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

Figure 10.2 shows separate plots of brain weight versus litter size and body weight, and litter size versus body weight. The plots indicate that when brain weight is high or low, body weight tends to be correspondingly high or low, whereas litter size tends to be the opposite. The relationship between litter size and body weight is similar to the relationship between brain weight and litter size. A numerical measure of the observed association between two variables is the correlation coefficient r. For completeness, we give the formula for r, which is

where n is the number of paired observations on X and Y.

The correlation coefficient is a number between –1 and 1; the value r = 0 indicates there is no linear relationship between the two variables. A statistical procedure has been developed to test the hypothesis of no association between X and Y, based on the observed correlation coefficient, but we do not intend to discuss it here. If r is negative, then X and Y are said to be negatively correlated, implying that low values of X tend to occur with high values of Y and vice-versa. This kind of association is illustrated by the relationship between brain weight and litter size, which have a correlation coefficient of –0.62. If r is positive, then X and Y have a relationship like that observed between brain weight and body weight. The correlation coefficient for these variables is 0.75.

Figure 10.3 contains eight plots which illustrate the degree of linear relationship between the values of two variables, X and Y, corresponding to various positive and negative values of the correlation coefficient. The plots are based on 50 (X, Y) pairs and have been artificially constructed so that, in each case, the 50 X values displayed in each plot are identical.

The correlation coefficient can be used in the initial examination of a data set to identify relationships which deserve further study. However, it is generally more useful to think of linking two variables via a regression equation. From a regression analysis we can see very directly how changes in one variable are associated with changes in the other outcome variable. The regressions of litter size on brain weight and body weight are summarized by the equations

LITŜIZ = 47.41 – 95.76(W) and LITŜIZ = 23.51 – 2.07(BODY).

Thus, for a specified value of brain weight or body weight, we could ‘predict’ a value for litter size.

Fig. 10.3. Scatterplots of 50 artificial (X, Y) measurement pairs illustrating various values of the estimated correlation coefficient r. a r = 0.71, b r = 0.99, c r = 0.53, d r = –0.01, e r = –0.27, f r = –0.51, g r = –0.75, h r = –0.92.

Another reason for generally preferring regression analysis is the fact that correlation analysis is applicable only to situations when both X and Y are random. In many studies, the selection of subjects will depend on the values of certain variables for these subjects. Such selection invalidates correlation analysis. For example, Armitage et al. [19] indicate that if, from a large population of individuals, selection restricts the values of one variable to a limited range, the absolute value of the correlation coefficient will decrease.

Two additional points deserve brief mention. The first concerns correlation measures which do not depend on the assumption that X and Y have normal distributions. Two of these measures are Spearman’s rank correlation coefficient and Kendall’s (tau). These are useful for analyzing non-normal data, but the general reservations concerning correlation analysis which we have already mentioned apply equally to any measure of correlation.

The second issue is the following. Although we advocate the use of regression models, it is important to realize that the estimated regression line of Y on X is not the same as that for X on Y. This can be seen by comparing the re-

Now that we have informally described the decomposition of the variance in W, we introduce squared differences, which are the correct representation.

average. The total variation in these data can be represented by

n

(Wi W)2 =(W1 W)2 +(W2 W)2 +... +(Wn W)2 ,

i=1

which is (n–1) times the sample variance for W. Although it is not obvious, it can be shown that

Thus, the total variation in W decomposes into two sums of squared differences. In view of the preceding discussion involving simple differences, it should not be difficult to accept that the second sum represents the component of total variation in W which is accounted for by the fitted regression model, while the first sum represents the residual variation. If we use SS to represent a sum of squares, then we can rewrite the above equation, symbolically, as

SSTotal = SSResidual + SSModel ;

the reasons for the subscripts should be quite obvious. Each sum of squares of the form

n

(Wi [estimated value of Wi ])2

i=1

has associated with it a quantity called its degrees of freedom (DF). This quantity is equal to the number of terms in the sum minus the number of values which must be calculated from the Wi’s in order to specify all the estimated values. For example, in SSTotal the estimated value for each Wi is the same,

namely W. Thus, we need one calculated value, and the degrees of freedom for SSTotal are (n – 1). For SSResidual, the estimated value of each Wi is equal to

j=1

where x1, …, xk are the values of the covariates corresponding to Wi. These n

ˆ ˆ

estimates require (k + 1) calculated values, â, b1, ..., bk; therefore, the degrees

of freedom for SSResidual are (n – k – 1).

Although we shall not attempt to justify the following result, it can be shown that

DFTotal = DFResidual + DFModel.

Table 10.5. An expanded ANOVA table for the regression analysis of brain weight

critical value for an F-distribution with 2 and 17 degrees of freedom is 3.59, we conclude that the regression is significant at the 5% level; the data contradict the hypothesis that both BODY and LITSIZ have no effect on brain weight. This result is consistent with the conclusion which we reached in §10.3 (cf. table 10.3) that the regression coefficients for BODY and LITSIZ are significantly different from zero.

Table 10.3 summarizes the results of tests concerning the effect of individual covariates when the other covariate was included in the regression model. For example, we determined whether LITSIZ had any influence on brain weight if BODY was already included in the model. By comparison, the F-test which we have discussed above is a joint test of the hypothesis that all the regression coefficients are zero. However, the ANOVA table which we have described can be generalized to address the question of the individual effect of each covariate.

If we had calculated an ANOVA table for the regression of W on BODY,

then the SSModel would have been 386.9 ! 10–5, and DFModel would have been one. This number, 386.9 ! 10–5, represents the component of the SSModel in table 10.4 which is due to BODY alone. If we next add LITSIZ to the regression

equation, so that we are fitting the model described by table 10.4, the SSModel increases by 65.2 ! 10–5, from 386.9 ! 10–5 to 452.1 ! 10–5. Thus, 65.2 !

10–5 is the component of SSModel which is accounted for by LITSIZ, when the model already includes BODY. Notice, also, that we could have performed

these calculations in the reverse order, i.e., LITSIZ first, followed by BODY.

In table 10.5, the SSModel is divided into two parts. One part is labelled BODY, and represents the component of SSModel which is due to BODY; the other part is labelled LITSIZ|BODY, and represents the component which is

due to LITSIZ in addition to BODY. The degrees of freedom for each compo-

nent in the sum of squares are one since DFModel equals two and the component of the SSModel which is due to BODY has one degree of freedom. Therefore, two F-ratios can be calculated, and these appear in the last column of table 10.5;

regardless of the information we have concerning an individual litter size, the variation of independent observations from litters of the same size can never be accounted for by the regression equation. Therefore, MSPure Error is a better

estimate of the natural variance of the observations than MSResidual. Recall that we are assuming we do not have any information apart from brain weight and

litter size. To calculate a SSPure Error for the regression summarized in table 10.5 would require independent observations on mice which have the same body

weight and were reared in litters of the same size. Such observations are not available, and therefore the pure error calculations cannot be carried out in that particular case.

The SSpure Error is one component of SSResidual, and the remainder is usually called the Lack of Fit component, since it constitutes a part of the SSTotal which cannot be accounted for, either by the regression model or by pure error.

The SSLack of Fit measures the potential for improvement in the model for W if additional covariate information is used, or possibly if an alternative form of

the regression equation is specified instead of the current version. A test that the lack of fit in the current model is significant can be based on the F-ratio

MSLack of Fit/MSpure Error. If the null hypothesis that there is no lack of fit in the current model is true, this ratio should have an F-distribution with degrees of

freedom DFLack of Fit = DFResidual – DFPure Error and DFPure Error. We will not dis-

cuss the question of lack of fit in detail; however, if there is a significant lack of fit, it is customary to plot the values of W – Ŵ for all the observations, to see if they appear to be normally distributed. If so, then the lack of fit is usually attributed to unavailable information rather than the existence of better alternative models which involve the same variates. In table 10.6, the test for Lack of Fit is not significant.

Despite its very obvious link to the methods of linear regression, ANOVA is frequently regarded by many investigators as a specialized statistical method. This is particularly the case when ANOVA is used to evaluate the results of carefully designed experiments in which the researcher fixes or controls the operational settings of certain explanatory variables, which are usually called factors. Although ANOVA, and the corresponding ANOVA tables that we have described in this section, do not play a role in the methods of analysis used with other types of regression models such as those involving response measurements that are binary or count data, the use of ANOVA is occurring with greater frequency in the medical literature. Therefore, we will return to the topic of ANOVA in chapter 15, but judge it best to introduce first the use of regression methods for other kinds of response measurements in chapters 11–14.