Estimating the ratio of two population parameters

If two sample means have a ratio of 1, this tells us only that the means are the same size in the sample. If the sample ratio is different from 1, you need to check whether this is simply due to chance, or if the difference is statistically significant – one mean is bigger than the other. You can do this with a 95 per cent confidence interval for the ratio of population means. And here’s the rule:

If the confidence interval for the ratio of two population parameters does not contain the value 1, then you can be 95 per cent confident that any difference in the size of the two measures is statistically significant.

Compare this with the rule for the difference between two population parameters, where that rule is that if the confidence interval does not contain zero, then any difference between the two parameters is statistically significant.

An example from practice

Look again at the last column in Table 9.1, which shows a number of outcomes from a randomised trial to compare integrated versus conventional care for asthma patients. The last column contains the 95 per cent confidence intervals for the ratio of population means for the treatment and control groups. You will see that all of the confidence intervals contain 1, indicating that the population mean number of bronchodilators used, the number of inhaled steroids prescribed and so on, was no larger (or smaller) in one population than in the other.

The sample ratio furthest away from 1 is 1.31, for the ratio of mean number of hospital admissions, i.e. the sample of integrated care group patients had 31 per cent more admissions than the conventionally treated control group patients. However, the 95 per cent confidence interval of (0.87 to 1.96) includes 1, which implies that this is generally not the case in the populations.

Confidence interval for a population risk ratio

Table 6.1 showed the contingency table for a cohort study into the risk of coronary heart disease (CHD) as an adult, among men who weighed 18 lbs or less at 12 months old (the risk factor). On p. 104 we derived a risk ratio of 1.93 from this sample cohort. In other words, men who weighed 18 lbs or less at one year, appear to have nearly twice the risk of CHD when an adult, as men who weighed more than 18 lbs at one year. But is this true in the population of such men, or no more than a chance departure from a population ratio of 1? You now know that you can answer this question by examining the 95 per cent confidence interval for this risk ratio.

The 95 per cent confidence interval for the CHD risk ratio turns out to be (0.793 to 4.697).1 Since this interval contains 1, you can conclude, that despite a sample risk ratio of nearly 2, that

1The calculation of confidence intervals for risk ratios and odds ratios is a step too far for this book. Those interested in doing the calculation by hand can consult Altman (1991) who gives the necessary formulae.

ratio = 0.41), and severely depressed (risk ratio = 0.39), compared to men with low levels of depression. Neither of the confidence intervals, (0.22 to 0.77) and (0.18 to 0.88), includes 1, indicating statistical significance. However, after adjusting for possible confounding variables, the adjusted risk ratios are 0.58 and 0.54, and are no longer statistically significant, because the confidence intervals for both risk ratios, for moderate depression (0.28 to 1.17), and severe depression (0.22 to 1.31), now include 1.

Exercise 11.1 Table 11.2 is from the same cohort study referred to in Exercise 8.9, to investigate dental disease, and risk of coronary heart disease (CHD) and mortality, involving over 20 000 men and women aged 25–74, who were followed up between 1971– 4 and 1986–7 (DeStefano et al. 1993).

The results give the risk ratios (called relative risks here) for CHD and mortality in those with a number of dental diseases compared to those without (the referent group), adjusted for a number of possible confounding variables (see table footnote for a list of the variables adjusted for).

Briefly summarise what the results show about dental disease as a risk factor for CHD and mortality. Note: the periodontal index (range from 0-8, higher is worse) measures the average degree of periodontal disease in all teeth present, and the oral hygiene index (range 0-6, higher is worse) measures the average degree of debris and calculus on the surfaces of six selected teeth.

Confidence intervals for a population odds ratio

Table 6.2 showed the data for the case-control study into exercise between the ages of 15 and 25, and stroke later in life. The risk factor was ‘not exercising’, and you calculated the sample crude odds ratio of 0.411 for a stroke, in those who hadn’t exercised compared to those who

Table 11.2 Adjusted risk ratios for CHD and mortality among those with dental disease compared to those without dental disease . Reproduced by permission of BMJ Publishing Group. (BMJ, 1993, Vol. 306, pages 688–691)

*Adjusted for age, sex, race, education, poverty index, marital state, systolic blood pressure, total cholesterol concentration, diabetes, body mass index, physical activity, alcohol consumption, and cigarette smoking.

†Excluding those with missing data for any variable and, for periodontal index and hygiene index, those who had no teeth.

had (see p. 105). So the exercising group appear to have under half the odds for a stroke as the non-exercising group. However, you need to examine the confidence interval for this odds ratio to see if it contains 1 or not, before you can come to a conclusion about the statistical significance of the population odds ratio.

SPSS produces an odds ratio of 0.411, with a 95 per cent confidence interval of (0.260 to 0.650). This does not contain 1, so you can be 95 per cent confident that the odds ratio for a stroke in the population of those who did exercise compared to the population of those who didn’t exercise is somewhere between 0.260 and 0.650. So early-life exercise does seem to reduce the odds for a stroke later on. Of course this is a crude, unadjusted odds ratio, which takes no account of the contribution, positive or negative, of any other relevant variables.

An example from practice

Table 11.3 shows the results from this same exercise/stroke study, where the authors provide both crude odds ratios and ratios adjusted for a number of different variables (Shinton and Sagar 1993).

We have been looking at exercise between the ages of 15 and 25, the first row of the table. Compared to the crude odds ratio calculated above of 0.411, the authors report an odds ratio for stroke, adjusted for age and sex, among those who exercised compared to those who didn’t exercise, as 0.33, with a 95 per cent confidence interval of (0.20 to 0.60). So even after the effects of any differences in age and sex between the two groups has been adjusted for, exercising remains a statistically significant ‘risk’ factor for stroke (although beneficial in this case). Adjustment for possible confounders is crucial if your results are to be of any use, and I will return to adjustment and how it can be achieved in Chapter 18.

Table 11.3 Odds ratios for stroke*, according to whether, and at what age, exercise was undertaken by patients, compared to controls without stroke. Reproduced by permission of BMJ Publishing Group. (BMJ, 1993, Vol. 307, pages 231–234)

*Adjusted for age and sex

Exercise 11.2. (a) Explain briefly why, in Table 11.3, age and sex differences between the groups have to be adjusted for. (b) What do the results indicate about exercise as a risk factor for stroke among the 25–40 years and 40–55 years groups?

Exercise 11.3. Refer back to Table 1.7, the results from a cross-section study into thrombotic risk during pregnancy. Identify and interpret any statistically significant odds ratios.

VI

Putting it to the Test