# Compare Percentiles

An alternative approach is to compare percentiles or quantiles across groups. Quantile regression is such an approach (Koenker and Bassett, 1978). It is used to estimate conditional quantile functions by minimizing the weighted sum function. Redden et al. (2004) recently developed a simplified quantile regression method using logistic regression which has an appropriate type I error rate and good power. For comparing maximum lifespan, one needs to first find the 90th percentile for all animals combined. Then the animals are categorized into two categories (i.e., constituting a binary variable): lifespan longer than the 90th percentile versus lifespan shorter than the 90th percentile. This binary variable can then be regressed on the predictors including the group assignment by logistic regression to test for differences in maximum lifespan. Although Redden et al.'s method performs very well when the sample size is reasonably large, this approach is not informative when there is a cell count of zero or near zero in the two-by-two contingency table, which is often likely given the sample sizes in animal experiments.

It should be noted that Redden et al.'s approach represents a form of categorical data analysis. The methods suitable for contingency tables with small sample sizes, such as Fisher's exact test, may be a better choice in this context. As shown by Wang et al. (2004), apart from Fisher's exact test, an ordinary chi-squared test, Boschloo's test, or a score test could also be used to test for significance. Mehrotra et al. (2003) thoroughly studied the properties of these four methods and found that when the sample size is small, the ordinary chi-squared test inflates the type I error rate; the Fisher's exact test is conservative but has good power; and the Boschloo's test and score test have relatively appropriate type I error rate and good power even when the sample sizes are as small as 50 per group (i.e., only 10 subjects in two groups having a lifespan equal to or longer than the 90th percentile). In summary, Fisher's exact test, Boschloo's test, and the score test are valid and suitable when testing for significant differences in maximum lifespan and they each possess reasonably good power. Wang et al.'s (2004) simulation study also provides practical guidelines for choosing the appropriate sample size for the given effect size and type I error rate. 