80

I am going to change the order of questions about. I've found textbooks and lecture notes frequently disagree, and would like a system to work through the choice that can safely be recommended as best practice, and especially a textbook or paper this can be cited to. Unfortunately, some discussions of this issue in books and so on rely on received wisdom. ...


44

I know non-parametric relies on the median instead of the mean Hardly any nonparametric tests actually "rely on" medians in this sense. I can only think of a couple... and the only one I expect you'd be likely to have even heard of would be the sign test. to compare...something. If they relied on medians, presumably it would be to compare medians. But -...


29

Wilcoxon is generally credited with being the original inventor of the test*, though Mann and Whitney's approach was a great stride forward, and they extended the cases for which the statistic was tabulated. My preference is to refer to the test as the Wilcoxon-Mann-Whitney, to recognize both contributions (Mann-Whitney-Wilcoxon is also seen; I don't mind ...


29

You should use the signed rank test when the data are paired. You'll find many definitions of pairing, but at heart the criterion is something that makes pairs of values at least somewhat positively dependent, while unpaired values are not dependent. Often the dependence-pairing occurs because they're observations on the same unit (repeated measures), but ...


28

In my view the principled approach recognizes that (1) tests and graphical assessments of normality have insufficient sensitivity and graph interpretation is frequently not objective, (2) multi-step procedures have uncertain operating characteristics, (3) many nonparametric tests have excellent operating characteristics under situations in which parametric ...


25

I wouldn't say the classic one sample (including paired) and two-sample equal variance t-tests are exactly obsolete, but there's a plethora of alternatives that have excellent properties and in many cases they should be used. Nor would I say the ability to rapidly perform Wilcoxon-Mann-Whitney tests on large samples – or even permutation tests – is recent, ...


24

I agree with @pikachu that the standard deviations are too large compared with the difference between means for a t test to find a significant difference. Thank you for posting your data. It is always a good idea to take a look at some graphic displays of the data before doing formal tests. Stripcharts of observations in the two groups do not show a ...


23

You should use Dunn's test$^{*}$. If one proceeds by moving from a rejection of Kruskal-Wallis to performing ordinary pair-wise rank sum tests (with or without multiple comparison adjustments), one runs into two problems: (1) the ranks that the pair-wise rank sum tests use are not the ranks used by the Kruskal-Wallis test; and (2) Dunn's test preserves a ...


23

The Kolmogorov-Smirnov test is the most common way to do this, but there are also some other options. The tests are based on the empirical cumulative distribution functions. The basic procedure is: Choose a way to measure the distance between the ECDFs. Since ECDFs are functions, the obvious candidates are the $L^p$ norms, which measure distance in ...


18

The Note in the help on the wilcox.test function clearly explains why R's value is smaller than yours: Note The literature is not unanimous about the definitions of the Wilcoxon rank sum and Mann-Whitney tests. The two most common definitions correspond to the sum of the ranks of the first sample with the minimum value subtracted or not: R subtracts ...


18

Suppose you and I are coaching track teams. Our athletes come from the same school, are similar ages, and the same gender (i.e., they're drawn from the same population), but I claim to have discovered a Revolutionary New Training System that will make my team members run much faster than yours. How can I convince you that it really does work? We have a race....


15

@HarveyMotulsky is right, you can use the Mann-Whitney U-test with unequal sample sizes. Note however, that your statistical power (i.e., the ability to detect a difference that really is there) will diminish as the group sizes become more unequal. For an example, I have a simulation (actually of a t-test, but the principle is the same) that demonstrates ...


15

No, you should not use the Mann-Whitney $U$ test in this circumstance. Here's why: Dunn's test is an appropriate post hoc test* following rejection of a Kruskal-Wallis test. If one proceeds by moving from a rejection of Kruskal-Wallis to performing ordinary pair-wise rank sum (i.e. Wilcoxon or Mann-Whitney) tests, then two problems obtain: (1) the ranks ...


14

Rand Wilcox in his publications and books make some very important points, many of which were listed by Frank Harrell and Glen_b in earlier posts. The mean is not necessarily the quantity we want to make inferences about. There maybe other quantities that better exemplifies a typical observation. For t-tests, power can be low even for small departures from ...


14

There is no way to do this by non-parametric paradigm, just think of it: the sampled distribution is a completely legit one, there is nothing preventing a single-population distribution from having two separate high density zones. But if you turn to parametric models, you may assume that your sub-populations are gaussian, and gaussian distribution has only ...


13

Frank Wilcoxon's 1945 paper [1] described two tests -- for "Unpaired Experiments" and "Paired Comparisons" which have come to be called the (Wilcoxon) rank sum test and the (Wilcoxon) signed rank test respectively. So the first test is for independent (unpaired) samples and the second is for paired samples*. * It can also be used for comparing single ...


13

If the original statement doesn't limit the conditions under which it applies pretty substantially, Field is just wrong on this. Responding to the quoted section: In effect, this means it does much the same as the Mann–Whitney test! No, it really doesn't. They really test for different kinds of things. As one example, if two close-to-symmetric ...


12

Wikipedia appears to have your answers. Here's an excerpt from the example statement of results: In reporting the results of a Mann–Whitney test, it is important to state: A measure of the central tendencies of the two groups (means or medians; since the Mann–Whitney is an ordinal test, medians are usually recommended) The value of U The ...


12

Steady on there! You have two very small samples there. Statistics is not taught at Hogwarts! No white magic for very small samples. Not rejecting the null on Shapiro-Wilk doesn't allow the description "is normally distributed", but rather a much more circumspect "not enough evidence to be clear that this isn't normally distributed". Let's look at graphs,...


11

Brief sketch of ARE for one-sample $t$-test, signed test and the signed-rank test I expect the long version of @Glen_b's answer includes detailed analysis for two-sample signed rank test along with the intuitive explanation of the ARE. So I'll skip most of the derivation. (one-sample case, you can find the missing details in Lehmann TSH). Testing Problem: ...


11

The Mann Whitney test does not require any specific N. However, what your instructor is probably talking about is power; that is, with a small N, differences are not going to be statistically significant unless they are really huge.


11

When you consider the difference between means you have to use a different unit than the simple absolute difference. Take into account that you are measuring the difference in means produced by two random sources. Theose random sources (whose outcomes are your two samples) contains variability. It is this variability which should be used to compare the ...


10

is a Mann Whitney test on data where assumptions aren't satisfied as or almost powerful as a t-test on data where assumptions are satisfied? A phrase like 'as powerful' doesn't really work as a general statement. Power isn't especially comparable across different distributional models. The size of a given effect has different meanings in different parts of ...


10

(Pulls Conover [1] off the bookshelf...) This idea is quite old; it dates back at least to van der Waerden (1952/1953) [2][3], who suggested a test that corresponds to the Kruskal Wallis but with ranks replaced by normal scores. (The idea of using ordered random normal values rather than an approximation of their expectation or their median - is perhaps ...


10

Anyone is allowed to give any possible name for the test statistic they design. Mann and Whitney (1947) decided to name the one they proposed as "$U$". In their paper they do not give any reasons why such name was chosen, but as from the very beginning of the text they compare their test to the one proposed by Wilcoxon (1945), named as "$T$", so I'd guess ...


10

In no way will the difference in sample sizes adversely affect the Mann-Whitney-Wilcoxon test. It's explicitly suitable for groups of different sizes, and how different doesn't impact the essential properties of the test. There's little more to say without some clearer indication of what your colleague thinks the problem is (aside from a burning desire to ...


10

2.2e-16 is the scientific notation of 0.00000000000000022, meaning it is very close to zero. Your statistical software probably uses this notation automatically for very small numbers. You may be able to change this in the settings. The notation alone is no reason to be suspicious. The result itself might be, but you will have to be the judge of that. < ...


10

Is the Wilcoxon rank-sum test a nonparametric alternative to the two sample t-test? Yes and no. (Go not to the elves for counsel...) Speaking broadly, any given test statistic has some power curve in relation to a given sequence of alternatives under some set of assumptions (sufficiently specified to have a unqiue value for power under any element in the ...


10

Let's start with terminology. Population in statistics is the "set of entities under study". When designing the study, we define the population of interest and then draw samples from this population. So sample cannot "consist" of multiple populations. More appropriate wording would be to talk about "groups", "clusters", or "subpopulations". To find clusters ...


9

Technically, the reference category and the direction of the test depend on the way the factor variable is encoded. With your toy data: > wilcox.test(x ~ y, data=data, alternative="greater") Wilcoxon rank sum test with continuity correction data: x by y W = 52, p-value = 1 alternative hypothesis: true location shift is greater than 0 > ...


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