# Comparison of frequency rates

Patients with chronic schizophrenia will take some antipsychotic drug during some period of time. The frequency of schizophrenia for each patient will be recorded before and after that period. I would like to determine the sample size to achieve a certain power. Is there a simple hypothesis test to assess the drug effect from which the power formula can be derived?

• What do you mean by the frequency of schizophrenia if all the subjects have chronic schizophrenia?
– swmo
Feb 26, 2015 at 12:04
• Example: Patient A had 2 episodes per month before treatment and 1 episode per month after treatment. The frequency has changed from 2 to 1. Feb 27, 2015 at 7:22

If patients can have more than one episode in a given period, then you are likely looking at a Poisson count distributed process*.

A Poisson regression framework allows you to calculate rates per unit of person-time (e.g. 2 events per month, 1 event per month) and also would allow each participant to have different amounts of follow up time, if needed (e.g. Patient A has four weeks of follow-up data; Patient B has six weeks.)

There are many papers out there that discuss sample size considerations for Poisson regression, but I have yet to see a "formal" calculator in any of the statistics packages that I use regularly (R/SAS/Stata), and generally I would approach this problem using simulation-based methods to determine a sample size to reach a particular level of estimation precision (e.g. confidence interval width on a rate or rate ratio) or power in its formal sense.

*Sometimes the patterning of outcomes can be more extreme, in which case alternative distributions like negative binomial might be more appropriate (e.g. most patients have 0 or 1 or 2 episodes; but substantial minority have 5/6/7/8...)

It seems to be that you are describing a test to determine a difference in proportions. Commonly, Fisher's exact test and the $\chi^2$ test are used in such scenarios. Other tests include McNemar's test and the binomial proportion test.

Generally the scenario is that one has two samples $y_1$ and $y_2$ and $\hat{y}_1$ and $\hat{y}_2$ are cases (i.e. schizophrenic in your instance) in each sample respectively. The null hypothesis is that the proportions, $p_i=\frac{\hat{y}_i}{y_i}$, are equal $H_0:p_1=p_2$. The differences amongst the tests have to do with approximations and other variations on the scenario (e.g. whether the number of samples were fixed ahead of time etc.).

Analytical derivations will depend on the test you chose (and may not exist). If you are using R, there are several libraries that could help you calculate the power including library(pwr) and library(Exact). Most statistical computing environments should have this ability.

• McNemar's test is a test for paired data. There's also a z test of proportions from independent samples. And in addition to Friedman's omnibus test for independent samples, there is Cochran's Q test for blocked samples with dichotomous outcomes. Mar 1, 2015 at 21:14
• @Alexis I suggested McNemar's test for this very reason as the patients would be paired pre- and post-treatment. Mar 2, 2015 at 1:29
• Also, see this previous answer for a great illustration of McNemar's test. Mar 2, 2015 at 1:33