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I'm trying to reconcile two seemingly opposite statements about robustness to non-normality of the Pearson's correlation test statistic (where the null means "no correlation").

This CV answer says:

Very non-robust.

This biostat handbook says:

[...] numerous simulation studies have shown that linear regression and correlation are not sensitive to non-normality; one or both measurement variables can be very non-normal, and the probability of a false positive (P<0.05, when the null hypothesis is true) is still about 0.05 (Edgell and Noon 1984, and references therein).

What am I missing?

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    $\begingroup$ The two different sources you cite seem to attach different meanings to the notion of 'departure from normality'. Is a sample obtained from a normal distribution but where a single observation is replaced by an arbitrary value considered to constitute an acceptable form of deviation from normality? If so, then clearly the biostat handbook (and the referenced Edgell and Noon paper) can quit easily be shown to be wrong. $\endgroup$ – user603 Apr 27 '16 at 11:23
  • $\begingroup$ @user603 Regression does not in any way require normal distribution of either or both variables: the assumption is built right into the mathematical formalism: $Y=\beta_{0}+\beta_{X}X+\varepsilon$ where $\varepsilon \sim \mathcal{N}(0,\sigma)$. Note that last part: it's the residuals not the variables that are distributed normally. Them empirically verify: (1) simulate $X$ using a uniform distribution from, oh, say 0 to 100; (2) simulate $Y=3 + 0.5 \times X+\mathcal{N}(0,1)$; (3) regress $Y$ on $X$ and recover $\beta_{0}\approx 3$,$\beta_{X}\approx 0.5$. Now view the histograms of $X$ and $Y$. $\endgroup$ – Alexis Apr 27 '16 at 18:47
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    $\begingroup$ @Alexis: I am not sure I understand the connection between your comment and mine. I do not think I claimed anything about regression (or normality) $\endgroup$ – user603 Apr 27 '16 at 18:53
  • $\begingroup$ @user603 Pretty sure you made a claim about the Edgell and Noon quote—particularly this bit: "numerous simulation studies have shown that linear regression and correlation are not sensitive to non-normality; one or both measurement variables can be very non-normal"— which is about precisely that. $\endgroup$ – Alexis Apr 28 '16 at 2:08
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The Edgell and Noon paper got it wrong.

Background

The paper describes result from simulated datasets $(x_i,y_i)$ with independent coordinates drawn from Normal, Exponential, Uniform, and Cauchy distributions. (Although it reports two "forms" of the Cauchy, they differed only in how the values were generated, which is an irrelevant distraction.) The dataset sizes $n$ ("sample size") ranged from $5$ to $100$. For each dataset the Pearson sample correlation coefficient $r$ was computed, converted into a $t$ statistic via

$$t = r \sqrt{\frac{n-2}{1-r^2}},$$

(see Equation (1)), and referred that to a Student $t$ distribution with $n-2$ degrees of freedom using a two-tailed calculation. The authors conducted $10,000$ independent simulations for each of the $10$ pairs of these distribution and each sample size, producing $10,000$ $t$ statistics in each. Finally, they tabulated the proportion of $t$ statistics that appeared to be significant at the $\alpha=0.05$ level: that is, the $t$ statistics in the outer $\alpha/2 = 0.025$ tails of the Student $t$ distribution.

Discussion

Before we proceed, notice that this study looks only at how robust a test of zero correlation might be to non-normality. That's not an error, but it's an important limitation to keep in mind.

There is an important strategic error in this study and a glaring technical error.

The strategic error is that these distributions aren't that non-normal. Neither the Normal nor the Uniform distributions are going to cause any trouble for correlation coefficients: the former by design and the latter because it cannot produce outliers (which is what causes the Pearson correlation not to be robust). (The Normal had to be included as a reference, though, to make sure everything was working properly.) None of these four distributions are good models for common situations where the data might be "contaminated" by values from a distribution with a different location altogether (such as when the subjects really come from distinct populations, unknown to the experimenter). The most severe test comes from the Cauchy but, because it is symmetric, does not probe the most likely sensitivity of the correlation coefficient to one-sided outliers.

The technical error is that the study did not examine the actual distributions of the p-values: it looked solely at the two-sided rates for $\alpha=0.05$.

(Although we can excuse much that happened 32 years ago due to limitations in computing technology, people were routinely examining contaminated distributions, slash distributions, Lognormal distributions, and other more serious forms of non-normality; and it has been routine for even longer to explore a wider range of test sizes rather than limiting studies to just one size.)

Correcting the Errors

Below, I provide R code that will completely reproduce this study (in less than a minute of computation). But it does something more: it displays the sample distributions of the p-values. This is quite revealing, so let's just jump in and look at those histograms.

First, here are histograms of large samples from the three distributions I looked at, so you can get a sense of how they are non-Normal.

Figure 1

The Exponential is skewed (but not terribly so); the Cauchy has long tails (in fact, some values out into the thousands were excluded from this plot so you can see its center); the Contaminated is a standard Normal with a 5% mixture of a standard Normal shifted out to $10$. They represent forms of non-Normality frequently encountered in data.

Because Edgell and Noon tabulated their results in rows corresponding to pairs of distributions and columns for sample sizes, I did the same. We don't need to look at the full range of sample sizes they used: the smallest ($5$), largest ($100$), and one intermediate value ($20$) will do fine. But instead of tabulating tail frequencies, I have plotted the distributions of the p-values.

Figure 2

Ideally, the p-values will have uniform distributions: the bars should all be close to a constant height of $1$, shown with a dashed gray line in each plot. In these plots there are 40 bars, at a constant spacing of $0.025$ A study of $\alpha=0.05$ will focus on the average height of the leftmost and rightmost bar (the "extreme bars"). Edgell and Noon compared these averages to the ideal frequency of $0.05$.

Because the departures from uniformity are prominent, not much commentary is needed, but before I provide some, look for yourself at the rest of the results. You can identify the sample sizes in the titles--they all run $5-20-100$ across each row--and you can read the pairs of distributions in the subtitles beneath each graphic.

Figure 3

Figure 4

What should strike you most is how different the extreme bars are from the rest of the distribution. A study of $\alpha=0.05$ is extraordinarily special! It doesn't really tell us how well the test will perform a other sizes; in fact, the results for $0.05$ are so special that they will deceive us concerning the characteristics of this test.

Second, notice that when the Contaminated distribution is involved--with its tendency to produce only high outliers--the distribution of p-values becomes asymmetric. One bar (which would be used for testing for positive correlation) is extremely high while its counterpart at the other end (which would be used for testing for negative correlation) is extremely low. On average, though, they nearly balance out: two huge errors cancel!

It is particularly alarming that the problems tend to get worse with larger sample sizes.

I also have some concerns about the accuracy of the results. Here are the summaries from $100,000$ iterations, ten times more than Edgell and Noon did:

                                5      20     100
Exponential-Exponential   0.05398 0.05048 0.04742
Exponential-Cauchy        0.05864 0.05780 0.05331
Exponential-Contaminated  0.05462 0.05213 0.04758
Cauchy-Cauchy             0.07256 0.06876 0.04515
Cauchy-Contaminated       0.06207 0.06366 0.06045
Contaminated-Contaminated 0.05637 0.06010 0.05460

Three of these--the ones not involving the Contaminated distribution--reproduce parts of the paper's table. Although they lead qualitatively to the same (bad) conclusions (namely, that these frequencies look pretty close to the target of $0.05$) they differ enough to call into question either my code or the paper's results. (The precision in the paper will be approximately $\sqrt{\alpha(1-\alpha)/n} \approx 0.0022$, but some of these results differ from the paper's by many times that.)

Conclusions

By failing to include non-Normal distributions that are likely to cause problems for correlation coefficients, and by not examining the simulations in detail, Edgell and Noon failed to identify a clear lack of robustness and missed an opportunity to characterize its nature. That they found robustness for two-sided tests at the $\alpha=0.05$ level appears to be almost purely an accident, an anomaly that is not shared by tests at other levels.


R Code

#
# Create one row (or cell) of the paper's table.
#
simulate <- function(F1, F2, sample.size, n.iter=1e4, alpha=0.05, ...) {
  p <- rep(NA, length(sample.size))
  i <- 0
  for (n in sample.size) {
    #
    # Create the data.
    #
    x <- array(cbind(matrix(F1(n*n.iter), nrow=n),
                     matrix(F2(n*n.iter), nrow=n)), dim=c(n, n.iter, 2))
    #
    # Compute the p-values.
    #
    r.hat <- apply(x, 2, cor)[2, ]
    t.stat <- r.hat * sqrt((n-2) / (1 - r.hat^2))
    p.values <- pt(t.stat, n-2)
    #
    # Plot the p-values.
    #
    hist(p.values, breaks=seq(0, 1, 1/40), freq=FALSE,
         xlab="p-values",
         main=paste("Sample size", n), ...)
    abline(h=1, lty=3, col="#a0a0a0")
    #
    # Store the frequency of p-values less than `alpha` (two-sided).
    #
    i <- i+1
    p[i] <- mean(1 - abs(1 - 2*p.values) <= alpha)
  }
  return(p)
}
#
# The paper's distributions.
#
distributions <- list(N=rnorm,
                      U=runif,
                      E=rexp,
                      C=function(n) rt(n, 1)
)
#
# A slightly better set of distributions.
#
# distributions <- list(Exponential=rexp,
#                       Cauchy=function(n) rt(n, 1),
#                       Contaminated=function(n) rnorm(n, rbinom(n, 1, 0.05)*10))
#
# Depict the distributions.
#
par(mfrow=c(1, length(distributions)))
for (s in names(distributions)) {
  x <- distributions[[s]](1e5)
  x <- x[abs(x) < 20]
  hist(x, breaks=seq(min(x), max(x), length.out=60),main=s, xlab="Value")
}
#
# Conduct the study.
#
set.seed(17)
sample.sizes <- c(5, 10, 15, 20, 30, 50, 100)
#sample.sizes <- c(5, 20, 100)

results <- matrix(numeric(0), nrow=0, ncol=length(sample.sizes))
colnames(results) <- sample.sizes
par(mfrow=c(2, length(sample.sizes)))
s <- names(distributions)
for (i1 in 1:length(distributions)) {
  s1 <- s[i1]
  F1 <- distributions[[s1]]
  for (i2 in i1:length(distributions)) {
    s2 <- s[i2]
    F2 <- distributions[[s2]]
    title <- paste(s1, s2, sep="-")
    p <- simulate(F1, F2, sample.sizes, sub=title)
    p <- matrix(p, nrow=1)
    rownames(p) <- title
    results <- rbind(results, p)
  }
}
#
# Display the table.
#
print(results)

Reference

Stephen E. Edgell and Sheila M. Noon, Effect of Violation of Normality on the $t$ Test of the Correlation Coefficient. Psychological Bulletin 1984, Vol., 95, No. 3, 576-583.

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  • $\begingroup$ Wow. So not only the two authors of the paper, but many people working in the field today (including the author of the blog and Handbook of Biostatistics I mentioned) have a really unfortunate misunderstanding of the technique that is really critical to their research. $\endgroup$ – max Apr 27 '16 at 17:36
  • $\begingroup$ So long as the research only involves one such test in each published article (so that multiple-comparisons corrections are unneeded), there's no chance of severe outliers, and $\alpha=0.05$ is the significance threshold, you might be OK. There are good reasons, though, that most textbooks on multiple regression and correlation written since the early 1980's have included major sections on identifying, detecting, and coping with non-normality. In fact, whole subfields of statistics (robust estimation and EDA) that were developed to deal with this situation have come and gone in the meantime. $\endgroup$ – whuber Apr 27 '16 at 17:42
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    $\begingroup$ +1 This is a great answer. A small nitpick: you saying that "a study of α=0.05 is extraordinarily special!" makes the impression that had the authors considered another $\alpha$, they would have observed radically different results (even following the identical methodology). But it's not clear from your histograms that this would be the case for e.g. $\alpha=0.01$ or $0.001$ or other common values, because there is not enough resolution. If the outcome for these alphas is around the same (actual test sizes from 0.4 to 0.8) then $\alpha=0.05$ is perhaps not "extraordinarily" special. $\endgroup$ – amoeba says Reinstate Monica Apr 27 '16 at 20:53
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    $\begingroup$ @amoeba You are quite correct: that's a good set of observations. I believe you will find, though, that the sharp trends emerging near the tails at this resolution become even stronger when shown at higher resolutions. Of course this will require larger simulations--at least 20 times larger. That's feasible for any interested person to carry out. $\endgroup$ – whuber Apr 27 '16 at 21:16
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Since whuber has given a comprehensive analysis of the behavior of the distributions of p-values under a null of zero-correlation, I'll focus my comments elsewhere.

  1. Robustness in relation to hypothesis tests doesn't only mean level-robustness (getting close to the desired significance level). Besides only looking at one level and only at two-sided tests, the study appears to have ignored impact on power. There's no much point saying that you're keeping close to a 5% rejection rate under the null if you also end up with a 5% rejection rate* for large deviations from the null.

    * (or maybe worse, if the test ends up biased under the non-normal distributions for some alternatives)

    Investigating power is considerably more involved. For a start, with these distributions you'd have to be looking at specifying some copula or copulas, presumably with close to a linear relationship in the untransformed variables, and certainly with close to some specified value for the population correlation coefficient. You'll have to look at several effect sizes (at least), and possibly both negative and positive dependence.

    Nevertheless, if one is to understand the properties of inference with the test in these situations, one cannot ignore the potential impact on power.

  2. It would seem odd to discuss that particular test of the Pearson correlation without examining alternative tests - for example, permutation tests of the Pearson correlation, rank tests like Kendall's tau and Spearman's rho (which not only have good performance when the normal assumptions hold, but which also have direct relevance to the issue with copulas needed for a power study that I mentioned before), perhaps robustified versions of the correlation coefficient, possibly also bootstrap tests.

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