Two-way ANOVA

Analysis of variance, or simply, ANOVA, is a type of parametric statistical test used to determine if there’s a significant difference between the means or averages of three or more groups.

And significance is normally defined by a p-value of less than 0.05 or 5%.

Now when doing any parametric test, there are three key assumptions that we have to make about the population.

First, the sample population must have been recruited randomly. Choosing names randomly ensures that the people included in the study will have similar characteristics to the target population. This is important because that ensures that the results of the t-test can be applied to the target population - meaning it has good external validity!

The second assumption is that each individual in the sample was recruited independently from other individuals in the sample. In other words, no individuals influenced whether or not any other individual was included in the study.

For example, if two friends decided to get their blood pressures measured on the same day, and they were both included in the study, these two individuals would not be independent of each other and the second assumption would not be met.

Like random sampling, independent recruitment of individuals is important because it ensures that the sample population approximates the target population.

The third assumption is that the sample size is large enough to approximate the target population, which usually means having more than 20 people.

If it’s impossible to get a large sample size, then the sample population must follow a normal bell-shaped distribution for the characteristic being studied because that’s what we would expect to see in the target population.

Okay, now let’s say there are three medications available for lowering systolic blood pressure, and you want to figure out if any of the medications work differently than the others. Additionally, you want to figure out if the medications work differently for males and females.

So, let’s say that you find 6 people - 3 males and 3 females - who take Medication A for 6 weeks, and that afterwards the mean systolic blood pressure is 130 mmHg for males and 125 mmHg for females. Then, you find another 3 males and 3 females who have been taking Medication B - and afterwards their mean systolic blood pressure is 138 for males and 126 for females, and finally you find 3 males and 3 females who have been taking Medication C - and afterwards their systolic blood pressure is 132 for males and 125 for females.

We can arrange the numbers in a table like this, with sex on the top and medication type on the side, and each cell represents a mean systolic blood pressure value.

To figure out if medication type and sex have an effect on systolic blood pressure, we can use an ANOVA test.