1. NEGATIVE BINOMIAL MODELS
THE POISSON
&
NEGATIVE BINOMIAL MODELS
By: ALVARD AYRAPETYAN

2. OUTLINE OF PRESENTATION
Poisson Regression
Model Assumptions, Assessment, and Interpretations
Applications in SAS and R
Quick Programming in SPSS and MINITAB
Negative Binomial
Model Assumptions, Assessment, and Interpretations
Quick Programming in SPSS

3. ASSUMPTIONS FOR POISSON MODEL
Number of events must occur at a fixed period of time
Number of events must occur at a constant rate
Events must be independent
Dependent variable’s conditional mean and variance must be equal
Dependent variable must be an integer

4. THE POISSON MODEL
Random Component: Poisson Distribution for the # of lead changes
Systematic Component:
Mass Function:
E(Y) = µ & V(Y)= µ
Link Function: g(µ) = log(µ)

5. EXAMPLES OF POISSON DISTRIBUTION
Number of earthquakes in a region
Number of accidents on a highway in a certain area in a specified time
Number of telephone calls received in one hour
Number of customers that enter a bank in one hour
Number of times an elderly person will fall in a month

6. INTEPRETING COEFFICIENTS
CONTINUOUS PREDICTOR
CATEGORICAL PREDICTOR
Keeping all constant, when is increased by one unit, Y increases/decreases (+/-) by
Keeping all constant, when is increased by one unit, the expected number of Y will go up/down (+/-) by
Keeping all constant, when , Y increases/decreases (+/-) by
Keeping all constant, when the expected number of Y will go up/down (+/-) by

7. POTENTIAL PROBLEM WITH POISSON
OVERDISPERSION-the variance is much larger than the mean
Negative Binomial is the solution!

8. THE DATA Trying to predict the number of field goal attempts in NBA
Extracted the top 100 highest scoring players in the NBA for the season
The following were used as predictors:
Number of games played (GP)
Number of defensive rebounds(DREB)
Number of assists (AST)
Number of steals (STL)
Number of blocks (BLK)
Number of turnovers (TOV)
Number of free throws made (FTM)

9. SAMPLE OF THE DATA Rank Player GP FGA DREB AST STL FTM TOV 1
Kevin Love (MIN) 15
268
146 68 13 95 41 2
Kevin Durant (OKC) 12
209 72 62 17
131 45 3
Monta Ellis (DAL) 14
235 42 76 22 85 55 4
Blake Griffin (LAC)
242
129 47 19 59 40 5
LeBron James (MIA)
201 67 88 71 49 6
Evan Turner (PHI)
272 53 56 7
Kevin Martin (MIN)
248 48 33 18 20 8
Paul George (IND)
231 23 70 9
LaMarcus Aldridge (POR)
285
105 35 54 34 10
Carmelo Anthony (NYK)
264 79 36 11
Kyrie Irving (CLE) 89
Klay Thompson (GSW)
212 30
Dirk Nowitzki (DAL)
206 82 16 74 25
James Harden (HOU)
195 65 21 91 52
Chris Paul (LAC)
208
188 81 44
Arron Afflalo (ORL)
197 61
Damian Lillard (POR)
225 64 31
DeMarcus Cousins (SAC)
230
103

10. POISSON-EXAMPLE WITH SAS
proc genmod data = nba;
model FGA= GP DREB AST STL TOV FTM /dist=poisson;
run;
/*check goodness of fit for model*/
data pvalue;
df = 93; chisq = ;
pvalue = 1 - probchi(chisq, df);
proc print data = pvalue noobs;
run; /*pvalue is NOT significant, model isnt good*; dispersion parameter >> 1, major overdipsersion/

11. EXAMPLE RESULTS-GOODNESS OF FIT
The GENMOD Procedure
Model Information
Data Set WORK.NBA
Distribution Poisson
Link Function Log
Dependent Variable FGA
Number of Observations Read
Number of Observations Used
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance
Scaled Deviance
Pearson Chi-Square
Scaled Pearson X
Log Likelihood
Full Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)   

12. RESULTS: Analysis of Maximum Likelihood Parameter EstimatesDF
ESTIMATE
STANDARD ERROR
WALD 95% CONFIDENCE LIMITS
WALD CHI-SQUARE
PR>CHISQ
Intercept 1
4.1864
0.0749
(4.0396,43332)
<.0001 GP
0.0422
0.0057
(0.0310,0.0534)
54.93
DREB
0.0004
0.0003
( ,0.0010)
1.55
0.2131
AST
( ,0.0005)
0.28
0.5995
STL
0.0028
0.0012
(0.0004,0.0052)
5.17
0.0230
TOV
0.0010
(0.0038,0.0077)
33.53
FTM
0.0040
(0.0032,0.0048)
98.23
Scale
1.000
(1.0, 1.0)

13. ASSESSMENT OF RESULTS
Ratio of Deviance/Df= >>>1==major overdispersion
Deviance= , not well fit because pvalue=1-prob(chisq,df) is NOT significant
Every term significant except for AST and DREB
False results possible if model is inaccurate
Must perform a NEGATIVE BINOMIAL

14. POISSON-EXAMPLE WITH R
nba <- read.csv("F:/STATS544/nba.cs v",header=TRUE)
poiss<-glm(FGA ~GP+DREB+AST+STL+TOV+FT M, family = "poisson", data = nba)
summary(poiss)

15. R-GOODNESS OF FITS
Deviance Residuals: Min 1Q Median 3Q Max (Dispersion parameter for poisson family taken to be 1) Null deviance: on 99 degrees of freedom Residual deviance: on 93 degrees of freedom AIC:

16. R-ANALYSIS OF PARAMETER ESTIMATES
Call: glm(formula = FGA ~ GP + DREB + AST + STL + TOV + FTM, family = "poisson", data = nba) Coefficients: Estimate Std. Error z value Pr(>|z|) --- Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
ESTIMATE
STD.ERROR
Z VALUE
PR(>|z|)
(Intercept)
55.902
< 2e-16 *** GP
7.411
1.25e-13 ***
DREB
1.245
0.213
AST
-0.525
0.600
STL
2.273
0.023 *
TOV
5.790
7.02e-09 ***
FTM
9.911

17. POISSON WITH SPSS & MINITAB
genlin FGA with GP DREB AST STL TOV FTM /model GP DREB AST STL TOV FTM INTERCEP=YES distribution = poisson link = log /print FIT SUMMARY SOLUTION.
Stat > Regression  > Poisson Regression > Fit Poisson Model.

18. Detecting over-dispersion with SAS
Poisson regression gives a ratio between DEVIANCE and DF >1.
proc genmod data = nba;
model FGA= GP DREB AST STL TOV FTM /dist=poisson;
run;
PROC MEANS--- the variance of FGA(Y) is much higher than its mean
proc means data = nba n mean var min max;
var FGA

19. Detecting over-dispersion with R
Poisson regression gives a ratio between RESIDUAL DEVIANCE and DF >1
poiss<-glm(FGA ~GP+DREB+AST+STL+TOV+FTM, family = "poisson", data = nba)
summary(poiss)
mean(nba$FGA)
[1]
var(nba$FGA)
[1]

20. NEGATIVE BINOMIAL REGRESSION
Generalization of Poisson regression
Used for over-dispersed count data
PMF:
E(Y)= m, V(Y) = m+ k*(m2)
K=dispersion parameter
As k0, the V(Y)m, NB approaches Poisson and V(Y)=E(Y)= m
Link Function same as Poisson: g(m) = log(m.)
Equation: Log(λ(X))= β0 + β1Χ1 + β2Χ2+……..+ βp-1Xp-1
Goodness Of fit Test-same as Poisson

21. NEGATIVE BINOMAL-EXAMPLE WITH SAS
proc genmod data = nba; model FGA= GP DREB AST STL TOV FTM /dist=negbin; (ONLY DIFFERENCE FROM POISSON) run; /*check goodness of fit for model*/ data pvalue; df = 93; chisq = ; pvalue = 1 - probchi(chisq, df); proc print data = pvalue noobs;

22. EXAMPLE RESULTS-GOODNESS OF FIT
Data Set WORK.NBA Distribution Negative Binomial Link Function Log Dependent Variable FGA Number of Observations Read 100 Number of Observations Used 100 Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance Scaled Deviance Pearson Chi-Square Scaled Pearson X Log Likelihood Full Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better)

23. RESULTS: Analysis of Maximum Likelihood Parameter EstimatesDF
ESTIMATE
STANDARD ERROR
WALD 95% CONFIDENCE LIMITS
WALD CHI-SQUARE
PR>CHI-SQ
INTERCEPT 1
4.1742
0.1641
(3.8525,4.4958)
647.01
<.0001 GP
0.0426
0.0125
(0.0181,0.0671)
11.62
0.0007
DREB
0.0003
( ,0.0016)
0.15
0.7028
AST
0.0008
( ,0.0014)
0.03
0.8619
STL
0.0024
0.0027
( ,0.0077)
0.78
0.3756
TOV
0.0060
0.0023
(0.0015,0.0105)
6.95
0.0084
FTM
0.0042
0.0010
(0.0023,0.0061)
19.32
DISPERSION
0.0230
0.0040
(0.0163,0.0325)

24. Assessment of Results
Ratio of Deviance/Df= ≈1 (over-dispersion fixed!)
Deviance= , now is well fit because pvalue=1- prob(chisq,df) IS significant
Extra parameter in the “Analysis of Maximum Likelihood Parameter Estimates” called “Dispersion” (aka ALPHA)
Accounts for the over-dispersion factor we came across in the Poisson regression
This estimate has a value of with a Wald Confidence Interval of (.0163, 0325). Based on the 95% Confidence Limits for our dispersion parameter, we can say that dispersion is significantly different from 0, justifying the negative binomial model is more appropriate
GP, TOV, & FTM only significant predictors
Notice the drastic change in number of significant predictors

25. NEGATIVE BINOMIAL-EXAMPLE WITH R
nba <- read.csv("F:/STATS544/nba.csv",header= TRUE)
install.packages('MASS')
library(MASS)
nb<-glm.nb(FGA ~GP+DREB+AST+STL+TOV+FTM, data = nba)
summary(nb)

26. EXAMPLE RESULTS-GOODNESS OF FIT
(Dispersion parameter for Negative Binomial( ) family taken to be 1) Null deviance: on 99 degrees of freedom Residual deviance: on 93 degrees of freedom AIC: Number of Fisher Scoring iterations: 1 Deviance Residuals: Min 1Q Median 3Q Max Theta: Std. Err.: x log-likelihood:

27. RESULTS: Analysis of Maximum Likelihood Parameter Estimates
Call: glm.nb(formula = FGA ~ GP + DREB + AST + STL + TOV + FTM, data = nba, init.theta = , link = log) Coefficients: Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
ESTIMATE
STD.ERROR
Z-VALUE
PR(>|Z|)
(Intercept)
25.663
< 2e-16 *** GP
3.438
***
DREB
0.383
AST
-0.176
STL
0.886
TOV
2.652 **
FTM
4.467
7.95e-06 ***

28. INTERPETATION OF SIGNIFICANT COEFFICIENTS
GP: Holding all other variables constant, for every one unit addition of games played, the expected log number of field goal attempts will go up by Or similarly, for every additional game played, the number of field goal attempts will increase by 4.35%
TOV: Holding all other variables constant, for every one extra TOV, the expected log number of field goal attempts will increase by Or similarly, for every additional turnover made, the number of field goal attempts will increase by 0.60%.
FTM: Holding all other variables constant, for every one unit addition of free throws made, the expected log number of field goal attempts will go up by Or similarly, for every additional free throw made, the number of field goal attempts will increase by 0.42%.

29. NEGATIVE BINOMIAL WITH SPSS & MINITAB
genlin FGA with GP DREB AST STL TOV FTM /model GP DREB AST STL TOV FTM INTERCEP=YES Distribution=negbin(mle) link = log /print FIT SUMMARY SOLUTION. NA

30. SUMMARY Use Poisson regression when dealing with COUNT data
If there’s Overdispersion, switch to Negative binomial
Assumptions for both Poisson and NB are the same
Both model coefficients are interpreted same manner
Can perform both regressions in SAS, R, & SPSS
Minitab only able to perform Poisson