Things which are equivalent to t-tests

If you have a continuous measurement and two groups you’d like to compare based on that measurement, what’s the first statistical test that comes to mind? Chances are it’s the two-sample t-test, sometimes known as Student’s t-test. It’s typically the first statistical test taught in an introductory statistics course, it’s well known and understood, and it has good theoretical properties—so if a t-test answers your research question, you should probably use it. (Actually, in practice, you should probably use Welch’s t-test which doesn’t assume equal variance within groups. For the rest of the post, I’m only going to consider the equal variance case.)

Every now and again, I find a client in this situation who has done something which is… not a t-test. Here are some other ways to do the same thing which turn out to be identical to the two-sample t-test:

  • One-way ANOVA: since ANOVA is often taught immediately after the t-test as “what do I do if I have more than two groups”, it should perhaps be no surprise that ANOVA gives identical p-values and confidence intervals to the t-test when you are only comparing two groups. This applies to both the overall F-test and the “post-hoc” pairwise t-test, which produce identical p-values in the two group scenario.

  • Linear regression with an indicator variable: by this I mean an explanatory variable which takes one numerical value for the first group and a different numerical value for the second group. For example, 0 for the first group and 1 for the second group. This also turns out to be equivalent to a t-test. If you’ve seen how ANOVA is implemented by most software as a linear model with indicator variables for categories, this might not surprise you either.

  • Linear regression with the indicator variable as the outcome and the measurement as the explanatory variable: as backwards as this may sound, this will also produce exactly the same p-value as a t-test. The hypothesis test for linear regression slope is the same regardless of which variable is the outcome and which is the predictor, although the regression equation is usually different.

  • Correlation with an indicator variable: this is also equivalent to a t-test. This follows from the above two because the hypothesis test for correlation being equal to zero is equivalent to the hypothesis test for linear regression slope being equal to zero.

  • Logistic regression with the group variable as the outcome and the continuous variable as a predictor: this is not quite the same as a t-test. However, if the measurements of the two groups are normally distributed, then the t-test and logistic regression are asymptotically equivalent—meaning that for a sufficiently large sample size, they will give the same p-values. The two methods also answer different research questions: the t-test is for a difference in means between two groups while logistic regression estimates the odds ratio of having a positive outcome (being in a particular group) for a given increase in the continuous predictor. The choice between the two methods should be based on which variable is your outcome and which variable is your explanatory variable; or whether you would prefer to discuss an odds ratio or a difference in means.

  • Pairwise comparisons from an ANOVA model with more than two groups are also not quite equivalent to a t-test. What’s the difference? Both use a test statistic that has a t distribution calculated from the same difference in means, but the ANOVA pairwise comparisons will have better power because they use a pooled standard deviation from all groups (which, if the assumption of equal standard deviation in every group is true, will be more a precise estimate) and a t distribution with more degrees of freedom (which can make a big difference with a small sample size). As the sample size increases, the advantage of ANOVA over individual t-tests diminishes.

  • Bonus: The Mann-Whitney test (or Wilcoxon rank sum test) is equivalent to proportional odds logistic regression. (This claim is made in Frank Harrell’s Regression Modeling Strategies text; unlike the other examples above, I haven’t proved or read a proof of this equivalence.)

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Cameron Patrick
Statistical Consultant

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