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Posted on Dec 22nd, 2020

*This post is an extended synopsis of Linde, M., Tendeiro, J. N., Selker, R., Wagenmakers, E.-J., & van Ravenzwaaij, D. (submitted). Decisions about equivalence: A comparison of TOST, HDI-ROPE, and the Bayes factor. Preprint available on PsyArXiv*: https://psyarxiv.com/bh8vu

Some important research questions require the ability to find evidence for two conditions being practically equivalent. This is impossible to accomplish within the traditional frequentist null hypothesis significance testing framework; hence, other methodologies must be utilized. We explain and illustrate three approaches for finding evidence for equivalence: The frequentist two one-sided tests procedure (TOST), the Bayesian highest density interval region of practical equivalence procedure (HDI-ROPE), and the Bayes factor interval null procedure (BF). We compare the classification performances of these three approaches for various plausible scenarios. The results indicate that the BF approach compares favorably to the other two approaches in terms of statistical power. Critically, compared to the BF procedure, the TOST procedure and the HDI-ROPE procedure have limited discrimination capabilities when the sample size is relatively small: specifically, in order to be practically useful, these two methods generally require over 250 cases within each condition when rather large equivalence margins of approximately 0.2 or 0.3 are used; for smaller equivalence margins even more cases are required. Because of these results, we recommend that researchers rely more on the BF approach for quantifying evidence for equivalence, especially for studies that are constrained on sample size.

Science is dominated by a quest for effects. Does a certain drug work better than a placebo? Are pictures containing animals more memorable than pictures without animals? These attempts to demonstrate the presence of effects are partly due to the statistical approach that is traditionally employed to make inferences. This framework – null hypothesis significance testing (NHST) – only allows researchers to find evidence against but not in favour of the null hypothesis that there is no effect. In certain situations, however, it is worthwhile to examine whether there is evidence for the absence of an effect. For example, biomedical sciences often seek to establish equal effectiveness of a new versus an existing drug or biologic. The new drug might have fewer side effects and would therefore be preferred even if it is only as effective as the old one. Answering questions about the absence of an effect requires other tools than classical NHST. We compared three such tools: The frequentist two one-sided tests approach (TOST; e.g., Schuirmann, 1987), the Bayesian highest density interval region of practical equivalence approach (HDI-ROPE; e.g., Kruschke, 2018), and the Bayes factor interval null approach (BF; e.g., Morey & Rouder, 2011).

We estimated statistical power and the type I error rate for various plausible scenarios using an analytical approach for TOST and a simulation approach for HDI-ROPE and BF. The scenarios were defined by three global parameters:

- Population effect size: δ = {0,0.01,…,0.5}
- Sample size per condition:
*n*= {50,100,250,500} - Standardized equivalence margin:
*m*= {0.1,0.2,0.3}

In addition, for the Bayesian approaches we placed a Cauchy prior on the population effect size with a scale parameter of *r *= {0.5/√2,1/√2,2/√2}. Lastly, for the BF approach specifically, we used Bayes factor thresholds of BF_{thr }= {3,10}.

The results for an equivalence margin of *m* = 0.2 are shown in Figure 1. The overall results for equivalence margins of* m* = 0.1 and *m* = 0.3 were similar and are therefore not shown here. Ideally, the proportion of equivalence decisions would be 1 when δ lies inside the equivalence interval and 0 when δ lies outside the equivalence interval. The results show that TOST and HDI-ROPE are maximally conservative to conclude equivalence when sample sizes are relatively small. In other words, these two approaches never make equivalence decisions, which means they have no statistical power but they also make no type I errors. With our choice of Bayes factor thresholds, the BF approach is more liberal to make equivalence decisions, displaying higher power but also a higher type I error rate. Although far from perfect, the BF approach has rudimentary discrimination abilities for relatively small sample sizes. As the sample size increases, the classification performances of all three approaches improve. In comparison to the BF approach, the other two approaches remain quite conservative.

*Figure 1.* Proportion of equivalence predictions with a standardized equivalence margin of *m* = 0.2. Panels contain results for different sample sizes. Colors denote different inferential approaches (and different decision thresholds within the BF approach). Line types denote different priors (for Bayesian metrics). Predictions of equivalence are correct if the population effect size (δ) lies within the equivalence interval (power), whereas predictions of equivalence are incorrect if δ lies outside the equivalence interval (Type I error rate).

Making decisions based on small samples should generally be avoided. If possible, more data should be collected before making decisions. However, sometimes sampling a relatively large number of cases is not feasible. In that case, the use of Bayes factors might be preferred because they display some discrimination capabilities. In contrast, TOST and HDI-ROPE are maximally conservative. For large sample sizes, all three approaches perform almost optimally when the population effect size is in the center of the equivalence interval or when it is very large (or low). However, the BF approach results in more balanced decisions at the decision boundary (i.e., where the population effect size is equal to the equivalence margin). In summary, we recommend the use of Bayes factors for making decisions about the equivalence of two groups.

Kruschke, J. K. (2018). Rejecting or accepting parameter values in Bayesian estimation. *Advances in Methods and Practices in Psychological Science*, *1*(2), 270–280.

Morey, R. D., & Rouder, J. N. (2011). Bayes factor approaches for testing interval null hypotheses. *Psychological Methods, 16*(4), 406–419.

Schuirmann, D. J. (1987). A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. *Journal of Pharmacokinetics and* *Biopharmaceutics, 15*(6), 657–680.

Maximilian Linde is PhD student at the Psychometrics & Statistics group at the University of Groningen.

Jorge N. Tendeiro is assistant professor at the Psychometrics & Statistics group at the University of Groningen.

Ravi Selker was PhD student at the Psychological Methods group at the University of Amsterdam (at the time of involvement in this project).

Eric-Jan (EJ) Wagenmakers is professor at the Psychological Methods Group at the University of Amsterdam.

Don van Ravenzwaaij is associate professor at the Psychometrics & Statistics group at the University of Groningen.

Posted on Dec 10th, 2020

Background: the 2018 article “Redefine Statistical Significance” suggested that it is prudent to treat p-values just below .05 with a grain of salt, as such p-values provide only weak evidence against the null. Here we provide another empirical demonstration of this fact. Specifically, we examine the degree to which recently published data provide evidence for the claim that students who are given a specific hypothesis to test are less likely to discover that the scatterplot of the data shows a gorilla waving at them (p=0.034).

In a recent experiment, Yanai & Lercher (2020; henceforth YL2020) constructed the statistical analogue of the famous Simons and Chabris demonstration of inattentional blindness , where a waving gorilla goes undetected when participants are instructed to count the number of passes with a basketball.

In YL2020, a total of 33 students were given a data set to analyze. They were told that it contained “the body mass index (BMI) of 1786 people, together with the number of steps each of them took on a particular day, in two files: one for men, one for women. (…) The students were placed into two groups. The students in the first group were asked to consider three specific hypotheses: (i) that there is a statistically significant difference in the average number of steps taken by men and women, (ii) that there is a negative correlation between the number of steps and the BMI for women, and (iii) that this correlation is positive for men. They were also asked if there was anything else they could conclude from the dataset. In the second, “hypothesis-free,” group, students were simply asked: What do you conclude from the dataset?”

*Figure 1. The artificial data set from YL2020 (available on https://osf.io/6y3cz/, courtesy of Yanai & Lercher via https://twitter.com/ItaiYanai/status/1324444744857038849). Figure from YL2020.*

The data were constructed such that the scatterplot displays a waving gorilla, invalidating any correlational analysis (cf. Anscombe’s quartet). The question of interest was whether students in the hypothesis-focused group would miss the gorilla more often than students in the hypothesis-free group. And indeed, in the hypothesis-focused group, 14 out of 19 (74%) students missed the gorilla, whereas this happened only for 5 out of 14 (36%) students in the hypothesis-free group. This is a large difference in proportions, but, on the other hand, the data are only binary and the sample size is small. YL2020 reported that “students without a specific hypothesis were almost five times more likely to discover the gorilla when analyzing this dataset (odds ratio = 4.8, P = 0.034, N = 33, Fisher’s exact test (…)). At least in this setting, the hypothesis indeed turned out to be a significant liability.”

*Table 1. Results from the YL2020 experiment. Table from YL2020.*

We like the idea to construct a statistical version of the gorilla experiment, we believe that the authors’ hypothesis is plausible, and we also feel that the data go against the null hypothesis. However, the middling p=0.034 does make us skeptical about the degree to which these data provide evidence against the null. To check our intuition we now carry out a Bayesian comparison of two proportions using the A/B test proposed by Kass & Vaidyanathan (1992) and implemented in R and JASP (Gronau, Raj, & Wagenmakers, in press).

For a comparison of two proportions, the Kass & Vaidyanathan method amounts to logistic regression with “group” coded as a dummy predictor. Under the no-effect model H0, the log odds ratio equals ψ=0, whereas under the positive-effect model H+, ψ is assigned a positive-only normal prior N+(μ,σ), reflecting the fact that the hypothesis of interest (i.e., focusing students on the hypothesis makes them more likely to miss the gorilla, not less likely) is directional. A default analysis (i.e., μ=0, σ=1) reveals that the data are 5.88 times more likely under H+ than under H0. If the alternative hypothesis is specified to be bi-directional (i.e., two-sided), this evidence drops to 2.999, just in Jeffreys’s lowest evidence category of “not worth more than a bare mention”.

Returning to the directional hypothesis, we can show how the evidence changes with the values for μ and σ. A few keyboard strokes in JASP yield the following heatmap robustness plot:

*Figure 2. Robustness analysis for the results from YL2020.*

This plot shows that the Bayes factor (i.e., the evidence) can exceed 10, but only when the prior is cherry-picked to have a location near the maximum likelihood estimate and a small variance. This kind of oracle prior is unrealistic. Realistic prior values for μ and σ generally produce Bayes factors lower than 6. Note that when both hypotheses are deemed equally likely a priori, a Bayes factor of 6 increases the prior plausibility for H+ from .50 to 6/7 = .86, leaving a non-negligible .14 for H0.

Finally, we can apply an estimation approach and estimate the log odds ratio using an unrestricted hypothesis. This yields the following “Prior and posterior” plot:

*Figure 3. Parameter estimation results for the data from YL2020.*

Figure 3 shows that there exists considerable uncertainty concerning the size of the effect: it may be massive, but it may also be modest, or miniscule. Even negative values are not quite out of contention.

In sum, our Bayesian reanalysis showed that the evidence that the data provide is relatively modest. A p-value of .034 (“reject the null hypothesis; off with its head!”) is seen to correspond to one-sided Bayes factors of around 6. This does constitute evidence in favor of the alternative hypothesis, but its strength is modest and does not warrant a public execution of the null. We do have high hopes that an experiment with more participants will conclusively demonstrate this phenomenon.

Benjamin, D. J. et al. (2018). Redefine statistical significance. *Nature Human Behaviour, 2, 6-10.*

Gronau, Q. F., Raj, K. N. A., & Wagenmakers, E.-J. (in press). Informed Bayesian inference for the A/B test. *Jo**urnal of Statistical Software*. Preprint: http://arxiv.org/abs/1905.02068

Kass, R. E., & Vaidyanathan, S. K. (1992). Approximate Bayes factors and orthogonal parameters, with application to testing equality of two binomial proportions. *Journal of the Royal Statistical Society: Series B (Methodological)*, *54*, 129-144.

Simons, D. J., & Chabris, C. F. (1999). Gorillas in our midst: Sustained inattentional blindness for dynamic events. *Perception, 28*, 1059-1074.

Yanai, I., & Lercher, M. (2020). A hypothesis is a liability. *Genome Biology,* 21:231.

Eric-Jan (EJ) Wagenmakers is professor at the Psychological Methods Group at the University of Amsterdam.

Quentin is a PhD candidate at the Psychological Methods Group of the University of Amsterdam.