Statistical Methods for Psychology

For what we are doing here (predicting the odds of surviving breast cancer), we will

work with the natural logarithm^16 of the odds, the result is called the log odds of survival.

For our example the log odds of being delinquent for a male with high testosterone,

The log odds will be positive for odds greater than 1 1 and negative for odds less than 1 1.

(They are undefined for odds 5 0.) You will sometimes see log odds referred to as the logit

and the transformation to log odds referred to as the logit transformation.

Returning to the cancer study, we will start with the simple prediction of Outcome on

the basis of SurvRate. Letting p 5 the probability of improvement and 1 2 p 5 the proba-

bility of nonimprovement, we will solve for an equation of the form:

Here will be the amount of increase in the log oddsfor a one unit increase in SurvRate.

It is important to keep in mind how the data were coded. For the Outcome variable, 1 5

improvement and 2 5 no change or worse. For SurvRate, a higher score represents a better

prognosis. So you might expect to see that SurvRate would have a positive coefficient, be-

ing associated with a better outcome. But with SPSS that will not be the case. SPSS will

transform Outcome 5 1 and 2 to 0 and 1, and then try to predict a 0 (better). Thus its coef-

ficient will be negative. (SAS would try to predict a 1, and its coefficient would be posi-

tive, though of exactly the same magnitude.)

In simple linear regression we had formulae for and and could use methods of

least squares to solve the equations with pencil and paper. Things are not quite so simple in

logistic regression, in part because our data consist of 0 and 1 for SurvRate, not the condi-

tional proportions of improvement. For logistic regression we are going to have to use max-

imum likelihood methods and solve for our regression coefficients iteratively.This means

that our computer program will begin with some starting values for and , see how well

the estimated log odds fit the data, adjust the coefficients, again examine the fit, and so on

until no further adjustments in the coefficients will lead to a better fit. This is not some-

thing you would attempt by hand.

In simple linear regression you also had standard Fand tstatistics testing the signifi-

cance of the relationship and the contribution of each predictor variable. We are going to

have something similar in logistic regression, although here we will use tests instead of

For t.

In Exhibit 15.4 you will see SPSS results of using SurvRate as our only predictor of

Outcome. I am beginning with only one predictor just to keep the example simple. We will

shortly move to the multiple predictor case, where nothing will really change except that

we have more predictors to discuss. The fundamental issues are the same regardless of the

number of predictors.

I will not discuss all of the statistics in Exhibit 15.4, because to do so would take us

away from the fundamental issues. For more extensive discussion of the various statistics

see Darlington (1990), Hosmer and Lemeshow (1989), and Lunneborg (1994). My purpose

here is to explain the basic problem and approach.

The first part of the printout is analogous to the first part of a multiple regression print-

out, where we have a test on whether the model (all predictors taken together) predicts the

dependent variable at greater than chance levels. For multiple regression we have an Ftest,

whereas here we have (several) x^2 tests.

x^2

b 0 b 1

b 0 b 1

b 1

log(p> 12 p)=log odds=b 01 b 1 SurvRate

> >

log odds=loge(odds)=ln(odds)=ln(0.293)=-0.228

566 Chapter 15 Multiple Regression

logit

logit
transformation

iteratively

(^16) The natural logarithm of Xis the logarithm to the base e of X. In other words, it is the power to which e must be
raised to produce X, where e is the base of the natural number system 5 2.71828.