Ординатура / Офтальмология / Английские материалы / Using and Understanding Medical Statistics_Matthews, Farewell_2007

of stratification, and typically facilitate a more precise comparison than might otherwise be achieved. Before and after measurements are perhaps the most common illustration of naturally paired data; for example, we might wish to investigate the efficacy of an anti-hypertensive drug by using measurements of a patient’s blood pressure before and after the drug is administered. In another circumstance, if we had identical twins who differed with respect to smoking status, we might consider measuring several aspects of lung function in each twin in order to investigate the effect of smoking on lung function.

When data involve natural pairing, a common method of analysis is based on the assumption that the differences between the paired observations are normally distributed. If we let the random variables D1, D2, …, Dn represent these differences, e.g., Before-After, then we are assuming that Di N( d, d2) for i = 1, 2, …, n; d is the mean difference in response, and d2 is the variance of the differences. Provided this assumption is reasonable, then all the methods for a single sample of normally distributed data can be used to analyze the differences. For example, a natural hypothesis to test in this situation would be H: d = 0; i.e., the mean difference in response is zero. If the data provide evidence to contradict this hypothesis, then we would conclude that the factor which varies within each pair, e.g., smoking status, or use of the anti-hyper- tensive drug, has a real effect. We might also want to evaluate a suitable confidence interval for d in order to estimate the magnitude and direction of the effect which has been detected.

The immunological data which we have previously discussed do involve natural pairing since, for each subject, we have one assay result at each concentration level. If we compute the High-Low differences of the assay results for each patient, we could separately investigate the effect of concentration among hemophiliacs, and also among the controls. Although this question is not of primary interest to the study, we shall treat the data for the controls as we have described in order to illustrate the analysis of paired observations.

The original assay results may be found in table 9.1. The difference, High – Low, was calculated for each control subject, yielding 33 observations. A histogram of this sample of differences is given in figure 9.2a. The distribution of the differences is quite spread out, and one might be reluctant to assume that the distribution is normal. An alternative approach would be to consider the differences in the logarithms of the assay results. This is equivalent to considering the logarithm of the ratio of the assay results. A histogram of the logarithm differences is given in figure 9.2b, and it can be seen that the shape of this histogram is considerably closer to the characteristic shape of a normal probability curve than that of the histogram in figure 9.2a. Therefore, we will analyze the differences between the logarithms of the assay results at the two concentrations, and assume that these differences are normally distributed.

Fig. 9.2. Histograms of the difference in immunological assay results at high and low concentrations for the control sample of 33 individuals. a Original data. b Logarithms of the original data.

For these logarithm differences we obtain d = 0.47 and s = 0.33. The observed value of T, the statistic for testing H: d = 0, is

As we might expect, this large observed value tells us that the data provide strong evidence to contradict the hypothesis that the mean logarithm difference is zero (p ! 0.001); we therefore conclude that concentration influences the assay result.

The magnitude of this effect, i.e., the change with concentration, is indicated by the estimated mean of 0.47, and also by the 95% confidence interval for d, which is (0.35, 0.59). In terms of the original assay measurements obtained on each control subject at the Low and High concentration levels, these results represent a corresponding estimate for the ratio of High to Low concentration assay results of e0.47 = 1.6; the corresponding 95% confidence interval is (e0.35, e0.59) = (1.42, 1.80).

In many circumstances, it is not possible to make comparisons which are based on naturally paired data. Methods which are appropriate for this more common situation are discussed in the next section.

9.4.2. Unpaired Data

If the data are not naturally paired, then the comparison of interest usually involves two separate, independent samples from two (assumed) normal distributions. We can portray this situation, symbolically, by using the random variables X1, X2, …, Xn to represent one sample and Y1, Y2, …, Ym to represent the second sample; then Xi N( x, 2) for i = 1, 2, …, n and Yj N( y, 2) for j = 1, 2, …, m. For example, the X’s may be the results of lung function tests for a random sample of smokers, and the Y’s may be a corresponding set of observations on nonsmokers. Notice that whereas the means, x and y, of the two distributions may differ, the corresponding variances are equal to the same value, 2. The primary question of interest in such a situation is usually a comparison of the means, x and y.

obvious quantity on which to base a comparison of x and y is X – Y, the differ-

ence in sample means. Conveniently, X – Y has a normal distribution with mean ( x – y) and variance 2(1/n + 1/m). When is known, a confidence interval for ( x – y) or a significance test concerning this difference can be based on the normal distribution of X – Y. In most cases, the value of will not be known. However, if we replace by a suitable estimate, s, then it can be shown that

X Y ( )

+x y t(n+m 2) .

s 1/n 1/m

Table 9.6. Comparing the mean immunological assay results, at low and high concentrations, in the hemophiliac and control populations

a Since the 5% critical value for t(45) is not given in table 9.4, we use the corresponding value fort(40) , which is 2.021.

The significance level for a test of the hypothesis that x = y is

Pr( t(45) 6 to), where to is the observed value of the test statistic. Since there is no row in table 9.4 corresponding to 45 degrees of freedom, we compare to with

the 5% critical values for both 40 and 60 degrees of freedom. These values are 2.021 and 2.00, respectively. For the low concentration results, the value of to is 2.00, which corresponds, approximately, to a significance level of 0.05. The observed value of the test statistic for the high concentration assays is 1.81, which lies between the 5 and 10% critical values for both 40 and 60 degrees of freedom; thus, the p-value associated with this test of significance is between 0.05 and 0.10. From these results we conclude that there is suggestive evidence, at both low and high concentrations, for different mean assay results in the two populations. The 95% confidence intervals for ( x – y) which are given in table 9.6 include zero near the upper endpoint of each interval, and are consistent with this conclusion concerning x and y.

Notice that the evidence concerning different population mean values is somewhat stronger in table 9.6 than that which might be derived from an informal comparison of the confidence intervals presented in table 9.5. This is largely because the scale factor 1/n + 1/m in the formula for the 95% confidence interval for ( x – y) is considerably smaller than the sum of the scale

Table 9.7a. Selected 5% critical values of the F-distribution; the table gives values of the number ro such that Pr(Fn,m 6 ro) = 0.05

Abridged from Pearson ES, Hartley HO: Biometrika Tables for Statisticians. London, Biometrika Trustees, Cambridge University Press, 1954, vol 1, p 171. It appears here with the kind permission of the publishers.

Table 9.8. Testing the assumption of equal variances for the distribution of immunological assay results, at low and high concentrations, in the hemophiliac and control populations

Pr(R 6 ro) = 2Pr(F32,13 6 1.04) > 0.10b

a The 5% critical values for F12,30 and F12,40 are 2.09 and 2.00, respectively. b The 5% critical values for F12,13 and FG,13 are 2.60 and 2.21, respectively.

tion given in table 9.6. Details of the calculations which are involved in testing the common variance assumption are presented in table 9.8. The equal variances assumption is not contradicted by the data at either concentration level.

If the data suggest that the variances in two unpaired samples are apparently different, then it is important to ask whether, in light of this information, a comparison of means is appropriate. A substantial difference in the variances may, in fact, be the most important finding concerning the data. Also, when the variances are different, the difference in means no longer adequately summarizes how the two groups differ.

If, after careful consideration, a comparison of means is still thought to be important, then the appropriate procedure is not uniformly agreed upon among statisticians. The reasons for this disagreement are beyond the scope of this book, but approximate methods should be adequate in most situations. Therefore, we suggest the simple approach of calculating a confidence interval for x and a second interval for y. If these intervals do not overlap, then the hypothesis that x is equal to y would also be contradicted by the data. Substantial overlap suggests consistency with the null hypothesis while minimal overlap corresponds to a problematic situation. In this latter case, or if the na-