Poisson Regression

1
Poisson Regression

• Lecture 8

2
Poisson Distribution

• Count Data are often modeled using the Poisson
  distribution
• The mean and variance of the count y equals ?
• ? is the rate parameter the number of events
  one would expect to see for a particular unit of
  time or space
• Often used as an approximation for the binomial
  distribution for rare events

3
Poisson Regression

• Poisson regression models expected counts as
  follows
• log(?)ß0ß1x
• Taking the exponential of both sides,
• What is the interpretation of ß?
• If ß0, then exp(ß)1 and
• If ßgt0, then exp(ß)gt1 and
• If ßlt0, then exp(ß)lt1 and

4
Ideal Poisson Example

• Data were simulated from a poisson regression
  model.

5
Seizure Example

• A study examined the effectiveness of progabide
  on the number of seizures experienced by
  epileptics. Upon beginning either placebo or
  progabide treatment, patients made four biweekly
  visits to the doctor. The data provided gives
  the number of seizures reported in the two weeks
  preceding the fourth visit. Other variables
  measured include the age of the patient as well
  as how many seizures the patient experienced in
  the 8 weeks prior to entering the study.

6
Results Seizure Example

• The prediction equation based on the Poisson
  Regression Model is
• Interpreting the parameters
• .0140
• .2693

7
Seizure E.G. SAS program

• DATA prog
• INPUT subject seizs visit trt base age
• IF visit ne 4 THEN delete
• IF subject207 THEN delete
• CARDS
• 104 5 1 0 11 31
• 104 3 2 0 11 31
• 104 3 3 0 11 31
• 104 3 4 0 11 31
• 106 3 1 0 11 30
• Etc.
• 236 2 4 1 12 37
• RUN
• PROC GENMOD
• CLASS trt
• MODEL seizs trt base age/distpoisson
  linklog
• RUN

8
Seizure E.G. SAS output

• The GENMOD Procedure
• Model Information
• Data Set
  WORK.PROG
• Distribution
  Poisson
• Link Function
  Log
• Dependent Variable
  seizs
• Observations Used
  58
• Criteria For Assessing
  Goodness Of Fit
• Criterion DF
  Value Value/DF
• Deviance 54
  147.0210 2.7226
• Scaled Deviance 54
  147.0210 2.7226
• Pearson Chi-Square 54
  136.6080 2.5298
• Scaled Pearson X2 54
  136.6080 2.5298
• Log Likelihood
  392.6703

9
Seizure E.G. SAS output, cont.

• Analysis Of Parameter Estimates
• Standard Wald
  95 Chi-
• Parameter DF Estimate Error Confidence
  Limits Square Pr gt ChiSq
• Intercept 1 0.5050 0.2638 -0.0119
  1.0220 3.67 0.0555
• trt 0 1 0.2693 0.1134 0.0470
  0.4916 5.64 0.0176
• trt 1 0 0.0000 0.0000 0.0000
  0.0000 . .
• base 1 0.0221 0.0017 0.0187
  0.0255 161.15 lt.0001
• age 1 0.0140 0.0086 -0.0028
  0.0309 2.66 0.1029
• Scale 0 1.0000 0.0000 1.0000
  1.0000
• NOTE The scale parameter was held fixed.

10
Are these results valid?

• Assumptions for Poisson Regression Model
• Random, Independent Sample
• Mean is equal to variance
• Standard deviation square root of mean
• Explanatory variable is linearly linked to the
  log of the mean response
• If a variable is continuous, then increasing it
  by 1 unit has a multiplicative effect on the
  expected response

11
Linearity with respect to log(µ)

• Need only check for continuous predictors
• For baseline counts and age

12
Checking Standard Deviations

• Easiest to do when looking at categorical
  variables.
• Group observations by categories.
• Are the means approximately equal to the
  variances for each group?
• From proc univariate,
• For the progabide group, mean4.83333, std
  dev.4.2838
• For the placebo group, mean7.9643, std
  dev.7.6278

13
Std devs more

• Schematic Plots
• 30
• 0
• 25 0
• 0
• 20
• 
  0
• 15
  0
• 
  

14
Checking model with Deviances

• Deviance defined as the difference between the
  -2log-likelihood of specified model and one that
  perfectly fits the data
• Specified model
• Perfect fit
• Deviance
• Deviance for model is 147.0210 with 54 degrees of
  freedom.
• Measure of fit is deviance/d.f. If gtgt1, theres
  a overdispersion. (Here 2.7226.)

15
Checking model with Pearson Chi-square

• Pearson goodness of fit statistic is
• Measure of fit is Pearson chi-square/d.f. If
  gtgt1, theres a overdispersion. (Here 2.7226.)

16
Model Fit Statistics - Implications

• Deviance and Pearson Chi-square can be large when
• All appropriate covariates are not included in
  model (correctly)
• When distributional assumptions are not correct
  (assumption variancemean)

17
Overdispersion

• Overdispersion occurs when the responses have
  greater variability than when expected given the
  Poisson distribution.
• In this case, standard errors for the regression
  parameters will be too small.
• We can adjust the standard errors to account for
  this extra variability in the response
• In particular, the overdispersion
  parameterX2/df.
• Multiply the standard errors by the square root
  of this.
• In the seizure example, square root(X2/df)1.6500.
  
• This indicates that we increase the standard
  errors by 65.

18
SAS program with overdispersion

• PROC GENMOD
• CLASS trt
• MODEL seizs trt base age/distpoisson
  linklog scalepearson
• RUN

19
SAS output with overdispersion

• The GENMOD Procedure
• Criteria For Assessing Goodness Of Fit
• Criterion DF
  Value Value/DF
• Deviance 54
  147.0210 2.7226
• Scaled Deviance 54
  58.1162 1.0762
• Pearson Chi-Square 54
  136.6080 2.5298
• Scaled Pearson X2 54
  54.0000 1.0000
• Log Likelihood
  155.2193
• Standard Wald
  95 Chi-
• Parameter DF Estimate Error Confidence
  Limits Square Pr gt ChiSq
• Intercept 1 0.5050 0.4195 -0.3172
  1.3273 1.45 0.2287
• trt 0 1 0.2693 0.1804 -0.0842
  0.6228 2.23 0.1355
• trt 1 0 0.0000 0.0000 0.0000
  0.0000 . .
• base 1 0.0221 0.0028 0.0167
  0.0275 63.70 lt.0001
• age 1 0.0140 0.0137 -0.0128
  0.0408 1.05 0.3051

20
Caution about overdispersion

• We can get deviance statistics to look good
  just by allowing for a overdispersion parameter.
• Should still examine linearity.
• Can sometimes examine linearity via the Pearson
  residuals.
• (observed-expected)/std.dev.
• proc sort
• by base
• run
• PROC GENMOD
• CLASS trt
• MODEL seizs trt base age/distpoisson
  linklog
• scalepearson
  residuals
• RUN

21
Interpreting

• Pearson and Deviance Residuals are given by
  reschi and resdev
• and
• Plots of residuals vs. continuous predictors

22
Adding quadratic terms for continuous predictors

• Analysis Of Parameter Estimates
• Standard Wald
  95 Chi-
• Parameter DF Estimate Error Confidence
  Limits Square Pr gt ChiSq
• Intercept 1 -3.8873 1.6566 -7.1341
  -0.6404 5.51 0.0189
• trt 0 1 0.2964 0.1537 -0.0049
  0.5978 3.72 0.0538
• trt 1 0 0.0000 0.0000 0.0000
  0.0000 . .
• base 1 0.0623 0.0096 0.0435
  0.0812 42.05 lt.0001
• age 1 0.2623 0.1104 0.0459
  0.4787 5.64 0.0175
• basesq 1 -0.0004 0.0001 -0.0005
  -0.0002 18.57 lt.0001
• agesq 1 -0.0041 0.0019 -0.0077
  -0.0005 4.90 0.0268
• Scale 0 1.3423 0.0000 1.3423
  1.3423
• NOTE The scale parameter was estimated by the
  square root of Pearson's
• Chi-Square/DOF.

23
Residuals for new model

• The quadratic terms are statistically
  significant.
• With the addition of these terms, the
  significance of the progabide treatment has gone
  from 0.1355 to 0.0538.

24
SAS Results for Simulated Data

• Criteria For Assessing Goodness Of Fit
• Criterion DF
  Value Value/DF
• Deviance 98
  110.1826 1.1243
• Scaled Deviance 98
  110.1826 1.1243
• Pearson Chi-Square 98
  107.7462 1.0995
• Scaled Pearson X2 98
  107.7462 1.0995
• Log Likelihood
  1044.2169
• Algorithm converged.
• Analysis Of Parameter
  Estimates
• Standard Wald
  95 Chi-
• Parameter DF Estimate Error Confidence
  Limits Square Pr gt ChiSq
• Intercept 1 0.9300 0.0973 0.7392
  1.1207 91.28 lt.0001
• x 1 1.0496 0.0719 0.9088
  1.1905 213.30 lt.0001
• Scale 0 1.0000 0.0000 1.0000
  1.0000

25
Residual Plot for Simulated Data
26
Poisson Regression for Rates

• We are typically interested in the number of
  events that happen within a particular unit of
  time.
• Suppose that we counted the number of times that
  a graduate student eats pizza during a week.
  However, some students were observed for 2 weeks
  and others for 3 weeks. How do we model the
  data?
• Suppose that ? represents the mean number of
  times that a student eats pizza during a one week
  period.
• Suppose that if we count over one week, this
  count would follow a Poisson distribution and
  have mean ?.
• A count over two weeks would also follow a
  Poisson distribution with mean 2?.
• A count over three weeks Poisson(3?)

27
Poisson for Rates, cont.

• Therefore, we model
• Or
• Or
• log(ti) is called the offset term.
• The offset term is added to a poisson regression
  in SAS by adding the option offsetvariable-name.
  

28
Poisson for Rates, cont.

• Suppose that you are modeling the number of
  people with asthma in a city. It is not fair to
  combine Los Angelos with Portland, Maine, unless
  you first somehow adjust for the sizes of the
  city.
• This could be done per square mile of city,
  however
• We could also count number of cases per
  ten-thousand people.

29
Melanoma

• See page 354 in Stokes et al.
• Gail 1978 and Koch, Imrey et al. 1985 reported
  the number of new melanoma cases reported in
  1969-1971 for white males in two regions.
  Researchers were interested in whether the rates
  varied across age groups or region (North/South)
• The observed rates for each age-group and region
• We also know the total number of people in each
  age group/region.
• We would like to directly model counts
  (count/total).

30
Melanoma, cont.

• The sample sizes used to calculate the rates in
  each category
• We cannot just run linear regression on rates.
• Heterogeneity of variances exists because
• Counts follow a Poisson distribution, where
  variance is related to mean
• Sample sizes vary between cells (Can think of
  modeling cases/1000. Thousands differ.)

31
Melanoma SAS saturated model

• First, we run the saturated model with age (as a
  nominal predictor) and region main effects AND
  the interaction.
• Model Information
• Data Set
  WORK.MEL
• Distribution
  Poisson
• Link Function
  Log
• Dependent Variable
  cases
• Offset Variable
  ltotal
• Observations Used
  12
• Criteria For Assessing Goodness Of Fit
• Criterion DF
  Value Value/DF
• Deviance 0
  0.0000 .
• Scaled Deviance 0
  0.0000 .
• Pearson Chi-Square 0
  0.0000 .
• Scaled Pearson X2 0
  0.0000 .
• Log Likelihood
  2698.0337

32
Melanoma Saturated parameters

• The GENMOD Procedure
• LR Statistics For Type 3
  Analysis
• Chi-
• Source DF Square
  Pr gt ChiSq
• age 5 715.99
  lt.0001
• region 1 108.19
  lt.0001
• ageregion 5 6.21
  0.2859
• The interaction is not significant. Therefore,
  we need to consider a simpler model.
• Type 3 LR statistics are obtained by adding the
  option type3 to the model statement.

33
Melanoma without interaction

• Criteria For Assessing Goodness Of Fit
• Criterion DF
  Value Value/DF
• Deviance 5
  6.2149 1.2430
• Scaled Deviance 5
  6.2149 1.2430
• Pearson Chi-Square 5
  6.1151 1.2230
• Scaled Pearson X2 5
  6.1151 1.2230
• Log Likelihood
  2694.9262
• LR Statistics For Type 3
  Analysis
• Chi-
• Source DF Square
  Pr gt ChiSq
• age 5 796.74
  lt.0001
• region 1 124.22
  lt.0001

34
Final Model Melanoma

• Age and region as predictors.
• The offset term was population size in number of
  10 thousands.
• What does the value 2.3162 represent?
• What does the value 1.0316 represent?
• What does the value 0.8195 represent?
• We can also predict the number of cases per 10
  thousand people for each age group and region
  combination.