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

ture of the practical problem makes this ad hoc approach inappropriate or inadequate, a statistician should be consulted.

In chapter 10, we will also be discussing a method for analyzing normally distributed data. In fact, the techniques which we have described in chapter 9 can be regarded as a special case of the methodology which is presented in chapter 10. We do not intend to elaborate further on this connection, since the purposes of these two chapters are quite different. Chapter 9 constitutes a brief introduction to material which, although important, is not a primary focus of this book. However, in chapter 10 we introduce a class of statistical techniques which will feature prominently in the remaining chapters. We therefore direct our attention, at this point, to the important topic of regression models.

10

U U U U U U U U U U U U U U U U U U U U U U U U U U U

Linear Regression Models for Medical Data

10.1. Introduction

Many medical studies investigate the association of a number of different factors with an outcome of interest. In some of the previous chapters, we discussed methods of studying the role of a single factor, perhaps with stratification according to other factors to account for heterogeneity in the study population. When there is simultaneous interest in more than one factor, these techniques have limited application.

Regression models are frequently used in such multi-factor situations. These models take a variety of forms, but their common aim is to study the joint effect of a number of different factors on an outcome variable. The quantitative nature of the outcome variable often determines the particular choice of regression model. One type of model will be discussed in this chapter. The five succeeding chapters will introduce other regression models. In all six chapters, we intend to emphasize the general nature of these models and the types of questions which they are designed to answer. We hope that the usefulness of regression models will become apparent as our discussion of them proceeds.

Before we launch into a description of linear regression models and how they are used in medical statistics, it is probably important to make a few brief comments about the concept of a statistical model. A relationship such as Einstein’s famous equation linking energy and mass, E = mc2, is an exact and true description of the nature of things. Statistical models, however, use equations quite differently. The equation for a statistical model is not expected to be exactly true; instead, it represents a useful framework within which the statistician is able to study relationships which are of interest, such as the association between survival and age, aggressiveness of disease and the treatment which a patient receives. There is frequently no particular biological support for statis-

Table 10.1. Average weight measurements from 20 litters of mice

Figure 10.2 consists of separate scatterplots of the average brain weight of the pups in a litter versus the corresponding average body weight and the litter size, as well as a third scatterplot of litter size versus average body weight. Notice that both body weight and litter size appear to be associated with the average brain weight for the mouse pups.

A simple linear regression model relates a response or outcome measurement such as brain weight to a single explanatory variable. Equation (10.1) is an example of a simple linear regression model. By comparison, a multiple linear regression model attempts to relate a measurement like brain weight to more than one explanatory variable. If we represent brain weight by W, then a multiple linear regression equation relating W to body weight and litter size would be

W = a + b1(BODY) + b2(LITSIZ).

At this point, it is helpful to introduce a little notation. If Y denotes a response variable and X1, X2, …, Xk denote explanatory variables, then a regression equation for Y in terms of X1, X2, …, Xk is

Y = a + b1X1 + b2X2 + … + bkXk

or

k

Y =a + bi Xi .

i=1

Remember, also, that if we wish to refer to specific values of the variables X1, X2, …, Xk, it is customary to use the lower case letters x1, x2, …, xk.

Table 10.2. Estimated regression coefficients and corresponding standard errors for simple linear regressions of brain weight on body weight and litter size

The variances of the different normal distributions corresponding to different sets of covariate values are assumed to be the same. Notice that a regression analysis makes no assumptions about the distribution of the explanatory variables X1, X2, …, Xk. In many designed experiments, values of the explanatory variables are, in fact, selected by the investigator in order to study the relationship between Y and X1, X2, …, Xk systematically.

In our example, two separate, simple linear regressions of brain weight on body weight and litter size can be performed. These lead to the two equations

Ŵ = 0.336 + 0.010(BODY) and Ŵ = 0.447 – 0.004(LITSIZ).

If the multiplier, or regression coefficient, for any variable was zero, then that variable would have no influence on the response variable W. It is very unlikely that the best estimate of a regression coefficient would be exactly zero. A more reasonable question to ask is whether the data provide evidence to contradict the hypothesis that the regression coefficient could be zero, thereby confirming a relationship between the explanatory and response variables.

Table 10.2 lists the estimated regression coefficients and their corresponding standard errors for the two simple linear regressions of brain weight on body weight and litter size. If there is no relationship between the explanatory

ˆ ˆ

variable and brain weight, then the ratio b/est. standard error (b) has a Student’s t distribution with the same number of degrees of freedom as the estimated standard error. The results of chapter 9, therefore, lead to the test statistic

to test the hypothesis that the regression coefficient b equals zero. To calculate the significance level of the test, the observed value of T is compared with the critical values of the modulus of a Student’s t distribution with the appropriate degrees of freedom (cf. table 9.4). Table 10.2 also includes the observed values of T and the associated significance levels for each regression coefficient.

Table 10.3 is consistent with the biological concept of brain sparing, whereby the nutritional deprivation represented by litter size has a proportionately smaller effect on brain weight than on body weight. In a simple linear regression, litter size is negatively related to brain weight because the larger litters tend to consist of the smaller mice. In the multiple linear regression, the effect of body size is taken into account and the positive regression coefficient associated with litter size indicates that mice from large litters will have larger brain weights than mice of comparable size from smaller litters.

The analysis which we have described in this section is approximate and can be improved upon. The results of a linear regression analysis are frequently presented, in summary form, in an analysis of variance (ANOVA) table. Since this type of presentation does not extend easily to other regression models (see chapters 11–14), it is not of primary importance to the aims of this book. Our presentation is consistent with the usual analysis of other regression models which are widely used in medical research, and therefore serves as a useful introduction to the general topic of regression models.

Before we consider other types of regression models, we propose to briefly discuss correlation analysis, a historical antecedent to regression analysis. Also, the final section of this chapter describes ANOVA tables. This material is not needed to understand chapters 11–14, but is included for completeness. The amount of detail which is necessary to explain ANOVA tables far exceeds the complexity of most other explanations appearing in this book. We therefore suggest that the reader bypass §10.5 on a first reading.

10.4. Correlation

Regression models presuppose there is an outcome or response variable of some importance, and that interest in other variables derives from their potential influence on the outcome variable. Historically, the development of regression analysis was preceded by another approach known as correlation analysis.

Consider the case of two variables, Y and X. In a regression analysis, we assume Y has a normal distribution, but X may take any value and no distributional assumptions about X are required. Thus, X may be determined along with Y, X values may be fixed by the experimenter, e.g., X represents treatment received in a randomized clinical trial, or any number of factors might influence the X values observed in the data. Correlation analysis is restricted to the situation when X and Y are both random variables, and commonly assumes that both variables have a normal distribution.