# How to test whether a ratio is different from one using bootstrap?

I have a dataset including two groups of individuals. I get the median of a measurement (say weight) for each group, say W1 and W2 for group 1 and 2, and then get the ratio of the medians W1/W2. How can I test whether the ratio W1/W2 differs from one using bootstrap?

Please do not offer alternative methods to compare weights of the two groups such as t-test or wilcoxon test; I use weight here just for simple description.

• Are you just asking how to use the bootstrap to test a hypothesis? – gung - Reinstate Monica Jun 18 '15 at 16:36
• @gung, not really, but bootstrap is the only I can think of for this test. Please let me know if you have different ways to test a ratio. Thanks. – ZZG Jun 19 '15 at 2:42

Set $\alpha=0.05$. Suppose there are $n_1$ and $n_2$ people in groups $1$ and $2$ respectively. Calculate $\frac{M_1}{M_2}_{obs}$ from your data. The hypotheses are: $$H_0: \frac{M_1}{M_2} = 1$$ $$H_a: \frac{M_1}{M_2} \neq 1$$
First ensure that the medians of both groups are equal. Draw a bootstrap sample of size $n_1$ from group $1$ and a bootstrap sample of size $n_2$ from group $2$. Calculate $\frac{M_{1}^{*}}{M_{2}^{*}}$ from these samples and repeat for $1000$ times. The $p$-value is:
$$\text{p-value} = \frac{\text{number of times} \frac{M_{1}^{*}}{M_{2}^{*}} > \frac{M_1}{M_{2}}_{obs} \ \text{and number of times} \ \frac{M_{1}^{*}}{M_{2}^{*}} < \frac{M_1}{M_{2}}_{obs}}{1000}$$
If $p < 0.05$ then reject $H_0$.
• Your solution always gets a p-value = 1, because one of the two relationships $M^∗_1/M^∗_2 > M_1/M_2$ and $M^∗_1/M^∗_2 < M_1/M_2$ must be always true, because of noise, the equality rarely appears. – ZZG Jun 19 '15 at 2:39