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Count Outcomes

Count Outcomes. Chapter 6. Count Outcomes. Counting (0, 1, 2, 3, …) Limited number of counts (categorical?) Many counts (continuous?) Poisson Examples —n umber of Steps taken Concussions Relapses Healthcare utilizations. Describing the Data. Numerical summaries: Event rates

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Count Outcomes

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  1. Count Outcomes Chapter 6

  2. Count Outcomes • Counting (0, 1, 2, 3, …) Limited number of counts (categorical?) Many counts (continuous?) Poisson • Examples—number of Steps taken Concussions Relapses Healthcare utilizations

  3. Describing the Data • Numerical summaries: Event rates Person-time • Graphical summaries: Bar graphs Line graphs

  4. Numerical Summaries • Previous chapters described summaries for continuous and categorical data. • Count data is similar to event data (did the event occur: yes/no), but the events can occur multiple times. Subjects can contribute multiple events. Subjects can have different amounts of “risk.”

  5. Person-time • It is not usually the case that all subjects are followed for the same amount of time. Accrual of patients Drop-ins/Drop-outs Availability of medication Missed visits Length of disease • How do you account for different amounts of contribution? Person-time

  6. Event Rates • Event rates: Numerator: Number of events (can be multiple) Denominator: Total amount at risk for event (each subject is not treated equally)

  7. Event Rates • The number of events divided by the accumulated person-time (person-years, subject-years) • Range from 0 to infinity • AKA Incidence density

  8. One-sample studies

  9. Why a One-Sample Study? • Obtaining an additional group or sample for comparisons may not be practical. Comparisons involve historical control(s).

  10. Inference for the One-Sample Study • Sometimes the event rate or the sample size is small, and the normal approximation cannot be used. • Can use the fact that the data come from a Poisson distribution and use exact methods.

  11. Multiple-sample Studies

  12. Rate Ratios • Often rates are compared between groups by considering the rate ratio. • The rate ratio is the ratio of two rates. Generally, the rate corresponding to the control group is denominator. If the rates of the two groups are the same, then the rate ratio is 1. A rate ratio that is larger than 1 means that the event rate in the numerator is larger than the event rate in the denominator. If the rate ratio is less than 1, then the event rate in the denominator is larger than the event rate in the numerator.

  13. Poisson Regression

  14. Poisson Regression • We want to estimate a count and determine its relationship to a set of explanatory variables. • Counts follow a Poisson distribution. • If the outcome is distributed as Poisson and has a mean (and variance) of μ and x is an explanatory variable ln(μ) =  + 1x1 + 2x2 + … + KxK

  15. Poisson Regression • Rarely do we have count data that have been collected all at once. usually count data are collected over time or space often more interested in modeling rates than counts • If exposure time = T, then the rate = count/T and E(Y/T) = μ/T.

  16. Poisson Regression • We can model the rate as ln(μ/T) =  + 1x1 + 2x2 + … + KxKor ln(μ) – ln(T) =  + 1x1 + 2x2 + … + KxK or ln(μ) =  + 1x1 + 2x2 + … + KxK+ ln(T)

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