As reviewer, editor and reader of research articles, I’m regularly annoyed by the low standards in correlation analyses. In my experience with such articles, typically:
- Pearson’s correlation, a non-robust measure of association, is used;
- R and p values are reported, but not confidence intervals;
- sample sizes tend to be small, leading to large estimation bias and inflated effect sizes in the literature;
- R values and confidence intervals are not considered when interpreting the results;
- instead, most analyses are reported as significant or non-significant (p<0.05), leading to the conclusion that an association exists or not (frequentist fallacy);
- often figures illustrating the correlations are absent;
- the explicit or implicit comparison of two correlations is done without a formal test (interaction fallacy).
To find out if my experience was in fact representative of the typical paper, I had a look at all papers published in 2017 in the European Journal of Neuroscience, where I’m a section editor. I care about the quality of the research published in EJN, so this is not an attempt at blaming a journal in particular, rather it’s a starting point to address a general problem. I really hope the results presented below will serve as a wake-up call for all involved and will lead to improvements in correlation analyses. Also, I bet if you look systematically at articles published in other neuroscience journals you’ll find the same problems. If you’re not convinced, go ahead, prove me wrong 😉
I proceeded like this: for all 2017 articles (volumes 45 and 46), I searched for “correl” and I scanned for figures of scatterplots. If either searches were negative, the article was categorised as not containing a correlation analysis, so I might have missed a few. When at least one correlation was present, I looked for these details:
- confidence interval
- p value
- consideration of effect sizes
- figure illustrating positive result
- figure illustrating negative result
- interaction test.
164 articles reported no correlation.
7 articles used regression analyses, with sample sizes as low as n=6, n=10, n=12 in 3 articles.
48 articles reported correlations.
The norm was to not report degrees of freedom or sample size along with the correlation analyses or their illustrations. In 7 articles, the sample sizes were very difficult or impossible to guess. In the others, sample sizes varied a lot, both within and between articles. To confirm sample sizes, I counted the observations in scatterplots when they were available and not too crowded – this was a tedious job and I probably got some estimations and checks wrong. Anyway, I shouldn’t have to do all these checks, so something went wrong during the reviewing process.
To simplify the presentation of the results, I collapsed the sample size estimates across articles. Here is the distribution:
The figure omits 3 outliers with n= 836, 1397, 1407, all from the same article.
The median sample size is 18, which is far too low to provide sufficiently precise estimation.
The issue with low sample sizes is made worse by the predominant use of Pearson’s correlation or the lack of consideration for the type of estimator. Indeed, 21 articles did not mention the estimator used at all, but presumably they used Pearson’s correlation.
Among the 27 articles that did mention which estimator was used:
- 11 used only Pearson’s correlation;
- 11 used only Spearman’s correlation;
- 4 used Pearson’s and Spearman’s correlations;
- 1 used Spearman’s and Kendall’s correlations.
So the majority of studies used an estimator that is well-known for its lack of robustness and its inaccurate confidence intervals and p values (Pernet, Wilcox & Rousselet, 2012).
R & p values
Most articles reported R and p values. Only 2 articles did not report R values. The same 2 articles also omitted p values, simply mentioning that the correlations were not significant. Another 3 articles did not report p values along with the R values.
Only 3 articles reported confidence intervals, without mentioning how they were computed. 1 article reported percentile bootstrap confidence intervals for Pearson’s correlations, which is the recommended procedure for this estimator (Pernet, Wilcox & Rousselet, 2012).
Consideration for effect sizes
Given the lack of interest for measurement uncertainty demonstrated by the absence of confidence intervals in most articles, it is not surprising that only 5 articles mentioned the size of the correlation when presenting the results. All other articles simply reported the correlations as significant or not.
In contrast with the absence of confidence intervals and consideration for effect sizes, 23 articles reported illustrations for positive results. 4 articles reported only negative results, which leaves us with 21 articles that failed to illustrate the correlation results.
Among the 40 articles that reported negative results, only 13 illustrated them, which suggests a strong bias towards positive results.
Finally, I looked for interaction fallacies (Nieuwenhuis, Forstmann & Wagenmakers 2011). In the context of correlation analyses, you commit an interaction fallacy when you present two correlations, one significant, the other not, implying that the 2 differ, but without explicitly testing the interaction. In other versions of the interaction fallacy, two significant correlations with the same sign are presented together, implying either that the 2 are similar, or that one is stronger than the other, without providing a confidence interval for the correlation difference. You can easily guess the other flavours…
10 articles presented only one correlation, so there was no scope for the interaction fallacy. Among the 38 articles that presented more than one correlation, only one provided an explicit test for the comparison of 2 correlations. However, the authors omitted the explicit test for their next comparison!
In conclusion, at least in 2017 EJN articles, the norm is to estimate associations using small sample sizes and a non-robust estimator, to not provide confidence intervals and to not consider effect sizes and measurement uncertainty when presenting the results. Also, positive results are more likely to be illustrated than negative ones. Finally, interaction fallacies are mainstream.
How can we do a better job?
If you want to do a correlation analysis, consider your sample size carefully to assess statistical power and even better, your long-term estimation precision. If you have a small n, I wouldn’t even look at the correlation.
Do not use Pearson’s correlation unless you have well-behaved and large samples, and you are only interested in linear relationships; otherwise explore robust measures of associations and techniques that provide valid confidence intervals (Pernet, Wilcox & Rousselet, 2012; Wilcox & Rousselet, 2018).
These details are essential in articles reporting correlation analyses:
- sample size for each correlation;
- estimator of association;
- R value;
- confidence interval;
- scatterplot illustration of every correlation, irrespective of the p value;
- explicit comparison test of all correlations explicitly or implicitly compared;
- consideration of effect sizes (R values) and their uncertainty (confidence intervals) in the interpretation of the results.
Report p values if you want but they are not essential and should not be given a special status (McShane et al. 2018).
Finally, are you sure you really want to compute a correlation?
“Why then are correlation coefficients so attractive? Only bad reasons seem to come to mind. Worst of all, probably, is the absence of any need to think about units for either variable. Given two perfectly meaningless variables, one is reminded of their meaninglessness when a regression coefficient is given, since one wonders how to interpret its value. A correlation coefficient is less likely to bring up the unpleasant truth—we think we know what r = —.7 means. Do we? How often? Sweeping things under the rug is the enemy of good data analysis. Often, using the correlation coefficient is “sweeping under the rug” with a vengeance. Being so disinterested in our variables that we do not care about their units can hardly be desirable.”
Analyzing data: Sanctification or detective work?
John W. Tukey.
American Psychologist, Vol 24(2), Feb 1969, 83-91. http://dx.doi.org/10.1037/h0027108
McShane, B.B., Gal, D., Gelman, A., Robert, C. & Tackett, J.L. (2018) Abandon Statistical Significance. arxiv.
Nieuwenhuis, S., Forstmann, B.U. & Wagenmakers, E.J. (2011) Erroneous analyses of interactions in neuroscience: a problem of significance. Nat Neurosci, 14, 1105-1107.
Pernet, C.R., Wilcox, R. & Rousselet, G.A. (2012) Robust correlation analyses: false positive and power validation using a new open source matlab toolbox. Front Psychol, 3, 606.
Rousselet, G.A. & Pernet, C.R. (2012) Improving standards in brain-behavior correlation analyses. Frontiers in human neuroscience, 6, 119.
Wilcox, R.R. & Rousselet, G.A. (2018) A Guide to Robust Statistical Methods in Neuroscience. Curr Protoc Neurosci, 82, 8 42 41-48 42 30.