# Testing statistical significance of three percentages with ANOVA?

I want to compare the mortality rates of three groups. Can I use ANOVA? If so, how will I deal with the percentages?

• The biggest problem will be that the variances won't be equal. There are more appropriate tests for count data (e.g. number of deaths are often modelled as Poisson or binomial) that will better deal with the variance issue, and when the expected counts are small, with the discreteness as well. For example, if the three groups are internally fairly homogeneous (not across widely differing ages, for example), perhaps a chi-squared test would be suitable; alternatively if you know the source(s) of heterogeneity (again, like age groups) you could again do a chi-squared test, or some GLM. Mar 26, 2016 at 7:39

## 1 Answer

Usually when people say ANOVA, they assume that the response being modeled has a Normal distribution, which mortality rates (or any quantity that is bounded between $0$ and $1$) does not. That being said, depending on the field, lots of people seem to ignore this failed assumption and proceed with ANOVA anyways.

If you would like to try to be formally correct, then the most straightforward approach (if I understand your situation correctly) would be to perform a logistic or generalized linear model regression. When you do this, you model the conditional mean of the response given the group that the particular response belongs to using a dummy coding.

So, based on your description (correct me if I am wrong), it seems that you have access to the number of members of three different groups and you have access to the number of members of each of those groups who died in a given time frame. If this is in fact the case, then we can let $Y_i$ be the $i$th observation which will be either a $1$ (indicating death) or a $0$. Furthermore let $X_{i1}$ be 1 if observation $i$ belongs in group $1$, and $0$ other wise, and $X_{i2}$ be 1 if observation $i$ belongs to group $2$ and $0$ otherwise (this would make group $3$ the reference group; there are also other valid dummy variable codings). Then in doing a logistic regression (which is implemented in most statistical software available) we assume that there is some link function g such that $$g(E[Y_i|\hbox{group membership of obs. i}]) =\beta_0 +X_{i1}\beta_1 +X_{i2}\beta_2$$ Because $Y_i$ can either be $0$ or $1$, its distribution will be Bernoulli and thus the conditional mean of $Y_i$ given which group observation $i$ belongs to will just be the proportion of deaths in that group. Additionally, because the response is bernoulli, the most common choice for $g$ is the logit function, i.e. $g(p) = log\left(p/(1-p)\right)$.

The mortality rates of the three groups would be compared by testing whether or not $\beta_1$ and/or $\beta_2$ were zero or not. If, for example your test fails to reject the hypothesis that $\beta_{i1}=0$, then that suggests that there really is no difference between the mortality rate of group $1$ and group $3$.

I hope that helps!