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There are many alternatives to the t-test as this is not the only choice for testing hypotheses about average expression levels. However, they all have similar problems. They work well with either independent samples or paired samples, but not both together.

• Permutation test
• bootstrap test
• Wilcoxon test

You have already seen the permutation test. The bootstrap test is similar to the permutation test but instead of permuting the labels, you put all of the observations into a single pool and simulate by selecting \(n_X\) samples to be X and \(n_Y\) samples to be Y with replacement.  This is done many times to obtain an estimate of the distribution of \(\bar{X}-\bar{Y}\) when the null hypothesis is true. The Wilcoxon test is often used when the underlying populations are not normal.  It proceeds by taking all the data and replacing the numbers by the ranks from 1 (for the lowest number) to \(x_X+n_Y\) for the largest number and then doing a t-test on the ranks.  However, although it is popular, it turns out not to perform well with high throughput data for 3 reasons:

1) It lacks flexible extensions to complex samples (e.g. with both dependent and independent samples)

2) Although it has similar power to the t-test in large samples, it lacks power in small samples leading to false negatives.

3) In many situations in which the Wilcoxon test is used, the t-test is robust to non-normality and is more powerful.

Tests of Proportion

The other kinds of tests are test of proportions and we will see this quite a lot. For example, when you do a GWAS study using SNPs or some other genetic marker and you have three genotypes, (major/major, minor/minor and major/minor alleles), patients and controls, you might want to know whether the distribution of genotypes is the same in the two groups or whether it differs. This will involve tests of two-way tables, and will be covered  later in the course.