AN
INTRODUCTION
TO LOGISTIC
REGRESSION
ENI SUMARMININGSIH, SSI, MM PROGRAM STUDI STATISTIKA JURUSAN MATEMATIKA
OUTLINE
Introduction and
Description
Some Potential
INTRODUCTION AND DESCRIPTION
Why use logistic regression?
Estimation by maximum
likelihood
Interpreting coefficients
Hypothesis testing
Evaluating the performance of
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
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
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
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
PROBLEMS:
Predicted Values outside the 0,1
range
Descriptive Statistics: FITS1
Variable N N* Mean StDev Minimum Q1 Median Q3
Maximum
HETEROSKEDASTICITY
Weight R ES I1 220 200 180 160 140 120 100 1.00 0.75 0.50 0.25 0.00 -0.25 -0.50THE 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
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
COMPARING LP AND LOGIT MODELS
0
1
LP Model
MAXIMUM LIKELIHOOD
ESTIMATION (MLE)
MLE is a statistical method for
INTERPRETING COEFFICIENTS
Since:
ln[p/(1-p)] =
+
X + e
The slope coefficient (
) is interpreted
as the rate of change in the "log
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
FROM MINITAB OUTPUT:
**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.
Logistic Regression Table
Odds 95% CI
Predictor Coef SE Coef Z P Ratio Lower Upper
Constant -1.98717 1.67930 -1.18 0.237
Smokes
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
EVALUATING THE PERFORMANCE OF THE
MODEL
Goodness-of-Fit Tests displays Pearson, deviance, and Hosmer-Lemeshow goodness-of-fit tests. If the p-value is less than
your accepted α-level, the test would reject the null hypothesis of an adequate fit.
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
