# 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.)