13: Lecture More on Binary Data
Link functions for Binomial models Link = g() identity logarithmic log logistic log 1 probit 1 () log-log log( log ) complementary log( log(1 )) log-log = g 1 () e
e 1+e
() exp(e ) 1 exp(e )
Comparison of Link Functions
Logit Probit CLL
g(p)
-4 0.0
-2
0
2
4
0.2
0.4 p
0.6
0.8
1.0
When g is the identity or logarithmic function, = g 1 () may lie outside the interval [0, 1]. Therefore, these link functions may not be the best choices. The logistic and probit are by far the most commonly used. 1
The probit model requires numerical integration in the computation of the MLEs (since does not have a closed form). Both the logistic and probit are symmetric about = 1/2, and produce very similar results in most analysis (e.g., beetle example) unless there are many small or large probabilities. A very large amount of data would be required to show that one was better than the other. The logistic, probit, and complementary log-log links are similar for small . The complementary log-log link is skewed (asymmetric). Advantages of the logistic model The logistic model provides direct interpretation as log-odds of a success. This interpretation is particularly nice in the context of case-control studies. Here, the exposure level (e.g. of a toxin) is compared between individuals who have a particular disease or condition (the cases) and those who do not have the disease (the controls). The logit link is the canonical link for binomial distribution, hence is mathematically convenient. Interpretation of the link functions: bioassays and dose-response models A biological assay is a method for estimating the potency of a material by means of the reaction which follows its application to living matter. An m-point assay is an assay where a lethal drug is administered in m doses d1 , . . . , dm where di = log(concentration) (usually). In the experiment, ni subjects get dose di , and yi of these subjects die. We model Yi Binomial(ni , i ), where i = P(death given dose i) = function of di .
2
Tolerance Distribution We think of dierent subjects as having dierent tolerances to a drug. Dene D to be the minimum dose required to produce a response in a subject. Then D is a random variable. Its distribution is called the tolerance distribution, and is denoted by FD . An animal is killed by dose di if and only if its tolerance is less than or equal to di , i.e., D di . Thus, the probability i of death at dose di is given by i = P(D < di ) = FD (di ). Probit Analysis Based on the assumption of a normal tolerance distribution, i.e.
2 D N (d , d ),
then i = P(D < di ) = where is the N (0, 1) cdf. Thus, 1 (i ) = where 0 = /d d and 1 = 1/d . Hence, the mean and standard deviation of the tolerance distribution can be estimated via the regression parameters in this model. Logistic analysis Assume now that the tolerance distribution is the logistic distribution, i.e. fD (d) = exp {(d d )/ } , [1 + exp {(d d )/ }]2 di d 0 + 1 di , d di d , d
where < d < , < d < , and > 0. Then E[D] = d , and Var[D] = 2 2 /3 (where = 3.1415 . . .). Under this assumption, i = FD (di ) = exp {(di d )/ } 1 + exp{(di d )/ } 3
log
i 1 i
=
di d = 0 + 1 di ,
where 0 = d / and 1 = 1/ . Complementary log-log analysis Similarly, the assumption of the extreme-value tolerance distribution FD (di ) = 1 ee
adi b
(where b < 0) leads to the complementary log-log model. Example: Beetle data, cont. Goal: Find the tolerance distribution. We chose a linear complementary log-log model for the probability of death at a given dose x. This implies that the tolerance distribution is the extreme value distribution. Therefore, FD (x) = = 1 exp [ exp(0 + 1 x)] The density function of D is given by f (x) = 1 exp [(0 + 1 x) exp(0 + 1 x)] . An estimate of this density can be obtained by substituting the estimated values 0 and 1 : f (x) = 0.1525 exp [(9.603 + 0.1525x) exp(9.603 + 0.1525x)] .
Probit-logistic relationship The normal and logistic distributions are often quite similar (for certain values of their parameters). Consider the case where the true tolerance distribution is similar to these distributions. In this case, we would expect the GLMs with probit and logistic links to t almost equally well. More specically,...