Monthly Archives: July 2019

The bootstrap-t technique

There are many types of bootstrap methods, but for most applications, two methods are most common: the percentile bootstrap, presented in an earlier post, and the bootstrap-t technique—also known as the percentile-t bootstrap or the studentized bootstrap (Efron & Tibshirani, 1994; Wilcox, 2017)​. For inferences on the population mean, the standard ​T-test and the percentile bootstrap can give unsatisfactory results when sampling from skewed distributions, especially when sample size is small. To illustrate the problem with the t-test, imagine that we sample from populations of increasing skewness.

Probability density functions for ​g&h​ distributions. Parameter ​g​ varies from 0 to 1. Parameter ​h=​0.

Here we use ​g&h​ distributions, in which parameter ​g​ controls the skewness, and parameter ​h​ controls the thickness of the tails—a normal distribution is obtained by setting ​g​=​h​=0 (Hoaglin, 1985; Yan & Genton, 2019)​. If we take many samples of size n=30 from these distributions, and for each sample we compute a ​T​ value, using the population mean as the null value, we obtain progressively more negatively skewed ​T​ value sampling distributions.

Sampling distributions of ​T​ values for different ​g​ values. Results are based on a simulation with 50,000 iterations and samples of size n=30.​

However, when we perform a ​T​-test, the ​T​ values are assumed to be symmetric, irrespective of sample size. This assumption leads to incorrect confidence intervals (CIs). The idea behind the bootstrap-t technique is to use the bootstrap (sampling with replacement) to compute a data-driven T​ distribution. In the presence of skewness, this ​T​ distribution could be skewed, as suggested by the data. Then, the appropriate quantile of the bootstrap ​T distribution is plugged into the standard CI equation to obtain a parametric bootstrap CI.

Bootstrap-t procedure

Let’s illustrate the procedure for a CI for the population mean. We start with a sample of 30 observations from a ​g&h​ distribution with ​g​=1 and ​h=​ 0.

Sample of size n=30 from a ​g&h​ distribution with ​g=1 and ​h​=0. The vertical line indicates the sample mean.

In a first step, we centre the distribution: for inferences on the mean, we subtract the mean from each observation in the sample, so that the mean of the centred distribution is now zero. This is a way to create a data-driven null distribution, in which there is no effect (the mean is zero), but the shape of the distribution and the absolute distance among observations are unaffected, as shown in the next figure. For inferences on a trimmed mean, we subtract the trimmed mean from each observation, so that the centred distribution now has a trimmed mean of zero.

Same distribution as in the previous figure, but the distribution has been mean centred, so that the sample mean is now zero.

In the next step, we sample with replacement from the centred distribution many times, and for each random sample we compute a ​T​ value. That way, we obtain a bootstrap distribution of ​T​ values expected by random sampling, under the hypothesis that the population has a mean (or trimmed mean) of zero, given the distribution of the data. Then, we use some quantile of the bootstrap ​T distribution in the standard CI equation. (Note that for trimmed means, the T-test equation is adjusted—see Tukey & McLaughlin, 1963).

5,000 bootstrap ​T​ values obtained by sampling with replacement from the mean centred data. In the asymmetric bootstrap-t technique, the quantiles (red vertical lines) of that distribution of ​T​ values are used to define the CI bounds. The insets contain the formulas for the lower (CI​lo)​ and upper bounds (CI​up)​ of the CI. Note that the lower ​T​ quantile is used to compute the upper bound (this not an error). In the symmetric bootstrap-t technique, one quantile of the distribution of absolute ​T​ values is used to define the CI bounds.​

Because the bootstrap distribution is potentially asymmetric, we have two choices of quantiles: for a 95% CI, either we use the 0.025 and the 0.975 quantiles of the signed ​T​ values to obtain a potentially asymmetric CI, also called an equal-tailed CI, or we use the 0.95 quantile of the absolute ​T​ values, thus leading to a symmetric CI.

In our example, for the mean the symmetric CI is [-0.4, 1.62] and the asymmetric CI is [0.08, 1.87]. If instead we use the 20% trimmed mean, the symmetric CI is [-0.36, 0.59] and the asymmetric CI is [-0.3, 0.67] (see Rousselet, Pernet & Wilcox, 2019). So clearly, confidence intervals can differ a lot depending on the estimator and method we use. In other words, a 20% trimmed mean is not a substitute for the mean, it asks a different question about the data.

Bootstrap samples

Why does the bootstrap-t approach work better than the standard ​T-test CI? Imagine we take multiple samples of size n=30 from a ​g&h​ distribution with ​g=​1 and ​h​=0.

Comparison of ​T​ distributions for ​g​=1 & h=0: the theoretical ​T distribution in red​ is the one used in the T-​test, the empirical ​T​ distribution in black was obtained by sampling with replacement multiple times from the g&h distribution. The red and black vertical lines indicate the ​T​ quantiles for a 95% CI. The grey lines show examples of 20 bootstrap sampling distributions, based on samples of size n=30 and 5000 bootstrap samples.

In the figure above, the standard ​T​-test assumes the sampling distribution in red, symmetric around zero. As we considered above, the sampling distribution is actually asymmetric, with negative skewness, as shown in black. However, the black empirical distribution is unobservable, unless we can perform thousands of experiments. So, with the bootstrap, we try to estimate this correct, yet unobservable, sampling distribution. The grey curves show examples of 20 simulated experiments: in each experiment, a sample of 30 observations is drawn, and then 5,000 bootstrap ​T​ values are computed. The resulting bootstrap sampling distributions are negatively skewed and are much closer to the empirical distribution in black than the theoretical symmetric distribution in red. Thus, it seems that using data-driven ​T​ distributions could help achieve better CIs than if we assumed symmetry.

How do these different methods perform? To find out we carry out simulations in which we draw samples from​ g&h distributions with the​ g​ parameter varying from 0 to 1, keeping ​h=0. For each sample, we compute a one-sample CI using the standard ​T-​ test, the two bootstrap-t methods just described (asymmetric and symmetric), and the percentile bootstrap. When estimating the population mean, for all four methods, coverage goes down with skewness.

Confidence interval coverage for the 4 methods applied to the mean. Results of a simulation with 20,000 iterations, sample sizes of n=30, and 599 bootstrap samples.​ You can see what happens for the 10% trimmed mean and the 20% trimmed mean in Rousselet, Pernet & Wilcox, 2019.

Among the parametric methods, the standard ​T-​test is the most affected by skewness, with coverage less than 90% for the most skewed condition. The asymmetric bootstrap-t CI seems to perform the best. The percentile bootstrap performs the worst in all situations, and has coverage systematically below 95%, including for normal distributions.

In addition to coverage, it is useful to consider the width of the CIs from the different techniques.

Confidence interval median width, based on the same simulation reported in the previous figure.

The width of a CI is its upper bound minus its lower bound. For each combination of parameters. the results are summarised by the median width across simulations. At low levels of asymmetry, for which the three parametric methods have roughly 95% coverage, the CIs also tend to be of similar widths. As asymmetry increases, all methods tend to produce larger CIs, but the ​T-test produces CIs that are too short, a problem that stems from the symmetric theoretical ​T​ distribution, which assumes T​ ​values too small. Compared to the parametric approaches, the percentile bootstrap produces the shortest CIs for all ​g​ values.

Confidence intervals: a closer look

We now have a closer look at the confidence intervals in the different situations considered above. We use a simulation with 20,000 iterations, sample size n=30, and 599 bootstrap samples.

Under normality

As we saw above, under normality the coverage is close to nominal (95%) for every method, although coverage for the percentile bootstrap is slightly too low, at 93.5%. Out of 20,000 simulated experiments, about 1,000 CI (roughly 5%) did not include the population value. About the same number of CIs were shifted to the left and to the right of the population value for all methods, and the CIs were of similar sizes:

We observed the same behaviour for several parametric methods in a previous post. Now, what happens when we sample from a skewed population?

In the presence of skewness (g=1, h=0)

Coverage is lower than the expected 95% for all methods. Coverage is about 88% for the standard and percentile bootstrap CIs, 92.3% for the asymmetric bootstrap-t CIs, and 91% for the symmetric bootstrap-t CIs. As we saw above, CIs are larger for the bootstrap-t CIs relative to the standard and percentile bootstrap CIs. CIs that did not include the population value tended to be shifted to the left of the population value, and more so for the standard CIs and the bootstrap-t symmetric CIs.

So when making inferences about the mean using the standard T-test, our CI coverage is lower than expected, and we are likely to underestimate the population value (the sample mean is median biased—Rousselet & Wilcox, 2019).

Relative to the other methods, the asymmetric bootstrap-t CIs are more evenly distributed on either side of the population and the right shifted CIs tends to be much larger and variable. The difference with the symmetric CIs is particularly striking and suggests that the asymmetric CIs could be misleading in certain situations. This intuition is confirmed by a simulation in which outliers are likely (h=0.2).

In the presence of skewness and outliers (g=1, h=0.2)

In the presence of outliers, the patterns observed in the previous figure are exacerbated. Some of the percentile bootstrap and asymmetric bootstrap-t intervals are ridiculously wide (x axis is truncated).

In such situation, inferences on trimmed means would greatly improve performance over the mean.


As we saw in a previous post, a good way to handle skewness and outliers is to make inferences about the population trimmed means. For instance, trimming 20% is efficient in many situations, even when using parametric methods that do not rely on the bootstrap. So what’s the point of the bootstrap-t? From the examples above, the bootstrap-t can perform much better than the standard Student’s approach and the percentile bootstrap when making inferences about the mean. So, in the presence of skewness and the population mean is of interest, the bootstrap-t is highly recommended. Whether to use the symmetric or asymmetric approach is not completely clear based on the literature (Wilcox, 2017). Intuition suggests that the asymmetric approach is preferable but our last example suggests that could be a bad idea when making inferences about the mean.

Symmetric or not, the bootstrap-t confidence intervals combined with the mean do not necessarily deal with skewness as well as other methods combined with trimmed means. But the bootstrap-t can be used to make inferences about trimmed means too! So which combination of approaches should we use? For instance, we could make inferences about the mean, the 10% trimmed mean or the 20% trimmed mean, in conjunction with a non-bootstrap parametric method, the percentile bootstrap or the bootstrap-t. We saw that for the mean, the bootstrap-t method is preferable in the presence of skewness. For inferences about trimmed means, the percentile bootstrap works well when trimming 20%. If we trim less, then the other methods should be considered, but a blanket recommendation cannot be provided. The choice of combination can also depend on the application. For instance, to correct for multiple comparisons in brain imaging analyses, cluster-based statistics are strongly recommended, in which case a bootstrap-t approach is very convenient. And the bootstrap-t is easily extended to factorial ANOVAs (Wilcox, 2017; Field & Wilcox, 2017).

What about the median? The bootstrap-t should not be used to make inferences about the median (50% trimming), because the standard error is not estimated correctly. Special parametric techniques have been developed for the median (Wilcox, 2017). The percentile bootstrap also works well for the median and other quantiles in some situations, providing sample sizes are sufficiently large (Rousselet, Pernet & Wilcox, 2019).


Efron, Bradley, and Robert Tibshirani. An Introduction to the Bootstrap. Chapman and Hall/CRC, 1994.

Field, Andy P., and Rand R. Wilcox. ‘Robust Statistical Methods: A Primer for Clinical Psychology and Experimental Psychopathology Researchers’. Behaviour Research and Therapy 98 (November 2017): 19–38.

Hesterberg, Tim C. ‘What Teachers Should Know About the Bootstrap: Resampling in the Undergraduate Statistics Curriculum’. The American Statistician 69, no. 4 (2 October 2015): 371–86.

Hoaglin, David C. ‘Summarizing Shape Numerically: The g-and-h Distributions’. In Exploring Data Tables, Trends, and Shapes, 461–513. John Wiley & Sons, Ltd, 1985.

Rousselet, Guillaume A., Cyril R. Pernet, and Rand R. Wilcox. ‘A Practical Introduction to the Bootstrap: A Versatile Method to Make Inferences by Using Data-Driven Simulations’. Preprint. PsyArXiv, 27 May 2019.

Rousselet, Guillaume A., and Rand R. Wilcox. ‘Reaction Times and Other Skewed Distributions: Problems with the Mean and the Median’. Preprint. PsyArXiv, 17 January 2019.

Tukey, John W., and Donald H. McLaughlin. ‘Less Vulnerable Confidence and Significance Procedures for Location Based on a Single Sample: Trimming/Winsorization 1’. Sankhyā: The Indian Journal of Statistics, Series A (1961-2002) 25, no. 3 (1963): 331–52.

Wilcox, Rand R. Introduction to Robust Estimation and Hypothesis Testing. 4th edition. Academic Press, 2017.

Wilcox, Rand R., and Guillaume A. Rousselet. ‘A Guide to Robust Statistical Methods in Neuroscience’. Current Protocols in Neuroscience 82, no. 1 (2018): 8.42.1-8.42.30.

Yan, Yuan, and Marc G. Genton. ‘The Tukey G-and-h Distribution’. Significance 16, no. 3 (2019): 12–13.

Comparing two independent Pearson’s correlations: confidence interval coverage

This post looks at the coverage of confidence intervals for the difference between two independent correlation coefficients. Previously, we saw how the standard Fisher’s r-to-z transform can lead to inflated false positive rates when sampling from non-normal bivariate distributions and the population correlation differs from zero. In this post, we look at a complementary perspective: using simulations, we’re going to determine how often confidence intervals include the population difference. As we saw in our previous post, because we compute say 95% confidence intervals does not mean that over the long run, 95% of such confidence intervals will include the population we’re trying to estimate. In some situations, the coverage is much lower than expected, which means we might fool ourselves more often that we thought (although in practice in most discussions I’ve ever read, authors behave as if their 95% confidence intervals were very narrow 100% confidence intervals — but that’s another story).

We look at confidence interval coverage for the difference between Pearsons’ correlations using Zou’s method (2007) and a modified percentile bootstrap method (Wilcox, 2009). We do the same for the comparison of Spearmans’ correlations using the standard percentile bootstrap. We used simulations with 4,000 iterations. Sampling is from bivariate g & h distributions (see illustrations here).

We consider 4 cases:

  • g = h = 0, difference = 0.1, vary rho
  • g = 1, h = 0, difference = 0.1, vary rho
  • rho = 0.3, difference = 0.2, vary g, h = 0
  • rho = 0.3, difference = 0.2, vary g, h = 0.2

g = h = 0, difference = 0.1, vary rho

That’s the standard normal bivariate distribution. Group 1 has values of rho1 = 0 to 0.8, in steps of 0.1. Group 2 has values of rho2 = rho1 + 0.1.

For normal bivariate distributions, coverage is at the nominal level for all methods, sample sizes and population correlations. (Here I only considered sample sizes of 50+ because otherwise power is far too low, so there is no point.)

The width of the CIs (upper bound minus lower bound) decreases with rho and with sample size. That’s expected from the sampling distributions of correlation coefficients

When CIs do not include the population value, are they located to the left or the right of the population? In the figure below, negative values indicate a preponderance of left shifts, positive values a preponderance of right shifts. A value of 1 = 100% right shifts, -1 = 100% left shifts. For Pearson, CIs not including the population value tend to be located evenly to the left and right of the population. For Spearman, there is a preponderance of left shifted CIs for rho1 = 0.8. This left shift implies a tendency to over-estimate the difference (the difference group 1 minus group 2 is negative).

g = 1, h = 0, vary rho

What happens when we sample from a skewed distribution?

The coverage is lower than the expected 95% for Zou’s method and the discrepancy worsens with increasing rho1 and with increasing sample size. The percentile bootstrap does a much better job. Spearman’s combined with the percentile bootstrap is spot on.

For CIs that did not include the population value, the pattern of shifts varies as a function of rho. For Pearson, CIs are more likely to be located to the right of the population (under-estimation of the population value or wrong sign) for rho = 0, whereas for rho = 0.8, CIs are more likely to be located to the left. Spearman + bootstrap produces much more balanced results.

To investigate the asymmetry, we look at CIs for g=1, a sample size of n = 200 and the extremes of the distributions, rho1 = 0 and rho2 = 0.8. The figure below shows the preponderance of right shifted CIs for the two Pearson methods. The vertical line marks the population difference of -0.1.

For rho1 = 0.8, the pattern changes to a preponderance of left shifts for all methods, which means that the CIs tended to over-estimate the population difference. CIs for differences between Spearman’s correlations were quite smaller than Pearson’s ones though, thus showing less bias and less uncertainty.  

rho=0.3, diff=0.2, vary g, h = 0

For another perspective on the three methods, we now look at a case with:

  • group 1: rho1 = 0.3
  • group 2: rho2 = 0.5
  • we vary g from 0 to 1.

For Pearson + Zou, coverage progressively decreases with increasing g, and to a much more limited extent with increasing sample size. Pearson + bootstrap is much more resilient to changes in g. And Spearman + bootstrap just doesn’t care about asymmetry!

The better coverage of Pearson + bootstrap seems to be achieved by producing wider CIs.

Matters only get’s worse for Pearson + Zou when outliers are likely (see notebook on GitHub).


Based on this new comparison of the 3 methods, I’d argue again that Spearman + bootstrap should be preferred over the two Pearson methods. But if the goal is to assess linear relationships, then Pearson + bootstrap is preferable to Zou’s method. I’ll report on other methods in another post.


Comparison of correlation coefficients

Zou, Guang Yong. Toward Using Confidence Intervals to Compare Correlations. Psychological Methods 12, no. 4 (2007): 399–413.

Wilcox, Rand R. Comparing Pearson Correlations: Dealing with Heteroscedasticity and Nonnormality. Communications in Statistics – Simulation and Computation 38, no. 10 (1 November 2009): 2220–34.

Baguley, Thom. Comparing correlations: independent and dependent (overlapping or non-overlapping)

Diedenhofen, Birk, and Jochen Musch. Cocor: A Comprehensive Solution for the Statistical Comparison of Correlations. PLoS ONE 10, no. 4 (2 April 2015).

g & h distributions

Hoaglin, David C. Summarizing Shape Numerically: The g-and-h Distributions. In Exploring Data Tables, Trends, and Shapes, 461–513. John Wiley & Sons, Ltd, 1985.

Yan, Yuan, and Marc G. Genton. The Tukey G-and-h Distribution. Significance 16, no. 3 (2019): 12–13.

When is a 95% confidence interval not a 95% confidence interval?

In previous posts, we saw how skewness and outliers can affect false positives (type I errors) and true positives (power) in one-sample tests. In particular, when making inferences about the population mean, skewness tends to inflate false positives, and skewness and outliers can destroy power. Here we investigate a complementary perspective, looking at how confidence intervals are affected by skewness and outliers.

Spoiler alert: 95% confidence intervals most likely do not have a coverage of 95%. In fact, I’ll show you an example in which a 95% CI for the mean has an 80% coverage…

The R code for this post is on GitHub.

Back to the title of the post. Seems like a weird question? Not if we consider the definition of a confidence interval (CI). Let say we conduct an experiment to estimate quantity x from a sample, where x could be the median or the mean for instance. Then a 95% CI for the population value of x refers to a procedure whose behaviour is defined in the long-run: CIs computed in the same way should contain the population value in 95% of exact replications of the experiment. For a single experiment, the particular CI does or does not contain the population value, there is no probability associated with it. A CI can also be described as the interval compatible with the data given our model — see definitions and common misinterpretations in Greenland et al. (2016).

So 95% refers to the (long-term) coverage of the CI; the exact values of the CI bounds vary across experiments. The CI procedure is associated with a certain coverage probability, in the long-run, given the model. Here the model refers to how we collected data, data cleaning procedures (e.g. outlier removal), assumptions about data distribution, and the methods used to compute the CI. Coverage can differ from the expected one if model assumptions are violated or the model is just plain wrong.

Wrong models are extremely common, for instance when applying a standard t-test CI to percent correct data (Kruschke, 2014; Jaeger, 2008) or Likert scale data (Bürkner & Vuorre, 2019; Liddell & Kruschke, 2019). 

For continuous data, CI coverage is not at the expected, nominal level, for instance when the model expects symmetric distributions and we’re actually sampling from skewed populations (which is the norm, not the exception, when we measure sizes, durations, latencies etc.). Here we explore this issue using g & h distributions that let us manipulate asymmetry.

Illustrate g & h distributions

All g & h distributions have a median of zero. The parameter g controls the asymmetry of the distribution, while the parameter h controls the thickness of the tails (Hoaglin, 1985; Yan & Genton, 2019). Let’s look at some illustrations to make things clear.

Examples in which we vary g from 0 to 1.

As g increases, the asymmetry of the distributions increases. Using negative g values would produce distributions with negative skewness.

Examples in which we vary h from 0 to 0.2.

As h increases, the tails are getting thicker, which means that outliers are more likely. 

Test with normal (g=h=0) distribution

Let’s run simulations to look at coverage probability in different situations and for different estimators. First, we sample with replacement from a normal population (g=h=0) 20,000 times (that’s 20,000 simulated experiments). Each sample has size n=30. Confidence intervals are computed for the mean, the 10% trimmed mean ™, the 20% trimmed mean and the median using standard parametric methods (see details in the code on GitHub, and references for equations in Wilcox & Rousselet, 2018). The trimmed mean and the median are robust measures of central tendency. To compute a 10% trimmed mean, observations are sorted, the 10% lowest and 10% largest values are discarded (20% in total), and the remaining values are averaged. In this context, the mean is a 0% trimmed mean and the median is a 50% trimmed mean. Trimming the data attenuates the influence of the tails of the distributions and thus the effects of asymmetry and outliers on confidence intervals.

First we look at coverage for the 4 estimators: we look at the proportion of simulated experiments in which the CIs included the population value for each estimator. As expected for the special case of a normal distribution, the coverage is close to nominal (95%) for every method:

Mean 10% tm 20% tm Median
0.949 0.948 0.943 0.947

In addition to coverage, we also look at the width of the CIs (upper bound minus lower bound). Across simulations, we summarise the results using the median width. CIs tends to be larger for trimmed means and median relative to the mean, which implies lower power under normality for these methods (Wilcox & Rousselet, 2018). 

Mean 10% tm 20% tm Median
0.737 0.761 0.793 0.889

For CIs that did not include the population, the distribution is fairly balanced between the left and the right of the population. To see this, I computed a shift index: if the CI was located to the left of the population value, it receives a score of -1, when it was located to the right, it receives a score of 1. The shift index was then computed by averaging the scores only for those CI excluding the population.

Mean 10% tm 20% tm Median
0.046 0.043 0.009 0.013

Illustrate CIs that did not include the population

Out of 20,000 simulated experiments, about 1,000 CI (roughly 5%) did not include the population value for each estimator. About the same number of CIs were shifted to the left and to the right of the population value, which is illustrated in the next figure. In each panel, the vertical line marks the population value (here it’s zero in all conditions because the population is symmetric). The CIs are plotted in the order of occurrence in the simulation. So the figure shows that if we miss the population value, we’re as likely to overshoot than undershoot our estimation.

Across panels, the figure also shows that the more we trim (10%, 20%, median) the larger the CIs get. So for a strictly normal population, we more precisely estimate the mean than trimmed means and the median.

Test with g=1 & h=0 distribution

What happens for a skewed population? Three things happen for the mean:

  • coverage goes down
  • width increases
  • CIs not including the population value tend to be shifted to the left (negative average shift values)

The same effects are observed for the trimmed means, but less so the more we trim, because trimming alleviates the effects of the tails.

Measure Mean 10% tm 20% tm Median
Coverage 0.880 0.936 0.935 0.947
Width 1.253 0.956 0.879 0.918
Shift -0.962 -0.708 -0.661 0.017
# left 2350 1101 1084 521
# right 45 188 221 539

Illustrate CIs that did not include the population

The figure illustrates the strong imbalance between left and right CI shifts. If we try to estimate the mean of a skewed population, our CIs are likely to miss it more than 5% of the time, and when that happens, the CIs are most likely to be shifted towards the bulky part of the distribution (here the left for a right skewed distribution). Also, the right shifted CIs vary a lot in width and can be very large.

As we trim, the imbalance is progressively resolved. With 20% trimming, when CIs do not contain the population value, the distribution of left and right shifts is more balanced, although with still far more left shifts. With the median we have roughly 50% left / 50% right shifts and CIs are narrower than for the mean.

Test with g=1 & h=0.2 distribution

What happens if we sample from a skewed distribution (g=1) in which outliers are likely (h=0.2)?

Measure Mean 10% tm 20% tm Median
Coverage 0.801 0.934 0.936 0.947
Width 1.729 1.080 0.934 0.944
Shift -0.995 -0.797 -0.709 0.018
# left 3967 1194 1086 521
# right 9 135 185 540

The results are similar to those observed for h=0, only exacerbated. Coverage for the mean is even lower, CIs are larger, and the shift imbalance even more severe. I have no idea how often such a situation occur, but I suspect if you study clinical populations that might be rather common. Anyway, the point is that it is a very bad idea to assume the distributions we study are normal, apply standard tools, and hope for the best. Reporting CIs as 95% or some other value, without checking, can be very misleading.

Simulations in which we vary g

We now explore CI properties as a function of g, which we vary from 0 to 1, in steps of 0.1. The parameter h is set to 0 (left column of next figure) or 0.2 (right column). Let’s look at column A first (h=0). For the median, coverage is unaffected by g. For the other estimators, there is a monotonic decrease in coverage with increasing g. The effect is much stronger for the mean than the trimmed means.

For all estimators, increasing g leads to monotonic increases in CI width. The effect is very subtle for the median and more pronounced the less we trim. Under normality, g=0, CIs are the shortest for the mean, explaining the larger power of mean based methods relative to trimmed means in this unusual situation.

In the third panel, the zero line represents an equal proportion of left and right shifts, relative to the population, for CIs that did not include the population value. The values are consistently above zero for the median, with a few more right shifts than left shifts for all values of g. For the other estimators, the preponderance of left shifts increases markedly with g.

Now we look at results in panel B (h=0.2). When outliers are likely, coverage drops faster with g for the mean. Other estimators are resistant to outliers.

When outliers are common, CIs for the population mean are larger than for all other estimators, irrespective of g.

Again, there is a constant over-representation of right shifted CIS for the median. For the other estimators, the left shifted CIs dominate more and more with increasing g. The trend is more pronounced for the mean relative to the h=0 situation, with a sharper monotonic downward trajectory.


The answer to the question in the title is: most of the time! Simply because our models are wrong most of the time. So I would take all published confidence intervals with a pinch of salt. [Some would actually go further and say that if the sampling and analysis plans for an experiment were not clearly stipulated before running the experiment, then confidence interval, like P values, are not even defined (Wagenmakers, 2007). That is, we can compute a CI, but the coverage is meaningless, because exact repeated sampling might be impossible or contingent on external factors that would need to be simulated.] The best way forward is probably not to advocate for the use of trimmed means or the median over the mean in all cases, because different estimators address different questions about the data. And there are more estimators of central tendency than means, trimmed means and medians. There are also more interesting questions to ask about the data than their central tendencies (Rousselet, Pernet & Wilcox, 2017). For these reasons, we need data sharing to be the default, so that other users can ask different questions using different tools. The idea that the one approach used in a paper is the best to address the problem at hand is just silly.

To see what happens when we use the percentile bootstrap or the bootstrap-t to build confidence intervals for the mean, see this more recent post.


Bürkner, Paul-Christian, and Matti Vuorre. ‘Ordinal Regression Models in Psychology: A Tutorial’. Advances in Methods and Practices in Psychological Science 2, no. 1 (1 March 2019): 77–101.

Greenland, Sander, Stephen J. Senn, Kenneth J. Rothman, John B. Carlin, Charles Poole, Steven N. Goodman, and Douglas G. Altman. ‘Statistical Tests, P Values, Confidence Intervals, and Power: A Guide to Misinterpretations’. European Journal of Epidemiology 31, no. 4 (1 April 2016): 337–50.

Hoaglin, David C. ‘Summarizing Shape Numerically: The g-and-h Distributions’. In Exploring Data Tables, Trends, and Shapes, 461–513. John Wiley & Sons, Ltd, 1985.

Jaeger, T. Florian. ‘Categorical Data Analysis: Away from ANOVAs (Transformation or Not) and towards Logit Mixed Models’. Journal of Memory and Language 59, no. 4 (November 2008): 434–46.

Kruschke, John K. Doing Bayesian Data Analysis. 2nd Edition. Academic Press, 2014.

Liddell, Torrin M., and John K. Kruschke. ‘Analyzing Ordinal Data with Metric Models: What Could Possibly Go Wrong?’ Journal of Experimental Social Psychology 79 (1 November 2018): 328–48.

Rousselet, Guillaume A., Cyril R. Pernet, and Rand R. Wilcox. ‘Beyond Differences in Means: Robust Graphical Methods to Compare Two Groups in Neuroscience’. European Journal of Neuroscience 46, no. 2 (1 July 2017): 1738–48.

Rousselet, Guillaume A., and Rand R. Wilcox. ‘Reaction Times and Other Skewed Distributions: Problems with the Mean and the Median’. Preprint. PsyArXiv, 17 January 2019.

Wagenmakers, Eric-Jan. ‘A Practical Solution to the Pervasive Problems of p Values’. Psychonomic Bulletin & Review 14, no. 5 (1 October 2007): 779–804.

Wilcox, Rand R., and Guillaume A. Rousselet. ‘A Guide to Robust Statistical Methods in Neuroscience’. Current Protocols in Neuroscience 82, no. 1 (2018): 8.42.1-8.42.30.

Yan, Yuan, and Marc G. Genton. ‘The Tukey G-and-h Distribution’. Significance 16, no. 3 (2019): 12–13.