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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.

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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.

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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 .

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4.3.2 Horseshoe Crab Mating Example

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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)

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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?

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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;

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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;

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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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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;

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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 .

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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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4.3.3 Overdispersion for Poisson GLMs

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Solution?

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4.3.4 Negative binomial GLMs

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/*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;

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4.3.6 Poisson GLM of independence in I × J contingence tables

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