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and figure 11.1 is actually a plot of this estimated dependence of Pr(Y = 1 d) on dose.

In many dose-response problems, the estimation of the median effective dose, which is denoted by ED50, is of particular interest. The ED50 value is the actual dose required to induce the designated response in 50% of the population. Since Pr(Y = 1 d = ED50) = Pr(Y = 0 d = ED50) = 0.5, it follows that

=0 =a +b(ED50 );

ˆ ˆ

therefore ED50 = –a/b and its estimated value is ED50 = –â/b. For the data analyzed in table 11.9 the estimated ED50 is –(–1.92)/2.78 = 0.69, which corresponds to a concentration of exp(0.69) = 1.99 mg/kg. The derivation of a 95% confidence interval for this concentration is somewhat complicated, since the ED50 estimate is itself a ratio of estimates. Readers are advised to consult a statistician if a confidence interval of this type is required. In this particular case, most approaches to the technical problem of deriving a confidence interval for ED50 would yield an interval which is approximately (0.43, 0.95). The 95% confidence interval for the corresponding concentration is therefore (e0.43, e0.95) = (1.54, 2.59). The derivation of estimates and confidence intervals for other doses or concentrations which are similarly defined is handled in a corresponding way.

To illustrate some special aspects of the use of logistic regression methods we now consider a combined analysis of the dose-response data for two of the patient groups described by Duncan et al. [20]. The first set of patients (Group 1) consists of 94 children who did not receive premedication, whereas the second set (Group 2) consists of the 137 children premedicated orally with TDP and atropine. The regression model which was fitted to these data involves three covariates, X1, X2 and X3. The binary variable X1 takes the value 0 if a patient did not receive premedication and 1 if the patient received TDP and atropine. The variable X2 corresponds to the logarithm of the concentration of thiopentone administered, i.e., the dosage d, while the covariate X3 represents an interaction term which is formed by multiplying X1 and X2. Thus, X3 is equal to the dosage, d, for patients in Group 2 and takes the value 0 for patients in Group 1. This means that the fitted model consists of two different forms, one for each group of patients. These two forms are

Table 11.10. Maximum likelihood fit of a binary logistic regression model to data on 231 children anaesthetized with thiopentone

A total of 137 children were premedicated with TDP and atropine; the remaining 94 did not receive premedication.

Notice that a and b2 in the model for Group 1 are replaced in the model for Group 2 by (a + b1) and (b2 + b3), respectively. The regression coefficients b1 and b3 reflect two kinds of differences between the patient groups. To understand these differences, let p1(d) and p2(d) be the probabilities of responding to dose level d of the anaesthetic in Groups 1 and 2, respectively. Then the odds ratio (OR) for Group 2 versus Group 1 is

If b1 and b3 are both zero, then this odds ratio is one and the probability of responding is identical in both groups. If b3 = 0, then the odds ratio at all dose levels has the same value, namely exp(b1). Thus, the dependence on dose is the same in both groups of patients and is described by the coefficient b2. However, if b3 0 0, then the dose dependence is different in the two groups, and therefore the odds ratio changes with dose by a factor exp(b3d).

Table 11.10 presents the estimation of model (11.3). The regression coefficients for X1 (Group 2) and X2 (dose) are both significant, but the coefficient for X3 is not significant. Thus, the dependence on dose of the probability of responding to thiopentone is the same in the two groups of patients. However, the actual probability of responding is higher for the group of patients who were premedicated with TDP and atropine. This difference between the two patient groups is reflected in the odds ratio, exp(b1), which has an estimated

ˆ

value of exp(b1) = exp(3.67) = 39.3.

Table 12.1. The results of a Poisson regression analysis of the number of focussed and/or open questions asked by a physician during a patient consultation

designed to study the effect on physician communication skills of an intensive three-day training course.

Here, we ignore additional trial complexity, and consider a comparison of 80 doctors who were randomized to attend the three-day course with 80 doctors who were randomly chosen not to attend. We also restrict attention to a single outcome measure, namely the number of focussed and/or open questions asked by a physician during a patient consultation that occurred three months after the course ended, or three months after randomization for those physicians who did not receive any communication skills training. The course was designed to increase the frequency of such questions. For each physician, data were available from two consultations; to avoid undue complexity, we ignore the expected correlation between counts for the same physician and simply assume we have 160 observed counts for both the treatment group, i.e., those physicians who received training, and the control group.

Table 12.1 summarizes the results of a Poisson regression analysis of the number of focussed and/or open questions asked. The regression model included three explanatory variables that coded course attendance (yes = 1, no = 0), physician sex (female = 1, male = 0) and physician seniority (senior = 1, junior = 0). Readers will observe immediately that the format of this table is similar to those we first introduced in chapter 10 and subsequently encountered in chapter 11.

12.2. The Model for Poisson Regression

The theoretical formula from which we can calculate probabilities for counts that follow a Poisson probability function is characterized by a single parameter that is usually represented by the Greek letter . Conveniently, turns out to be the theoretical mean of the corresponding Poisson distribution,

so that if we have an estimated value for , we can immediately calculate the corresponding probability that a count equal to y is observed in a Poisson distribution with mean . Since is the only adjustable parameter in this Poisson model for the variation in observed counts, it is natural to link to the values of explanatory variables of interest. Because the mean, , of a Poisson distribution must be greater than zero, it would be unsuitable simply to assume that

= a + b1X1 + ... + bk Xk = a + ik=1 bi Xi ,

where X1, …, Xk represent the values of various explanatory variables, such as coding for the sex of a physician. Unless we restrict the values of a and the regression coefficients b1, ..., bk, the right-hand side of this equation for could sometimes be a negative value.

The logarithmic transformation is a remedy for this dilemma; the sign and magnitude of log is completely unrestricted, making the logarithm of the Poisson mean a natural choice to equate to the expression, a + ki=1 biXi, the component of the Poisson regression model which is the same as that which occurs in other regression models. Thus, if X1, …, Xk are potential explanatory variables whose values we wish to use to model variability in a response measurement, Y, that is thought to follow a Poisson distribution, then using the equation

log = a + b1X1 + … + bkXk

is a natural way of allowing the measured values of these explanatory variables to account for the variability in observed values of Y.

As in other regression models that we have previously considered, if a particular regression coefficient, say bi, is zero, then the corresponding explanatory variable, Xi is not associated with the response, Y. Thus, if there is no evidence to contradict the hypothesis that bi equals 0, then we probably can omit Xi from a Poisson regression model for the observed data. As we discussed in §11.2 for the case of logistic regression, a suitable statistic for testing the hypothesis that the regression coefficient, bi, equals zero is

ˆ

T = | bi | ˆ . est. standard error(bi )

The results of an analysis may also be presented in terms of the ratio

ˆ

bi ˆ , est. standard error(bi )

which is equal to T, apart from the sign. The latter is the ratio found in table 12.1. Whichever version of this test statistic is used, the conclusion regarding the associated explanatory variable, Xi, is the same.

The explanatory variables used in fitting the Poisson regression model summarized in table 12.1 are all binary ones that encode whether or not a physician in the study attended the training course, was a female, and was more senior.

There is considerable evidence in the study data that the estimated regression coefficient associated with attending the course is significantly different from zero, establishing a behavioural effect that is associated with the training provided. There is also some evidence of an effect associated with a physician’s sex, but there is no evidence of different behaviour patterns between senior and junior physicians. The signs of the estimated regression coefficients for course attendance and physician sex each indicate that the estimated Poisson mean is larger if the physician is female or if he or she attended the communication skills training.

The results of an analysis based on a Poisson regression model can also be described in terms of a rate ratio or ‘relative rate’. If bj is the regression coefficient associated with a particular binary explanatory variable, such as course attendance, exp(bj) represents the ratio of the rate at which the events of interest occur among physicians who received the skills training compared to those who did not. Thus, the key feature of the analysis that we can distill from table 12.1 is that the rate at which physicians asked focussed and/or open questions, adjusted for sex and seniority, is exp(0.24) = 1.27 times greater after attending

ˆ

the training course. And if we use the estimated standard error for b1 of 0.05 to derive the 95% confidence interval 0.24 8 1.96(0.05), i.e., (0.14, 0.34), for b1, then a corresponding 95% confidence interval for the relative rate is (exp(0.14), exp(0.34)) or (1.15, 1.40).

Of course, the importance of an effect may be linked to its absolute rather than relative size. In this communication skills study, the relative rate effect of 1.26 was associated with a mean number of 6.54 focussed and/or open questions asked during a patient consultation by a physician in the trained group compared with a mean of 5.14 in the control group. Providing such information, in addition to the Poisson regression results listed in table 12.1, is sensible and informative.

As in the case of logistic regression, the calculations involved in fitting a Poisson regression model to observed data are known as maximum likelihood estimation, the details of which are beyond the intended scope of this book. Even though many software packages that are now available will fit Poisson regression models, there are some aspects of these models that may require careful attention in any particular analysis. Thus, readers may wish to consult a statistician when the use of Poisson regression seems appropriate. However, we hope that our brief introduction to this regression model for count data has been informative, and will enable readers to understand the use of this statistical methodology in published papers.

Although the preceding discussion is the standard approach to motivating, and characterizing, the Poisson regression model, there are at least two other ways in which a Poisson distribution might be thought appropriate for a given set of data, even when the response measurements are not obviously observed counts. This same technique is often used to analyze cohort studies when there are counts of events, such as deaths, but the length of total observation time for both the deaths and those who survive is important. In such a situation, cumulative exposure time is used as a denominator for the observed counts, e.g., the Standardized Mortality Ratio and Cumulative Mortality Figure. Interested readers can find a discussion of both these measures in Breslow and Day [23], and a related example is discussed in chapter 21. Likewise, a Poisson regression model can arise as the limiting approximation to what might otherwise constitute logistic regression with a very large sample size, n, and a very small probability, p, of observing the event of interest. This situation typically occurs when the probability that an individual subject experiences an outcome of interest is very small and, therefore, a substantial number of subjects are enrolled in the study so that a reasonable number of events can be observed during follow-up.

The example of Poisson regression that we consider in the following section represents yet another situation, and provides additional evidence that Poisson regression models are useful in a variety of different settings.

12.3. An Experimental Study of Cellular Differentiation

Trinchieri et al. [24] describe an experiment to investigate the immunoactivating ability of two agents – tumour necrosis factor (TNF) and immune interferon- (IFN) – to induce monocytic cell differentiation in human promyelocytes, which are precursor leukocytic cells. Following exposure to none, only one, or both agents, individual cells were classified as exhibiting, or lacking, markers of differentiation. The study design involved 16 different dose combinations of TNF and IFN; after exposure and subsequent incubation for five days, between 200 and 250 cells were individually examined and classified, although the precise numbers of cells involved were not available. Consequently, the observed data are simply counts of differentiated cells, rather than the observed numbers of both types of cells after exposure to one of the 16 dose combinations of the two agents.

If we knew the number of undifferentiated cells counted in each of the 16 experimental treatment combinations, we could consider using a logistic regression dose-response model to study the possible systematic relationship between the dose levels of TNF/IFN administered, and the observed proportion

of differentiated cells. Since the undifferentiated cell totals were not available, we will use a Poisson regression model to investigate the same questions, treating the differentiated cell counts as the observed responses. In doing so, we are implicitly assuming that any variation in the total number of cells examined for each dose combination of TNF and IFN is unimportant.

As we have previously noted, dose-response studies such as this one often use a logarithmic concentration as the measurement scale of the experimental agent. In this instance, we chose to use base-10 logarithms, since the concentrations of TNF (in units per ml) used in the study design were 0, 1, 10 and 100, i.e., mostly ten-fold increases in concentration. To avoid numerical problems in fitting the Poisson regression model, we also replaced the absence of TNF, i.e., a concentration of 0 U/ml, by 0.1 U/ml, and the complete absence of IFN, i.e., 0 U/ml of IFN, by 0.1. Thus, the concentration levels 0, 1, 10 and 100 U/ml of TNF used in the experiment became the four doses –1, 0, 1 and 2 log U/ml, respectively. Likewise, the concentrations 0, 4, 20 and 100 U/ml of IFN used in the study corresponded to the four dose levels –1, 0.602, 1.301 and 2 log U/ml. For convenience, we will refer to these two dose measurements as the explanatory variables X1 and X2.

Through their study, Trinchieri et al. [24] sought to establish whether the two agents acted independently, or synergistically, to stimulate a differentiation response in cells. In the context of a Poisson regression model, this can be examined by creating a third explanatory variable, X3 = X1 ! X2, which is called the interaction of X1 and X2. Another example of an interaction term was used in §11.5. If the regression coefficient, b3, associated with X3 is nonzero, then the effects of X1 and X2 cannot be treated separately in the sense that we would simply add the individual terms, b1X1 and b2X2, to the regression model. Instead, the event rates depend on the specific combination of X1 and X2 values because X3 = X1 ! X2 is present in the model.

Accordingly, the equation for the mean of the Poisson regression model that we fitted to the number of differentiated cells observed for each of the 16 treatment combinations was

log{ (X1, X2, X3)} = a + b1X1 + b2X2 + b3X3.

If the regression coefficient b3 is equal to zero, changing the dose of IFN by one unit will have the same effect on the mean of the Poisson distribution, regardless of the dose of TNF used. However, if b3 is different from zero, the effect on the mean of the Poisson distribution of a one-unit change in IFN will depend on how much TNF is present, and on the sign and value of b3.

Table 12.2 summarizes the results of fitting this model involving the explanatory variables X1, X2 and X3 to the observed data on cell differentiation.

Fig. 12.1. Graphic display of the Poisson regression model log{ (X1, X2, X3)} = 3.32 + 0.773X1 + 0.434X2 – 0.101X3 fitted to the cell differentiation data; observed responses versus fitted values.

Table 12.2. The results of a Poisson regression analysis of the relationship between cellular differentiation and the dose of TNF or IFN administered

Although a thorough evaluation and interpretation of the results summarized in table 12.2 would be appropriate, we will simply note that all three explanatory variables are statistically important sources of systematic variation in the observed counts of the number of differentiating cells. Since the regression coefficient associated with the interaction term, X3, is significantly differ-