# Hypothesis testing of two standard mortality ratios

Probably this is an easy question, but I don’t seem to find an answer.

The standard mortality ratio (SMR) is defined as “the ratio of actual number of deaths in the group under study to the expected number of deaths based on the standard population rates that were applied to the study group”.

SMR = Observed death/Expected Death

Let’s say that I have SMR-1 and SMR-2 be two independent standard mortality ratios. I’m interested to construct hypothesis to test the difference between them. That’s

H0: SMR-1 = SMR-2 vs. H0: SMR-1 != SMR-2

Any idea?

This problem is about the hypothesis testing of the difference between two proportions.

The first thing we should do is to find the standard error of the difference between the two sample proportions.

Since the null hypothesis states that p1 = p2, we use a pooled sample proportion(p) to compute the standard error. The formulas are:

p = (p1*n1+p2*n2)/(n1+n2)

SE = sqrt[p(1-p) * (1/n1 + 1/n2)]

Then, we calculate the z which we want to test:

z = (p1 - p2)/SE

There should be a specific level of significance given by the question. You can find the z-critical in the z-test table.

Then compare the z we got with the z-crit. (You can also compare the p-value)

If our z is >= the positive z-crit or <= the negative z-crit, we can reject the hypothesis. (vice versa)

Hope this would help.

• This seems to be assuming the SMR cannot be larger than 1! May 14, 2020 at 11:14

While the answer above might be accurate for proportions, the complicating factor with SMR's is that the definition of the variable as a 'ratio' must allow for SMR's greater than 1.0. In other words if one hospital has less deaths than expected then the SMR will be <1.0. However if another hospital has more deaths than expected then the SMR is >1.0 and the formula for proportions is not acceptable nor will it work.