification?
I should have said "No adjustment to n1 alone". The computations for the first stratum are dubious anyway in my opinion. Under the assumption that P1 = 0.01, there's a 96% chance that none of the four observations would be diseased ($0.99^4=0.961$). I would have estimated the number of cases in Stratum 1 from  others' expert knowledge and combined this number with the estimates in Strata 2 & 3. I believe Deming has an example of this kind of calculation for a two-phase prevalence survey in his book Sampling Design in Business Research, Wiley, 1960.
Thanks, in this case do you think a transformation would be a better approach?
"Legality" really gets to the heart of the issue: it concerns whether this way of viewing the data is appropriate for your analysis.  I cannot comment on that because you haven't disclosed enough details about the nature of these observations.  Scalability might be poor if you had to create physical records for each observation.  However, it should suffice to record counts in each 4D cell: for each combination of subject, category, X value, and Y value, indicate the total number of times it has been observed.
Note you probably don't want to (see e.g. [here](http://stats.stackexchange.com/questions/66448/should-covariates-that-are-not-statistically-significant-be-kept-in-when-creat/66454)).
There's number of things wrong here, but I'll just mention one for the moment. To get a *t* distribution requires independence of the numerator and denominator. You don't have that, so you have no solid basis for using a t-test. On the other hand, under the null, you know $\\sigma$, so you do have a basis for an asymptotic z-test.
My earlier comment still stands: combining correlated sources of error may produce correlated estimates of position, which will affect the calculations. Although in practice the magnitude of the resulting error may be inconsequential, some analysis of it ought to be offered to justify the solution. Incidentally, this question has no unique answer unless you provide a prior distribution for the ship locations (or at least their mutual distance): that may be why a solution has been eluding you.
Finally had a chance to have a look at this. Looks almost perfect for my needs, but doesn't include the p values for those eta2 values. I'm going to leave it as the answer as I didn't originally ask for a package that gave the p values on the partial R2! I'm guessing maybe when my R skills get good enough I can modify the package to display this info as well, as it already includes partial SS and SSR.
(+1) I get the feeling the OP's proof is being proved for him  on this site in somewhat random order. For example, this seems to be essentially one side of the argument one would need to show the $n=2$ case from [one of his other questions](http://stats.stackexchange.com/questions/29052).
I've seen $\\mathbb{E}(\\cdot)$ used as an *operator*, while $\\mu$ represents a parameter from a population. Also, in a regression context, we generally consider $\\mathbb{E}(Y\\mid X)$ (the expectation is conditional on the predictor(s)).
Maybe a starting point can be found here : http://grokbase.com/t/r/r-help/10b80sxv1s/r-sample-size-calculation-for-differences-between-two-very-small-proportions-fishers-exact-test-or-others
Unfortunately, this thread is about comparing 2 proportions. I am a bit confused: how could this be applied to my case?
[The Coupon collector's problem](http://en.wikipedia.org/wiki/Coupon_collector%27s_problem) also [see](http://www.math.uah.edu/stat/urn/Coupon.html) -- googling will give you many more hits and more info. Also try searching on that [here at stats.SE](http://stats.stackexchange.com/search?q=coupon+collector).
A large part of your problem is that you have a highly nonlinear function (in the parameter in question) and not much data.  You might want to try the experiment of using $n=20, 40, 80, 160, 320$ and plotting the posteriors for a given weakly informative prior, to see how fast they become "reasonable."  Plots of the series of generated estimates and the `acf` function will help with determining burn in times and thinning rates.
NP doesn't stand for Nonlinear Programming. Could that be a point of confusion?
+1 to stackoverflow [r]
Single arm, uncontrolled, retrospective.
Thank you all for your responses. The baseline variables that differ from one another based on the multiple t-tests are the baseline levels of some of the outcome variables (e.g., baseline depression scores differed and depression at 1 and 4 months is one of the outcome measures).
@user26091 That explanation was mistaken, sorry. Replaced it with the correct explanation.
I considerably reworded your question after I believed I got it correctly. Please check is it alright for you now.
@Ronald, I believe the answers to your questions are no and yes, respectively.  You can find out by modifying the example in this reply: change the coefficients (0.05 and 0.01) in the model and the number of cases ($n=32$) to see what happens.
> yes it does local quantile regression as well (see the vignette (http://cran.r-project.org/package=quantreg).
Hi James, Thanks for the answer. Can you give me an intuitive explanation of multiple testing correction?
Rather than ask for a solution using a particular package, you could ask for a general solution to the problem.  As it stands, people unfamiliar with your area may have a hard time getting a grasp on what your question really is.
Thanks a lot for your help @gung
Thanks. I've updated my question to include a summary for the number of "close neighbors" per object. I'm not sure if this means I have a lot of outliers... but what can I do do protect hclust / k-means / pam (partition around medoids) against outliers? On association rules, would the same idea be captured by fuzzy clusters?
+1 for the asymptotic result.
Could you elaborate a little on that last sentence? For instance, consider an iid sample of size $2n$. I will estimate the median of the population by using a coin flip to choose among the $n$th and $n+1$st largest values in the sample. According to your definition this is not a statistic, because it is not a "deterministic" procedure (although it *is* a statistic according to a common more general definition). It also is a reasonably good estimator. So I am wondering just what kind of object you have in mind when you refer to an "estimator" that is not a "statistic."
"*the $σ$ of $n$ samples follows $χ^2(n)$ distribution*" --- you mean something else there; like $(n-1)s^2/\\sigma^2$ follows a $\\chi^2_{n-1}$. Obviously the CLT often applies to sample variances, since (aside a minor handwave about $n$ vs $n-1$) it's an average. Under some circumstances (at least anecdotally) a sum of squares can behave like a scaled chi-square but I'm not sure of where to point you for that, nor how to work out what the scaling factor might be, nor exactly the circumstances in which it can be expected to work.
(+1) This answer would be more complete if you explain in text why if the correlation between A and B is higher the power of the test is lower (instead of burying it in code).
Where did $b$ come from?
"Good fit" is not necessarily synonymous with "significant", -- perhaps oddly, this can be the case even when your significance test is called a goodness of fit.
Hey, thx again for your time and answer. I have reread all the theory again. Are you sure a random walk with drift is the same as a trend stationary process?
This is related to the concept of "ecological correlation" - see, for example, the Wikipedia page http://en.m.wikipedia.org/wiki/Ecological_correlation
It's not a *principle*, it's advice, and it's not about time-series, but duration data.
@ArthurSmall done, We define $u$ to be a threshold value for the difference between the highest and the second highest probability (a threshold for $P_1$-P2); thus, if $P_1-P_2 > u$ then we can (with some cost) ask for the true class of $x$ (let's note it $c_x$).
Could you explain some more about why you're doing this.  Are you looking at lots of different people to find a correlation between, or do you want to provide individual information back to individual people ?
A zero variance estimate does not necessarily mean that there's something bad (see @Glen_b's answer), but the warning about the hessian does  (see http://support.sas.com/resources/papers/proceedings12/332-2012.pdf). We could be more helpful if you posted the data.
This is a different way of explaining these ideas than I'm used to seeing, but really nicely done. +1
can you give an example? Such as plotting data from a 4-dimensional multivariate normal distribution?
Do you have a particular application domain in mind?  I suspect the techniques might depend upon the application domain.  There's a lot of work in the computer science literature which may be relevant, under the phrase "anomaly detection".
Since the $X_i$ are all supported on the interval $[0,1]$, *all* of the probability of $Y$ must be contained within the same interval.  Because your answer (even after it is properly normalized) assigns positive probability to values larger than $1$, it cannot be correct. Taking logs is a good strategy, but why don't you first show that each $-\\log(X_i)$ has an Exponential distribution, then use the fact that sums of iid Exponentials have Gamma distributions? That makes it clear that $-\\log(Y)$ has a Gamma distribution and you're practically done.
@Peter Notice, for instance, that the (usual) formula for the CI of a mean is proportional to $n^{-1/2}$ and to an estimated SD. The latter will be stable for such large samples, whereas $n^{-1/2}$ decreases with sample size. Although CI formulas for other estimators and other situations differ, the widths still typically show the $n^{-1/2}$ scaling. Intuitively, when you take a smaller sample you have to expect wider intervals, *caeteris paribus.* (Note that I am not referring to *estimation bias,* which is something altogether different.)
Thanks for the interesting list of examples. Given the objective of the question: To contribute to the debate on the review of our statistics courses, we will try to get more details on the implementation of testing in modern devices, can be a great motivation for our engineering students.
This entirely is dependent on the program you use. Many regressional procedures nowadays create dummies internally (so you may just input a categorical variable and tell the program that it is nominal).
Thanks, I want to be able to say for example, when tail lesions are present, there is a 20% chance that lameness will also be present
With the recent rise of deep learning, NNs can be considere "more in" than SVMs, I'd say.
Could you give some more details about what kind of model you fit and what your predictors are?
At present, this question is only about how an R function works. Thus, it is off-topic for CV (see our [help page](http://stats.stackexchange.com/help)). Can you edit this Q to make it on-topic (eg, what are different splitting criteria, & their pros & cons). If not, this Q may need to be closed. (If the answer below is not enough to address your needs, you could also try the [r-help-listserv](https://stat.ethz.ch/mailman/listinfo/r-help).)
I am not sure I follow how to do this.  Do you think you could just show the last step of what the likelihood should be?
There's (currently) a nice free course on coursera that is dedicated to your question.   https://www.coursera.org/course/neuralnets
The data is too large to do that. The counts have been presented as numbers like 5000 --- 2000 etc. Wouldn't be able to type each one in.
When the scale and offset parameters are the same, this convolution is just another Gamma distribution. If that's the case, you only need to estimate the Gamma parameters. There are many ways, though: which are good depend on how you intend to use those estimates.
@Glen_b has provided a good answer that addresses the main point. I just thought I'd add that the number of resampled observations does not *have* to be the same as the number of elements in the original sample. For most of the underlying asymptotic theory to work, all that is required (from memory) is that the number of resampled observations and the number of observations in the original sample grow at the same rate.
Good point. I just wanted to flag that the Bayesian option seems like the way to go. Of course @Kingsford has now provided a more rigorous explanation.
(+1) I believe that for SPSS key papers are from [Jacqueline J. Meulman](http://bit.ly/KQmzSg), see e.g. [Nonlinear principal components analysis: Introduction and application](http://bit.ly/LURRoJ) (Psychol Methods. 2007 12(3):336-58) or [PCA with nonlinear optimal scaling transformations for ordinal and nominal data](http://bit.ly/HZhe6U) (SAGE Handbook of Quantitative Methodology for the Social Sciences, 2004). For R, there is [homals](http://bit.ly/LUSIWB) and a JSS paper, [Gifi Methods for Optimal Scaling in R: The Package homals](http://www.jstatsoft.org/v31/i04).
$\\sigma_0^2$ and $\\sigma_1^2$ have to be covariance matrices for their definitions to be sensical. Is this what you meant?
@probability You can obtain an expression for the integrand simply by splitting the integral into regions $\\mu \\ge 0$ and $\\mu \\lt 0$; no absolute values are needed.  But the result is a messy rational function of $x$, $exp(-x^2)$, and error functions, and so is unlikely to be integrable in closed form.
In what sense is the median a "symmetrical" statistic?  It's certainly not the case that distributions tend to be symmetrically distributed about their medians (nor about their means).  If you mean merely what you wrote in another comment that the "median splits the population in half" (which *defines* the median), your argument sounds circular: the median is good because the median is the median!
See http://stats.stackexchange.com/q/5504/919 .
No prob! Note that although it's true the t-test assumes normality, it is also robust to minor deviations, and seeing your boxplots, there's no clear evidence of non-normality. So I'd feel totally comfortable using the t procedure.
Yes.  The book covers the `Design` package which is now replaced with `rms` and used the same way except for how you get partial effect plots and the final plotting step for nomograms - see http://biostat.mc.vanderbilt.edu/Rrms.  For $D_{xy}$ google is your friend.  I got several hits.  This one looks good: http://www.stata-journal.com/sjpdf.html?articlenum=st0007
+1 The point about convexity of $-\\log(L)$ is good, although perhaps overspecialized (it does not apply to many likelihoods). But I believe the answer goes much deeper than that: as statisticians, we are (or at least should be) at least as interested in the possible *errors* in our estimates as we are in the estimates themselves. Those errors are computed from relatively simple properties of $-\\log(L)$ and *not* from properties of $L$ itself.
Your response is not an example, but a repetition of the original claim. Who says that, and where? Kriging is a form of smoothing, certainly. However, kriging is not a '[Gaussian process](http://en.wikipedia.org/wiki/Gaussian_process)'; it's an operation that is applied to what is assumed to be a Gaussian process.
Not all errors are equally bad. The table is the result of classification on a ordinal scale. So the further from the prediction is from the actual class, the worse the error is.
Is the omitted variable unknown or do you have a candidate in mind?
We need more information on why you are interested in this to be able to give any advice on what analysis to do.  Also it isn't clear what you mean by an "event", and certainly not what might be meant by the probability distribution of the incident of event.  And occurence of sessions - what do you need to know?  Is the task one of summarising, presenting, or inference to some population or model?