AN INTRODUCTION TO LOGISTIC REGRESSION ENI SUMARMININGSIH, SSI, MM PROGRAM STUDI STATISTIKA JURUSAN MATEMATIKA UNIVERSITAS BRAWIJAYA
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
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
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.
AN EXAMPLE
THE DATARestingPulse 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 RESULTSResultsRegression Analysis: Tekanan Darah versus Weight, Merokok The regression equation isTekanan Darah = 0.745 - 0.00392 Weight + 0.210 Merokok Predictor Coef SE Coef T PConstant 0.7449 0.2715 2.74 0.007Weight -0.003925 0.001876 -2.09 0.039Merokok 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* Mean StDev Minimum Q1 Median Q3 MaximumFITS1 92 0 0.2391 0.1204 -0.0989 0.1562 0.2347 0.3132 0.5309
HETEROSKEDASTICITY
Weight
RES
I1
220200180160140120100
1.00
0.75
0.50
0.25
0.00
-0.25
-0.50
Scatterplot of RESI1 vs Weight
THE LOGISTIC REGRESSION MODELThe "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
0
1
LP Model
Logit Model
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:
**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% CIPredictor Coef SE Coef Z P Ratio Lower UpperConstant -1.98717 1.67930 -1.18 0.237Smokes Yes -1.19297 0.552980 -2.16 0.031 0.30 0.10 0.90Weight 0.0250226 0.0122551 2.04 0.041 1.03 1.00 1.05
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 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.
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.