10

Estimating the difference between two population parameters

Learning objectives

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

Give some examples of situations where there is a need to estimate the difference between two population parameters.

Very briefly outline the basis of estimation of the difference between two population means using methods based on the two-sample t test1 (for independent populations) and the matched-pairs t test (for matched populations).

Very briefly outline the basis of estimation of the difference between two population medians using methods based on the Mann-Whitney test (for independent populations) and the Wilcoxon test (for matched populations).

Interpret results from studies that estimate the difference between two population means, two percentages or two medians.

Demonstrate an awareness of any assumptions that must be satisfied when estimating the difference between two population parameters.

1Throughout this chapter we will be looking at methods of estimation based on various hypothesis tests. I will begin to discuss hypothesis tests properly in Chapter 12.

Medical Statistics from Scratch, Second Edition David Bowers

C 2008 John Wiley & Sons, Ltd

Figure 10.1 SPSS output (abridged) for 95 per cent confidence interval (last two columns) for the difference between two independent population mean birthweights, using samples of 30 infants born in maternity units and 30 at home (data in Table 10.1)

In other words, if you want to know if there is a statistically significant difference between two population means, calculate the 95 per cent confidence interval for the difference and see if it contains zero.

It is possible to calculate these confidence intervals by hand, but the process is timeconsuming and tedious. Fortunately, most statistics programs will do it for you. Since difference between independent population means is one of the most commonly used approaches in clinical research, you might find it helpful to see some of the output from SPSS and Minitab for this procedure.

With SPSS

Using the birthweight data in Table 10.1, SPSS produces the results (abridged3) shown in Figure 10.1. These tell us that the difference in the two sample mean birthweights is −82.17 g. The sign in front of this value depends on which variable you select first in the SPSS dialogue box. SPSS subtracts the second variable selected (home births in this case) from the first (maternity unit births). This result means that the sample mean birthweight was 82.17 g higher in the home birth infants.

SPSS calculates two confidence intervals, one with standard deviations4 assumed to be equal, and one with them not equal. The 95 per cent confidence interval shown in the last two columns

3I’ve removed material that is not relevant.

4Both Minitab and SPSS refer to equality of variances.

is (−114.9 to 279.2)g, the same in both cases. SPSS tests for equality of the standard deviations (or variances), using Levene’s test. The assumption is that they are the same. We will discuss tests in Chapter 12.

Since this confidence interval includes zero, you can conclude that there is no statistically significant difference in population mean birthweights of infants born in a maternity unit and infants born at home.

With Minitab The Minitab output, which confirms that from SPSS, is shown in Figure 10.2. The 95 per cent confidence interval is in the second row up.

An example from practice

Table 10.2 is from a cohort study of maternal smoking during pregnancy and infant growth after birth (Conter et al 1995). The subjects were 12 987 babies who were followed up for three years after birth. Of these, 10 238 had non-smoking mothers, 2276 had mothers who had

Figure 10.2 Minitab output for 95 per cent confidence interval for the difference between two independent population mean birthweights, using samples of 30 infants born in maternity units and 30 at home. Note that Minitab uses the word ‘mu’ to denote the population mean, normally designated as Greek μ