Because the data are censored to provide only success or failure, we have to fit our model
somewhat differently.
The horizontal line across the plot in Figure 15.8 represents a critical value. Anyone
scoring above that line would be classed as improved, and anyone below it would be classed
as not improved. As you can see, the proportion improving, as given by the shaded area of
each curve, changes slowly at first, then much more rapidly, and then slowly again as we
move from left to right. This should remind you of the sigmoid curve we saw in Figure 15.9,
because this is what gives rise to that curve. The regression line that you see in Figure 15.10
is the linear regression of the continuousmeasure of outcome against SurvRate, and it goes
through the mean of each distribution. If we had the continuous measure, we could solve for
this line. But we have censored data, containing only the dichotomous values, and for that
we are much better off solving for the sigmoidal function in Figure 15.9.
We have seen that although our hypothetical continuous variable is a linear function
of SurvRate, our censored dichotomous variable (or the probability of improvement) is
not. But a simple transformation from p(improvement) to odds(improvement) to log
odds(improvement) will give us a variable that isa linear function of SurvRate. Therefore
we can convert p(improvement) to log odds(improvement) and get back to a linear func-
tion. An excellent discussion of what we are doing here can be found in Allison (1999).
Although that manual was written for people using SAS, it is one of the nicest descrip-
tions that I know and is useful whether you use SAS or not.
Dabbs and Morris (1990) ran an interesting study in which they classified male mili-
tary personnel as High or Normal in testosterone, and as either having, or not having, a his-
tory of delinquency. The results follow:
Delinquent
Yes No TotalTestosterone
Normal 402 3614 4016
High 101 345 446
503 3959 4462For these data, the odds of being delinquent if you are in the Normal group are
(frequency delinquent)/(frequency not delinquent). (Using probabilities instead of frequen-
cies, this comes down to pdelinquentpnot delinquent 5 p(delinquent) (1 2 p(delinquent).) For
the Normal testosterone group the odds of being delinquent are 402/3614 5 .1001 The
odds of being not delinquent if you are in the Normal group is the reciprocal of this, which
is 3614 402 5 8.990. This last statistic can be read as meaning that if you are a male with
normal testosterone levels you are nearly 9 times more likely to be not delinquent than
delinquent (or, if you prefer, 9 times less likelyto be delinquent than not delinquent). If we
look at the High testosterone group, however, the odds of being delinquent are 101 345 5
.293, and the odds of being not delinquent are 345 101 5 3.416. Both groups of males are
more likely to be not delinquent than delinquent, but that isn’t saying much, because we
would hope that most people are not delinquent. But notice that as you move from the Nor-
mal to the High group, your odds of being delinquent nearly triple, going from .111 to .293.
If we form the ratio of these odds we get .293 .111 5 2.64, which is the odds ratio. For
these data you are 2.64 more likely to be delinquent if you have high testosterone levels
than if you have normal levels. That is a pretty impressive statistic.
We will set aside the odds ratio for a moment and just look at odds. With our cancer
data we will focus on the odds of survival. (We can return to odds ratios any time we wish
simply by forming the ratio of the odds of survival and non-survival for each of two differ-
ent levels of SurvRate.)
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15.15 Logistic Regression 565