13

Testing hypotheses about the ratio of two population parameters

Learning objectives

When you have finished this chapter you should be able to:

Describe the usual form of the null hypothesis in the context of testing the ratio of two population parameters

Outline the differences between tests of ratios and tests of differences. Interpret published results on tests of risk and odds ratios.

Testing the risk ratio

In Chapter 11 you saw that if the confidence interval for a sample risk ratio contains 1, then the population risk ratio is most probably not statistically significant, i.e. not significantly different from 1. This in turn means that the risk factor in question is not a statistically significant risk. You can also use the hypothesis test approach to find out whether any departure in the sample risk ratio from 1 is statistically significant, or is more likely due to chance. The null hypothesis

Medical Statistics from Scratch, Second Edition David Bowers

C 2008 John Wiley & Sons, Ltd

156 CH 13 TESTING HYPOTHESES ABOUT THE RATIO OF TWO POPULATION PARAMETERS

is that the population risk ratio equals 1, the alternate hypothesis is that it isn’t equal to 1. That is:

H0: population risk ratio = 1

H1: population risk ratio =1

In other words, if H0 is true, the risk factor in question does not significantly increase or decrease the risk for the condition or disease. If the associated p value is less than 0.05 (or 0.01), you can reject the null hypothesis H0, and conclude that the population risk ratio in question is statistically significant, and the risk factor in question is a statistically significant risk.

An example from practice

Table 13.1 is from a randomised trial into the efficacy of long-term treatment with subcutaneous heparin in unstable coronary-artery disease (FRISC II Investigators 1999), and shows the risk ratios, 95 per cent confidence intervals and p values for a number of clinical outcomes, in two independent groups, one group given heparin, the other a placebo.

As you can see from the p values in the last column, three out of the six risk ratios were statistically significant: death, myocardial infarction or both, at one month (p value = 0.048); death, myocardial infarction or revascularisation, at one month (p value = 0.001); and death, myocardial infarction or revascularisation, at three months (p value = 0.031). All three of these p values are less than 0.05, the remaining three are all greater than 0.05. Notice that these results are confirmed by the corresponding 95 per cent confidence intervals.

Table 13.1 Risk ratios, 95 per cent confidence intervals, and p-values, for a number of clinical outcomes, at 1 month, 3 months and 6 months, in two independent groups, one group given heparin and the other group a placebo. Reprinted courtesy of Elsevier (The Lancet, 1999, Vol No. 354, page 701–7)

MI = myocardial infarction.

Dalteparin (n = 1115), placebo (n = 1103).

158 CH 13 TESTING HYPOTHESES ABOUT THE RATIO OF TWO POPULATION PARAMETERS

Testing the odds ratio

Here the null hypothesis is that the population odds ratio is not significantly different from 1. That is:

H0: population odds ratio = 1

H1: population odds ratio =1

In other words, in the population, if H0 is true the risk factor in question does not significantly increase or decrease the odds for the condition or disease. Only if the p value for the sample odds ratio is less than 0.05, can you reject the null hypothesis, and conclude that the risk factor is statistically significant.

An example from practice

Table 13.3 is from an unmatched case-control study into the effect of passive smoking as a risk factor for coronary heart disease (CHD), in Chinese women who had never smoked (He et al. 1994). The cases were patients with CHD, the controls women without CHD. The study looked at both passive smoking at home from husbands who smoked, and at work from smoking co-workers. The null hypotheses were that the population odds ratio was equal to 1, both at home and at work, i.e. passive smoking has no effect on the odds for CHD. The

Table 13.3 Odds ratios, 95 per cent confidence intervals and p values, from an unmatched case-control study into the effect of passive smoking as a risk factor for coronary heart disease. The cases were patients with coronary heart disease, the controls individuals without coronary heart disease. Reproduced from BMJ, 308, 380–4, courtesy of BMJ Publishing Group

Adjusted for the other variables in the final model.

†Adjusted for the first five varibles above; odds ratios for these variables in the other models were essentially the same as those shown above and are not shown.