## Sample Size calculator for the Cochran-Armitage trend test in proportions

The usual chi-squared test statistic, assuming no ordinal ordering of the treatment groups, can be partitioned into a component explained by the ordering (through the Cochran-Armitage test statistic) and a remainder (Agresti, 2002). Thus when a significant portion of the ordinary chi-square is explained by the ordering then the Cochran-Armitage test is a more powerful test of an effect of treatment than the ordinary chi-square.

The Cochran-Armitage trend test, tests for a non-zero slope in a relationship of the form *p _{i }= a + bd_{i}*, relating the proportion

*p*responding within a group to a numeric

_{i}*d*associated with the groups being studied. The Cochran-Armitage test is useful when you have increasing dose level groups and one could use the dose levels as the

_{i}*d*. Alternatively one could use a nominally ordered variate such as

_{i}*d*=

_{i}*i*for

*i = 0*to

*k-1*(

*i=0*for the control group with a total of k groups). The nominally ordered variate can also be used when treatment groups can be ordered in terms of increasing efficacy such as an ordering on increasingly potent combinations of multiple drugs when looking at tumor response in oncology.

To design an appropriately sized study to assess a trend in proportions, one needs estimates of anticipated proportions responding in each of the ordered treatment arms, the sample size in each group as a multiple of that in the control and the numeric representing the dose/group level. One needs a false positive (alpha) rate or the probability of concluding in favor of a trend when there is none and one needs a false negative rate (beta) which is the probability of concluding in favor of no trend in proportions when there is a trend.

The default example in the calculator involves an alpha level of 5%, a one sided test (trend tests by definition are one-sided but one could use alpha = 2.5% for a two-sided test), a beta of 10% (or power of 90%) and probabilities of response in the control, 100 mcg and 200 mcg dose groups of 5%, 10% and 15%. Randomization is assumed to be 1:1:1.

Entering these inputs in the blue boxes of the calculator produces the results in the two columns labeled “Sample Size per group –uncorrected” and “Sample Size per Group with Continuity Correction”. The continuity corrected results are a correction for the discrete dose levels. The continuity corrected sample sizes are valid only for equally spaced dose levels/group levels. This study would need 162 patients per group.

For some theory on the calculations from Nam(1987) see the following attachment.

**Edit the blue cells in the spreadsheet and enter your data and the calculations in the spreadsheet will refresh. You can enter as many as 10 groups. For fewer groups, as in the default example, enter details, in order, leaving the remaining cells blank.**