1. 1 STA 517 – Chp4 Introduction to Generalized Linear Models 4.3 GENERALIZED LINEAR MODELS FOR COUNTS  count data - assume a Poisson distribution  counts in contingency tables with categorical response variables.  modeling count or rate data for a single discrete response variable.

2. 2 STA 517 – Chp4 Introduction to Generalized Linear Models 4.3.1 Poisson Loglinear Models  The Poisson distribution has a positive mean µ.  Although a GLM can model a positive mean using the identity link, it is more common to model the log of the mean.  Like the linear predictor, the log mean can take any real value.  The log mean is the natural parameter for the Poisson distribution, and the log link is the canonical link for a Poisson GLM.  A Poisson loglinear GLM assumes a Poisson distribution for Y and uses the log link.

3. 3 STA 517 – Chp4 Introduction to Generalized Linear Models Log linear model  The Poisson loglinear model with explanatory variable X is  For this model, the mean satisfies the exponential relationship x  A 1-unit increase in x has a multiplicative impact of on µ  The mean at x+1 equals the mean at x multiplied by.

4. 4 STA 517 – Chp4 Introduction to Generalized Linear Models 4.3.2 Horseshoe Crab Mating Example

5. 5 STA 517 – Chp4 Introduction to Generalized Linear Models

6. 6 4.3.2 Horseshoe Crab Mating Example  a study of nesting horseshoe crabs.  Each female horseshoe crab had a male crab resident in her nest.  AIM: factors affecting whether the female crab had any other males, called satellites, residing nearby.  Explanatory variables are :  C - the female crab’s color,  S - spine condition,  Wt - weight,  W - carapace width.  Outcome: number of satellites (Sa) of a female crab.  For now, we only study W (carapace width)

7. 7 STA 517 – Chp4 Introduction to Generalized Linear Models number of satellites (Sa) = f (W)  Scatter plot – weakly linear ? (N=173)  Grouped plot: To get a clearer picture, we grouped the female crabs into width categories and calculated the sample mean number of satellites for female crabs in each category.  Figure 4.4 plots these sample means against the sample mean width for crabs in each category.  The sample means show a strong increasing trend. WHY?

8. 8 STA 517 – Chp4 Introduction to Generalized Linear Models

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10. 10 STA 517 – Chp4 Introduction to Generalized Linear Models

11. 11 STA 517 – Chp4 Introduction to Generalized Linear Models

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14. 14 STA 517 – Chp4 Introduction to Generalized Linear Models SAS code data table4_3; input C S W Wt Sa@@; cards; 2 3 28.3 3.05 8 3 3 22.5 … ; proc genmod data=table4_3; model Sa=W/dist=poisson link=identity; ods output ParameterEstimates=PE1; run; proc genmod data=table4_3; model Sa=w/dist=poisson link=log; ods output ParameterEstimates=PE2; run;

15. 15 STA 517 – Chp4 Introduction to Generalized Linear Models data _NULL_; set PE1; if Parameter="Intercept" then call symput("intercp1", Estimate); if Parameter="W" then call symput("b1", Estimate); data _NULL_; set PE2; if Parameter="Intercept" then call symput("intercp2", Estimate); if Parameter="W" then call symput("b2", Estimate); run; data tmp; do W=22 to 32 by 0.01; mu1=&intercp1 + &b1*W; mu2=exp(&intercp2 + &b2*W); output; end; run;

16. 16 STA 517 – Chp4 Introduction to Generalized Linear Models Graphs proc sort data=table4_3; by W; data tmp1; merge table4_3 tmp; by W; run; symbol1 i=join line=1 color=green value=none; symbol2 i=join line=2 color=red value=none; symbol3 i=none line=3 value=circle; proc gplot data=tmp1; plot mu1*W mu2*W Sa*W / overlay; run;

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18. 18 STA 517 – Chp4 Introduction to Generalized Linear Models Group data /*group data*/ data table4_3a; set table4_3; W_g=round(W-0.75)+0.75; *if W<23.25 then W_g=22.5; *if W>29.25 then W_g=30.5; run; proc sql; create table table4_3g as select W_g, count(W_g) as Num_of_Cases, sum(Sa) as Num_of_Satellites, mean(Sa) as Sa_g, var(sa) as Var_SA from table4_3a group by W_g; quit; proc print; run;

19. 19 STA 517 – Chp4 Introduction to Generalized Linear Models SAS output Num_of_ Num_of_ Obs W_g Cases Satellites Sa_g Var_SA 1 20.75 1 0 0.00000. 2 21.75 1 0 0.00000. 3 22.75 12 14 1.16667 3.0606 4 23.75 14 20 1.42857 8.8791 5 24.75 28 67 2.39286 6.5437 6 25.75 39 105 2.69231 11.3765 7 26.75 22 63 2.86364 6.8853 8 27.75 24 93 3.87500 8.8098 9 28.75 18 71 3.94444 16.8791 10 29.75 9 53 5.88889 9.8611 11 30.75 2 6 3.00000 0.0000 12 31.75 2 6 3.00000 2.0000 13 33.75 1 7 7.00000.

20. 20 STA 517 – Chp4 Introduction to Generalized Linear Models Graphs data tmp2; merge table4_3g(rename=(W_g=W)) tmp; by W; run; symbol1 i=join line=1 color=green value=none; symbol2 i=join line=2 color=red value=none; symbol3 i=none line=3 value=circle; proc gplot data=tmp2; plot mu1*W mu2*W Sa_g*W / overlay; run;

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22. 22 STA 517 – Chp4 Introduction to Generalized Linear Models 4.3.3 Overdispersion for Poisson GLMs

23. 23 STA 517 – Chp4 Introduction to Generalized Linear Models Solution?

24. 24 STA 517 – Chp4 Introduction to Generalized Linear Models 4.3.4 Negative binomial GLMs

25. 25 STA 517 – Chp4 Introduction to Generalized Linear Models

26. 26 STA 517 – Chp4 Introduction to Generalized Linear Models /*fit negative binomial with identical link to count for overdispersion*/ proc genmod data=table4_3; model Sa=W/dist=NEGBIN link=identity; ods output ParameterEstimates=PE3; run;

27. 27 STA 517 – Chp4 Introduction to Generalized Linear Models 4.3.6 Poisson GLM of independence in I × J contingence tables

28. 28 STA 517 – Chp4 Introduction to Generalized Linear Models