1 STA 617 – Chp9 STA 617 – Chp9 Loglinear/Logit Models Loglinear/Logit Models 9.7 Poisson regressions for rates In Section 4.3 we introduced Poisson regression for modeling counts. When outcomes occur over time, space, or some other index of size, it is more relevant to model their rate of occurrence than their raw number. We use GLM with log link, Poisson distribution, log(index) as offset
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1 STA 617 – Chp9 Loglinear/Logit Models 9.7 Poisson regressions for rates In Section 4.3 we introduced Poisson regression for modeling counts. When outcomes.
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In Section 4.3 we introduced Poisson regression for modeling counts. When outcomes occur over time, space, or some other index of size, it is more relevant to model their rate of occurrence than their raw number.
We use GLM with log link, Poisson distribution, log(index) as offset
9.7.1 Analyzing Rates Using Loglinear Models with Offsets
When a response count ni has index equal to ti , the sample rate is ni/ti. Its expected value is µi/ti.
With an explanatory variable x, a loglinear model for the expected rate has form
This model has equivalent representation
The adjustment term, -log ti , to the log link of the mean is called an offset. The fit correspond to using log ti as a predictor on the right-hand side and forcing its coefficient to equal 1.0.
The time at risk for a subject is their follow-up time of observation.
For a given age and valve type, the total time at risk is the sum of the times at risk for all subjects in that cell (those who died and those censored).
2004 birth vital statistics merged to death data in Florida
The predictors: smoking, drinking, education, marital status, Medicaid.
The response: infant death
Purpose: to indentify the maternal characteristics of Medicaid beneficiaries that are significantly associated with infant death so that health care and related services can be focused on risk factors that contribute to the adverse outcome
Effect smoking edu ms med _smoking _edu _ms _med Estimate Std Errsmoking No Yes -0.27 0.0875edu <HS >HS 0.2856 0.0787edu <HS HS 0.0156 0.0691edu >HS HS -0.27 0.069ms No Yes 0.4503 0.0637med No Yes -0.0169 0.0653ms*med No No No Yes 0.1082 0.0921ms*med No No Yes No 0.5754 0.0983ms*med No No Yes Yes 0.4334 0.105ms*med No Yes Yes No 0.4672 0.0751ms*med No Yes Yes Yes 0.3252 0.0786ms*med Yes No Yes Yes -0.1419 0.0881
smoking No Yes 0.763 0.643 0.906edu <HS >HS 1.331 1.14 1.552edu <HS HS 1.016 0.887 1.163edu >HS HS 0.763 0.667 0.874ms No Yes 1.569 1.385 1.778med No Yes 0.983 0.865 1.118ms*med No No No Yes 1.114 0.93 1.335ms*med No No Yes No 1.778 1.466 2.155ms*med No No Yes Yes 1.543 1.256 1.895ms*med No Yes Yes No 1.596 1.377 1.848ms*med No Yes Yes Yes 1.384 1.187 1.615ms*med Yes No Yes Yes 0.868 0.73 1.031