Solved – Repeated measure ANOVA vs independent 2-sample t-test

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I have this data where I have a control and an intervention group. For both of these two groups, I have a continuous outcome measured at baseline $(y_0)$ and at 1 month $(y_1)$. To test for the difference between the control and intervention group, I can create an outcome by $y_1 – y_0$ and run a 2 sample independent t-test. However, I was suggested to use a repeated measure ANOVA instead. Could anybody tell me if there is an advantage using an RM-ANOVA in this case? Thank you in advance.

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This problem you've described gets at the heart of an epistemological problem in the natural and social sciences that pits nomothetic knowledge versus ideographic knowledge. The ideographic knowledge you can gain from an RM-ANVOA is how individuals change over time. The nomothetic knowledge you can gain from an RM-ANOVA is how the difference between treatments will generalize to other samples. A t-Test only provides nomothetic knowledge.

An Independent Samples t-Test and ANOVA are both part of the General Linear Model (GLM). For a t-test, the null and alternative hypotheses are that $H_0:\mu_1-\mu_2=0;H_1:\mu_1-\mu_2\ne0$. For an ANOVA, the null and alternative hpyotheses are $H_0:\mu_1=\mu_2;H_1:\mu_1\ne\mu_2$.

A t-Test is represented by the formula $\hat{Y}=\beta_0+\beta_1(X_1)+$ and the ANOVA is also represented by the formula $\hat{Y}=\beta_0+\beta_1(X_1)$, so they are essentially the same, and that is because they are both part of the GLM. However, in this case, you have a new and rich data point called 'time,' and time is a really neat concept. Thus, you don't have just a plain old ANOVA, you have an RM-ANOVA.

The formula for an RM-ANOVA includes the time as a predictor (independent variable): $\hat{Y}=\beta_0+\beta_1(Group)+\beta_2(Time)$. That means that each of these predictors could exhibit statistical significance and both yield a measure of effect size. Using standard ANOVA table, you would get an $\eta^2$ effect size for each and using multiple linear regression, you would get an $R^2$ effect size. However, you might even consider Hierarchical Linear Modelling where time is a level-1 predictor and group is a level-2 predictor, but this is normally done with more than two measurements.

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Solved – ANOVA vs. T-test for two groups

Usually, we use ANOVA if there are more than two groups. But you also can use ANOVA with two groups, as you describe. In that case ANOVA will result in the same conclusion as an Student's t test, where $t^{2} = F$. See this R code:

# Makes example reproducible
set.seed(1)
# define sample size
n <- 100
# generate a group
group <- sample(0:1, n, replace= TRUE)
# generate a dependent variable that varies between groups
y <- rnorm(n) + group
# run a t test (variance must be assumed as being equel, otherwise the results do differ to those from ANOVA)
t.test(y ~ group, var.equal= TRUE)
# run ANOVA
summary(aov(y ~ group))
# F= 32.97
# See if t^2= F
t.test(y ~ group, var.equal= TRUE)$statistic**2
# t^2= 32.97

This is only true for the Student's t test where equal variances are assumed, while the results of a Welch t test differ to those of ANOVA. Further, since you have multiple dependent variables, maybe you want to have a look at a MANOVA, although I do not know whether this makes sense in your situation. Regardless what analysis you use you should check the assumptions.

If you do not have R you can simply put the code here.

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Solved – ANOVA vs. T-test for two groups

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