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November 5, 2008
Logistic and Poisson Regression: Modeling Binary and Count Data
LISA Short Course Series
Logistic and Poisson Regression: Modeling Binary and Count Data
LISA Short Course Series
Mark Seiss, Dept. of Statistics
Presentation Outline
1. Introduction to Generalized Linear Models
2. Binary Response Data -
Logistic Regression Model
3. Count Response Data -
Poisson Regression Model
Reference Material
Categorical Data Analysis – Alan Agresti
Examples found with SAS Code at www.stat.ufl.edu/~aa/cda/cda.html
Presentation and Data from Examples
www.stat.vt.edu/consult/short_courses.html
Generalized Linear Models
• Generalized linear models (GLM) extend ordinary regression to non-normal response distributions.
• 3 Components• Random – identifies response Y and its probability distribution
• Systematic – explanatory variables in a linear predictor function (Xβ)
• Link function – function (g(.)) that links the mean of the response (E[Yi]=μi) to the systematic
component.
• Model• for i = 1 to n ij
jjx ig
Generalized Linear Models
• Why do we use GLM’s?• Linear regression assumes that the response is
distributed normally• GLM’s allow us to analyze the linear relationship
between predictor variables and the mean of the response variable when it is not reasonable to assume the data is distributed normally.
Generalized Linear Models• Predictor Variables
• Two Types: Continuous and Categorical
• Continuous Predictor Variables• Examples – Time, Grade Point Average, Test Score, etc.
• Coded with one parameter – βixi
• Categorical Predictor Variables• Examples – Sex, Political Affiliation, Marital Status, etc.• Actual value assigned to Category not important• Ex) Sex - Male/Female, M/F, 1/2, 0/1, etc.• Coded Differently than continuous variables
Generalized Linear Models• Categorical Predictor Variables cont.
• Consider a categorical predictor variable with L categories
• One category selected as reference category • Assignment of Reference Category is arbitrary
• Variable represented by L-1 dummy variables • Model Identifiability
• Two types of coding – Dummy and Effect
Generalized Linear Models• Categorical Predictor Variables cont.
• Dummy Coding (Used in R)
• xk = 1 if predictor variable is equal to category k
0 otherwise
• xk = 0 for all k if predictor variable equals category I
• Effect Coding (Used in JMP)
• xk = 1 if predictor variable is equal to category k
0 otherwise
• xk = -1 for all k if predictor variable equals category I
Generalized Linear Models
• Saturated Model• Contains a separate indicator parameter for each
observation• Perfect fit μ = y• Not useful since there is no data reduction, i.e.
number of parameters equals number of observations.
• Maximum achievable log likelihood – baseline for comparison to other model fits
Generalized Linear Models
• Deviance
• Let L(μ|y) = maximum of the log likelihood for the model
L(y|y) = maximum of the log likelihood for the saturated model
• Deviance = D(y| μ) = -2 [L(μ|y) - L(y|y) ]
• Likelihood Ratio Statistic for testing the null hypothesis that the model is a good alternative to the saturated model
• Likelihood ratio statistic has an asymptotic chi-squared distribution with N – p degrees of freedom, where p is the number of parameters in the model.
• Allows for the comparison of one model to another using the likelihood ratio test.
Generalized Linear Models
• Nested Models
• Model 1 - model with p predictor variables {X1, X2, X3,….,Xp} and vector of fitted values μ1
• Model 2 - model with q<p predictor variables {X1, X2, X3,….,Xq} and vector of fitted values μ2
• Model 2 is nested within Model 1 if all predictor variables found in Model 2 are included in Model 1.
• i.e. the set of predictor variables in Model 2 are a subset of the set of predictor variables in Model 1
• Model 2 is a special case of Model 1 - all the coefficients associated with Xp+1, Xp+2, Xp+3,….,Xq are equal to zero
q2p1ppp110 0…+0+0++…+ = g(u) XXXXX
Generalized Linear Models
• Likelihood Ratio Test• Null Hypothesis: There is not a significant difference between the fit
of two models.
• Null Hypothesis for Nested Models: The predictor variables in Model 1 that are not found in Model 2 are not significant to the model fit.
• Alternate Hypothesis for Nested Models - The predictor variables in Model 1 that are not found in Model 2 are significant to the model fit.
• Likelihood Ratio Statistic = -2* [L(y,u2)-L(y,u1)]
= D(y,μ2) - D(y, μ1)
Difference of the deviances of the two models
• Always D(y,μ2) > D(y,μ1) implies LRT > 0
• LRT is distributed Chi-Squared with p-q degrees of freedom
Generalized Linear Models
• Likelihood Ratio Test cont.• Later, we will use the Likelihood Ratio Test to test the significance of
variables in Logistic and Poisson regression models.
Generalized Linear Models
• Theoretical Example of Likelihood Ratio Test
• 3 predictor variables – 1 Continuous (X1), 1 Categorical with 4 Categories (X2, X3, X4), 1 Categorical with 1 Category (X5)
• Model 1 - predictor variables {X1, X2, X3, X4, X5}
• Model 2 - predictor variables {X1, X5}
• Null Hypothesis – Variables with 4 categories is not significant to the model (β2 = β3 = β4 = 0)
• Alternate Hypothesis - Variable with 4 categories is significant
• Likelihood Ratio Statistic = D(y,μ2) - D(y, μ1)
• Difference of the deviance statistics from the two models
• Chi-Squared Distribution with 5-2=3 degrees of freedom
Generalized Linear Models• Model Selection
• 2 Goals: Complex enough to fit the data well
Simple to interpret, does not overfit the data
• Study the effect of each predictor on the response Y• Continuous Predictor – Graph P[Y=1] versus X• Discrete Predictor - Contingency Table of P[Y=1] versus
categories of X
• Unbalance Data – Few responses of one type• Guideline – 10 outcomes of each type for each X terms• Example – Y=1 for only 30 observations out of 1000
Model should contain no more than 3 X terms
Generalized Linear Models• Model Selection cont.
• Multicollinearity• Correlations among predictors resulting in an increase in
variance• Reduces the significance value of the variable • Occurs when several predictor variables are used in the
model
• Determining Model Fit• Other criteria besides significance tests (i.e. Likelihood
Ratio Test) can be used to select a model
Generalized Linear Models• Model Selection cont.
• Determining Model Fit cont.• Akaike Information Criterion (AIC)
– Penalizes model for having many parameters– AIC = Deviance+2*p where p is the number of
parameters in model
• Bayesian Information Criterion (BIC)– BIC = -2 Log L + ln(n)*p where p is the number of
parameters in model and n is the number of observations
Generalized Linear Models• Model Selection cont.
• Selection Algorithms• Best subset – Tests all combinations of predictor variables
to find best subset• Algorithmic – Forward, Backward and Stepwise
Procedures
Generalized Linear Models• Best Subsets Procedure
• Run model with all possible combinations of the predictor variables
• Number of possible models equal to 2p where p is the number of predictor variables
• Dummy Variables for categorical predictors considered together
• Ex) For a set of predictors {X1, X2, X3}
• runs models with sets of predictors {X1, X2, X3}, {X1, X2},
{X2, X3}, {X1, X3}, {X1}, {X2}, {X3}, and no predictor variables.
• 23 = 8 possible models
• Most programs only allow for a small set of predictor variables
• Cannot be run in a reasonable amount of time
• 210 = 1024 models run for a set of 10 predictor variables
Generalized Linear Models• Forward Selection
• Idea: Start with no variables in the model and add one at a time
• Step One: Fit model with single predictor variable and determine fit
• Step Two: Select predictor variable with best fit and add to model
• Step Three: Add each variable to the model one at a time and determine fit
• Step Four: If at least one variable produces better fit, return to step two
If no variables produce better fit, use model
• Drawback: Variables Added to the model cannot be taken out.
Generalized Linear Models• Backward Selection
• Idea: Start with all variables in the model and take out one at a time
• Step One: Fit all predictor variables in model and determine fit
• Step Two: Delete one variable at a time and determine fit
• Step Three: If the deletion of at least one variable produces better fit, remove variable that produces best
fit when deleted and return to step 2
If the deletion of a variable does not produce a better fit, use model
• Drawback: Variables taken out of model cannot be added back in.
Generalized Linear Models• Stepwise Selection
• Idea: Combination of forward and backward selection
• Forward Step then backward step • Step One: Fit each predictor variable as a single predictor variable
and determine fit
• Step Two: Select variable that produces best fit and add to model.
• Step Three: Add each predictor variable one at a time to the model and determine fit
• Step Four: Select variable that produces best fit and add to the model
• Step Five: Delete each variable in the model one at a time and determine fit
• Step Six: Remove variable that produces best fit when deleted
• Step Seven: Return to Step Two
• Loop until no variables added or deleted improve the fit.
Generalized Linear Models
• Summary
• 3 Components of the GLM• Random (Y)• Link Function (g(E[Y]))• Systematic (xtβ)
• Continuous and Categorical Predictor Variables• Coding Categorical Variables – Effect and Dummy Coding
• Likelihood Ratio Test for Nested Models• Test the significance of a predictor variable or set of
predictor variables in the model.
• Model Selection – Best Subset, Forward, Backward, Stepwise
Generalized Linear Models
• Questions/Comments
Logistic Regression• Consider a binary response variable.
• Variable with two outcomes
• One outcome represented by a 1 and the other represented by a 0
• Examples:
Does the person have a disease? Yes or No
Who is the person voting for? McCain or Obama
Outcome of a baseball game? Win or loss
Logistic Regression• Logistic Regression Example Data Set
• Response Variable –> Admission to Grad School (Admit)• 0 if admitted, 1 if not admitted
• Predictor Variables• GRE Score (gre)
– Continuous• University Prestige (topnotch)
– 1 if prestigious, 0 otherwise • Grade Point Average (gpa)
– Continuous
Logistic Regression• First 10 Observations of the Data Set
ADMIT GRE TOPNOTCH GPA
1 380 0 3.61
0 660 1 3.67
0 800 1 4
0 640 0 3.19
1 520 0 2.93
0 760 0 3
0 560 0 2.98
1 400 0 3.08
0 540 0 3.39
1 700 1 3.92
Logistic Regression• Consider the linear probability model
where yi = response for observation i
xi = 1x(p+1) matrix of covariates for observation i
p = number of covariates
• GLM with binomial random component and identity link g(μ) = μ
• Issue: π(Xi) can take on values less than 0 or greater than 0
• Issue: Predicted probability for some subjects fall outside of the [0,1] range.
iiiii xxxYPYE )()|0(
Logistic Regression• Consider the logistic regression model
• GLM with binomial random component and identity link g(μ) = logit(μ)
• Range of values for π(Xi) is 0 to 1
i
iiiii x
xxxYPYE
exp1
exp)()|0(
ii
ii x
x
xxit
1
loglog
Logistic Regression• Consider the logistic regression model
And the linear probability model
Then the graph of the predicted probabilities for different grade point averages:
Important Note: JMP models P(Y=0) and effect coding is used for categorical variables
ii gpaxit *log
ii gpax *)(
Logistic Regression
Logistic Regression• Interpretation of Coefficient β – Odds Ratio
• The odds ratio is a statistic that measures the odds of an event compared to the odds of another event.
• Say the probability of Event 1 is π1 and the probability of Event 2 is π2 . Then the odds ratio of Event 1 to Event 2 is:
• Value of Odds Ratio range from 0 to Infinity
• Value between 0 and 1 indicate the odds of Event 2 are greater
• Value between 1 and infinity indicate odds of Event 1 are greater
• Value equal to 1 indicates events are equally likely
2
2
1
1
1
1
2
1
)(
)(_
Odds
OddsRatioOdds
Logistic Regression• Interpretation of Coefficient β – Odds Ratio cont.
• Link to Logistic Regression :
• Thus the odds ratio between two events is)()()()()_( 2111 2
2
1
1
LogitLogitLogLogRatioOddsLog
)}()(exp{_ 12 LogitLogitRatioOdds
Logistic Regression• Interpretation of Coefficient β – Odds Ratio cont.
• Consider Event 1 is Y=0 given X and Event 2 is Y=0 given X+1
• From our logistic regression model
• Thus the ratio of the odds of Y=0 for X and X+1 is
))|0(())1|0(()_( XYPLogitXYPLogitRatioOddsLog
)())1(( XX
)exp(_ RatioOdds
Logistic Regression• Single Continuous Predictor Variable - GPA
Generalized Linear Model Fit
Response: Admit
Modeling P(Admit=0)
Distribution: Binomial
Link: Logit
Observations (or Sum Wgts) = 400
Whole Model Test
Model -LogLikelihood L-R ChiSquare DF Prob>ChiSq
Difference 6.50444839 13.0089 1 0.0003
Full 243.48381
Reduced 249.988259
Goodness Of Fit Statistic ChiSquare DF Prob>ChiSq
Pearson 401.1706 398 0.4460 398 0.4460
Deviance 486.9676 398 0.0015 398 0.0015
Logistic Regression• Single Continuous Predictor Variable – GPA cont.
Effect Tests
Source DF L-R ChiSquare Prob>ChiSq
GPA 1 13.008897 0.0003
Parameter Estimates
Term Estimate Std Error L-R ChiSquare Prob>ChiSq Lower CL Upper CL
Intercept -4.357587 1.0353175 19.117873 <.0001 -6.433355 -2.367383
GPA 1.0511087 0.2988695 13.008897 0.0003 0.4742176 1.6479411
Interpretation of the Parameter Estimate:
Exp{1.0511087} = 2.86 = odds ratio between the odds at x+1 and odds at x for all x
The ratio of the odds of being admitted between a person with a 3.0 gpa and 2.0 gpa is equal to 2.86 or equivalently the odds of the person with the 3.0 is 2.86 times the odds of the person with the 2.0.
Logistic Regression• Single Categorical Predictor Variable – Top Notch
Generalized Linear Model Fit
Response: Admit
Modeling P(Admit=0)
Distribution: Binomial
Link: Logit
Observations (or Sum Wgts) = 400
Whole Model Test
Model -LogLikelihood L-R ChiSquare DF Prob>ChiSq
Difference 3.53984692 7.0797 1 0.0078
Full 246.448412
Reduced 249.988259
Goodness Of Fit Statistic ChiSquare DF Prob>ChiSq
Pearson 400.0000 398 0.4624
Deviance 492.8968 398 0.0008
I
Logistic Regression• Single Categorical Predictor Variable – Top Notch cont.
Effect Tests
Source DF L-R ChiSquare Prob>ChiSq
TOPNOTCH 1 7.0796939 0.0078
Parameter Estimates
Term Estimate Std Error L-R ChiSquare Prob>ChiSq Lower CL Upper CL
Intercept -0.525855 0.138217 14.446085 0.0001 -0.799265 -0.255667
TOPNOTCH[0] -0.371705 0.138217 7.0796938 0.0078 -0.642635 -0.099011
Interpretation of the Parameter Estimate:
Exp{2*-.371705} = 0.4755 = odds ratio between the odds of admittance for a student at a less prestigous university and the odds of admittance for a student from a more prestigous university.
The odds of being admitted from a less prestigous university is .48 times the odds of being admitted from a more prestigous university.
I
Logistic Regression• Variable Selection– Likelihood Ratio Test
• Consider the model with GPA, GRE, and Top Notch as predictor variablesGeneralized Linear Model Fit
Response: Admit
Modeling P(Admit=0)
Distribution: Binomial
Link: Logit
Observations (or Sum Wgts) = 400
Whole Model Test
Model -LogLikelihood L-R ChiSquare DF Prob>ChiSq
Difference 10.9234504 21.84693 <.0001
Full 239.064808
Reduced 249.988259
Goodness Of Fit Statistic ChiSquare DF Prob>ChiSq
Pearson 396.9196 396 0.4775
Deviance 478.1296 396 0.0029
•
Logistic Regression• Variable Selection– Likelihood Ratio Test cont.
Effect Tests
Source DF L-R ChiSquare Prob>ChiSq
TOPNOTCH 1 2.2143635 0.1367
GPA 1 4.2909753 0.0383
GRE 1 5.4555484 0.0195
Parameter Estimates
Term Estimate Std Error L-R ChiSquare Prob>ChiSq Lower CL Upper CL
Intercept -4.382202 1.1352224 15.917859 <.0001 -6.657167 -2.197805
TOPNOTCH[0] -0.218612 0.1459266 2.2143635 0.1367 -0.503583 0.070142
GPA 0.6675556 0.3252593 4.2909753 0.0383 0.0356956 1.3133755
GRE 0.0024768 0.0010702 5.4555484 0.0195 0.0003962 0.0046006
Logistic Regression• Model Selection – Forward
Stepwise Fit
Response:
Admit
Stepwise Regression Control
Prob to Enter 0.250
Prob to Leave 0.100
Direction:
Rules:
Current Estimates
-LogLikelihood RSquare
239.06481 0.0437
Logistic Regression• Model Selection – Forward cont.
Parameter Estimate nDF Wald/Score ChiSq "Sig Prob"
Intercept[1] -4.3821986 1 0 1.0000
GRE 0.00247683 1 5.356022 0.0207
GPA 0.66755511 1 4.212258 0.0401
TOPNOTCH{1-0} 0.21861181 1 2.244286 0.1341
Step History
Step Parameter Action L-R ChiSquare "Sig Prob" RSquare p
1 GRE Entered 13.92038 0.0002 0.0278 2
2 GPA Entered 5.712157 0.0168 0.0393 3
3 TOPNOTCH{1-0} Entered 2.214363 0.1367 0.0437 4
Logistic Regression• Model Selection – Backward
• Start by selecting to enter all variables into the model
Stepwise Fit
Response: Admit
Stepwise Regression Control
Prob to Enter 0.250
Prob to Leave 0.100
Direction: Backward
Rules: Combine
Logistic Regression• Model Selection – Backward cont.
Current Estimates
-LogLikelihood RSquare
240.17199 0.0393
Parameter Estimate nDF Wald/Score ChiSq "Sig Prob"
Intercept[1] -4.9493751 1 0 1.0000
GRE 0.00269068 1 6.473978 0.0109
GPA 0.75468641 1 5.576461 0.0182
TOPNOTCH{1-0} 0 1 2.259729 0.1328
Step History
Step Parameter Action L-R ChiSquare "Sig Prob" RSquare p
1 TOPNOTCH{1-0} Removed 2.214363 0.1367 0.0393 3
Logistic Regression• Variable Selection – Stepwise
Stepwise Fit
Response:
Admit
Stepwise Regression Control
Prob to Enter 0.250
Prob to Leave 0.250
Direction: Mixed
Rules: Combine
Current Estimates
-LogLikelihood RSquare
239.06481 0.0437
Logistic Regression• Variable Selection – Stepwise cont.
Parameter Estimate nDF Wald/Score ChiSq "Sig Prob"
Intercept[1] -4.3821986 1 0 1.0000
GRE 0.00247683 1 5.356022 0.0207
GPA 0.66755511 1 4.212258 0.0401
TOPNOTCH{1-0} 0.21861181 1 2.244286 0.1341
Step History
Step Parameter Action L-R ChiSquare "Sig Prob" Rsquare p
1 GRE Entered 13.92038 0.0002 0.0278 2
2 GPA Entered 5.712157 0.0168 0.0393 3
3 TOPNOTCH{1-0} Entered 2.214363 0.1367 0.0437 4
Logistic Regression• Summary
• Introduction to the Logistic Regression Model
• Interpretation of the Parameter Estimates β – Odds Ratio
• Variable Significance – Likelihood Ratio Test
• Model Selection • Forward• Backward• Stepwise
Logistic Regression
• Questions/Comments
Poisson Regression• Consider a count response variable.
• Response variable is the number of occurrences in a given time frame.
• Outcomes equal to 0, 1, 2, ….
• Examples:
Number of penalties during a football game.
Number of customers shop at a store on a given day.
Number of car accidents at an intersection.
Poisson Regression• Poisson Regression Example Data Set
• Response Variable –> Number of Days Absent – Integer
• Predictor Variables• Gender- 1 if Female, 2 if Male• Ethnicity – 6 Ethnic Categories• School – 1 if School, 2 if School 2• Math Test Score – Continuous• Language Test Score – Continuous• Bilingual Status – 6 Bilingual Categories
Poisson Regression• First 10 Observations from the Poisson Regression Example
Data SetGENDER ethnicity school.1.or.2 ctbs.math.nce ctbs.lang.nce bilingual.status number.days.absent
1 2 4 1 56.988830 42.45086 2 4
2 2 4 1 37.094160 46.82059 2 4
3 1 4 1 32.275460 43.56657 2 2
4 1 4 1 29.056720 43.56657 2 3
5 1 4 1 6.748048 27.24847 3 3
6 1 4 1 61.654280 48.41482 0 13
7 1 4 1 56.988830 40.73543 2 11
8 2 4 1 10.390490 15.35938 2 7
9 2 4 1 50.527950 52.11514 2 10
10 2 6 1 49.472050 42.45086 0 9
Poisson Regression• Consider the model
where Yi = response for observation i
xi = 1x(p+1) matrix of covariates for observation i
p = number of covariates
μi = expected number of events given xi
• GLM with poisson random component and identity link g(μ) = μ
• Issue: Predicted values range from -∞ to +∞
iii xYE
Poisson Regression• Consider the Poisson log-linear model
• GLM with poisson random component and log link g(μ) = log(μ)
• Predicted response values fall between 0 and +∞
• In the case of a single predictor, An increase of one unit of x results an increase of exp(β) in μ
iiii xxYE exp|
ii x log
Poisson Regression• Consider the Poisson log-linear model
And the Poisson linear model
Then a graph of the predicted values from the model:
ii ScoreMath _*log
ii ScoreMathx _*
Poisson Regression
Poisson Regression• Single Continuous Predictor Variable – Math Score
> fitline<-glm(number.days.absent~ctbs.math.nce,data=poisson_data,family=poisson(link=log))
> summary(fitline)
Call:
glm(formula = number.days.absent ~ ctbs.math.nce, family = poisson(link = log), data = poisson_data)
Deviance Residuals:
Min 1Q Median 3Q Max
-4.4451 -2.5583 -1.0842 0.6647 12.4431
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.302100 0.062776 36.671 <2e-16 ***
ctbs.math.nce -0.011568 0.001294 -8.939 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Poisson Regression• Single Continuous Predictor Variable – Math Score
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 2409.8 on 315 degrees of freedom
Residual deviance: 2330.6 on 314 degrees of freedom
AIC: 3196
Number of Fisher Scoring iterations: 6
Interpretation of the parameter estimate:
Exp{-0.011568} = .98 = multiplicative effect on the expected number of days absent for an increase of 1 in the Math Score
Fabricated Example – If a student is expected to miss 5 days with a math of 50, then another student with a math score of 51 is expected to miss 5*.98 = 4.9 days
Poisson Regression• Single Continuous Predictor Variable – Gender
> fitline<-glm(number.days.absent~factor(GENDER),data=poisson_data,family=poisson(link=log))
> summary(fitline)
Call:
glm(formula = number.days.absent ~ factor(GENDER), family = poisson(link = log), data = poisson_data)
Deviance Residuals:
Min 1Q Median 3Q Max
-3.660 -2.755 -1.128 0.902 9.738
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.90174 0.03036 62.644 < 2e-16 ***
factor(GENDER)2 -0.31729 0.04747 -6.684 2.32e-11 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Poisson Regression• Single Continuous Predictor Variable – Gender
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 2409.8 on 315 degrees of freedom
Residual deviance: 2364.5 on 314 degrees of freedom
AIC: 3229.9
Number of Fisher Scoring iterations: 5
Important Note: The function factor(categorical variable) uses the dummy coding
Interpretation of the parameter estimate:
Exp{-0.31729} = 0.7289 = multiplicative effect on the expected number of days absent of being male rather than female
If a female student is expected to miss X days, then a male student is expected to miss 0.7289*X.
Poisson Regression• Variable Selection – Likelihood Ratio Test
• Model with all variables > fitline<-glm(number.days.absent~factor(GENDER)+factor(school.1.or.2)+ctbs.math.nce+ctbs.lang.nce+factor(bilingual.status)+
factor(ethnicity),data=poisson_data,family=poisson(link=log))
summary(fitline)
Call:
glm(formula = number.days.absent ~ factor(GENDER) + factor(school.1.or.2) +
ctbs.math.nce + ctbs.lang.nce + factor(bilingual.status) +
factor(ethnicity), family = poisson(link = log), data = poisson_data)
Deviance Residuals:
Min 1Q Median 3Q Max
-4.5222 -2.1863 -0.9622 0.7454 10.4077
Poisson Regression• Variable Selection – Likelihood Ratio Test
• Model with all variables Cont> Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.972325 0.424645 7.000 2.57e-12 ***
factor(GENDER)2 -0.401980 0.048954 -8.211 < 2e-16 ***
factor(school.1.or.2)2 -0.582321 0.070717 -8.235 < 2e-16 ***
ctbs.math.nce -0.001043 0.001845 -0.565 0.57181
ctbs.lang.nce -0.003048 0.002003 -1.521 0.12822
factor(bilingual.status)1 -0.344696 0.083754 -4.116 3.86e-05 ***
factor(bilingual.status)2 -0.282194 0.070846 -3.983 6.80e-05 ***
factor(bilingual.status)3 -0.053406 0.081850 -0.652 0.51409
factor(ethnicity)2 -0.131202 0.420704 -0.312 0.75515
factor(ethnicity)3 -0.434061 0.418013 -1.038 0.29909
factor(ethnicity)4 -0.326230 0.419158 -0.778 0.43639
factor(ethnicity)5 -0.876270 0.416398 -2.104 0.03534 *
factor(ethnicity)6 -1.188835 0.457470 -2.599 0.00936 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Poisson Regression• Variable Selection – Likelihood Ratio Test
• Model with all variables Cont(Dispersion parameter for poisson family taken to be 1)
Null deviance: 2409.8 on 315 degrees of freedom
Residual deviance: 1909.2 on 303 degrees of freedom
AIC: 2796.6
Number of Fisher Scoring iterations: 6
Poisson Regression• Variable Selection – Likelihood Ratio Test
• Model with all variables except Ethnicity>fitline<glm(number.days.absent~factor(GENDER)+factor(school.1.or.2)+ctbs.math.nce+ctbs.lang.nce+factor(bilingual.status),
data=poisson_data,family=poisson(link=log))
> summary(fitline)
Call:
glm(formula = number.days.absent ~ factor(GENDER) + factor(school.1.or.2) + ctbs.math.nce + ctbs.lang.nce + factor(bilingual.status),
family = poisson(link = log), data = poisson_data)
Deviance Residuals:
Min 1Q Median 3Q Max
-4.6955 -2.3130 -0.9115 0.7527 11.4247
Poisson Regression• Variable Selection – Likelihood Ratio Test
• Model with all variables except Ethnicity
Coefficients: Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.5741133 0.0838754 30.690 < 2e-16 ***
factor(GENDER)2 -0.4212841 0.0484383 -8.697 < 2e-16 ***
factor(school.1.or.2)2 -0.8242109 0.0570241 -14.454 < 2e-16 ***
ctbs.math.nce 0.0008193 0.0018278 0.448 0.65398
ctbs.lang.nce -0.0050753 0.0019380 -2.619 0.00882 **
factor(bilingual.status)1 -0.3080131 0.0762534 -4.039 5.36e-05 ***
factor(bilingual.status)2 -0.1815997 0.0581877 -3.121 0.00180 **
factor(bilingual.status)3 0.0363656 0.0686396 0.530 0.59625
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Poisson Regression• Variable Selection – Likelihood Ratio Test
• Model with all variables except Ethnicity
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 2409.8 on 315 degrees of freedom
Residual deviance: 1984.1 on 308 degrees of freedom
AIC: 2861.5
Number of Fisher Scoring iterations: 6
Poisson Regression• Variable Selection – Likelihood Ratio Test
• Model 1 with All Variables – Deviance = -2 Log L = 1909.2 with
df = 303
• Model 2 without Ethnicity - Deviance = -2 Log L = 1984.1 with
df = 308
• Likelihood Ratio Test = Deviance (Model 2) – Deviance (Model 1)
= 1984.1 – 1909.2= 74.9
• Likelihood Ratio Test ~ Chi Square with 308-303 = 5 degrees of freedom
• P-Value < .0001
• There is significant evidence to conclude that ethnicity is a significant predictor variable.
Poisson Regression• Model Selection
• Forward Selection> fitline<-glm(number.days.absent~1,data=data1,family=poisson(link=log))
> step(fitline,scope = list(upper = ~factor(GENDER)+factor(school.1.or.2)+ctbs.math.nce+ctbs.lang.nce+factor(bilingual.status)+factor(ethnicity), lower = ~1),direction="forward")
Start: AIC=3273.22
number.days.absent ~ 1
Df Deviance AIC
+ factor(school.1.or.2) 1 2103.7 2969.1
+ factor(ethnicity) 5 2095.9 2969.3
+ ctbs.lang.nce 1 2311.7 3177.0
+ ctbs.math.nce 1 2330.6 3196.0
+ factor(bilingual.status) 3 2339.2 3208.6
+ factor(GENDER) 1 2364.5 3229.9
<none> 2409.8 3273.2
Poisson Regression• Model Selection
• Forward Selection cont.Step: AIC=2969.12
number.days.absent ~ factor(school.1.or.2)
Df Deviance AIC
+ factor(ethnicity) 5 2018.7 2894.1
+ factor(GENDER) 1 2029.3 2896.7
+ factor(bilingual.status) 3 2066.0 2937.4
+ ctbs.lang.nce 1 2092.7 2960.1
+ ctbs.math.nce 1 2096.7 2964.1
<none> 2103.7 2969.1
-
Poisson Regression• Model Selection
• Forward Selection cont.Step: AIC=2894.07
number.days.absent ~ factor(school.1.or.2) + factor(ethnicity)
Df Deviance AIC
+ factor(GENDER) 1 1951.3 2828.7
+ factor(bilingual.status) 3 1981.6 2863.0
+ ctbs.math.nce 1 2011.1 2888.5
+ ctbs.lang.nce 1 2012.5 2889.9
<none> 2018.7 2894.1
Step: AIC=2828.67
number.days.absent ~ factor(school.1.or.2) + factor(ethnicity) + factor(GENDER)
Df Deviance AIC
+ factor(bilingual.status) 3 1915.3 2798.8
+ ctbs.lang.nce 1 1938.5 2817.8
+ ctbs.math.nce 1 1942.3 2821.7
<none> 1951.3 2828.7
Poisson Regression• Model Selection
• Forward Selection cont.
Step: AIC=2798.75
number.days.absent ~ factor(school.1.or.2) + factor(ethnicity) + factor(GENDER) + factor(bilingual.status)
Df Deviance AIC
+ ctbs.lang.nce 1 1909.5 2794.9
+ ctbs.math.nce 1 1911.5 2796.9
<none> 1915.3 2798.8
Step: AIC=2794.89
number.days.absent ~ factor(school.1.or.2) + factor(ethnicity) + factor(GENDER) + factor(bilingual.status) + ctbs.lang.nce
Df Deviance AIC
<none> 1909.5 2794.9
+ ctbs.math.nce 1 1909.2 2796.6
Poisson Regression• Model Selection
• Forward Selection cont.
Call: glm(formula = number.days.absent ~ factor(school.1.or.2) + factor(ethnicity) + factor(GENDER) + factor(bilingual.status) + ctbs.lang.nce, family = poisson(link = log), data = data1)
Coefficients:
(Intercept) factor(school.1.or.2)2 factor(ethnicity)2 factor(ethnicity)3 factor(ethnicity)4
2.948689 -0.586678 -0.126806 -0.423376 -0.313360
factor(ethnicity)5 factor(ethnicity)6 factor(GENDER)2 factor(bilingual.status)1 factor(bilingual.status)2
-0.862743 -1.175574 -0.404215 -0.343907 -0.284027
factor(bilingual.status)3 ctbs.lang.nce
-0.051558 -0.003763
Degrees of Freedom: 315 Total (i.e. Null); 304 Residual
Null Deviance: 2410
Poisson Regression• Model Selection cont.
• Backward Selection> fitline<-glm(number.days.absent~factor(GENDER)
+factor(school.1.or.2)+ctbs.math.nce+ctbs.lang.nce+factor(bilingual.status)+
factor(ethnicity),data=poisson_data,family=poisson(link=log))
> backwards<-step(fitline,direction="backward")
Start: AIC=2796.57
number.days.absent ~ factor(GENDER) + factor(school.1.or.2) + ctbs.math.nce + ctbs.lang.nce + factor(bilingual.status) +
factor(ethnicity)
Df Deviance AIC
- ctbs.math.nce 1 1909.5 2794.9
<none> 1909.2 2796.6
- ctbs.lang.nce 1 1911.5 2796.9
- factor(bilingual.status) 3 1937.8 2819.2
- factor(ethnicity) 5 1984.1 2861.5
- factor(GENDER) 1 1977.8 2863.2
- factor(school.1.or.2) 1 1983.6 2869.0
Poisson Regression• Model Selection cont.
• Backward Selection cont.Step: AIC=2794.89
number.days.absent ~ factor(GENDER) + factor(school.1.or.2) + ctbs.lang.nce + factor(bilingual.status) + factor(ethnicity)
Df Deviance AIC
<none> 1909.5 2794.9
- ctbs.lang.nce 1 1915.3 2798.8
- factor(bilingual.status) 3 1938.5 2817.8
- factor(ethnicity) 5 1984.3 2859.7
- factor(GENDER) 1 1979.4 2862.8
- factor(school.1.or.2) 1 1986.5 2869.9
Poisson Regression• Model Selection cont.
• Stepwise Selection cont.> fitline<-glm(number.days.absent~1,data=data1,family=poisson(link=log))
> step(fitline,scope = list(upper=~factor(GENDER)+factor(school.1.or.2)+ctbs.math.nce+ctbs.lang.nce+factor(bilingual.status)+factor(ethnicity), lower = ~1),direction="both")
Start: AIC=3273.22
number.days.absent ~ 1
Df Deviance AIC
+ factor(school.1.or.2) 1 2103.7 2969.1
+ factor(ethnicity) 5 2095.9 2969.3
+ ctbs.lang.nce 1 2311.7 3177.0
+ ctbs.math.nce 1 2330.6 3196.0
+ factor(bilingual.status) 3 2339.2 3208.6
+ factor(GENDER) 1 2364.5 3229.9
<none> 2409.8 3273.2
Poisson Regression• Model Selection cont.
• Stepwise Selection cont.
Step: AIC=2969.12
number.days.absent ~ factor(school.1.or.2)
Df Deviance AIC
+ factor(ethnicity) 5 2018.7 2894.1
+ factor(GENDER) 1 2029.3 2896.7
+ factor(bilingual.status) 3 2066.0 2937.4
+ ctbs.lang.nce 1 2092.7 2960.1
+ ctbs.math.nce 1 2096.7 2964.1
<none> 2103.7 2969.1
- factor(school.1.or.2) 1 2409.8 3273.2
Poisson Regression• Model Selection cont.
• Stepwise Selection cont.
• Step: AIC=2894.07
• number.days.absent ~ factor(school.1.or.2) + factor(ethnicity)
• Df Deviance AIC
• + factor(GENDER) 1 1951.3 2828.7
• + factor(bilingual.status) 3 1981.6 2863.0
• + ctbs.math.nce 1 2011.1 2888.5
• + ctbs.lang.nce 1 2012.5 2889.9
• <none> 2018.7 2894.1
• - factor(ethnicity) 5 2103.7 2969.1
• - factor(school.1.or.2) 1 2095.9 2969.3
Poisson Regression• Model Selection cont.
• Stepwise Selection cont.
Step: AIC=2828.67
number.days.absent ~ factor(school.1.or.2) + factor(ethnicity) + factor(GENDER)
Df Deviance AIC
+ factor(bilingual.status) 3 1915.3 2798.8
+ ctbs.lang.nce 1 1938.5 2817.8
+ ctbs.math.nce 1 1942.3 2821.7
<none> 1951.3 2828.7
- factor(GENDER) 1 2018.7 2894.1
- factor(ethnicity) 5 2029.3 2896.7
- factor(school.1.or.2) 1 2050.5 2925.9
Poisson Regression• Model Selection cont.
• Stepwise Selection cont.
Step: AIC=2798.75
number.days.absent ~ factor(school.1.or.2) + factor(ethnicity) + factor(GENDER) + factor(bilingual.status)
Df Deviance AIC
+ ctbs.lang.nce 1 1909.5 2794.9
+ ctbs.math.nce 1 1911.5 2796.9
<none> 1915.3 2798.8
- factor(bilingual.status) 3 1951.3 2828.7
- factor(GENDER) 1 1981.6 2863.0
- factor(ethnicity) 5 1993.4 2866.8
- factor(school.1.or.2) 1 2003.4 2884.8
Poisson Regression• Model Selection cont.
• Stepwise Selection cont.
• Step: AIC=2794.89
• number.days.absent ~ factor(school.1.or.2) + factor(ethnicity) + factor(GENDER) + factor(bilingual.status) + ctbs.lang.nce
Df Deviance AIC
<none> 1909.5 2794.9
+ ctbs.math.nce 1 1909.2 2796.6
- ctbs.lang.nce 1 1915.3 2798.8
- factor(bilingual.status) 3 1938.5 2817.8
- factor(ethnicity) 5 1984.3 2859.7
- factor(GENDER) 1 1979.4 2862.8
- factor(school.1.or.2) 1 1986.5 2869.9
Poisson Regression• Model Selection cont.
• Stepwise Selection cont.
Call: glm(formula = number.days.absent ~ factor(school.1.or.2) + factor(ethnicity) + factor(GENDER) + factor(bilingual.status) + ctbs.lang.nce, family = poisson(link = log), data = data1)
Coefficients:
(Intercept) factor(school.1.or.2)2 factor(ethnicity)2 factor(ethnicity)3 factor(ethnicity)4
2.948689 -0.586678 -0.126806 -0.423376 -0.313360
factor(ethnicity)5 factor(ethnicity)6 factor(GENDER)2 factor(bilingual.status)1 factor(bilingual.status)2
-0.862743 -1.175574 -0.404215 -0.343907 -0.284027
factor(bilingual.status)3 ctbs.lang.nce
-0.051558 -0.003763
Degrees of Freedom: 315 Total (i.e. Null); 304 Residual
Null Deviance: 2410
Residual Deviance: 1909 AIC: 2795
Poisson Regression• Lets look back at the Poisson log-linear model
• Taking the sample mean and sample variance of the response for intervals of Math Scores
ii ScoreMath _*log
Math Score Sample Mean Sample Standard Deviation
0-20 11.66666667 10.64397095
20-40 6.453333333 6.595029523
40-60 5.270072993 7.382913152
60-80 4.324675325 5.434881392
80-100 9.666666667 14.50861813
Poisson Regression
• Overdispersion for Poisson Regression Models• For Yi~Poisson(λi), E [Yi] = Var [Yi] = λi
• The variance of the response is much larger than the mean.
• Larger variance known as overdispersion• Consequences: Parameter estimates are still
consistent
Standard errors are inconsistent• Remedy: Negative Binomial model
Poisson Regression• Summary
• Introduction to the Poisson Regression Model
• Interpretation of β
• Variable Significance – Likelihood Ratio Test
• Model Selection • Forward• Backward• Stepwise
• Overdispersion
Poisson Regression
• Questions/Comments