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

ent from 0, the differentiated cell counts represent evidence that the two agents, TNF and IFN- , do not independently stimulate cell differentiation; instead, the effect of either agent depends on the level of the other. In this particular model, a negative sign for the estimated value of b3 means that the effect of a high concentration of one agent is reduced if the other agent is also present at a high concentration.

Figure 12.1 displays a plot of the observed numbers of differentiated cells, based on the fitted Poisson regression model, against the predicted counts from the experiments. This graphical display, as well as other plots that we have chosen not to reproduce here, indicate that the fitted model provides a very satisfactory statistical account of the apparent systematic dependence of the response measurement on the concentration of the two agents used in the study.

and the distributions tend to vary widely from one disease to another. Either the statistical models must incorporate a wide class of distributions that may be less readily applied than the normal distribution, or we must use methods that do not assume a specific distribution at all. Our discussion of proportional hazards regression concentrates on methodology that adopts the latter approach.

In chapters 6 and 7, we discussed the survival experience of 64 patients with Stages II, III or IV non-Hodgkin’s lymphoma. A comparison was made between those whose disease presented with clinical symptoms (B symptoms) and those whose disease was discovered indirectly (A symptoms). A strong survival advantage was attributed to those patients with A symptoms.

A natural question to ask is whether this difference could arise because these two patient groups differed with respect to other important prognostic factors. For illustration, we will consider two such factors, stage of disease and the presence of a bulky abdominal mass.

13.2. A Statistical Model for the Death Rate

When analyzing survival time, it is convenient to think in terms of the death rate. More generally, a failure rate can be defined for investigating the time to any other type of ‘failure’. The technical name for a failure rate is the hazard rate; historically, the death rate was called the force of mortality.

The death rate is defined, in mathematical terms, as the probability of dying at a specified time t when it is known that the individual did not die before t. This definition may seem somewhat artificial. However, it is a convenient, general way of representing the same type of information as that implied by the question ‘What is the probability of surviving two years after therapy if a patient has already survived one year?’.

We will denote a death rate at time t by the function d(t). To develop a regression model, we need to incorporate information which is coded as covariates X1, X2, ..., Xk. As before, it is convenient to refer to the set of covariates as

The death rate function, d0(t), in equation (13.1) represents the death rate at time t for an individual whose covariate values are all zero, i.e., X1 = 0, X2 =

0, ..., Xk = 0. It is not important that a patient having all values of the covariates equal to zero be realistic. Rather, d0(t) is simply a reference point, and serves the same function as the coefficient a in the expression a + bixi which we used in previous regression models. The difference which d0(t) incorporates is that d0(t) changes as t varies, whereas the coefficient a was a single numerical constant. This model was introduced by Cox [25] and is frequently referred to as the Cox regression model.

A survival curve is a graph of the function Pr(T 1 t), the probability of survival beyond time t. For those who remember some calculus, we remark that

Otherwise, it suffices to say that by specifying the death rate, d(t), we identify the function Pr(T 1 t). Moreover, in terms of Pr(T 1 t), it can be shown that the regression equation (13.1) is equivalent to

where Pr0 (T 1 t) is the survival curve for a patient whose covariate values are all zero.

Once again, the most important aspect of the regression model (13.1) concerns the regression coefficients. If bi is zero, then the associated covariate is not related to survival, or does not contain information on survival, when adjustment is made for the other covariates included in the model.

The model is also valuable, however, because it easily accommodates the two specific characteristics of survival data which we mentioned in §13.1. When we estimate the coefficients, the bi’s, it is easy, and is in fact preferable, not to assume anything about the death rate function d0(t). Thus, we do not need to specify any particular form for the distribution of the survival times, and the model is applicable in a wide variety of settings. Secondly, it turns out that it is very easy to incorporate the information that an individual has survived beyond time t, but the exact survival time is unknown, into the estimation of the regression coefficients. The details of the actual calculations are beyond the scope of this book but, as before, the model estimation can be summarized as a table of estimated regression coefficients and their corresponding estimated standard errors.

Table 13.1. The results of a proportional hazards regression analysis of survival time based on 64 advanced stage non-Hodgkin’s lymphoma patients

13.3. The Lymphoma Example

Table 13.1 records the estimated regression coefficients and standard errors from a Cox regression analysis of survival time for the 64 lymphoma patients described in chapter 7. Three covariates, X1, X2 and X3 were defined as follows:

Coincidentally, these covariates are all binary, although this is not a formal requirement of the proportional hazards regression model.

From table 13.1, we see that all three covariates are related to survival, even after adjustment for the other covariates. Thus, for example, we can say that the prognostic importance of B symptoms is not due to differences in stage and abdominal mass status among patients in the two symptom groups.

With a proportional hazards or Cox regression model and binary covariates, the results of an analysis can easily be described in terms of relative risk.

ˆ

For example, the death rate for Stage IV patients is exp(b1) = exp(1.38) = 3.97 times as large as that for Stage II or III patients. The relative risk associated with Stage IV disease is therefore 3.97. The relative risks for B symptoms and bulky disease are exp(1.10) = 3.00 and exp(1.74) = 5.70, respectively.

In this regression model, the relative risks need to be calculated relative to a baseline set of patient characteristics. For our model with three binary covariates, it is convenient to take X1 = 0, X2 = 0 and X3 = 0 as the baseline char-

acteristics. Then the estimated relative risk for an individual with X1 = x1, X2 = x2 and X3 = x3 is

For example, the relative risk for a Stage IV patient with B symptoms and bulky disease is exp(1.38 + 1.10 + 1.74) = 68.0 compared to a Stage II or III patient with A symptoms and no abdominal mass. Estimated relative risks for such extreme comparisons should be interpreted with some skepticism, since the associated standard errors are frequently very large. It is also possible to examine whether the model specified in equation (13.3), which sums the three regression components, is, in fact, reasonable. To do this, we would code additional interaction covariates which represent patients having two or three of the characteristics of interest. If these extra covariates have non-zero regression coefficients, then they should be included in a revised model, and their inclusion will alter the additive relationship in equation (13.3).

For example, if we code a covariate X4 as

then the estimated regression model including X1, X2, X3 and X4 has regression

has an associated p-value of 0.36. From this model, the estimated relative risk for a Stage IV patient with B symptoms and bulky disease is exp(2.17 + 1.98 + 1.66 – 1.07) = 114.4. This estimate is presented solely for the purposes of illustration, since the p-value of 0.36, which is associated with a test of the hypothesis b4 = 0, does not support the use of a model incorporating X4. In fact, with a small data set, a model involving a covariate which represents several factors combined usually would not be considered.

The recognition that regression coefficients are only estimates is more important than the particular estimates of relative risk which can be calculated. As we indicated in chapter 8, we can calculate confidence intervals for regression coefficients. The 95% confidence interval for bi is defined to be

If we represent this interval by (biL, biH), then, for a Cox regression model, we can also obtain a 95% confidence interval for the relative risk, exp(bi), namely

(exp(biL), exp(biH))·

Table 13.2 presents estimated regression coefficients, relative risks and confidence intervals for the lymphoma example. Since there are only 64 patients,

Table 13.2. Estimates of the 95% confidence intervals for regression coefficients and relative risks corresponding to the model analyzed in table 13.1

the confidence intervals are very wide; therefore, little importance should be attached to any of the estimates. Any discussion of estimation based on regression models should clearly indicate the potential error in the estimates. Remember, also, that a ‘significant’ regression coefficient (p ! 0.05) merely means that a 95% confidence interval for the regression coefficient excludes the value zero.

It is not possible to discuss, in one brief chapter, all the intricacies of a proportional hazards regression model. We would suggest that a statistician be involved if this model is used extensively in the analysis of any data set. However, there is one remaining feature of the model which merits some discussion.

Any regression model is based on a particular, assumed relationship between the dependent variable and the explanatory covariates. It is important to examine the validity of this assumed relationship. It need not have a biological basis, but it must be a reasonable, empirical description of the observed data. Most regression models allow the model assumptions to be examined.

The assumption of the Cox regression model which is discussed most often is that of proportional hazards. Equation (13.1) implies that

which specifies that the death rate for an individual with covariate values x is

˜

a constant multiple, exp{ bixi}, of the baseline death rate at all times. Thus, although the death rate d0(t) can change with time, the ratio of the death rates,

d(t; x)/do(t), is always equal to exp{ bixi}.

˜

The proportional hazards regression model is frequently criticized for this assumption but, in fact, it is very easy to generalize the model to accommodate a time-dependent hazard or death rate ratio. In the following discussion, we indicate a simple approach which is useful in many situations.

Let us suppose that we do not want to assume that the death rate for Stage IV patients is a constant multiple of that for Stage II or III patients, and we do

Table 13.3. The results of a proportional hazards regression analysis of survival time, stratified by stage of disease; the analysis is based on 64 advanced stage non-Hodg- kin’s lymphoma patients

not have any information concerning the actual nature of the relationship. In this case, two regression equations can be defined as follows:

log{d1(t; x2, x3)} = log{d01(t)} + b2x2 + b3x3

and

log{d2(t; x2, x3)} = log{d02(t)} + b2x2 + b3x3,

where d1(t; x2, x3) is the death rate for Stage II or III patients and d2(t; x2, x3) is the death rate for Stage IV patients. Notice that the regression coefficients for B symptoms and bulky disease are assumed to be the same in the two regression equations. This is called a stratified version of Cox’s regression model, and it is an extremely useful generalization of the basic proportional hazards regression model.

Table 13.3 presents estimates and standard errors of b2 and b3 based on this stratified model. In table 13.1, the test that b2 or b3 could be equal to zero was adjusted for any possible confounding effects of disease stage by the inclusion of the covariate X1 in the regression model. In table 13.3, the corresponding tests are adjusted because the model is stratified by disease stage. This latter adjustment is more general, in that it does not assume there is any specific relationship between d1(t; x2, x3) and d2(t; x2, x3), whereas, in table 13.1, the assumption has been made that

d2(t; x2, x3) = d1(t; x2, x3) exp(b1).

In this example, the conclusions with respect to b2 and b3 are hardly altered by the use of the stratified model. This will not always be the case, of course. When a stratified model is used, it is possible to estimate the baseline functions d01(t) and d02(t). This means, in our particular example, that we can estimate survival curves for patients with A symptoms and no bulky disease, subdivided by disease stage (II or III versus IV). Figure 13.1 presents these estimates graphically. The method of estimation will not be discussed here. It is a generalization of the techniques which we discussed in chapter 6. If we then

1

Fig. 13.1. The estimated survival curves for non-Hodgkin’s lymphoma patients with A symptoms and no bulky disease, stratified by disease stage.

Fig. 13.2. The estimated survival curves for non-Hodgkin’s lymphoma patients with B symptoms and bulky disease, stratified by disease stage.

Fig. 13.3. The estimated cumulative hazard functions for the estimated survival curves shown in Figure 13.1, plotted using logarithmic scaling on both axes.

the corresponding survival curves for patients with covariate values X2 = x2 and X3 = x3. For example, figure 13.2 presents the curves for patients with B symptoms and bulky disease.

If we wish to examine whether a proportional hazard assumption is reasonable for representing the effect of disease stage (II or III versus IV), then figure 13.1 can be replotted, displaying the estimated cumulative hazard (–1 times the logarithm of the survival probability) versus time and using logarithmic scales on both axes. This is done in figure 13.3. If the two estimated survival curves are parallel, then the ratio of the death rates for the two patient groups is constant over time. In figure 13.3, the two curves are roughly parallel; therefore, it seems appropriate to include X1 in an unstratified regression model.

13.4. The Use of Time-Dependent Covariates

The lymphoma example which was discussed in the previous section introduces most of the basic features of proportional hazards regression, with one notable exception. All the explanatory variables in the regression model