Equivalence Testing and the Second Generation P-Value

Recently Blume, D’Agostino McGowan, Dupont, & Greevy (2018) published an article titled: “Second-generation p-values: Improved rigor, reproducibility, & transparency in statistical analyses”. As it happens, I would greatly appreciate more rigor, reproducibility, and transparency in statistical analyses, so my interest was piqued. On Twitter I saw the following slide, promising a updated version of the p-value that can support null-hypotheses, takes practical significance into account, has a straightforward interpretation, and ideally never needs adjustments for multiple comparisons. Now it sounded like someone found the goose that lays the golden eggs.

Upon reading the manuscript, I noticed the statistic is surprisingly similar to equivalence testing, which I’ve written about recently and created an R package for (Lakens, 2017). The second generation p-value (SGPV) relies on specifying an equivalence range of values around the null-hypothesis that are practically equivalent to zero (e.g., 0 ± 0.3). If the estimation interval falls completely within the equivalence range, the SGPV is 1. If the confidence interval lies completely outside of the equivalence range, the SGPV is 0. Otherwise the SGPV is a value between 0 and 1 that expresses the overlap of the confidence interval with the equivalence bound, divided by the total width of the confidence interval.

Testing whether the confidence interval falls completely within the equivalence bounds is equivalent to the two one-sided tests (TOST) procedure, where the data is tested against the lower equivalence bound in the first one-sided test, and against the upper equivalence bound in the second one-sided test. If both tests allow you to reject an effect as extreme or more extreme than the equivalence bound, you can reject the presence of an effect large enough to be meaningful, and conclude the observed effect is practically equivalent to zero. You can also simply check if a 90% confidence interval falls completely within the equivalence bounds. Note that testing whether the 95% confidence interval falls completely outside of the equivalence range is known as a minimum-effect test (Murphy, Myors, & Wolach, 2014).

So together with my collaborator Marie Delacre we compared the two approaches, to truly understand how second generation p-values accomplished what they were advertised to do, and what they could contribute to our statistical toolbox.

To examine the relation between the TOST p-value and the SGPV we can calculate both statistics across a range of observed effect sizes. In Figure 1 p-values are plotted for the TOST procedure and the SGPV. The statistics are calculated for hypothetical one-sample t-tests for all means that can be observed in studies ranging from 140 to 150 (on the x-axis). The equivalence range is set to 145 ± 2 (i.e., an equivalence range from 143 to 147), the observed standard deviation is assumed to be 2, and the sample size is 100. The SGPV treats the equivalence range as the null-hypothesis, while the TOST procedure treats the values outside of the equivalence range as the null-hypothesis. For ease of comparison we can reverse the SGPV (by calculating 1-SGPV), which is used in the plot below.

 

 

Figure 1: Comparison of p-values from TOST (black line) and 1-SGPV (dotted grey line) across a range of observed sample means (x-axis) tested against a mean of 145 in a one-sample t-test with a sample size of 30 and a standard deviation of 2.

It is clear the SGPV and the p-value from TOST are very closely related. The situation in Figure 1 is not an exception – in our pre-print we describe how the SGPV and the p-value from the TOST procedure are always directly related when confidence intervals are symmetrical. You can play around with this Shiny app as confirm this for yourself: http://shiny.ieis.tue.nl/TOST_vs_SGPV/.

There are 3 situations where the p-value from the TOST procedure and the SGPV are not directly related. The SGPV is 1 when the confidence interval falls completely within the equivalence bounds. P-values from the TOST procedure continue to differentiate and will for example distinguish between a p = 0.048 and p = 0.002. The same happens when the SGPV is 0 (and p-values fall between 0.975 and 1).

The third situation when the TOST and SGPV differ is when the ‘small sample correction’ is at play in the SGPV. This “correction” kicks in whenever the confidence interval is wider than the equivalence range. However, it is not a correction in the typical sense of the word, since the SGPV is not adjusted to any ‘correct’ value. When the normal calculation would be ‘misleading’ (i.e., the SGPV would be small, which normally would suggest support for the alternative hypothesis, when all values in the equivalence range are also supported), the SGPV is set to 0.5 which according to Blume and colleagues signal the SGPV is ‘uninformative’.In all three situations the p-value from equivalence tests distinguishes between scenarios where the SGPV yields the same result.

We can examine this situation by calculating the SGPV and performing the TOST for a situation where sample sizes are small and the equivalence range is narrow, such that the CI is more than twice as large as the equivalence range.

 

Figure 2: Comparison of p-values from TOST (black line) and SGPV (dotted grey line) across a range of observed sample means (x-axis). Because the sample size is small (n = 10) and the CI is more than twice as wide as the equivalence range (set to -0.4 to 0.4), the SGPV is set to 0.5 (horizontal light grey line) across a range of observed means.

The main novelty of the SGPV is that it is meant to be used as a descriptive statistic. However, we show that the SGPV is difficult to interpret when confidence intervals are asymmetric, and when the 'small sample correction' is operating. For an extreme example, see Figure 3 where the SGPV's are plotted for a correlation (where confidence intervals are asymmetric). 

Figure 3: Comparison of p-values from TOST (black line) and 1-SGPV (dotted grey curve) across a range of observed sample correlations (x-axis) tested against equivalence bounds of r = 0.4 and r = 0.8 with n = 10 and an alpha of 0.05.

Even under ideal circumstances, the SGPV is mainly meaningful when it is either 1, 0, or inconclusive (see all examples in Blume et al., 2018). But to categorize your results into one of these three outcomes you don’t need to calculate anything – you can just look at whether the confidence interval falls inside, outside, or overlaps with the equivalence bound, and thus the SGPV loses its value as a descriptive statistic. 

When discussing the lack of a need for error correction, Blume and colleagues compare the SGPV to null-hypothesis tests. However, the more meaningful comparison is with the TOST procedure, and given the direct relationship, not correcting for multiple comparisons will inflate the probability of concluding the absence of a meaningful effect in exactly the same way as when calculating p-values for an equivalence test. Equivalence tests provide an easier and more formal way to control both Type I error rates (by setting the alpha level) and the Type II error rate (by performing an a-priori power analysis, see Lakens, Scheele, & Isager, 2018).

Conclusion

There are strong similarities between p-values from the TOST procedure and the SGPV, and in all situations where the statistics yield different results, the behavior of the p-value from the TOST procedure is more consistent and easier to interpret. More details can be found in our pre-print (where you can also leave comments or suggestions for improvement using hypothes.is). Our comparisons show that when proposing alternatives to null-hypothesis tests, it is important to compare new proposals to already existing procedures. We believe equivalence tests achieve the goals of the second generation p-value while allowing users to more easily control error rates, and while yielding more consistent statistical outcomes.

References

Blume, J. D., D’Agostino McGowan, L., Dupont, W. D., & Greevy, R. A. (2018). Second-generation p-values: Improved rigor, reproducibility, & transparency in statistical analyses. PLOS ONE, 13(3), e0188299. https://doi.org/10.1371/journal.pone.0188299

Lakens, D. (2017). Equivalence Tests: A Practical Primer for t Tests, Correlations, and Meta-Analyses. Social Psychological and Personality Science, 8(4), 355–362. https://doi.org/10.1177/1948550617697177

Lakens, D., Scheel, A. M., & Isager, P. M. (2018). Equivalence Testing for Psychological Research: A Tutorial. Advances in Methods and Practices in Psychological Science, 2515245918770963. https://doi.org/10.1177/2515245918770963.

Murphy, K. R., Myors, B., & Wolach, A. H. (2014). Statistical power analysis: a simple and general model for traditional and modern hypothesis tests (Fourth edition). New York: Routledge, Taylor & Francis Group.

ROPE and Equivalence Testing: Practically Equivalent?

In a previous post, I compared equivalence tests to Bayes factors, and pointed out several benefits of equivalence tests. But a much more logical comparison, and one I did not give enough attention to so far, is the ROPE procedure using Bayesian estimation. I’d like to thank John Kruschke for feedback on a draft of this blog post. Check out his own recent blog comparing ROPE to Bayes factors here. 


When we perform a study, we would like to conclude there is an effect, when there is an effect. But it is just as important to be able to conclude there is no effect, when there is no effect. I’ve recently published a paper that makes Frequentist equivalence tests (using the two-one-sided tests, or TOST, approach) as easy as possible (Lakens, 2017). Equivalence tests allow you to reject the presence of any effect you care about. In Bayesian estimation, one way to argue for the absence of a meaningful effect is the Region of Practical Equivalence (ROPE) procedure (Kruschke, 2014, chapter 12), which is “somewhat analogous to frequentist equivalence testing” (Kruschke & Liddell, 2017).

In the ROPE procedure, a 95% Highest Density Interval (HDI) is calculated based on a posterior distribution (which is calculated based on a prior and the data). Kruschke suggests that: “if the 95 % HDI falls entirely inside the ROPE then we decide to accept the ROPE’d value for practical purposes”. Note that the same HDI can also be used to reject the null hypothesis, where in Frequentist statistics, even though the confidence interval plays a similar role, you would still perform both a traditional t-test and the TOST procedure.

The only real difference with equivalence testing is that instead of using a confidence interval, a Bayesian Highest Density Interval is used. If the prior used by Kruschke was perfectly uniform, ROPE and equivalence testing would identical, barring philosophical differences in how the numbers should be interpreted. The BEST package by default uses a ‘broad’ prior, and therefore the 95% CI and 95% HDI are not exactly the same, but they are very close, for single comparisons. When multiple comparisons are made, (for example when using sequential analyses, Lakens, 2014), the CI needs to be adjusted to maintain the desired error rate, but in Bayesian statistics, error rates are not directly controlled (they are limited due to ‘shrinkage’, but can be inflated beyond 5%, and often considerably so).

In the code below, I randomly generate random normally distributed data (with means of 0 and a sd of 1) and perform the ROPE procedure and the TOST. The 95% HDI is from -0.10 to 0.42, and the 95% CI is from -0.11 to 0.41, with mean differences of 0.17 or 0.15.

Indeed, if you will forgive me the pun, you might say these two approaches are practically equivalent. But there are some subtle differences between ROPE and TOST

95% HDI vs 90% CI

Kruschke (2014, Chapter 5) writes: “How should we define “reasonably credible”? One way is by saying that any points within the 95% HDI are reasonably credible.” There is not a strong justification for the use of a 95% HDI over a 96% of 93% HDI, except that it mirrors the familiar use of a 95% CI in Frequentist statistics. In Frequentist statistics, the 95% confidence interval is directly related to the 5% alpha level that is commonly deemed acceptable for a maximum Type 1 error rate (even though this alpha level is in itself a convention without strong justification).

But here’s the catch: The TOST equivalence testing procedure does not use a 95% CI, but a 90% CI. The reason for this is that two one-sided tests are performed. Each of these tests has a 5% error rate. You might intuitively think that doing two tests with a 5% error rate will increase the overall Type 1 error rate, but in this case, that’s not true. You could easily replace the two tests, with just one test, testing the observed effect against the equivalence bound (upper or lower) closest to it. If this test is statistically significant, so is the other – and thus, there is no alpha inflation in this specific case. That’s why the TOST procedure uses a 90% CI to have a 5% error rate, while the same researcher would use a 95% CI in a traditional two-sided t-test to examine whether the observed effect is statistically different from 0, while maintaining a 5% error rate (see also Senn, 2007, section 22.2.4)

This nicely illustrates the difference between estimation (where you just want to have a certain level of accuracy, such as 95%), and Frequentist hypothesis testing, where you want to distinguish between signal and noise, and not be wrong more than 5% of the time when you declare there is a signal. ROPE keeps the accuracy the same across tests, Frequentist approaches keep the error rate constant. From a Frequentist perspective, ROPE is more conservative than TOST, like the use of alpha = 0.025 is more conservative than the use of alpha = 0.05.

Power analysis

For an equivalence test, power analysis can be performed based on closed functions, and the calculations take just a fraction of a second. I find that useful, for example in my role in our ethics board, where we evaluate proposals that have to justify their sample size, and we often check power calculations. Kruschke has an excellent R package (BEST) that can do power analyses for the ROPE procedure. This is great work – but the simulations take a while (a little bit over an hour for 1000 simulations).

Because the BESTpower function relies on simulations, you need to specify the sample size, and it will calculate the power. That’s actually the reverse of what you typically want in a power analysis (you want to input the desired power, and see which sample size you need). This means you most likely need to run multiple simulations in BESTpower, before you have determined the sample size that will yield good power. Furthermore, the software requires your to specify the expected means and standard deviations, instead of simply an expected effect size. Instead of Frequentist power analysis, where the hypothesized effect size is a point value (e.g., d = 0.4), Bayesian power analysis models the alternative as a distribution, acknowledging there is uncertainty.

In the end, however, the result of a power analysis for ROPE and for TOST is actually remarkably similar. Using the code below to perform the power analysis for ROPE, we see that 100 participants in each group give us approximately 88.4% power (with 2000 simulations, this estimate is still a bit uncertain) to get a 95% HDI that falls within our ROPE of -0.5 to 0.5, assuming standard deviations of 1.

We can use the powerTOSTtwo.raw function in the TOSTER package (using an alpha of 0.025 instead of 0.05, to mirror to 95% HDI) to calculate the sample size we would need to achieve 88.4% power for independent t-test (using equivalence bounds of -0.5 and 0.5, and standard deviations of 1):

powerTOSTtwo.raw(alpha=0.025,statistical_power=0.875,low_eqbound=-0.5,high_eqbound=0.5,sdpooled=1)

The outcome is 100 as well. So if you use a broad prior, it seems you can save yourself some time by using the power analysis for equivalence tests, without severe consequences.

Use of prior information

The biggest benefit of ROPE over TOST is that is allows you to incorporate prior information in your data analysis. If you have reliable prior information, ROPE can use this information, which is especially useful if you don’t have a lot of data. If you use priors, it is typically advised to check the robustness of the posterior against reasonable changes in the prior (Kruschke, 2013).

Conclusion

Using the ROPE procedure or the TOST procedure will most likely lead to very similar inferences. For all practical purposes, the differences are small. It’s quite a lot easier to perform a power analysis for TOST, and by default, TOST has greater statistical power because it uses 90% CI. But power analysis is possible for ROPE (which is a rare pleasure to see for Bayesian analyses), and you could choose to use a 90% HDI, or any other value that matches your goals. TOST will be easier and more familiar because it is just a twist on the classic t-test, but ROPE might be a great way to dip your toes in Bayesian waters and explore the many more things you can do with Bayesian posterior distributions.

References

Kruschke, J. (2013). Bayesian estimation supersedes the t test. Journal of Experimental Psychology: General, 142(2), 573–603. https://doi.org/10.1037/a0029146

Kruschke, J. (2014). Doing Bayesian Data Analysis, Second Edition: A Tutorial with R, JAGS, and Stan (2 edition). Boston: Academic Press.

Kruschke, J., & Liddell, T. M. (2017). The Bayesian New Statistics: Hypothesis testing, estimation, meta-analysis, and power analysis from a Bayesian perspective. Psychonomic Bulletin & Review. https://doi.org/10.3758/s13423-016-1221-4

Lakens, D. (2014). Performing high-powered studies efficiently with sequential analyses: Sequential analyses. European Journal of Social Psychology, 44(7), 701–710. https://doi.org/10.1002/ejsp.2023

Lakens, D. (2017). Equivalence tests: A practical primer for t-tests, correlations, and meta-analyses. Social Psychological and Personality Science.

Senn, S. (2007). Statistical issues in drug development (2nd ed). Chichester, England ; Hoboken, NJ: John Wiley & Sons.

Examining Non-Significant Results with Bayes Factors and Equivalence Tests

In this blog, I’ll compare two ways of interpreting non-significant effects: Bayes factors and TOST equivalence tests. I’ll explain why reporting more than only Bayes factors makes sense, and highlight some benefits of equivalence testing over Bayes factors. I’d like to say a big thank you to Bill (Lihan) Chen and Victoria Savalei for helping me out super-quickly with my questions as I was re-analyzing their data.

Does volunteering improve well being? A recent article by Ashley Whillans, Scott Seider, Lihan Chen, Ryan Dwyer, Sarah Novick, Kathryn Gramigna, Brittany Mitchell, Victoria Savalei, Sally Dickerson & Elizabeth W. Dunn suggests the answer is: Not so much. The study was published in Comprehensive Results in Social Psychology, one of the highest quality journals in social psychology, which peer-reviews pre-registrations of studies before they are performed.

People were randomly assigned to a volunteering program for 6 months, or to a control condition. Before and after, a wide range of well-being measures were collected. Bayes factors support the null for all measures. The main results (and indeed, except for some manipulation checks, the only results – not even means or standard deviations are provided in the article) are communicated in the form of Bayes factors in Table 2.

The Bayes factors were calculated using the Bayes factor calculator by Zoltan Dienes, who has a great open access paper in Frontiers, cited more than 200 times since 2014, on how to use Bayes to get most out of non-significant results. I won’t try to explain in detail how these Bayes factors are calculated – too many Bayesians on Twitter have told me I am too stupid to understand the math behind Bayes factors, and how I should have taken calculus in high school. They are right on both accounts, so just read Dienes (2014) for an explanation.

As Dienes (2014) discusses, you can also interpret non-significant results using Frequentist statistics. In a TOST equivalence test, which consists of two simple one-sided t-tests, you determine whether an effect falls between equivalence bounds set to the smallest effect size you care about (for an introduction, see Lakens, 2017). Dienes (2014) says it can be difficult to determine what this smallest effect size of interest is, but for me, if anything, it is easier to determine a smallest effect size of interest than to specify an alternative model in Bayesian statistics.

The authors examined whether well-being was improved by volunteering, and specified an alternative model (what would a true effect of improved well-being look like?) as follows (page 9): “Because our goal was to contrast the null hypothesis to an alternative hypothesis that the effect is moderate in size, we used a normal distribution prior with a mean of 0.50 and a standard deviation of 0.15 for the standardized effect size (e.g. the difference score between standardized T2 and T1 measures).”

It is interesting to see the authors wanted to specify their alternative in terms of a ‘standardized effect size’. I fully agree that using standardized effect sizes is currently the easiest way to think about the alternative hypothesis, and it is the reason my spreadsheet and R package “TOSTER” allow you to specify equivalence bounds in standardized effect sizes when performing an equivalence test.

In equivalence testing, we can test whether the observed data is surprisingly smaller than anything we would expect. The authors seem to find a true effect of d = 0.5 a realistic alternative model. So, a good start is to try to reject an effect of d = 0.5. We can just fill in the means, standard deviations, and sample sizes from both groups, and test against the equivalence bound of d = 0.5 (see the code at the bottom of the post). Note that the authors perform a two-sided test (even though they have a one-sided hypothesis, as indicated in the title “Does volunteering improve well-being?”, but following the authors, I will test whether the effect is statistically smaller than d = 0.5, and statistically larger than d = -0.5, instead of only testing whether the effect is smaller than d = 0.5). The most important results are summarized in the Figure below:

Testing the effect for WSB, one of the well-being measures, the standardized effect size of 0.5 equals a raw effect of 0.762 in scale points on the original measure. Because the 90% confidence interval around the mean difference does not contain -0.762 or 0.762, the observed data is surprising (a.k.a statistically significant), if there was a true effect of d = -0.5 or d = 0.5 (see Lakens, 2017, for a detailed explanation). We can reject the hypothesis that d = -0.5 or d = 0.5, and if we do this, given our alpha of 0.05, we would be wrong a maximum of 5% of the time, in the long run. Other people might find smaller effects still of interest. They can collect more data, and perform an equivalence test in a meta-analysis. 

We could write: Using a TOST procedure to test the data against equivalence bounds of d = -0.5 and d = 0.5, the observed results were statistically equivalent to zero, t(78.24) = -2.86, p = 0.003. The mean difference was -0.094, 90% CI[-0.483; 0.295].

Benefits of equivalence tests compared to Bayes factors.

If we perform equivalence tests, we see that we can conclude statistical equivalence for all nine measures. You might wonder about whether we need to correct for the fact that we perform nine tests for all the different well-being measures. Would we conclude that volunteering has a positive effect on well-being, if any single one of these tests showed a significant effect? If so, we should indeed correct for multiple comparisons to control our overall Type 1 error rate, and you can do this in equivalence testing. There is no easy way to control error rates in Bayesian statistics. Some Bayesians simply don’t care about error control, and I don’t exactly know what Bayesian who care about error control do. I care about error control, and the attention p-hacking is getting suggests I am not alone. In equivalence testing, you can control the Type 1 error rate simply by adjusting the alpha level, which is one benefit of equivalence testing over Bayes factors.

To calculate a Bayes factor, you need to specify your prior by providing the mean and standard deviation of the alternative. Bayes factors are quite sensitive to how you specify these priors, and for this reason, not every Bayesian statistician would recommend the use of Bayes factors. Andrew Gelman, a widely known Bayesian statistician, recently co-authored a paper in which Bayes factors were used as one of three Bayesian approaches to re-analyze data. In footnote 3 it is written: “Andrew Gelman wishes to state that he hates Bayes factors” – mainly because of this sensitivity to priors. So not everyone likes Bayes factors (just like not everyone likes p-values!). You can discuss the sensitivity to priors in a sensitivity analysis, which would mean plotting Bayes factors for alternative models with a range of means and standard deviations and different distributions, but I rarely see this done in practice. Equivalence tests also depend on the choice of the equivalence bounds. But it is very easy to see the effect of different equivalence bounds on the test result – you can just check if the equivalence bound you would have chosen falls within the 90% confidence interval. So that is a second benefit of equivalence testing.

The authors used a power analysis to determine the sample size they needed (page 7): "To achieve 80% power to detect an effect size of r = 0.21 (d = 0.40), we required at least 180 participants to detect significant effects of volunteering on our SWB measures of interest." But what was the power of the study to support the null? Although you can simulate everything in R, there is no software to perform power analysis for Bayes factors (indeed, 'power' is a Frequentist concept). When performing an equivalence test, you can easily perform a power analysis to make sure you have a well-powered study if there is an effect, and when there is no effect (and the spreadsheet and R package allow you to do this). When pre-registering a study, you need to justify your sample size, both with an eye for when the alternative hypothesis is true, as when the null hypothesis is true. The ease with which you can perform power calculations is another benefit of equivalence tests.

A final benefit I’d like to discuss concerns the assumptions of statistical tests. You should not perform tests when their assumptions are violated. The authors in the paper examining the effect of volunteering on well-being correctly report Welch’s t-tests, because they have unequal sample sizes in each group, and the equal variances assumption is violated. This is excellent practice. I don’t know how Bayes factors deal with unequal variances (I think they don’t, and simply assume equal variances, but I’m sure the answer will appear in the comments, if there is one). My TOST equivalence test spreadsheet and R code use Welch’s t-test by default (just as R does), so unequal variances is no longer a problem. The equal variances assumption is not very plausible in many research questions in psychology (Delacre, Lakens, & Leys, under review), so not having to assume equal variances is another benefit of equivalence testing compared to Bayes factors.

Conclusion

Only reporting Bayes factors seems, to me, an incomplete description of the data. I think it makes sense to report an effect size, the mean difference, and the confidence interval around it. And if you do that, and have determined a smallest effect size of interest, then performing the TOST equivalence testing procedure is nothing more than checking and reporting whether the p-value for the TOST procedure is smaller than your alpha level to conclude the effect is statistically equivalent. And you can still add a Bayes factor, if you want.

All approaches to statistical inferences have strengths and weaknesses. In most situations, both Bayes factors and equivalence tests lead to conclusions that have the same practical consequences. Whenever they do not, it is never the case that one approach is correct, and one is wrong – the answers differ because the tests have different assumptions, and you will have to think about your data more, which is never a bad thing. In the end, as long as you share the data of your paper online, as the current authors did, anyone can calculate the statistics they like. But only reporting Bayes factors is not really enough to describe your data. You might want to at least report means and standard deviations, so that people who want to include the effect size in a meta-analysis don’t need to re-analyze your data. And you might want to try out equivalence tests next time you interpret null results.