1 B. The log-rate model Statistical analysis of occurrence-exposure rates.

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1 B. The log-rate model Statistical analysis of occurrence-exposure rates

2 Laird, N. and D. Olivier (1981) Covariance analysis of censored survival data using log-linear analysis techniques. Journal of the American Statistical Institute, 76(374): Holford, T.R. (1980) The analysis of rates and survivorship using log-linear models. Biometrics, 36: Yamaguchi, K. (1991) Event history analysis. Sage, Newbury Park, Chapter 4:’Log-rate models for piecewise constant rates’ References

3 Data: leaving parental home

4 The log-rate model: the occurrence matrix and the exposure matrix Occurrences: Number leaving home by age and sex, 1961 birth cohort: n ij Exposures: number of months living at home (includes censored observations): PM ij

5 The log-rate model offset The log-rate model is a log-linear model with OFFSET (constant term) ij = E[N ij ] PM ij fixed

6 The log-rate model  Ln(PM): offset  : linear predictor The log-rate model is a log-linear model with OFFSET (constant term) Multiplicative form Addititive form

7 The log-rate model in two steps Use the model to predict the counts (predict counts from marginal distribution of occurrences and from exposures): IPF Estimate parameters of log-rate model from predicted values using conventional log-linear modeling The model:

8

9

10 The log-rate model in SPSS: unsaturated model Model and Design Information: unsaturated model Model: Poisson Design: Constant + SEX + TIMING Parameter Estimates Asymptotic 95% CI Parameter Estimate SE Lower Upper

11 The log-rate model in SPSS: unsaturated model PM *exp[ ] = RATE 9114*exp[ ] = *exp[ ] = *exp[ ] = *exp[ ] =

12 The log-rate model in GLIM: unsaturated model Occ = Exp * exp[overall + sex] DATA: Occurrence matrix and exposure matrix (2*2) [i] $fit +sex$ [o] scaled deviance = (change = ) at cycle 4 [o] d.f. = 2 (change = -1 ) [o] [i] $d e$ [o] estimate s.e. parameter [o] [o] SEX(2) [o] scale parameter taken as Females 278 = * exp[-4.275] RATE = exp[-4.275] = Males 252 = * exp [ ] RATE = exp [ ] = [i] $d r$ [o] unit observed fitted residual [o] [o] [o] [o]

13 The log-rate model in GLIM: unsaturated model Occ = Exp * exp[overall + sex + timing]

14 The log-rate model in GLIM: unsaturated model

15 Related models Poisson distribution: counts have Poisson distribution (total number not fixed) Poisson regression Log-linear model: model of count data (log of counts) Binomial and multinomial distributions: counts follow multinomial distribution (total number is fixed) Logit model: model of proportions [and odds (log of odds)] Logistic regression Log-rate model: log-linear model with OFFSET (constant term) Parameters of these models are related

16 I. The unsaturated model Similarity with log-rate model

17 The unsaturated log-linear model Assume: two-way classification; counts unknown but marginal totals given Predict the expected counts (cell entries)

18

19 The unsaturated log-linear model as a log-rate model Odds ratio = 1

20 With PM ij = 1

21 II. Update a table Similarity with log-rate model

22 Updating a table: THE LOG-RATE MODEL IN TWO STEPS Odds ratio =

23 Updating a table: THE LOG-RATE MODEL IN TWO STEPS