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

Table 21.9. Values of the explanatory variable, X6, representing an approximate interaction between age and smoking status for the Poisson regression analysis of the DollHill cohort study

0.11, to derive the 95% confidence interval 0.36 8 1.96(0.11), i.e., (0.14, 0.58), for b1, then a corresponding 95% confidence interval for the relative risk associated with smoking is (exp(0.14), exp(0.58)) or (1.15, 1.79).

Readers will notice that in table 21.8 we have only cited one p-value in the column labelled ‘Significance level’ for the set of four explanatory variables that encode the information about age. Although each of the four regression coefficients for age could be examined separately, the hypothesis summarizing the notion that a physician’s age is not an important systematic effect in the regression model is best expressed as b2 = b3 = b4 = b5 = 0, simultaneously. To investigate the plausibility of this joint hypothesis, we use a likelihood ratio test, which we previously discussed in §16.6. The observed value of the test statistic for these coronary mortality data is 893.8, which is calibrated against the sampling variability summarized in the 2 distribution with four degrees of freedom. As the results for the age-adjusted Poisson regression indicate, the significance level of this test that b2 = b3 = b4 = b5 = 0 is very small, underscoring the importance of adjusting for age in our analysis of the relative risk of coronary mortality associated with being a male doctor who smokes.

As well as being a potential confounding factor, age would be an effect modifier if the data collected by Doll and Hill provided evidence that the relative risk of smoking somehow depended on age.

To investigate this possibility requires the exercise of a little skill, since the age-adjusted regression model, which is summarized in table 21.8, already involves six explanatory variables with corresponding estimated regression coefficients. In total, we only have 10 independent observations; they are the observed numbers of deaths in the various subject groups defined by smoking status and the five ten-year age groups. However, with a moderate degree of creativity, we can introduce a simple smoking-age interaction via a single extra explanatory variable, X6. The possible values for this new variable are given in table 21.9, and depend on age and smoking status.

Table 21.10. The results of a Poisson regression model that includes an interaction between smoking status and age

cIncluding age as an effect modifier

1 Likelihood ratio test .

Assigning these values amounts to adopting a linear measurement scale with the values 1, 2, 3, 4 and 5 for the five increasing ten-year age groups, and an indicator variable for smoking status which equals 0 if a doctor was not a smoker, and 1 otherwise. The numerical value of X6 represents the interaction of smoking status and age group, and is calculated by first centering each component – age group around the value 3 and smoking status around 0.5 – and then calculating their product to get the values shown in the table. By centering each component in the product (interaction) variable, we reduce the correlation between the explanatory variable X6 and the corresponding main effect variables X1 through X5, and should thereby improve the numerical properties of the fitted model.

Table 21.10 summarizes the results of fitting a third Poisson regression model, labelled c, which adds the interaction term X6 to the age-adjusted regression model, i.e., b, in table 21.8. The coefficient for this interaction term represents a measure of the effect-modifying aspects of age on the risk of coronary death associated with smoking.

The results displayed in table 21.8 indicate that both smoking status and age are important explanatory variables, not only statistically but also in terms of the size of the epidemiological effects. Moreover, as the estimated regression coefficient for X6 and its corresponding standard error indicate (see table 21.10), age is not just a potential confounding factor with respect to coronary mortality; it also appears to be an effect modifier. This conclusion is supported by the significance level of the hypothesis test that b6 is zero. Since the estimated value of b6 is noticeably different from zero, the effect of a doctor’s

Table 21.11. The estimated relative risks of coronary mortality by smoking status and ten-year age bands, based on the Poisson regression model summarized in table 21.10

smoking status on mortality due to coronary heart disease depends systematically on the age group to which he belongs. In particular, since the estimated value of b6 is negative, the relative risk associated with smoking declines with age.

Perhaps the best summary of the results of our Poisson regression analysis involving smoking status, age, and the modelled interaction between these two explanatory variables is a set of estimated relative risks. Table 21.11 shows these estimates for smokers and non-smokers in the different age groups. The values are all compared to the experience of doctors in the 35–44 years age group who didn’t smoke.

Although this introduction to a regression analysis of the Doll and Hill cohort study of coronary mortality has involved some modelling subtleties that not every reader may have entirely grasped, we hope everyone has been able to appreciate the potential usefulness of Poisson regression methods in analyzing cohort studies.

21.8. Clinical Epidemiology

At the beginning of this chapter, we remarked that previous chapters were primarily concerned with clinical studies, whereas epidemiology concentrates on the incidence of disease. There is also now increasing use of the term ‘clinical epidemiology’.

Certain areas of clinical investigation, such as diagnostic tests which we will discuss in chapter 22, are sometimes particularly associated with clinical epidemiology. This term is also used by some writers to refer to the application of epidemiological methods to clinical medicine. However, we are sympathet-

ic to the more general definition given by Weiss [73]. Weiss argues that ‘epidemiology is the study of variation in the occurrence of disease and of the reasons for that variation’. He defines clinical epidemiology in a parallel way as ‘the study of variation in the outcome of illness and of the reasons for that variation’.

Whether a specific term for this activity is required may be debatable, but whatever one’s reaction, semantically, to the term clinical epidemiology, the area of study defined by Weiss is of obvious importance and would encompass many of the examples used in other chapters of this book. We trust, therefore, that this book will be useful to readers who regard themselves as interested in clinical epidemiology.

sity measurements derived from the same specimen, but assayed on different ELISA plates, would necessarily be identical. However, if the two measurements obtained from the same specimen were assayed on the same ELISA plate, it seems reasonable to suppose that the resulting optical densities would be quite similar. In that case, a plot like the one displayed in figure 22.1 should show that virtually all of the points displayed lie very close to the reference line of equal values running from the lower left corner to the upper right corner of the diagram. If repeated measurements of the same characteristic of a fixed specimen or unit, obtained under the same conditions, have virtually the same numerical value, they are said to be repeatable. Since the optical density measurements for each specimen represented in figure 22.1 were derived from two ELISAs evaluated on the same plate, the testing kit clearly lacked repeatability. Incidentally, a logarithmic scale was used on each axis of the plot in order to enhance simultaneous visualization of all 1,762 paired optical density measurements.

Fortunately, test characteristics such as measurement repeatability and reproducibility are investigated during the licensing or approval process of an appropriate regulatory authority such as the US Food and Drug Administration. Clinical chemistry, which is probably not appreciated as much as it deserves, plays a crucial role in first developing, and then improving and maintaining, the complex measurement systems that underly virtually all of the diagnostic tests used in modern clinical practice.

As we previously indicated, many diagnostic test results are not reported as measurements like the optical density values displayed in figure 22.1, but rather as binary outcomes, e.g., reactive/non-reactive, normal/abnormal, positive/negative or perhaps diseased/disease-free. This classification of the underlying physical measurement is based on characteristics of the measurement system that the manufacturer has incorporated into the test protocol. If we suppose that the population in which the test is approved for use can be subdivided into the two groups identified by the binary outcomes, say diseased and disease-free, then individuals in the diseased group will have a distribution for their test measurements.

For convenience, we will assume that this distribution looks like the normal distribution. Likewise, the individuals who are disease-free will have a separate distribution for their test measurements. If it also resembles the normal distribution, we might have a situation similar to the one depicted in the upper panel of figure 22.2. Readers can no doubt see immediately that this

Fig. 22.2. A pictorial framework for diagnostic testing. a The ideal world; b, c more realistic settings.

would be an ideal world, because test measurements that turned out to be positive would always correspond to diseased individuals (so-called true positives), and test measurements that were negative values would indicate persons that were disease-free (so-called true negatives). Fortuitously, the sign of the test measurement would suffice to classify subjects without ever making an error.

Regrettably, the world of diagnostic testing is rarely so unequivocally ordered. For example, consider the prostate-specific antigen (PSA) test that is routinely used to screen for prostate cancer. Most men who are middle-aged or older will have some detectable antigen in their blood, say 0.5 ng/ml, and those individuals who have advanced prostate cancer will have high concentrations, perhaps in excess of 20 ng/ml. However, as Catalona et al. [74] report, a concentration of 6.8 ng/ml may be observed in an individual who is disease-free, or in someone with early cancer. As we have depicted in figure 22.2b and c, the distributions of test measurements in the diseased and disease-free groups overlap to some extent. Regardless of where the test outcome threshold is situated on the measurement scale, some diseased individuals will be incorrectly classified as ‘Negative’, and hence give rise to the kind of diagnostic error known as a false-negative outcome. Likewise, disease-free individuals whose test measurement is larger than the outcome threshold (see fig. 22.2c) will be identifed as ‘Positive’, which represents the other kind of diagnostic error – one known as a false-positive outcome. And provided the distributions of test measurements derived from individuals in the diseased and disease-free groups overlap, neither type of diagnostic error can be avoided, or eliminated. Because the diagnostic test has a single outcome threshold, moving it to the right will reduce the false-positive error rate, but simultaneously increase the false-neg- ative error rate. Similarly, moving the single outcome threshold to the left will reduce the false-negative error rate; however, the false-positive error rate automatically increases. Only by changing the distributions of test measurements in the diseased and disease-free groups, so that they begin to approximate what is depicted in figure 22.2a, can investigators make simultaneous reductions in the rates of both types of diagnostic test errors.

22.3. Sensitivity, Specificity, and Post-Test Probabilities

Many readers will recognize that the true status of a patient, and the outcome of a diagnostic test for the presence of a particular condition or disease, can be represented conveniently in the cells of a 2 ! 2 table, such as the one depicted in table 22.1. The probability that a diseased individual is correctly identified as ‘Positive’ is often called the sensitivity of the diagnostic test. Ob-

Table 22.1. A 2 2 table summarizing the four possible outcomes of a diagnostic test for the presence of a condition or disease

viously, one desirable characteristic of a good diagnostic test is that it has a sensitivity value close to 1. Likewise, the probability that a disease-free individual is correctly identified by the diagnostic test as ‘Negative’ is often called the test specificity. Another preferred characteristic of a diagnostic test is that it should be highly specific, i.e., have a specificity that is close to 1.

To make these two notions more concrete, we can refer to figure 22.2, especially the middle and lower panels. Since sensitivity represents the probability that a diseased individual is correctly identified as ‘Positive’, it corresponds to the fraction of the area under the probability curve for test measurements obtained from diseased individuals that lies to the right of the test outcome threshold (see fig. 22.2b). Thus, a highly sensitive test is one for which virtually all of the test measurements in the population of diseased individuals lie on the side of the test outcome threshold designated as a positive (or diseased) outcome. The remaining fraction of the area under the same probability curve represents the false-negative error rate. And specificity, which is the probability that a disease-free individual is correctly identified as ‘Negative’, corresponds to the fraction of the area under the probability curve for test measurements from disease-free individuals that lies to the left of the test outcome threshold (see fig. 22.2c). If virtually all the test measurements that might be obtained in the disease-free population lie on the side of the test outcome threshold corresponding to a negative (or disease-free) outcome, then the test is said to be highly specific. Of course, the false-positive error rate is equal to the remaining fraction of the area under the same probability curve for measurements from disease-free individuals.

These two characteristics of all diagnostic tests – sensitivity and specificity – are estimated during the approval process. However, it should be noted that the reported estimates may be overly optimistic due to inadequacies in the design of the study through which the estimates were obtained. Ransohoff and Feinstein [75] refer to the ‘spectrum’ or range of characteristics represented in subjects participating in any study used to investigate the characteristics of a putative diagnostic test. They cite various examples from the medical litera-