An Introduction to Logistic Regression

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AN INTRODUCTION TO
LOGISTIC REGRESSION
ENI SUMARMININGSIH, SSI, MM
PROGRAM STUDI STATISTIKA
JURUSAN MATEMATIKA
UNIVERSITAS BRAWIJAYA
OUTLINE
 Introduction and
Description
 Some Potential Problems
and Solutions
INTRODUCTION AND DESCRIPTION
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Why use logistic regression?
Estimation by maximum likelihood
Interpreting coefficients
Hypothesis testing
Evaluating the performance of the model
WHY USE LOGISTIC REGRESSION?
There are many important research topics for which
the dependent variable is "limited."
For example: voting, morbidity or mortality, and
participation data is not continuous or distributed
normally.
Binary logistic regression is a type of regression
analysis where the dependent variable is a dummy
variable: coded 0 (did not vote) or 1(did vote)
THE LINEAR PROBABILITY MODEL
In the OLS regression:
Y =  + X + e ; where Y = (0, 1)
 The error terms are heteroskedastic
 e is not normally distributed because Y
takes on only two values
 The predicted probabilities can be greater
than 1 or less than 0
AN EXAMPLE
You are a researcher who is interested in
understanding the effect of smoking and weight upon
resting pulse rate. Because you have categorized the
response-pulse rate-into low and high, a binary logistic
regression analysis is appropriate to investigate the
effects of smoking and weight upon pulse rate.
THE DATA
RestingPulse
Smokes
Weight
Low
No
140
Low
No
145
Low
Yes
160
Low
Yes
190
Low
No
155
Low
No
165
High
No
150
Low
No
190
Low
No
195
⁞
⁞
Low
No
110
High
No
150
Low
No
108
⁞
OLS RESULTS
Results
Regression Analysis: Tekanan Darah versus Weight, Merokok
The regression equation is
Tekanan Darah = 0.745 - 0.00392 Weight + 0.210 Merokok
Predictor
Coef
SE Coef
T
P
Constant
0.7449
0.2715
2.74 0.007
Weight
-0.003925 0.001876 -2.09 0.039
Merokok
0.20989 0.09626 2.18 0.032
S = 0.416246 R-Sq = 7.9% R-Sq(adj) = 5.8%
PROBLEMS:
Predicted Values outside the 0,1
range
Descriptive Statistics: FITS1
Variable N N*
FITS1
92 0
Mean StDev Minimum
Q1 Median
Q3
Maximum
0.2391 0.1204 -0.0989
0.1562 0.2347 0.3132 0.5309
HETEROSKEDASTICITY
Scatterplot of RESI1 vs Weight
1.00
0.75
RESI1
0.50
0.25
0.00
-0.25
-0.50
100
120
140
160
Weight
180
200
220
THE LOGISTIC REGRESSION
MODEL
The "logit" model solves these problems:
ln[p/(1-p)] =  + X + e
 p is the probability that the event Y occurs,
p(Y=1)
 p/(1-p) is the "odds ratio"
 ln[p/(1-p)] is the log odds ratio, or "logit"
More:
 The logistic distribution constrains the estimated
probabilities to lie between 0 and 1.
 The estimated probability is:
p = 1/[1 + exp(- -  X)]
 if you let  +  X =0, then p = .50
 as  +  X gets really big, p approaches 1
 as  +  X gets really small, p approaches 0
COMPARING LP AND LOGIT MODELS
LP Model
1
Logit Model
0
MAXIMUM LIKELIHOOD
ESTIMATION (MLE)
 MLE is a statistical method for estimating the
coefficients of a model.
INTERPRETING COEFFICIENTS
 Since:
ln[p/(1-p)] =  + X + e
The slope coefficient () is interpreted as the rate
of change in the "log odds" as X changes … not
very useful.
 An interpretation of the logit
coefficient which is usually more
intuitive is the "odds ratio"

Since:
[p/(1-p)] = exp( + X)
exp() is the effect of the independent
variable on the "odds ratio"
FROM MINITAB OUTPUT:
Logistic Regression Table
Predictor
Coef
SE Coef
Z
Constant -1.98717 1.67930 -1.18
Smokes
Yes
-1.19297
0.552980 -2.16
Weight
0.0250226 0.0122551 2.04
P
0.237
Odds
95% CI
Ratio Lower Upper
0.031 0.30 0.10 0.90
0.041 1.03 1.00 1.05
**Although there is evidence that the estimated coefficient for Weight
is not zero, the odds ratio is very close to one (1.03), indicating that a
one pound increase in weight minimally effects a person's resting
pulse rate
**Given that subjects have the same weight, the odds ratio can be
interpreted as the odds of smokers in the sample having a low pulse
being 30% of the odds of non-smokers having a low pulse.
HYPOTHESIS TESTING

The Wald statistic for the  coefficient is:
Wald (Z)= [ /s.e.B]2
which is distributed chi-square with 1 degree of freedom.

The last Log-Likelihood from the maximum likelihood iterations is displayed along
with the statistic G. This statistic tests the null hypothesis that all the coefficients
associated with predictors equal zero versus these coefficients not all being equal to
zero. In this example, G = 7.574, with a p-value of 0.023, indicating that there is
sufficient evidence that at least one of the coefficients is different from zero, given
that your accepted level is greater than 0.023.
EVALUATING THE PERFORMANCE OF THE
MODEL
Goodness-of-Fit Tests displays Pearson, deviance, and Hosmer-Lemeshow goodnessof-fit tests. If the p-value is less than your accepted α-level, the test would reject the
null hypothesis of an adequate fit.
The goodness-of-fit tests, with p-values ranging from 0.312 to 0.724, indicate that
there is insufficient evidence to claim that the model does not fit the data
adequately
MULTICOLLINEARITY
 The presence of multicollinearity will not lead to biased
coefficients.
 But the standard errors of the coefficients will be inflated.
 If a variable which you think should be statistically
significant is not, consult the correlation coefficients.
 If two variables are correlated at a rate greater than .6, .7,
.8, etc. then try dropping the least theoretically important
of the two.

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