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

To characterize the progression of a patient’s disease during the study, investigators chose to measure the number of joints that were permanently damaged. Since this outcome variable was evaluated at each clinic visit, dependence between observations on the same study participant is once again a prominent feature of the design of this study. Although a Liang-Zeger regression model could be used to analyze these data, sometimes it is desirable to model temporal changes in the outcome variable more directly than the simple inclusion of a correlation parameter affords. In this case, for example, the number of damaged joints can only increase in time, and an appropriate statistical model should reflect this fact.

The study investigators measured a number of different explanatory variables whose relationship with disease progression could be examined. In the following discussion, we will focus on two of these measures. The first, X1, indicates the number of effusions, i.e., inflamed joints, while the second, X2, is derived from the laboratory measurement of erythrocyte sedimentation rate (ESR). The binary variable, X1, is equal to one if the number of effusions exceeds four, and is coded zero otherwise. Likewise, if the ESR rate is less than 15 mm/h then the binary covariate, X2, is equal to one; otherwise, X2 = 0. The values of these covariates were determined for each study participant at the time of the first clinic visit since the primary goal in this study was to determine if patients at an increased risk of disease progression could be identified early in the course of their disease. The identification of such patients would facilitate earlier treatment intervention and also might allow physicians to avoid the unnecessary use of aggressive treatment in patients expected to experience a more benign course of the disease. In other study settings, interest might focus on the relationship between outcome and explanatory variates where both types of observations are measured repeatedly over time. In principle, the methods that we describe in the remainder of this section can be extended to include this more complicated type of analysis.

14.4.2. A Multi-State Statistical Model

The method of analyzing longitudinal data that we intend to use for this study of psoriatic arthritis patients is called multi-state modelling. This approach is possible when, at any point in time, each study subject can be classified into exactly one of a set of possible states. For example, carefully defined conditions such as ‘well’, ‘ill’, and ‘dead’ could represent the set of possible states of interest in a longitudinal study. In other investigations, including this study of psoriatic arthritis, the states may represent different values of an outcome variable of interest. In this study, the states in the model are determined by the number of damaged joints. The state labelled 1 denotes no damaged

Fig. 14.1. Schematic representation of the multi-state model used to analyze the psoriatic arthritis data.

joints; similarly, States 2, 3, and 4 correspond to one to four, five to nine, and ten or more damaged joints, respectively.

The analysis of multi-state models is based on movements, that we will call transitions, from one state to another. Each transition is modelled, statistically, in a manner similar to that used in chapter 13 to analyze survival data. Therefore, a multi-state model involves a rate for each possible transition between states. Since joint damage is a progressive condition, the only possible transitions are from State 1 to State 2, State 2 to State 3, and State 3 to State 4. We will denote a transition rate at time t by the function ri(t); the subscript i indicates that ri(t) is the transition rate from the state labelled i to the state labelled i+1. Therefore, the multi-state model for the psoriatic arthritis study involves the three rate functions r1(t), r2(t),and r3(t).

Figure 14.1 summarizes the structure of this multi-state model of the study in a schematic way. The boxes represent the different states and arrows indicate the various transitions that can occur; the corresponding transition rates are indicated above the arrows.

In the model for a hazard rate used in chapter 13 we incorporated the possible effects of various covariates. To achieve the same goal in this multi-state model for the psoriatic arthritis study, we represent the transition rate between State i and State i+1 for a subject with observed values of the covariates X1 =

x1, …, Xk = xk, or x = {x1, x2, ..., xk}, by the equation

˜

log{r (t; x)} =log(a )+b x +b x + ... +b x .

i i i1 1 i2 2 ik k

In contrast to the hazard rate models that we used in chapter 13, this transition rate model has more than one regression coefficient associated with each covariate. In fact, for each covariate, Xj say, there is a different regression coefficient for each transition rate function in the multi-state model. This is why each regression coefficient, bij, has two subscripts. The first subscript, i, identifies the state from which the transition occurs and the second subscript, j, denotes the corresponding covariate, Xj. Although the number of regression coefficients associated with each covariate has increased, the interpretation of regression coefficients is just as we have previously described for all other regression models. A non-zero value for bij indicates that the covariate, Xj, is as-

sociated with the occurrence of transitions from State i to State i+1. For the transition from State i to State i+1, the relative risk, or ratio of transition rates, for an individual with Xj = 1 compared to an individual with Xj = 0 is

exp(bij).

This multi-state model does not incorporate the correlations between observations on the same subject directly. However, the correlation has little impact in this situation because the analysis is based on non-overlapping portions of the follow-up period. To estimate the regression coefficients associated with transitions from State 2 to State 3, say, only the information pertaining to subjects who occupy State 2 is used. What happens to a subject before or after that individual occupies State 2 does not contribute to the estimation of the regression coefficients b21, b22, ..., b2k.

Notice also that the term ai does not depend on the length of time a study subject has occupied State i. This assumption is not necessary, but it simplifies the analysis considerably; moreover, it is frequently possible to define states in such a way that this simplifying assumption is a reasonable approximation.

The technical name for data that are used to estimate the regression coefficients in this multi-state model is panel data. Individuals are observed on different occasions and, at each of these times, the state that each subject occupies is recorded. The requirements for panel data do not dictate that the observation times must be equally spaced. The information that is used to estimate the regression coefficients in the model for the transition rate, ri(t), is whether, between any two consecutive observation times, an individual has moved from State i to another state or remained in State i. If a transition has occurred, the estimation procedure takes into account all possible intermediate states through which the subject might have passed. Additional details concerning the analysis of panel data may be found in Kalbfleisch and Lawless [31].

If it is reasonable to assume that the regression coefficients associated with a single covariate are all the same, the analysis of this multi-state model can be simplified considerably. In the psoriatic arthritis study, this assumption would correspond to specifying that b1j = b2j = b3j for each covariate, Xj, j = 1, 2, ..., k. However, this assumption must be tested and an appropriate significance test can be devised for this purpose. There is no standard terminology for such a test; we will refer to it as a test for common covariate effects. The corresponding assumption of common covariate effects is frequently made in the early stages of an investigation when the effects of many covariates are being investigated; the primary aim of such a procedure is to screen for covariates that warrant further attention. At the latter stages of an analysis, the test for common covariate effects is used to simplify models that involve multiple covariates.

Table 14.3. Observed changes in damage states between consecutive clinic visits for the psoriatic arthritis study

14.4.3. Illustrative Results

When the data from the psoriatic arthritis study were compiled for analysis, 305 patients who had fewer than ten damaged joints when they became patients were registered at the clinic. As we indicated in the previous subsection, estimation of the regression coefficients in the multi-state model is based on the number of transitions that occurred between the various states in the model during the follow-up period for each subject. In this study, patients entered the clinic in States 1, 2, and 3. Some patients did not experience a change in state after becoming clinic patients; other individuals experienced more than one change of state. Table 14.3 summarizes the various transitions that occurred between consecutive visits to the clinic.

Because subjects occupied different states when they became clinic patients and entered the study, it would be desirable to incorporate the possibility of differences between patients who enter the study at different stages in the disease course. The model that we outlined in the previous section can be generalized to accommodate this possibility by stratifying the analysis according to the entry state for each subject. The resulting stratified multi-state model is analogous to the stratified regression models for the hazard function that we previously discussed in chapter 13. To stratify our analysis of the psoriatic arthritis data according to the states occupied by subjects when they became clinic patients, we simply need to allow the ai terms in the regression model for

ri(t; x) to depend on the entry state. We will continue to assume that the effects

˜

of the various covariates on the rate of transitions from State i are common across the strata. Hereafter, we will focus our attention solely on the regression coefficients, i.e., the bijs. However, in each analysis that we describe, the stra-

Table 14.4. The results of a stratified, multi-state regression analysis of disease progression in psoriatic arthritis patients

tum-specific effects that are represented by three ai terms, one for each possible entry state in the study, also have been estimated.

Table 14.4 summarizes the estimated regression coefficients and corresponding standard errors for a stratified multi-state model that incorporates the two explanatory variables X1 and X2. Recall that X1 is a binary covariate indicating that the subject had more than four joints with effusions at the first clinic visit, and X2 is another binary covariate that indicates the subject had an ESR that was less than 15 mm/h. Three regression coefficients were estimated for each covariate; these coefficients correspond to the effects of the covariate on transitions from States 1, 2, and 3.

According to the results presented in table 14.4, both X1 and X2 are associated with one or more transition rates. The estimated regression coefficients for X1, the number of effusions, are roughly comparable. For X2, the covariate indicating ESR at the first clinic visit, the estimated regression coefficients corresponding to transitions from State 1 to State 2 and State 2 to State 3 are comparable whereas the estimated regression coefficient for transitions from State 3 to State 4 has a different sign and is not significantly different from zero.

The results of this preliminary analysis of the psoriatic arthritis data suggest that a simpler model may fit the observed data as well as the preliminary model summarized in table 14.4. This simpler model incorporates a common effect for X1 across all transitions and a common effect for X2 for transitions from States 1 and 2 only. When this simpler model is used to analyze the psoriatic arthritis data, the estimated regression coefficient for X1 is 0.53 with a corresponding standard error of 0.14. The estimated regression coefficient associated with X2 is –0.52; its estimated standard error is 0.16. Although it is beyond the scope of this book, a significance test can be carried out to evaluate

Table 14.5. Estimated relative risks for the simpler version of the multi-state regression model analyzed in table 14.4

whether, when the two models are compared, the simpler model provides an adequate summary of the data. This test leads to an observed value for the test statistic that is equal to 2.8. If the null hypothesis concerning the adequacy of the simpler model is true, this test statistic should follow a 24 distribution. Therefore, the significance level of the aforementioned test is approximately 0.60, and we conclude that the simpler multi-state model adequately represents the psoriatic arthritis data.

To summarize the results of fitting this multi-state model, it may be useful to tabulate the estimated relative risks corresponding to the effects of the explanatory variables; see table 14.5. Often, relative risks can be interpreted more readily than the corresponding estimated regression coefficients.

Based on the results of this simplified analysis of the psoriatic arthritis data, the findings of this longitudinal study would be that evidence of extensive inflammation at the first clinic visit is associated with more rapid progression of joint damage in the future. However, a low sedimentation rate when the patient is first evaluated is related to a lower rate of disease progression subsequently, provided the number of damaged joints at the first visit is small.

14.4.4. Conclusions

The use of multi-state models to analyze longitudinal medical studies is increasing. A number of features of the model we have used in this section go beyond what we can describe adequately in this book. However, we hope that our description of the psoriatic arthritis study and its analysis has provided some indication of how multi-state models are related to simpler regression models and of their potential value in the statistical analysis of longitudinal data.

In the last five chapters, we have presented a brief introduction to the concept of regression models. A much longer exposition of these topics would be required to enable readers to fully master these important techniques. Nevertheless, we trust that our discussion has helped to provide some understanding of these methods, and has rendered their use in research papers somewhat less mysterious.

Table 15.1. ANOVA table corresponding to a regression of the logarithmic international normalized ratio (INR) from 354 specimens on explanatory variables representing machine and reagent study conditions

ment for a diverse spectrum of patients, using the INR methodology first adopted by the WHO in 1973.

Using only the insight garnered from our previous discussion of such summary tables in §10.5, we might infer that two of the three terms in the fitted regression model, i.e., the ones called Machine and Reagent, constitute important, systematic sources of variation in the log INR measurements obtained from the 354 specimens tested. However, these are clearly categorical variables, involving obvious subjectivity in how they might be labelled, rather than quantitative measurements like the body weight of a mouse pup. To be able to understand table 15.1 and how it relates to a corresponding regression model better, we need to know how information such as the lab machine used to obtain an INR measurement can be suitably represented in a regression model equation.

The methodology known as ANOVA can be used to analyze very complex experimental studies; indeed, entire books have been written on the subject. Our goal in this chapter will be a very modest one. By building on the understanding of multiple regression that readers have already acquired, we aim to furnish a wider perspective on this oft-used set of tools which could form the basis for a more general appreciation of its use and for additional reading on the topic if desired.

15.2. Representing Categorical Information in Regression Models

Although we have not previously used categorical information like Machine or Reagent details in any of the regression models that we considered in chapter 10, a brief glance at figure 15.1 will convince most readers that the INR

6

5

4

3

INR value

2

Treatment combination

Fig. 15.1. Dependence of the prothrombin time international normalized ratio (INR) measurements on the six experimental conditions determined by the machine and thromboplastin reagent used. The horizontal lines indicate the average INR measurement for that combination.

measurement for any particular specimen surely depends on aspects of the measurement process, e.g., the choices of Machine and Reagent used, in some systematic way. If we can identify and estimate these effects, we would begin to understand the measurement system that produces INR values, and perhaps gain some insights concerning how that process might be maintained, or even improved. We could adopt the representations used in figure 15.1 and label the machines 1, 2 and 3, and the reagents 1 and 2, but these assigned numbers are purely convenient labels. With no less justification, we could in-

Table 15.2. Settings of the indicator variables, X1, X2, and X3, that identify machine and reagent information in the INR study

stead label the machines A, B and C, and the reagents M and N. If labels for categories are so arbitrary, how can we incorporate categorical information into a multiple regression model as one or more explanatory variables, since the values of all the variables used in the multiple regression calculations must be numbers?

One way to achieve our goal involves using sets of appropriate indicator variables, i.e., variables that equal either 0 or 1 according to well-defined rules. For example, a single indicator variable would suffice to identify whether an INR measurement was obtained using the reagent labelled 1. For convenience, let’s call that indicator variable X1. So if the value of X1 associated with a particular INR measurement is 1, we know immediately that reagent 1 was used to obtain that INR measurement. And if the value assigned to X1 happens to be 0, then we know that reagent 2 was used instead. Hence the values that we assign to X1 for each INR measurement will identify which reagent was used to obtain each of the 354 observations in the study.

To identify the three separate machines used in the investigation we require two additional indicator variables; let’s call them X2 and X3. Table 15.2 shows how these two variables, together with X1, could unequivocally encode the machine and reagent information associated with any particular INR measurement.

Table 15.2 implicitly defines how X2 and X3 encode the lab equipment details. The combination X2 = 0, X3 = 0 identifies machine 1, X2 = 1, X3 = 0 corresponds to machine 2, and machine 3 is represented by the pair of values

X2 = 0, X3 = 1.

It is worth noting that the encoding summarized in table 15.2 is not unique. For example, we could switch assignment of the values 0 and 1 for X1