Question

A common tool in laboratory medicine and biology is the microarray, a device that can measure the expression levels of thousands of genes at once. So, for example, an investigator might collect samples from normal subjects and subjects with cancer in the hopes of finding genes that are significantly associated with cancer. The investigator is therefore testing a null hypothesis for every gene on the array. Suppose that a given array has 2,000 genes, of which 20 are truly associated with cancer. Suppose further that the investigator’s hypothesis tests have a Type I error rate of 5% and a Type II error rate of 20%.

a.Out of the 2,000 hypothesis tests that the investigator carries out, how many are type I errors?

b.How many are type II errors?

c.How many times did the investigator correctly reject the null hypothesis?

d.What was the investigator’s false discovery rate? (i.e. what percentage of hypothesis tests that they rejected were correctly rejected?)

e.If, for each gene, a 95% confidence interval was calculated for the association between the gene and cancer status, how many of those confidence intervals would contain the true association for that gene?

Answer 1

Number of genes = 2000

a. Number of hypothesis tests having type I errors = 2000 * 5% = 100

b. Number of hypothesis tests having type II errors = 2000*20% = 400

c. Number of times investigator correctly reject the null hypothesis = Total - Number of hypothesis tests having type II error = 2000 - 400 = 1600

d. Percentage of hypothesis tests that they rejected were correctly rejected = 1 - type II error

= 1 - 0.20 = 0.80

e. Number of confidence interval would contain the true association for that gene = Confidence interval * Total number of gene = 0.95*200 = 190