# Significance test for skewed distribution

I have evaluated two methods on a test set with $n=4252$ samples. Each sample gets a score from the two methods and the scores are in the range $[0,1]$.

The distribution of the scores of method 1 and method 2 are plotted in the following picture, along with the mean and median values: The sample means of method 1 and 2 are calculated as $\mu_1=0.75$, $\mu_2=0.824$ and the sample standard deviations are $\sigma_1=0.236$, $\sigma_2=0.194$.

I have not done many significance tests before, so I am a bit lost how to proceed here and which method to apply; especially because the distributions do not seem to follow a normal distribution. My null hypothesis $H_0$ is that "Method 1 and Method 2 do not differ significantly"; the alternative hypothesis $H_a$ is that "Method 2 is better than Method 1".

How can I test this?

Edit: I think I need to be a bit more explicit here. I have the ground truth data for all the $n$ samples. Method 1 and Method 2 do something to every single sample and use a score function $f$ to compare the result with the ground truth. A score of 1 means that the method worked perfectly on this sample, a score of 0 means that the method failed completely. I think the details of how $f$ works and that it needs ground truth data for the comparison with the method's result is not relevant here. If it helps, let us just say that some wise entity tells us with absolute certainty for each sample how well method 1 and method 2 worked, that is, this entity gives us for each sample two numbers between 0 and 1 and this is all we work with.

Now I run the two methods separately on all $n$ samples so that I get $n$ scores for method 1 and $n$ scores for method 2. The distribution of the scores is plotted in the figure above.

One of the two methods should be used in a production environment, so I have to decide which one is better. The ideal method would return a score of 1 for all samples. So if for example method 1 would return a score of 0.1 for all samples and method to would always give 0.9 it would be easy to say that method 2 is favorable. But how can I go on here? I honestly dont care if medians, means or whole distributions are compared. All I want is to find out is which method performs better.

Link to scores of method 1: https://pastebin.com/jkdBEJM1

Link to scores of method 2: https://pastebin.com/1S4zYAaa

Edit 2 Here is the difference plot @NickCox suggested: For each sample $i\in[1,n]$ the two scores $s_1^{(i)}$ and $s_2^{(i)}$ of method 1 and method 2 are used to calculate the difference $d_i=s_2^{(i)} - s_1^{(i)}$ and the mean $m_i=(s_1^{(i)}+s_2^{(i)})\cdot 0.5$ The point $(m_i/d_i)$ is then plotted.

However, this plot doesn't make me much smarter.

• The word "significantly" does not belong comfortable in the phrasing of $H_0$, which is just that parameters are equal. Significance levels figure in your assessment of that hypothesis, naturally. Sep 28, 2017 at 14:35
• One might be inclined to interpret "do not differ" as asking whether after some kind of calibration is performed the two methods yield the same information about the subjects being sampled. For instance, it's possible you could find a function $g$ such that with $(x,y)$ being the two scores, $g(x)=y$ in every case. Arguably then the methods would merely be different ways of expressing identical results. One would expect the match not to be that perfect, so there remain questions of (a) how to estimate $g$ and (b) how to assess any remaining discrepancies $y-g(x)$. Is this what you need?
– whuber
Sep 28, 2017 at 17:42
• Incidentally, how could you tell which method is better? You haven't indicated the existence of any data telling you how "correct" or "accurate" any particular score might be. All you can do with the data you describe is to compare the two methods in some way to each other.
– whuber
Sep 28, 2017 at 17:44
• You're comparing distributions, but as I understand it Method 1 and Method 2 are both applied to each individual, so the scatter plot is as or more relevant to comparing the methods. It might be that plotting (method 2 $=$ method 1) versus their mean is also helpful. Sep 28, 2017 at 21:31
• The peculiar shape of the difference vs mean plot follows from the definitions used. Both scores are in $[0, 1]$. A mean of $1$ requires both to be $1$; their difference is then necessarily $0$. Similarly a mean of $0$ requires both scores to be $0$; their difference is also $0$. Conversely, differences of as much as $\pm 1$ are possible but if and only if one score is $1$ and the other $0$; in both such cases the mean must be $0.5$. So points must lie inside what might be called a unit diamond. Sep 29, 2017 at 17:34

By and large, with this kind of sample size marginal distributions hardly matter to comparisons of means (and the fact that you have paired observations helps too, as you are looking at the pattern of differences).

Backing up first with some discursive remarks: It is common in many fields to compare measurements using two or more methods. When there is some gold standard or benchmark measurement taken to be true, we can say which method is best by comparison. Otherwise we can always look at the extent of agreement.

What I think requires some adjustment of thought here is that we have, as I understand it, not the original measurements (declared irrelevant!) but scores of how well each method did. So agreement of those scores doesn't necessarily mean that the methods agree, just that they are equally good or bad. That difference seems to make some methods for comparing methods less useful than they are with original measurements, or at least to imply that they need to be thought about differently.

We are comparing two distributions but values are paired and that structure needs to be respected. Assuming that I have interpreted the data files correctly, we can draw scatter plots too. Here are the original data:

It is hard to see what is going on there, even with some use of transparency, but the results are naturally consistent with the strongly skewed distributions shown by the OP.

I followed up my own suggestion of fourth powers (which arose from some earlier simulations with loosely similar beta distributions). The fourth power respects the interval $[0, 1]$ as $0^4 = 0$, $1^4 = 1$ and the transformation does not fall over if either extreme ($0$ or $1$) is met. More crucially, it does a reasonable job of making the distributions more symmetric and so a little easier to think about. People might say this transformation is ad hoc and they would be right: translated in 21st century terms, that means "fit for purpose".

Again, it is hard to see much structure there. Although not shown here, various smoothers don't help either. This is good news in so far as all the useful information is implied to be in distribution plots, to which we now turn. Here I use quantile-box plots in which quantile plots (each value against its rank) have median and quartile boxes superimposed. I also add lines for means.

As before we look at transformed scales too.

In short, the conclusion that method 2 is systematically better than method 1 is confirmed as robust under use of mean or median and under use of original and strongly transformed scales.

On the question of significance tests, I suggest that a general solution is to convert this to a different but related problem: find a confidence interval for the mean difference, that mean being something we can calculate on whatever scale we please.

I did this in Stata. The code should seem not too cryptic to those enamoured of other software.

* original measurements: score 1 - score 2
. bootstrap, reps(1000) nodots: mean sdiff
(running mean on estimation sample)

Mean estimation                   Number of obs   =      4,252
Replications    =      1,000

--------------------------------------------------------------
|   Observed   Bootstrap         Normal-based
|       Mean   Std. Err.     [95% Conf. Interval]
-------------+------------------------------------------------
sdiff |    .074044   .0047372      .0647593    .0833286
--------------------------------------------------------------

. estat bootstrap, all

Mean estimation                                 Number of obs     =      4,252
Replications      =       1000

------------------------------------------------------------------------------
|    Observed               Bootstrap
|        Mean       Bias    Std. Err.  [95% Conf. Interval]
-------------+----------------------------------------------------------------
sdiff |   .07404398  -.0000625   .00473716    .0647593   .0833286   (N)
|                                       .0649868   .0836285   (P)
|                                       .0650588   .0838786  (BC)
------------------------------------------------------------------------------
(N)    normal confidence interval
(P)    percentile confidence interval
(BC)   bias-corrected confidence interval

* difference of 4th powers
. gen pdiff = p4_score2 - p4_score1

. bootstrap, reps(1000) nodots : mean pdiff
(running mean on estimation sample)

Mean estimation                   Number of obs   =      4,252
Replications    =      1,000

--------------------------------------------------------------
|   Observed   Bootstrap         Normal-based
|       Mean   Std. Err.     [95% Conf. Interval]
-------------+------------------------------------------------
pdiff |   .1058628   .0075672      .0910314    .1206942
--------------------------------------------------------------

. estat bootstrap, all

Mean estimation                                 Number of obs     =      4,252
Replications      =       1000

------------------------------------------------------------------------------
|    Observed               Bootstrap
|        Mean       Bias    Std. Err.  [95% Conf. Interval]
-------------+----------------------------------------------------------------
pdiff |    .1058628   .0004516   .00756718    .0910314   .1206942   (N)
|                                       .0910895   .1211185   (P)
|                                       .0905725   .1203873  (BC)
------------------------------------------------------------------------------
(N)    normal confidence interval
(P)    percentile confidence interval
(BC)   bias-corrected confidence interval


Firing all possible shotguns, it can be seen that no reasonable confidence interval goes anywhere near 0, so even extreme sceptics should be convinced that means differ.

EDIT: These comments appeared in an earlier version but now seem off-target. They are kept for anyone trying to make sense of earlier discussion.

Feed your values to a paired $t$-test and to more appropriate beta regressions and to generalised linear models with binomial distributions, power links and robust standard errors. My prediction is that figures of merit (e.g. $t$ or $z$ statistics) will remain about the same.

What is more diverting is that your plot of differences suggests much more structure than just method 2 scoring higher. If that's true, then comparison of means alone is a very limited basis for comparison. I would want to see method 2 smoothed as a function of method 1 and vice versa. There are also pertinent exploratory methods that treat the outcomes symmetrically. See e.g. this paper.

• I think I still don't understand how I can reject or accept my null hypothesis that method 1 and 2 perform equally well. Do you suggest to use a $t$-test and compare it to other tests? And you expect that the results of the tests will be similar, no matter if I use a t-test or more fancy methods? What do you imply with "with this kind of sample size". Is the sample size to small? Sep 28, 2017 at 19:55
• No; the sample size is large enough that virtually any test will confirm significance at conventional levels. Sep 28, 2017 at 20:19
• Okay, so a $t$-test is a good measure here, even though the distributions do not seem "normal"? Sep 28, 2017 at 20:50
• Not quite, as any test whose assumptions are more accurate than the t test would be better. A beta regression or an appropriate generalized model is more likely to be better. So, I am not saying "good", just "likely to work almost as well in practice". I am more concerned that you are fixated on comparing means and missing other structure. Sep 28, 2017 at 21:00
• It also seems possible that you would see comparisons more clearly on a transformed scale. It is a little ad hoc, but fourth powers might help. Sep 28, 2017 at 21:01

One way to do significance testing is to measure the tails not by the x axis variable, but by probabilities. So you take 2.5% from each tail to get 5% significance.

Instead of measuring the distance from mean in standard deviations you simply build CDF then look at 2.5% and 97.5% quantiles which give you the 95% confidence interval.

Example. Let's say you have 1000 data points from a standard lognormal distribution. You look at the 2.5% percentile, and it comes up at 0.1442; then you look at 97.5%, which is 7.2485. This is your confidence interval that you can use for inference.

x = lognrnd(0,1,1000,1);
p = prctile(x, [2.5 97.5])
hist(x,100)
line([p(1) p(1) ],[0 250])
line([p(2) p(2) ],[0 250])

p =

0.1442    7.2485


UPDATE:

One more thing: the mean is often not a good measure of centrality for heavily skewed distribution. That's one reason why you were struggling when trying to apply the mean and standard deviation here. Hence, the method that I suggested works better with the median, which happens to be symmetrical in relation to the way the tails are defined

• Could you please elaborate more how to do this or give me a good reference where this procedure is displayed on an example? Sep 28, 2017 at 13:00
• Quintiles (fifths, in effect) at most correspond to cumulative probabilities 0(0.2)1. So I edited quintiles to quantiles. I trust you agree. Sep 28, 2017 at 14:14
• The bigger deal is how this relates to the question, because the central 95% of a sample doesn't correspond to a 95% confidence interval for any summary of level. Sep 28, 2017 at 14:16