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

Table 20.2. Summary data from 11 randomized trials of prolonged antiplatelet therapy

Adapted from Peto et al. [64]. It appears here with the kind permission of the pub-

lisher.

An appropriate stratified logistic regression model that is similar to equation (11.2) would correspond to the equation

where the subscript i indexes the 11 studies. This version of the logistic regression model assumes that the overall death rate can vary arbitrarily from study to study; however, the effect of the treatment, which is measured by b, is the same in all studies. Such a stratified logistic regression model is foundational for a meta-analysis because allowing for variation in the pattern of outcomes from study to study is essential in order to avoid the type of problem that we identified in §11.4. Such study heterogeneity is seldom of particular interest itself.

If we adopt this model to analyze the data summarized in table 20.2, the estimated value of b is –0.28, with a standard error of 0.04. The corresponding estimated odds ratio is exp(–0.28) = 0.75. Since the magnitude of the estimate is seven times larger than its standard error, a test of the null hypothesis b = 0, i.e., antiplatelet therapy has no effect on the death rate, is highly significant. The 95% confidence interval for b is (–0.37, –0.20), and the corresponding in-

terval for the odds ratio associated with treatment is (0.69, 0.82). All the values in this interval represent a substantial treatment benefit.

This foundational logistic regression model is sometimes called a ‘fixed effect model’ because b, the effect of interest, is assumed to be constant, or fixed, across all 11 studies. We can relax this assumption by adopting the model

exp(ai + bi x) Pr(Y =1|x)= 1+exp(ai +bi x) ,

where the treatment effect parameter b is allowed to vary from study to study. This added flexibility is achieved by adding a subscript i on b as well as on the parameter a.

Such a model has two primary uses. The first involves using the model as a means of testing for heterogeneity of the treatment effect from study to study. This action corresponds to testing the null hypothesis that all the bis are the same, represented by the hypothesis bi = b0 for all i; the value b0 represents a common, unknown constant. Typically a test of this type would be calibrated against a 2 distribution with degrees of freedom equal to one less than the total number of studies. For a logistic regression model, this hypothesis can be investigated using a likelihood ratio test similar to the one mentioned in §16.6. Based on the data summarized in table 20.2, such a likelihood ratio test statistic has an observed value of 14.18 and should be compared with a 210 distribution. The resulting significance level is 0.17. Thus, the data do not provide any evidence for heterogeneity of the treatment effect between studies. We will not go into any greater detail here, since the issues concerning heterogeneity were discussed in the previous section.

The second use of this modified logistic regression model is based on assuming that the treatment effect, bi, is random. We previously introduced this random effects assumption in §14.3, although in that instance it was the intercept term, ai, that we assumed was random. In this case, it is the bis that are random and come from a common distribution; the resulting model is another type of ‘random effects model’. The normal distribution is a frequent choice for that common distribution of some random effect; in this case, we might suppose that the associated, unknown mean and variance are b and 2, respectively. Then the primary result stemming from the meta-analysis is an estimate of b, the mean of this common distribution for the random treatment effects. This estimated mean is then regarded as the ‘overall’ estimate of the treatment effect from the meta-analysis.

There has been considerable debate amongst statisticians as to whether a fixed or random effects model is more appropriate for a meta-analysis. Essentially, a fixed effect analysis will produce an estimate of the overall treatment

effect for all studies by weighting the information from each study on the basis of its total size. The random effects analysis explicitly introduces study-to- study variability; the estimated overall treatment effect is then some sort of mean parameter. The fixed and random effects overall estimates can differ because the random effects model will allocate a larger weight to smaller studies than would be justified solely on the basis of their sample size. Perhaps the more important difference between these two approaches is that the extra variability in the random effects model will be reflected in any confidence interval for the overall estimate. Therefore, such an interval for the overall estimated outcome measure from a random effects model will be wider, and hence more conservative.

For the data in table 20.2, the overall treatment effect estimated via a random effects model corresponds to an odds ratio of 0.73 with associated 95% confidence interval (0.66, 0.81). In this case, the interval estimate is only slightly larger than its fixed effects model counterpart, and the two estimates of the odds ratio are almost identical.

We will not express a preference for one method or the other in general, but hope that we have at least indicated the nature of the difference between these two approaches to meta-analysis. If any reader is forced to make a choice between them, perhaps our discussion will prove helpful. In many situations, if there is doubt, then undertaking both analyses to determine the practical consequences of such a choice is be wise.

20.5. Graphical Displays

Graphical displays are the most common method of presenting the results of a meta-analysis. A typical figure would include effect estimates and corresponding confidence intervals from all studies incorporated in the meta-anal- ysis, as well as the overall estimate of the effect measure and associated confidence interval.

Figure 20.1 presents two such displays, called caterpillar plots, for the fixed and random effects meta-analyses of the 11 studies summarized in table 20.2. These plots show estimated odds ratios and 95% confidence intervals for each individual study, as well as the overall estimate and associated 95% confidence interval. The first caterpillar plot is based on the results of a fixed effect analysis, and the second plot is derived from the random effects model that we discussed in the previous section. The only difference between the results from the two analyses occurs in the assigned weights and therefore also in the overall estimated odds ratio and its corresponding 95% confidence interval.

Fig. 20.1. Caterpillar plots of the meta-analysis of 11 randomized trials of prolonged antiplatelet therapy. a Fixed effects analysis; b random effects analysis.

The estimated odds ratios and confidence intervals from the individual studies are listed beside each caterpillar plot, along with the weight assigned to each individual study in the calculation of the overall estimate. These weights make clear the difference between the two analyses. For example, study 5 is

assigned a weight of 30% in the fixed effect analysis, but only 20% in the random effects analysis. The smallest trial, study 11, was assigned 0.7 and 1.1% weights in these two analyses, respectively.

20.6. Sensitivity

Our final comment concerning meta-analyses is that they typically involve a number of ‘choices’, and the robustness of any final conclusions to the decisions taken deserves to be investigated. The most critical of these choices probably relates to selecting individual studies to include in the meta-analysis, and the actual method of analysis to use. Carrying out a variety of analyses by varying these choices is certainly the simplest way to investigate the confidence that one can attach to any preferred meta-analysis.

A special case of cohort data is population-based, cause-specific incidence and mortality data which are routinely collected for surveillance purposes in many parts of the world. These data are usually stratified by year, age at death (or incidence) and sex and have been used to study geographic and temporal variations in disease. In certain instances, aggregate data on explanatory variables such as fat consumption or smoking habits may also be available for incorporation into the analysis of these vital data.

For many diseases, the collection of cohort data is both time-consuming and expensive. Therefore, one of the most important study designs in epidemiology is the case-control study. This design involves the selection of a random sample of incident cases of the study disease in a defined population during a specified case accession period. Corresponding comparison individuals (the controls) are randomly selected from those members of the same population, or a specified subset of it, who are disease-free during the case accession period. Information on the values of explanatory variables during the time period prior to case or control ascertainment is obtained at the time of ascertainment. These retrospective data are usually subject to more error in measurement than the prospective data of the cohort study; however, a case-control study can be completed in a much shorter period of time. The case-control design facilitates comparisons of disease rates in different subsets of the study population but, since the number of cases and controls sampled is fixed by the design, it cannot provide an estimate of the actual disease rates. An important variation in case-control designs involves the degree of matching of cases to controls, particularly with respect to primary time variables such as subject age. In §21.4 we present an example of a case-control study in which logistic regression is used to analyze the data.

21.3. Relative Risk Models

In chapter 13, we introduced proportional hazards regression as a method for modelling a death rate function. This same method can also be used to model the rate of disease incidence. If we denote the disease incidence rate at time t by r(t), then we can rewrite equation (13.1) as

where x = {x1, x2, ..., xk} refers to a set of explanatory variables to be related to

˜

the incidence rate. The function r0(t) represents the disease incidence rate at time t for an individual whose explanatory variables are all equal to zero.

To illustrate how the regression model (21.1) can be used in epidemiology, consider the cohort study reported by Prentice et al. [66]. This cohort study was based on nearly 20,000 residents of Hiroshima and Nagasaki, identified by population census, and actively followed by the Radiation Effects Research Foundation from 1958. Information concerning systolic and diastolic blood pressure, as well as a number of other cardiovascular disease risk factors, was obtained during biennial examinations. During the period 1958–1974, 16,711 subjects were examined at least once. In total, 108 incident cases of cerebral hemorrhage, 469 incident cases of cerebral infarction and 218 incident cases of coronary heart disease were observed during follow-up. The determination of the relative importance of systolic and diastolic blood pressure as risk indicators for these three major cardiovascular disease categories was a primary objective of the analysis of these data. The time variable t was defined to be the examination cycle (i.e., t = 1 in 1958–1960, t = 2 in 1960–1962, etc.). The use of this discrete time scale introduces some minor technical issues which need not concern us here.

Two generalizations of model (21.1) which were introduced in chapter 13 are of particular importance to the application of the model to cohort studies. Frequently, there are key variables for which the principal comparison in the analysis must be adjusted. For example, in the study of blood pressure and cardiovascular disease it is important to adjust for the explanatory variables age and sex. A very general adjustment is possible through the stratified version of Cox’s regression model, for which the defining equation is

Equation (21.2) specifies models for J separate strata, each of which has an unspecified ‘baseline’ incidence rate rj0(t), where j indexes the strata. As in the lymphoma example discussed in §13.3, the regression coefficients are assumed to be the same in all strata. Prentice et al. [66] used 32 strata defined on the basis of sex and 16 five-year age categories.

The second generalization of model (21.1) involves the use of time-depen- dent covariates. In this case the equation for the analysis model is

In their analysis of the cohort data, Prentice et al. used blood pressure measurements at examinations undertaken before time t as covariates. Thus, for example, table 21.1 presents regression coefficients for a covariate vector, X(t)

˜

= {S(t–1), D(t–1)}, which represents the systolic and diastolic blood pressure measurements, respectively, in examination cycle t – 1. The table includes the

Table 21.1. The results of a relative risk regression analysis of cardiovascular disease incidence in relation to previous examination cycle systolic and diastolic blood pressure measurements; the analyses stratify on age and sex

mates.

b The values in parentheses are significance levels for testing the hypothesis b = 0. Adapted from Prentice et al. [66]. It appears here with the permission of the pub-

lisher.

corresponding significance levels for testing the hypothesis bi = 0. Separate analyses are presented for each cardiovascular disease classification. The incidence of other disease classifications is regarded as censoring in each analysis. This is consistent with the assumption that the overall risk of cardiovascular disease is the sum of the three separate risks.

In table 21.1, the diastolic blood pressure during the previous examination cycle is the important disease risk predictor for cerebral hemorrhage, whereas the corresponding systolic blood pressure is the more important predictor for cerebral infarction and for coronary heart disease. From table 21.1, the estimated risk of cerebral hemorrhage for an individual with a diastolic blood pressure of 100 is

exp{0.0548(100–80)} = 2.99

times that of an individual whose diastolic pressure is 80. Relative risks comparing any two individuals can be calculated in a similar way for each of the disease classifications.

The assumption that the blood pressure readings during the previous examination cycle are indeed the important predictors can be tested by defining new regression vectors X(t) = {D(t – 1), D(t – 2), D(t – 3)} for the cerebral hem-

˜

orrhage analysis and X(t) = {S(t – 1), S(t – 2), S(t – 3)} for cerebral infarction

˜

and coronary heart disease. For a subject to contribute to these analyses at an incidence time t, all three previous biennial examinations must have been at-

Table 21.2. The results of a relative risk regression analysis of cardiovascular disease incidence in relation to blood pressure measurements from the three preceding examination cycles; the analyses stratify on age and sex

mates.

b The values in parentheses are significance levels for testing the hypothesis b = 0. Adapted from Prentice et al. [66]. It appears here with the permission of the pub-

lisher.

tended; therefore, the analyses presented in table 21.2 involve fewer cases than the results reported in table 21.1. The more extensive analysis presented in table 21.2 suggests that the most recent systolic blood pressure measurement, S(t – 1), is strongly associated with the risk of cerebral infarction, while the next most recent, S(t – 2), shows a much weaker association. On the other hand, after adjusting for the levels of systolic blood pressure in the two preceding cycles, a recent elevated systolic blood pressure measurement is negatively associated, although marginally, with the risk of coronary heart disease. One possible explanation for this result would be the suggestion that hypertensive medication achieves blood pressure control without a corresponding reduction in coronary heart disease risk. The analysis for cerebral hemorrhage indicates that both elevated diastolic blood pressure and the duration of elevation are strongly related to the incidence of cerebral hemorrhage.