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LINEAR PREDICTORS AND FITTED VALUES
Biostat/Epi 536 Discussion Session 2 – October 9, 2007
TERMINOLOGY
Fitted value: p ˆ i
Linear predictor: βˆ o^ +βˆ 1 x 1 i +...+ βˆ KxKi
FITTED VALUE LINEAR PREDICTOR
Given the value of p ˆ i , we can calculate βˆ o + βˆ 1 x 1 i +...+ βˆ KxKi using:
i
i i K Ki i p
p x x p 1 ˆ
ˆ 0 ˆ 11 ... ˆ logit ˆ log −
LINEAR PREDICTOR FITTED VALUE
Given the value of βˆ o + βˆ 1 x 1 i +...+ βˆ KxKi , we can calculate p ˆ i using:
1 exp(ˆ ˆ ... ˆ )
exp(ˆ ˆ ... ˆ ) ˆ 0 11
0 11 i K Ki
i K Ki i x x
x x p
EXERCISE
Using the dataset http://students.washington.edu/gardc/ille975.dta, we wish to model the probability of being a case (cc=1), adjusting for alcohol use (alc) and tobacco use (tob), which are defined as follows:
alc = 1 0-39 grams/day 2 40- 3 80- 4 120+
tob = 1 0-9 grams/day 2 10- 3 20- 4 30+
STATA
. xi: logit cc i.alc i.tob i.alc _Ialc_1-4 (naturally coded; _Ialc_1 omitted) i.tob _Itob_1-4 (naturally coded; _Itob_1 omitted)
Iteration 0: log likelihood = -494. Iteration 1: log likelihood = -422. Iteration 2: log likelihood = -415. Iteration 3: log likelihood = -415. Iteration 4: log likelihood = -415.
Logistic regression Number of obs = 975 LR chi2(6) = 159. Prob > chi2 = 0. Log likelihood = -415.18 Pseudo R2 = 0.
cc | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- _Ialc_2 | 1.224567 .2344119 5.22 0.000 .7651279 1. _Ialc_3 | 1.99794 .264346 7.56 0.000 1.479832 2. _Ialc_4 | 3.179962 .3273925 9.71 0.000 2.538285 3. _Itob_2 | .3861617 .2116419 1.82 0.068 -.0286488. _Itob_3 | .4251191 .2562347 1.66 0.097 -.0770916. _Itob_4 | .9877336 .2893924 3.41 0.001 .420535 1. _cons | -2.813544 .2111281 -13.33 0.000 -3.227347 -2.
What is the equation of the linear predictor?
2 3 4 2 3 4
0 2 2 3 3 4 4 2 2 3 3 4 4
logitˆ log
logitˆ log
a a a t t t i
i i
a a a a a a t t t t t t i
i i
x x x x x x p
p p
x x x x x x p
p p
What is the value of the linear predictor for a person who smokes 25 grams/day and drinks 50 grams/day?
What is the fitted value for a person who smokes 25 grams/day and drinks 50 grams/day?
