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.

Specifically, we would use a two-way ANOVA test, because we are testing two factors - medication type and sex - that each have multiple groups within them.

There are three groups in the medication factor - which are A, B, and C - and we’ll use two groups for sex - male and female.

A two-way ANOVA test has six hypotheses. The first three are null hypotheses.

The first null hypothesis says that there’s no difference in systolic blood pressure for people taking different medication types.

The second null hypothesis says that there is no difference in systolic blood pressure for males and females.

The third null hypothesis says that there is no interaction between medication type and sex.

Interaction means that one factor influences the relationship between the second factor and the outcome. So, in this example, interaction would mean that the effect of medication on systolic blood pressure is different for males and females.

The alternate hypotheses are the opposite of the null hypotheses.

The first one says that there is a difference in systolic blood pressure for people taking different medication types; the second one says that there is a difference in systolic blood pressure for males and females; and the third one says that there is an interaction between medication type and sex.

Now, there are seven steps to test these hypotheses.

The first step is to calculate the mean of each factor, so the mean for all the medication types and the mean for both sexes.

To find the overall row mean systolic blood pressure for the medication A group, we add up the mean blood pressures for both males and females in that group, and divide by 2. So, 130 plus 125, divided by 2, is 127.5. We can do the same thing for the medication B and C groups. So, 138 plus 126, divided by 2, is 132. And 132 plus 125, divided by 2, is 128.5.

Finally, we can find the grand mean - which is the mean blood pressure measurements for all the medication groups - by adding up each row mean and dividing by 3. So 127.5 plus 132 plus 128.5, divided by 3, gives us a grand mean of 129.

To find the overall column means for the second factor, we add up the mean blood pressures in each medication group and divide by 3. So for males, 130 plus 138 plus 132, divided by 3, is 133.3. And for females, 125 plus 126 plus 125, divided by 3, equals 125.3. Notice that if we add up the column means for both sexes and divide by 2, it equals 129, which is the grand mean.

The second step is to find the total sum of squares, which tells you how much variation there is in the dependent variables. In other words, it compares every single individual in the study to the grand mean.

If the total sum of squares is large, it means that the individuals in the study, at every time point, are very spread out from one another; and if the total sum of squares is small, it means that the individuals in the study, at every time point, are clustered together.

To get the sum of squares, we start by subtracting the grand mean from each individual’s systolic blood pressure measurement, and then squaring it, which is called the squared difference. Then, we add up all the squared differences to get the total sum of squares. There are 18 people in this study, so we will have 18 squared differences, but to save time, let’s just do the first three.

The first person in the study is a female who took Medication A, and her systolic blood pressure is 124. So, we subtract the grand mean, which is 129, from 124, which equals negative 5. Then we square it, so 5-squared is 25. The second person is also a female who took Medication A, and her blood pressure is 126. So, 126 minus 129 is negative 3, squared, is 9. For the third person, who is also a female that took Medication A, we do 125 minus 129, which is negative 4, and then square it, which is 16. So, if we add up all of the squared differences for everyone in the study, we get a total sum of squares of 743.

The third step is to find the between-group variation, which is also called the sum of squares-between, or the SSB.

The sum of squares-between is a measure of how similar each group’s mean is to the grand mean. And, since we have two factors, we have to find the sum of squares-between for each factor and them add them together to get the total sum of squares-between.

To get the sum of squares-between for the medication factor, we start by subtracting the grand mean from each group’s row mean and squaring it, to get the squared difference. Then, we multiply the squared difference by the number of people in that group.

So, let’s start with the medication factor. For medication 1, the row mean is 127.5. So, the squared difference is 127.5 minus 129, squared, which is 2.25. Then, since there are 6 people in the medication group, we multiply 2.25 by 6, which equals 13.5. We do that same thing for the other two medication groups - so, 132 minus 129, squared, times 6, is 54; and 128.5 minus 129, squared, times 6, is 1.5. When we add up all the squared differences for all the medications, we get 13.5 plus 54 plus 1.5, which equals 69.

We do a similar process to get the sum of squares-between for the sex factor, but this time we use the column means.

For males, we subtract the grand mean from the column mean, then square it. So, 133.3 minus 129 is 4.3, and 4.3-squared is 18.5. Then we multiply it by the number of males in the study, which is 9, so 18.5 times 9 is 166.5. For females, the equation is the same, but we substitute in their column mean, which is 125.3. So 125.3 minus 129, squared, times 9, is 123. And when we add up both the sum of squares-between for males and females we get 289.5.

The fourth step in the ANOVA calculation is to find the within-group variation, which is also called the sum of squares-within, or SSW.

The sum of squares-within is a measure of how similar each individual blood pressure measurement is from its own group mean.

The sum of squares-within is sometimes called the sum of squared-error, because it’s the variation in systolic blood pressure that can’t be explained by either the different medication or the different sex, so it’s sort of this unexplained error in the blood pressure measurement. Basically, it’s the variation caused by individual characteristics, like age or genetic differences.

To find the sum of squares-within, we divide the individual blood pressure measurements into two tables - one for males and one for females. Now, let’s work with the male table first. For the sum of squares-within, you start by finding the squared differences for each person in one group, and to do this, you take each individual blood pressure measurement and subtract that group’s mean - which is the row mean - and then square it. For example, let’s just take the first 3 systolic blood pressure, which are the measurements in the Medication A group. So, it’s 122, 136, and 133. Since the row mean for Medication A is 130, you subtract 130 from each individual measurement, so 122 minus 130 is negative 8, 136 minus 130 is 6, and 133 minus 130 is 3. Then you square each number and add them all together to get the squared difference. So, when you add up negative 8-squared, or 64, and 6-squared, or 36, and 3-squared, or 9, you get 109.