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Overview of Logistics Regression and its SAS implementation
•
Logistics regression is widely used nowadays in finance, marketing research and clinical
studies when the dependent variable is dichotomous, representing an event or a nonevent. However, because ordinary linear regression was routinely used before we had
the modern statistical packages for analyzing logit, we will compare the statistical
assumptions of logistic regression with that of ordinary least square linear regression.
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Next we will examine PROC LOGISTICS implemented in SAS and discuss the basic
statistic output for understanding the logistic regression results.
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We will then discuss how to setup and understand logistics regression when the
dependent variable has more than two outcomes.
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We will conclude the presentation by comparing PROC LOGISTICS with other SAS
procedures that can also perform logistics regression.
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Examples of discrete responses:
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Examples of discrete responses:
–
–
–
–
–
Getting decease vs. not getting decease
Good, medium and bad credit risks
Responders vs. non – responders (both in marketing or clinical trial studies)
Married vs. unmarried
Guilty vs. not guilty
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Comparing linear and logistic regression
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Linear Probability Model
p i    xi
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Logit Model
 pi 
log 
    1 xi1   2 xi 2  ..  1 xik
1  pi 
pi 
1
1  exp(   1 xi1   2 xi 2  ..  1 xik )
3
Why can linear regression work reasonable well on binary dependent
variables ?
Assumptions
1
yi  a  bxi  i
2
E (i )  0
3
Var (i )   i
4
Cov(i , j )  0
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i ~ Noraml
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Consequence of violations
Notes
Biased parameter estimates
Parameter’s meaning hard to interpret,
except a linear approximation to
nonlinear functions. Prediction can be
<0, >1
Biased intercept estimate
2
^
^
Unbiased estimates but biased Variance of b
Biased confidence interval b
Same as 3
Unable for us to use t , F statistical tests for
regression models.
The estimates may still be normal if
sample size is large.
If 1) and 2) are true, it can be shown that 3) and 5) are necessarily false. However, the consequences
may not be as serious as you expect.
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Logistic regression for binary response variables
Basic Syntax:
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proc logistic data=chdage1 outest=parms descending;
model chd = age /
selection = stepwise
ctable pprob = (0 to 1 by 0.1)
outroc=roc1;
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proc score data=chdage1 score = parms out=scored type=parms;
var age;
run;
In the events/trials syntax, you specify two variables that contain count data for a binomial experiment. These two variables
are separated by a slash. The value of the first variable, events, is the number of positive responses (or events). The value of
the second variable, trials, is the number of trials.
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Interpretation of SAS output - continued
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Model Selection Criteria:
–
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Convergence - difference in parameter estimates is small enough.
Model Fit Statistics Criteria:
n
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Likelihood Function:
L   pi i (1  pi )1 yi
y
i 1
–
–
–
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– 2 * log (likelihood )
AIC = – 2 * log ( max likelihood ) + 2 * k
SIC = – 2 * log ( max likelihood ) + log (N) * k
Testing Global Null Hypothesis: BETA=0
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Likelihood ratio: ln(L intercept)- ln(L int + covariates),
–
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Score: 1st and 2nd derivative of Log(L)
Wald: (coefficient / std error)2
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Interpretation of SAS output - continued
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Analysis of Maximum Likelihood Estimates
–
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Parameter estimates and significance test
Odds Ratio Estimates
k
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Odds:
Oi   exp(  j xij )
j 0
–
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Association of Predicted Probabilities and Observed Responses
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–
–
–
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Odds ratio: Oi / Oj per unit change in covariate.
Pairs: 43 (event) * 57 (non event) = 2451
Concordant (0- lower prob vs. 1- higher prob)
Discordant (0- higher prob vs. 1- lower prob)
Tie – all other
ROC used to visualize model model prediction strength.
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Interpretation of SAS output - continued
Classification Table:
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The model classifies an observation as an event if its estimated probability is greater than
or equal to a given probability cutpoints.
Prob. Level
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Correct
Incorrect
Percentages (%)
Event Non Event Event Non Event Correct
Sensitivity Specificity
FALSE POS FALSE NEG
57
0
43
0
57
100
0
43 .
57
1
42
0
58
100
2.3
42.4
0
55
7
36
2
62
96.5
16.3
39.6
22.2
51
19
24
6
70
89.5
44.2
32
24
50
25
18
7
75
87.7
58.1
26.5
21.9
45
27
16
12
72
78.9
62.8
26.2
30.8
41
32
11
16
73
71.9
74.4
21.2
33.3
32
36
7
25
68
56.1
83.7
17.9
41
24
39
4
33
63
42.1
90.7
14.3
45.8
6
42
1
51
48
10.5
97.7
14.3
54.8
0
43
0
57
43
0
100 .
57
Tot
Correct /
Total
Item
a
b
c
d
Correct
Correct
Event/ Tot N.Event/ Tot F.Pos /
Event
N.Event
(F.Pos+Pos)
(a+b) /
(a+b+c+d) a / (a+d)
b / (b+c)
c / (a+c)
F.Neg /
(F.Neg+Neg)
d / (b+d)
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Logistic regression for polychotomous response variables
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Example: Three outcomes
p1  pr (Y  1 x)
p2  pr (Y  2 x)
p3  pr (Y  3 x)  1  p1  p2
•
•
The cumulative probability model
log(
p1
)  1  x
1  p1
log(
p1  p2
)   2  x
1  p1  p2
The assumption:
– A common slope parameter associated with the predictor.
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Logistic regression for polychotomous response variables
Examples:
• proc logistic data=diabetes descending;
model group=glutest;
output out=probs predicted=prob xbeta=logit;
format group gp.;
run;
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Other SAS Procedures for Logistic Regression Models
Proc
Model Options
Logistics
Notes
Event/Trial format only works for
binary Response
GENMOD
Dist = Binormial
Link=logit
PROBIT
Dist=Logistic
CATMOD
Proc catmod data=diabetes;
direct glutest;
response logits / out =
cat_prob;
model group = glutest;
run;
Allow for individual parameters:
proc phreg data=diabetes;
model t * group = glutest;
run;
Trick: events occur at time one, non
events occur at a later times (censored).
* t is dummy time var,
group is censoring var
PHREG
One of General linear models
log(
p1
)  1  1 x
p2
log(
p2
)  2  2x
p3
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References
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Hosmer, D.W, Jr. and Lemeshow, S. (1989), Applied Logistic Regression, New
York: John Wiley & Sons, Inc.
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SAS Institute Inc. (1995), Logistic Regression Examples Using the SAS
System, Cary, NC: SAS Institute Inc.
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Paul D. Allison (1999) Logistic Regression Using the SAS System: Theory and
Application, BBU Press and John Wiley Sons Inc.
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