# 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.

# Conclusion

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.

# References

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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. https://doi.org/10.1007/s10654-016-0149-3.

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

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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. https://doi.org/10.1111/ejn.13610.

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.

Wagenmakers, Eric-Jan. ‘A Practical Solution to the Pervasive Problems of p Values’. Psychonomic Bulletin & Review 14, no. 5 (1 October 2007): 779–804. https://doi.org/10.3758/BF03194105.

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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.