hypothesis test on non normal data

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13.10: Testing Non-normal Data with Wilcoxon Tests

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• Danielle Navarro
• University of New South Wales

Okay, suppose your data turn out to be pretty substantially non-normal, but you still want to run something like a t-test? This situation occurs a lot in real life: for the AFL winning margins data, for instance, the Shapiro-Wilk test made it very clear that the normality assumption is violated. This is the situation where you want to use Wilcoxon tests.

Like the t-test, the Wilcoxon test comes in two forms, one-sample and two-sample, and they’re used in more or less the exact same situations as the corresponding t-tests. Unlike the t-test, the Wilcoxon test doesn’t assume normality, which is nice. In fact, they don’t make any assumptions about what kind of distribution is involved: in statistical jargon, this makes them nonparametric tests . While avoiding the normality assumption is nice, there’s a drawback: the Wilcoxon test is usually less powerful than the t-test (i.e., higher Type II error rate). I won’t discuss the Wilcoxon tests in as much detail as the t-tests, but I’ll give you a brief overview.

sample Wilcoxon test

I’ll start by describing the two sample Wilcoxon test (also known as the Mann-Whitney test), since it’s actually simpler than the one sample version. Suppose we’re looking at the scores of 10 people on some test. Since my imagination has now failed me completely, let’s pretend it’s a “test of awesomeness”, and there are two groups of people, “A” and “B”. I’m curious to know which group is more awesome. The data are included in the file awesome.Rdata , and like many of the data sets I’ve been using, it contains only a single data frame, in this case called awesome . Here’s the data:

As long as there are no ties (i.e., people with the exact same awesomeness score), then the test that we want to do is surprisingly simple. All we have to do is construct a table that compares every observation in group A against every observation in group B. Whenever the group A datum is larger, we place a check mark in the table:

We then count up the number of checkmarks. This is our test statistic, W. 200 The actual sampling distribution for W is somewhat complicated, and I’ll skip the details. For our purposes, it’s sufficient to note that the interpretation of W is qualitatively the same as the interpretation of t or z. That is, if we want a two-sided test, then we reject the null hypothesis when W is very large or very small; but if we have a directional (i.e., one-sided) hypothesis, then we only use one or the other.

The structure of the wilcox.test() function should feel very familiar to you by now. When you have your data organised in terms of an outcome variable and a grouping variable, then you use the formula and data arguments, so your command looks like this:

Just like we saw with the t.test() function, there is an alternative argument that you can use to switch between two-sided tests and one-sided tests, plus a few other arguments that we don’t need to worry too much about at an introductory level. Similarly, the wilcox.test() function allows you to use the x and y arguments when you have your data stored separately for each group. For instance, suppose we use the data from the awesome2.Rdata file:

When your data are organised like this, then you would use a command like this:

The output that R produces is pretty much the same as last time.

What about the one sample Wilcoxon test (or equivalently, the paired samples Wilcoxon test)? Suppose I’m interested in finding out whether taking a statistics class has any effect on the happiness of students. Here’s my data:

What I’ve measured here is the happiness of each student before taking the class and after taking the class; the change score is the difference between the two. Just like we saw with the t-test, there’s no fundamental difference between doing a paired-samples test using before and after , versus doing a one-sample test using the change scores. As before, the simplest way to think about the test is to construct a tabulation. The way to do it this time is to take those change scores that are positive valued, and tabulate them against all the complete sample. What you end up with is a table that looks like this:

Counting up the tick marks this time, we get a test statistic of V=7. As before, if our test is two sided, then we reject the null hypothesis when V is very large or very small. As far of running it in R goes, it’s pretty much what you’d expect. For the one-sample version, the command you would use is

As this shows, we have a significant effect. Evidently, taking a statistics class does have an effect on your happiness. Switching to a paired samples version of the test won’t give us different answers, of course; but here’s the command to do it:

What to do with nonnormal data

You have several options when you want to perform a hypothesis test with nonnormal data.

Proceed with the analysis if the sample is large enough

Although many hypothesis tests are formally based on the assumption of normality, you can still obtain good results with nonnormal data if your sample is large enough. The amount of data you need depends on how nonnormal your data are but a sample size of 20 is often adequate. The relationship between robustness to normality and sample size is based on the central limit theorem . This theorem proves that the distribution of the mean of data from any distribution approaches the normal distribution as the sample size increases. Therefore, if you're interested in making an inference about a population mean the normality assumption is not critical so long as your sample is large enough.

Use a nonparametric test

Nonparametric tests do not assume a specific distribution for the population. Minitab provides several nonparametric tests that you can use instead of tests that assume normality. These tests can be especially useful when you have a small sample that is skewed or a sample that contains several outliers.

Nonparametric tests are not completely free of assumptions about your data: for example, they still require the data to be an independent random sample.

Transform the data

Sometimes you can transform your data by applying a function to make your data fit a normal distribution, so that you can finish your analysis.

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2.6 - non-normal data.

So far, all of our discussion has been on finding a confidence interval for the population mean \(\mu\) when the data are normally distributed. That is, the \(t\)-interval for \(\mu\) (and \(Z\)-interval, for that matter) is derived assuming that the data \(X_1, X_2, \ldots, X_n\) are normally distributed. What happens if our data are skewed, and therefore clearly not normally distributed?

Well, it is helpful to note that as the sample size \(n\) increases, the \(T\) ratio:

\(T=\dfrac{\bar{X}-\mu}{\frac{S}{\sqrt{n}}}\)

approaches an approximate normal distribution regardless of the distribution of the original data. The implication, therefore, is that the \(t\)-interval for \(\mu\):

\(\bar{x}\pm t_{\alpha/2,n-1}\left(\dfrac{s}{\sqrt{n}}\right)\)

and the \(Z\)-interval for \(\mu\):

\(\bar{x}\pm z_{\alpha/2}\left(\dfrac{s}{\sqrt{n}}\right)\)

(with the sample standard deviation s replacing the unknown population standard deviation \(\sigma\)!) yield similar results for large samples. This result suggests that we should adhere to the following guidelines in practice.

In practice! Section  

Use \(\bar{x}\pm t_{\alpha/2,n-1}\left(\dfrac{s}{\sqrt{n}}\right)\) if the data are normally distributed.

If you have reason to believe that the data are not normally distributed, then make sure you have a large enough sample ( \(n\ge 30\) generally suffices, but recall that it depends on the skewness of the distribution.) Then:

\(\bar{x}\pm t_{\alpha/2,n-1}\left(\dfrac{s}{\sqrt{n}}\right)\) and \(\bar{x}\pm z_{\alpha/2}\left(\dfrac{s}{\sqrt{n}}\right)\)

will give similar results.

If the data are not normally distributed and you have a small sample, use:

with extreme caution and/or use a nonparametric confidence interval for the median (which we'll learn about later in this course).

Example 2-3 Section  

A random sample of 64 guinea pigs yielded the following survival times (in days):

What is the mean survival time (in days) of the population of guinea pigs? (Data from K. Doksum, Annals of Statistics , 2(1974): 267-277.)

Because the data points on the normally probability plot do not adhere well to a straight line:

it suggests that the survival times are not normally distributed. We have a large sample though ( \(n=64\)). Therefore, we should be able to use the \(t\)-interval for the mean without worry. Asking Minitab to calculate the interval for us, we get:

One-Sample T:   guinea

That is, we can be 95% confident that the mean survival time for the population of guinea pigs is between 289.7 and 400.7 days.

Incidentally, as the following Minitab output suggests, the \(Z\)-interval for the mean is quite close to that of the \(t\)-interval for the mean:

One-Sample Z:   guinea

The assumed sigma = 222.2.

as we would expect, because the sample is quite large.

The Stats Geek

The t-test and robustness to non-normality

So, as constructed, the two-sample t-test assumes normality of the variable X in the two groups. On the face of it then, we would worry if, upon inspection of our data, say using histograms, we were to find that our data looked non-normal. In particular, we would worry that the t-test will not perform as it should – i.e. that if the null hypothesis is true, it will falsely reject the null 5% of the time (I’m assuming we are using the usual significance level).

What does this mean in practice? Provided our sample size isn’t too small, we shouldn’t be overly concerned if our data appear to violate the normal assumption. Also, for the same reasons, the 95% confidence interval for the difference in group means will have correct coverage, even when X is not normal (again, when the sample size is sufficiently large). Of course, for small samples, or highly skewed distributions, the above asymptotic result may not give a very good approximation, and so the type 1 error rate may deviate from the nominal 5% level.

Of course if X isn’t normally distributed, even if the type 1 error rate for the t-test assuming normality is close to 5%, the test will not be optimally powerful. That is, there will exist alternative tests of the null hypothesis which have greater power to detect alternative hypotheses.

For more on the large sample properties of hypothesis tests, robustness, and power, I would recommend looking at Chapter 3 of ’Elements of Large-Sample Theory’ by Lehmann. For more on the specific question of the t-test and robustness to non-normality, I’d recommend looking at this paper by Lumley and colleagues.

Addition – 1st May 2017 Below Teddy Warner queries in a comment whether the t-test ‘assumes’ normality of the individual observations. The following image is from the book Statistical Inference by Casella and Berger , and is provided just to illustrate the point that the t-test is, by its construction, based on assuming normality for the individual (population) values:

You may also be interested in:

• Wilcoxon-Mann-Whitney as an alternative to the t-test

9 thoughts on “The t-test and robustness to non-normality”

This website’s description and explanation of the “normality assumption” of t-tests (and by extension to ANOVA and MANOVA and regression) is simply incorrect. Parametric tests do not assume normality of sample scores nor even of the underlying population of scores from which samples scores are taken. Parametric tests assume that the sampling distribution of the statistic being test is normally distributed – that is, the sampling distribution of the mean or difference in means. Sampling distributions approach normality as sample size increases as shown by the Central Limit Theorem, and the consensus among experts is that with means that are based on sample sizes as small as 30 that the sampling distribution of means is so close to being a normal distribution, that the differences in probabilities in the tails of the normal and almost normal sampling distributions are not worth attention. Also, for any symmetrical distribution of scores, the sampling distribution will be normal. The normality assumption for Pearson correlation coefficients is also commonly misstated in MANY, if not most, online and textbook sources as requiring the variables being correlated to be normally distributed. X & Y variables do not require normal distributions, but only require that the sampling distribution of r (or t that is used to test the statistical significance of r) be normally distributed. In addition, many sources also misstate the normality assumption for regression and multiple regression as requiring that scores for predictor variables to be normally distributed. Regression only assumes that the residuals of the regression model being fit be normally distributed. Moreover, the assumption of normality for any statistical test is only relevant if one tests the null hypothesis because the sampling distribution has no effect on the parameter being estimated by the sample – that is, the sample mean and the sample correlation and the sample regression weights are the best unbiased estimates of their corresponding parameters, assuming there are no other “problems” with the data such as outliers, thus not affecting effect size estimates such as the values of r, R-square, b or Cohen’s d. Outliers usually should be trimmed, scores transformed or Winsorized because outliers can distort estimates of any parametric model. Of course, other assumptions may be important to consider beyond the normality assumption – such as heterscadasticity, independence of observations, and homogeneity of variances. But telling people that their data need to be normally distributed to use t-tests, Pearson correlations, ANOVA, MANOVA, etc. erroneously leads many researchers to use sometimes less powerful non-parametric statistics and thus not to take advantage of more sophisticated parametric models in understanding their data. For example non-parametric models generally cannot deal with interaction effects, but ANOVA, MANOVA and multiple regression can.

Thanks for your comment Teddy. I do believe however that the t-test referred to as the t-test, by its construction , and as I wrote, assumes normality of the underlying observations in the population from which your sample is drawn (see the image I have now included in the bottom of the post, which is from Casella and Berger’s book Statistical Inference). From this it follows that the sampling distribution of the test statistic follows a t-distribution. As you say, and as I wrote and was illustrating, this distribution gets closer and closer to a normal as the sample size increases, as a result of the central limit theorem. This means that provided the sample sizes in the two groups are not too small, the normality assumption of the individual observations is not a concern regarding the test’s size (type 1 error).

You make an interesting point about the assumption being about the distribution of the test statistic, rather than the distribution of the individual observations. I think the point is however that stating that the assumption of a test is regarding the distribution of the test statistic is not easy to exploit. In contrast, assumptions about individual observations (e.g. normality) is something which an analyst can consider and perhaps easily assess from the observed data. In the t-test case, one can examine or consider the plausibility of the observations being normal. But if you tell someone they can only use a test if the test statistic follows a given distribution in repeated samples, how are they do know if and when this will be satisfied?

You wrote “Parametric tests assume that the sampling distribution of the statistic being test is normally distributed” – this of course is not true in general – there are many tests that have sampling distributions other than the normal, for example the F-distribution, or mixtures of chi squared distribution in the case of testing variance components in mixed models.

Teddy is right, here. The t-test doesn’t assume normality. Only in small samples are non-parametric tests necessary. http://www.annualreviews.org/doi/pdf/10.1146/annurev.publhealth.23.100901.140546

y’all are both wrong and the author is right. in support of your criticism you mention the central limit theorem, but the central limit theorem shows that (under certain conditions) sample means converge in distribution to a NORMAL distribution. they do not converge to a T-DISTRIBUTION. the only common situation that will result in a sample mean having a t-distribution is when the population follows a normal distribution. and in that case, it’s not a statement of convergence. the distribution of a sample mean of repeated realizations from a normal distribution IS a t-distribution

The author is right :normality is the condition for which you can have a t-student distribution for the statistic used in the T-test . To have a Student, you must have at least independence between the experimental mean in the numerator and the experimental variance in the denominator, which induces normality. See theorem of Geary. There is also an another important reason to suppose normality: to have the most powerful test under certain conditions (look to generalization of Pearson theorem).

The author is wrong. The reason this “works” is because as N becomes larger so does the t-stat which means every test you run will be statistically significant. Use the formula provided by the author to confirm this. For large populations, z-score should be used because it doesn’t depend on sample size. In my work, I am using samples of thousands or millions so the t-test becomes irrelevant. Also, if you use z-score to calculate your own confidence interval then you don’t have to worry about the distribution. As long as mean of sample B is greater than 95% of the values of sample A your result is statistically significant.

Can the commenters who claim the author is wrong provide a scientifically robust reference rather than a paper from the field of public health?

According to Montgomery’s “Design and Analysis of Experiments”, the t statistic is defined as the ratio of a z statistic and the square root of a chi-square statistic, the latter divided by the d.f. under the radical. This gives rise to the d.f. for the t statistic. The chi-square is defined as the sum of squared independent z-draws, which are achieved in the t-statistic by dividing the corrected sum of squares of the samples by sigma-squared, the underlying variance of the samples. Hence, the t-statistic assumes the samples are drawn from a normal distribution and may not rely on the central limit theorem to achieve that compliance. That said, the t-test is pretty robust to departures from that assumption.

I am not familiar with the phrasing “a sample mean of repeated realizations from a normal distribution.”

As the distribution of sample means converge to a normal distribution under the CLT as sample size increases, the distribution of the standardized t-score of sample mean (x-bar – mu) / (s/sqrt(n)) converges to a t-distribution. I don’t believe the sample means ever form a t-distribution.

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Non-normal Distribution

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About this lesson

Non-normal data often has a pattern. Knowing that pattern can help to transform the data to normal data and can aid in the selection of an appropriate hypothesis test.

Exercise files

Download this lesson’s related exercise files.

Quick reference

Non-normal data can occur from stable physical systems. Hypothesis tests can be done using non-normal data.

When to use

Prior to actually conducting the hypothesis test, if the data set parameters are a continuous “Y” and a discrete “X,” the data should be checked to determine if it is normal or non-normal so as to be able to choose the correct test.

Instructions

Non-normal data is often created by stable physical systems. The non-normality is often due to constraints in the system or environment. There are hypothesis tests that are structured to accept non-normal data sets. However, different tests are best suited to different types of non-normality. The non-normal hypothesis test lessons describe which method to use for different types of non-normality. It is often desirable to graph or plot the data so as to determine the nature of the non-normality.  

Normality is determined using basic descriptive statistics of the data sample.  In particular, non-normal tests usually use either the median or the variance. Descriptive statistics can also provide measures of the level of non-normality. When doing a descriptive statistics test, several parameters are determined:

• Median – the midpoint of the data points.  This is often used in Hypothesis tests with non-normal data.
• Variance – a measure of the spread or width of the distribution.  This measure is calculated by squaring the standard deviation of the data set.
• Skewness – this is a measure of symmetry.  A symmetrical distribution will have a skewness value of zero.  The distribution is considered normal as long as the value is between -.8 and +.8. Beyond that, the distribution is non-normal.
• Kurtosis – this is a measure of the tails of the distribution to indicate if they are “heavy” or “light.”   There are three types of Kurtosis. Leptokurtic is heavy tails. There are many points near the upper and lower bounds of the data. Mesokurtic is associated with the normal curve. Platycurtic is the condition when the tails are light, they rapidly drop to near zero on the upper and lower edges of the distribution.  Kurtosis can be measured in several ways.  The method used in Excel is “Sample Excess Kurtosis.”  This measure has the advantage that a Normal curve score will be zero – just like with Skewness.  In this case, values from -0.8 to +0.8 are still considered Normal.  Minitab uses the true Kurtosis scale which places the midpoint at 3.0.
• Multi-modal – this occurs when there are multiple datasets combined into the same set being investigated. This data set being investigated will often have multiple “peaks” representing each of the constituent data sets.

Granularity – this occurs when the measurement system resolution is too coarse for the data. All the data is lumped together in just a few slices.

Hints & tips

• If the Data Analysis Menu does not show on your Data ribbon in Excel, you need to add the Analysis ToolPak Add-in.  Go to the File menu, select Options, then select Add-in.  Enable the Analysis ToolPak add-in.  This is a free feature that is already in Excel, you just need to enable it.  You may need to close and reopen Excel for the menu to appear.
• If you don’t have Minitab, consider downloading the free trial.  Minitab normally has a 30-day free trial period.  All the non-normal tests we discuss are available in Minitab.
• If your data is non-normal, place your data points in a column or row of Excel and select the graph function to determine the shape of your data. The selection of a non-normal hypothesis test is based on the nature of the non-normality.
• 00:04 Hi, I'm Ray Sheen.
• 00:06 We just looked at how to determine whether or not data is normal.
• 00:10 Let's take a minute and
• 00:11 talk about the implications of non-normal data when doing hypothesis testing.
• 00:16 >> So what do we mean by non-normal variation?
• 00:20 Sometimes the data is not normally distributed.
• 00:23 That doesn't mean that there is a special cause variation.
• 00:25 It may mean that there is some aspect of the physical system that prevents the data
• 00:30 from being characterized with a normal distribution.
• 00:33 The good news is that the hypothesis testing does not require the data to be
• 00:37 normally distributed.
• 00:39 Now, the statistical analysis with non-normal data is different, and
• 00:43 often the math is much more complex.
• 00:46 Back when the analysis was being done by hand,
• 00:48 we wanted to use normal data because it was easier to do the analysis.
• 00:53 But it turns out that with computers to help us,
• 00:55 we can do the non-normal analysis math without too much difficulty.
• 01:00 It seems that computers are actually pretty good at doing math, so
• 01:03 we'll let them.
• 01:05 Now a recommendation for using the non-normal tests.
• 01:08 Different tests are suited to different types of non-normality.
• 01:12 So I recommend that you graph your data first so
• 01:14 that you can see what type of non-normality you're dealing with.
• 01:18 This will make it easier to select the best test for your application.
• 01:23 One more thing about test selection.
• 01:25 Excel does not have the non-normal tests in the data analysis function.
• 01:29 So you will need to be using Minitab or another statistical software package for
• 01:34 those types of tests.
• 01:36 Obviously, you want to select the test that best suits your data.
• 01:40 So if you're limited to using Excel,
• 01:42 you'll need to transform your non-normal data to normal.
• 01:45 We'll talk about that later.
• 01:47 So let's talk about what we mean when we say data is not normal.
• 01:51 First, let's acknowledge that there are many things in the physical world, or
• 01:55 the process and
• 01:56 product design that prevent a data distribution from being normal.
• 02:00 Examples include extreme points that disrupt edges of a distribution.
• 02:05 Now, granted, the sudden incidence of many of these
• 02:08 points is an indication of a special cause occurring.
• 02:11 But an occasional one can occur, and
• 02:13 it may skew your data if you have a small distribution.
• 02:17 Also, there may be physical limits.
• 02:20 For instance, a parameter may be limited so that it cannot be less than 0.
• 02:24 Another thing that we're often trying to test with our hypothesis is whether or
• 02:29 not we have a combination of two or more processes within
• 02:32 the same dataset that will normally create a non-normal distribution.
• 02:37 Now, that's not an exhaustive list.
• 02:38 It's just an illustrative one.
• 02:40 First, there's skewness.
• 02:41 In this case, the data is not symmetrical.
• 02:44 The data is weighted, one towards one side or the other.
• 02:48 And typically this occurs when there is a physical limit,
• 02:51 either a natural one such as temperature hitting a boiling or
• 02:54 freezing point that changes how the system performs or an artificial one,
• 02:58 such as a machine limit that saturates a capacitor at a certain level.
• 03:03 Next is kurtosis.
• 03:05 This is the shape measure of the distribution, and
• 03:08 is focused on what happens at the edges or tails.
• 03:12 We have three types of Kurtosis, leptokurtosis, which looks like
• 03:16 heavy tails, many extreme points, and sometimes, has the look of a bathtub.
• 03:22 This is often due to many outliers.
• 03:25 Leptokurtosis in Minitab is a value that is >3.
• 03:29 Excel uses a measure known as Excess Kurtosis which subtracts the value of
• 03:34 3 from the actual Kurtosis number.
• 03:36 So in Excel, leptokurtosis occurs when the number is greater than 0.
• 03:43 Mesokurtosis is the normal curve.
• 03:46 In this case, kurtosis does equal 3 in Minitab, or Excess Kurtosis =0 in Excel.
• 03:52 Now however, you may remember that when we were talking about what is normal
• 03:57 variation that we said we would consider everything from -⁠0.8,
• 04:02 to + 0.8, to be normal within the Mesokurtosis range for Excel, and so on.
• 04:07 Minitab, that would be from 2.2 to 3.8.
• 04:13 Then finally Platykurtosis, is very short tails.
• 04:17 Instead of heavy tails, they're very short.
• 04:19 And think of the platykurtosis as essentially a flat peak with sharp sides
• 04:23 on the edges.
• 04:25 There are very few outliers.
• 04:27 Kurtosis is < 3 and Excess Kurtosis is < 0.
• 04:31 This often occurs when there is either physical limits or
• 04:35 rework or tampering within the data set so that anything that was outside limits
• 04:40 was reworked to be brought back within the central zone.
• 04:45 Another type of non-normality is when there are multiple modes reflected in
• 04:49 the data.
• 04:50 This can occur when the data as collected,
• 04:52 actually has several processes represented in it.
• 04:56 If the modes are widely separated, it's easy to see,
• 04:59 then there will be several distinct peaks when we plot the data.
• 05:03 When the modes are close together, this can often take on a skewness or
• 05:07 kurtotic effect, and it's a little bit more difficult to find.
• 05:12 Finally, there's the issue of granularity.
• 05:14 This is the case when the variable data is not smooth, but
• 05:19 rather seems to come and go in steps or chunks.
• 05:22 This normally means that you have a measurement system problem.
• 05:26 The resolution is not fine enough to distinguish between the different values
• 05:30 in the distribution.
• 05:31 The other possibility is a machine function with step level changes.
• 05:36 Think of it like a gearbox on a car, and
• 05:39 the data shows that you just shifted from first to third.
• 05:44 Let's wrap this up with some principles of hypothesis testing with non-normality.
• 05:51 First, if your hypothesis test is to compare data from multiple data sets.
• 05:55 If the data is in any of them is non-normal,
• 05:57 you must use a non-normal analysis.
• 06:00 Generally speaking, the non-normal tests don't have any trouble with normal
• 06:05 data but the normal tests can have some real problems with non-normal data.
• 06:09 Non-normal data often uses the median for
• 06:12 central tendency rather than the mean that we use with normal tests.
• 06:17 This is because of a skewness effect.
• 06:19 The mean or average value will not be a good indication of central tendency.
• 06:24 Also non-normal data often relies on variance, which is the standard deviation
• 06:28 squared, rather than means or medians, when comparing datasets for similarity.
• 06:34 And finally, when working with skewed data,
• 06:37 you want a few more data points than you would have had with normal data.
• 06:41 The number of data points we need will be based upon the confidence interval,
• 06:45 which we discussed in a different lesson.
• 06:47 Start with the number of data points from the confidence interval calculation,
• 06:51 then divide that by 0.86, or it's actually probably easier to multiply it by 1.16.
• 06:56 This is the minimum number of data points needed with skewed data.
• 07:01 >> Non-normal variation occurs frequently in the real world.
• 07:05 When that happens, determine the nature of the non-normality and
• 07:10 then select the best hypothesis test for that data.

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Statistics > Machine Learning

Title: non-convex robust hypothesis testing using sinkhorn uncertainty sets.

Abstract: We present a new framework to address the non-convex robust hypothesis testing problem, wherein the goal is to seek the optimal detector that minimizes the maximum of worst-case type-I and type-II risk functions. The distributional uncertainty sets are constructed to center around the empirical distribution derived from samples based on Sinkhorn discrepancy. Given that the objective involves non-convex, non-smooth probabilistic functions that are often intractable to optimize, existing methods resort to approximations rather than exact solutions. To tackle the challenge, we introduce an exact mixed-integer exponential conic reformulation of the problem, which can be solved into a global optimum with a moderate amount of input data. Subsequently, we propose a convex approximation, demonstrating its superiority over current state-of-the-art methodologies in literature. Furthermore, we establish connections between robust hypothesis testing and regularized formulations of non-robust risk functions, offering insightful interpretations. Our numerical study highlights the satisfactory testing performance and computational efficiency of the proposed framework.

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Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .