MULTIPLE LOGISTIC REGRESSION

LOGISTIC REGRESSION

182

Ž .

tests H₀:␤s0 for the linear trend in the proportions 5.6 . The test of independence using this statistic is called theCochran᎐Armitage trend test.

This analysis seems unrelated to the linear logit model. However, the Cochran᎐Armitage statistic is equivalent to the score statistic for testing H₀:␤s0 in that model. Moreover, this statistic relates to the statistic M² in Ž3.15 used to test for a linear trend in an. I=J table; namely, it equals M²

Ž .

applied when Js2, except with ny1 replaced by n. When Is2,

2Ž . ^{2 2}Ž .

X L s0 and z sX I .

2Ž .

For Table 5.3 on alcohol consumption and malformation, X I s12.1.

Using the same scores as in the linear logit model, the Cochran᎐Armitage

2 Ž .

trend test has z s6.6 P-values0.010 . The test suggests strong evidence of a positive slope. In addition,

X²ŽI.s12.1s6.6q5.5,

2Ž . Ž .

where X L s5.5 dfs3 shows only slight evidence of departure of the proportions from linearity. The trend test agrees with M² for the sample

Ž .

correlation of rs0.014 for ns32,573 Section 3.4.5 . For the chosen scores, the correlation seems weak. However, r has limited use as a descrip-tive measure for tables that are highly discrete and unbalanced.

Ž .

The Cochran᎐Armitage trend test i.e., the score test usually gives results similar to the Wald or likelihood-ratio test of H₀:␤s0 in the linear logit

4

model. The asymptotics work well even for quite small nwhen n_i are equal

4

and x_i are equally spaced. With Table 5.3, the Wald statistic equals

ˆ ^{2 2}

Ž␤rSE. sŽ0.317r0.125. s6.4 ŽPs0.012 and the likelihood-ratio statis-.

Ž .

tic equals 4.25 Ps0.039 . The highly unbalanced counts suggest that it is safest to use the likelihood function through the likelihood-ratio approach.

This is also true for estimation. The profile likelihood 95% confidence

Ž .

interval of 0.02, 0.52 for ␤ reported in Table 5.4 is preferable to the Wald

Ž . Ž .

interval of 0.317"1.96 0.125 s 0.07, 0.56 . Even though n is very large, exact inference based on small-sample methods presented in Section 6.7.4 is relevant here.

MULTIPLE LOGISTIC REGRESSION 183 The alternative formula, directly specifying ␲Ž .x, is

exp

Ž

␣q␤1x1q␤2x2q⭈⭈⭈q␤pxp

.

␲Ž .x s . Ž5.9.

1qexp

Ž

␣q␤1x1q␤2x2q⭈⭈⭈q␤pxp

.

The parameter ␤_i refers to the effect of x_i on the log odds that Ys1, Ž .

controlling the other x_j. For instance, exp ␤_i is the multiplicative effect on the odds of a 1-unit increase in x_i, at fixed levels of other x_j. An explanatory variable can be qualitative, using dummy variables for categories.

5.4.1 Logit Models for Multiway Contingency Tables

When all variables are categorical, a multiway contingency table displays the data. We illustrate ideas with binary predictors X and Z. We treat the

Ž .

sample size at given combinations i,k of X and Z as fixed and regard the two counts onY at each setting as binomial, with different binomials treated

Ž .

as independent. Denote the two categories for each variable by 0, 1 , and let dummy variables for X and Z have x₁sz₁s1 and x₂sz₂s0. The model logit P YŽ s1. s␣q␤₁x_iq␤₂z_k Ž5.10. has main effects for X and Z but assumes an absence of interaction. The effect of one factor is the same at each level of the other.

At a fixed level zk of Z, the effect on the logit of changing categories of X is

␣q␤₁Ž .1 q␤₂z_k y ␣q␤₁Ž .0 q␤₂z_k s␤₁. Ž5.11. This logit difference equals the difference of log odds, which is the log odds

Ž .

ratio between X andY, fixing Z. Thus, exp ␤₁ is the conditional odds ratio between X and Y. Controlling for Z, the odds of success when Xs1 equal

Ž .

exp ␤₁ times the odds when Xs0. This conditional odds ratio is the same at each level of Z; that is, there is homogeneous XY association ŽSection

. Ž .

2.3.5 . The lack of an interaction term in 5.10 implies a common odds ratio for the partial tables. When ␤₁s0, that common odds ratio equals 1. Then X and Y are independent in each partial table, or conditionally independent,

Ž .

gi®en Z Section 2.3.4 .

Additivity on the logit scale is the generally accepted definition of no interaction for categorical variables. However, one could, instead, define it as additivity on some other scale, such as with probit or identity link. Significant interaction can occur on one scale when there is none on another scale. In some applications, a particular definition may be natural. For instance, theory might assume an underlying normal distribution and predict that the probit is an additive function of predictor effects.

LOGISTIC REGRESSION

184

A factor with I categories needs Iy1 dummy variables, as we showed in Section 5.3.2. An alternative representation of such factors resembles the way that ANOVA models often express them. The model formula

X Z

logit P YŽ s1. s␣q␤_i q␤_k Ž5.11. ^X4

represents effects of X with parameters ␤_i and effects of Z with ^Z4 Ž

ters ␤_k . The X and Z superscripts are merely labels and do not represent

. Ž .

powers. Model form 5.11 applies for any number of categories for X and Z. The parameter ␤_i^X denotes the effect on the logit of classification in category i of X. Conditional independence between X and Y, given Z,

X X X

Ž .

corresponds to ␤₁ s␤₂ s ⭈⭈⭈ s␤_I , whereby P Ys1 does not change as i changes.

Ž .

For each factor, one parameter in 5.11 is redundant. Fixing one at 0, such as ␤_I^Xs␤_K^Zs0, represents the category not having its own dummy variable. When X and Z have two categories, the parameterization in model Ž5.11 then corresponds to that in model 5.10 with. Ž . ␤₁^Xs ␤₁ and ␤₂^Xs0, and with ␤₁^Zs␤₂ and ␤₂^Zs0.

5.4.2 AIDS and AZT Example

Table 5.5 is from a study on the effects of AZT in slowing the development of AIDS symptoms. In the study, 338 veterans whose immune systems were beginning to falter after infection with the AIDS virus were randomly assigned either to receive AZT immediately or to wait until their T cells showed severe immune weakness. Table 5.5 cross-classifies the veterans’ race, whether they received AZT immediately, and whether they developed AIDS symptoms during the 3-year study.

Ž . Ž

In model 5.10 , we identify X with AZT treatment x₁s1 for immediate

. Ž

AZT use, x₂s0 otherwise and Z with race z₁s1 for whites, z₂s0 for blacks , for predicting the probability that AIDS symptoms developed. Thus,.

␣ is the log odds of developing AIDS symptoms for black subjects without immediate AZT use, ␤₁ is the increment to the log odds for those with immediate AZT use, and ␤₂ is the increment to the log odds for white

TABLE 5.5 Development of AIDS Symptoms by AZT Use and Race Symptoms

Race AZT Use Yes No

White Yes 14 93

No 32 81

Black Yes 11 52

No 12 43

Source: New York Times, Feb. 15, 1991.

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