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.

Conclusion

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).

References

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. https://doi.org/10.1016/j.brat.2017.05.013.

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. https://doi.org/10.1080/00031305.2015.1089789.

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. https://doi.org/10.1002/9781118150702.ch11.

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. https://doi.org/10.31234/osf.io/h8ft7.

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. https://doi.org/10.31234/osf.io/3y54r.

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. https://doi.org/10.1002/cpns.41.

Yan, Yuan, and Marc G. Genton. ‘The Tukey G-and-h Distribution’. Significance 16, no. 3 (2019): 12–13. https://doi.org/10.1111/j.1740-9713.2019.01273.x.

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  1. Pingback: When is a 95% confidence interval not a 95% confidence interval? | basic statistics

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