1. The log-rate model Statistical analysis of occurrence-exposure rates
16 January 2019
The log-rate model Statistical analysis of occurrence-exposure rates

2. References
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’

3. Data: leaving parental home
Leaving home

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

5. ij = E[Nij] The log-rate model PMij fixed offset
The log-rate model is a log-linear model with OFFSET
(constant term)

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

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 (Iterative proportional fitting)
Estimate parameters of log-rate model from predicted values using conventional log-linear modeling
The model:

8. Leaving home

9. Leaving home

10. The log-rate model in SPSS: unsaturated model
Leaving home
The log-rate model in SPSS: unsaturated model
Model and Design Information: unsaturated model
Model: Poisson
Design: Constant + SEX + TIMING
Ref. cat
Ref. cat
Parameter Estimates
Asymptotic 95% CI
Parameter Estimate SE Lower Upper
ln 170/9114 (ref.cat)
[ln 151/4876]
[ln 82/16202]

11. The log-rate model in SPSS: unsaturated model
Leaving home
The log-rate model in SPSS: unsaturated model
PM *exp[ ] = RATE
9114*exp[ ] =
16202*exp[ ] =
15113*exp[ ] =
4876*exp[ ] =

12. The log-rate model in SPSS: unsaturated model
SEX TIMING NUMBER EXPOSURE
GENLOG
timing sex /CSTRUCTURE=exposure
/MODEL=POISSON
/PRINT FREQ ESTIM CORR COV
/CRITERIA =CIN(95) ITERATE(20) CONVERGE(.001) DELTA(0)
/DESIGN sex timing
/SAVE PRED .

13. Leaving home
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. = (change = )
[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 = * exp [ ]
RATE = exp [ ] =
[i] $d r$
[o] unit observed fitted residual
[o]
[o]
[o]
[o]

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

15. The log-rate model in GLIM: unsaturated model
Leaving home
The log-rate model in GLIM: unsaturated model

16. Leaving home
The log-rate model in TDA The basic exponential model with time-constant covariates (Blossfeld and Rohwer, pp. 87ff) Occ = Exp * exp[overall + sex]
SN Org Des Episodes Weighted Duration TS Min TF Max Excl
Sum
Number of episodes: 583
Successfully created new episode data.
Idx SN Org Des MT Variable Coeff Error C/Error Signif
A Constant
A SEX
Log likelihood (starting values):
Log likelihood (final estimates):
command file: ehd21.cf
data file: test.dat (micro data)

17. LOG-RATE MODEL IN TDA: PROGRAMME
Leaving home
LOG-RATE MODEL IN TDA: PROGRAMME
# ehd2.cf Basic exponential model with covariate SEX
nvar( dfile = test.dat, # data file
ID = c1, # identification number
SN = c2, # spell number
TF = c3, # TIME LEAVING HOME (=ENDING TIME)
# measured from age 0!!!!
TF15 = TF-180, # measured from age 15
SEX = c4, # sex REASON = c5, # reason
SEX1 = SEX[1], # see boek p. 61 SEX1 = 1 for females and 0 for males
# MALES ref.cat
SEX2 = SEX[2], # = 1 for females
DES = if eq(REASON,4) then 0 else 1, # destination
TFP = TF15, # Blossfeld: TF+1 !!!!!! );
edef( # define single episode data
ts = 0, # starting time
tf = TFP, # ending time
org = 0, # origin state
des = DES, # destination state
# BASIC exponential model (Blossfeld-Rohwer p )
rate(
xa (0,1) = SEX1,
pres = ehd21.res,
) = 2;

18. Related models Parameters of these models are related
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

19. The unsaturated model Similarity with log-rate model

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

21. Leaving home

22. Odds ratio = 1 The unsaturated log-linear model as a log-rate model
Leaving home
The unsaturated log-linear model as a log-rate model
Odds ratio = 1

23. Leaving home
With PMij = 1

24. Update table
Update a table Similarity with log-rate model Illustration: migration analysis with incomplete data
Migration is a realisation of a Poisson process
Literature: “Indirect estimation of migration”, Special issue of Mathematical Population Studies, A. Rogers ed. Vol 7, no 3 (1999)

25. Update table
Updating a table: THE LOG-RATE MODEL IN TWO STEPS
Odds ratio =

26. Updating a table: THE LOG-RATE MODEL IN TWO STEPS
Update table
Updating a table: THE LOG-RATE MODEL IN TWO STEPS

27. Update table

28. Log-rate model: rate = events/exposure
Update table
Log-rate model: rate = events/exposure
Gravity / spatial interaction model
i and j are balancing factors

29. IPF and biproportional adjustment
Update table
IPF and biproportional adjustment
Log-likelihood function:

30. Biproportional adjustment method
Update table
Biproportional adjustment method
RAS method (Richard A. Stone: Input-output models, 1962)
DSF procedure (DSF = Deming, Stephan, Furness) (Sen and Smith, 1995, p. 374)
See e.g. Willekens (1983) Log-linear analysis of spatial interaction

31. Biproportional adjustment
Update table
Biproportional adjustment
Step 0: s (Step) = 0
Step 1
Step 2
Step 3: go to Step 1 unless convergence criteria is reached. The stopping criterion is reached when the change is the adjustment factors is less than
10-6 for all x and j.

32. Likelihood equations may be written as:
Update table
Likelihood equations may be written as:
Marginal totals are sufficient statistics

33. A different way of writing the spatial interaction model:
Update table
A different way of writing the spatial interaction model:
Link Poisson - Multinomial

34. The gravity model is a log-linear model
Update table
The gravity model is a log-linear model
The entropy model is a log-linear model
The RAS model is as log-linear (log-rate) model

35. Update table
Parameter estimation
Maximise (log) likelihood function: probability that the model predicts the data
Expectation: predict E[Nrs] = rs given the model and initial parameter estimates.
Maximisation: maximise the ‘complete-data’ log-likelihood.

36. The log-rate model Piecewise constant hazard model
Kidney Transplant Histocompatibility Study
The data describe the survival of the kidney graft (organ) following kidney transplant operations. The risk factor 'donor relationship' has two categories, cadaveric nonrelated donor (CAD) and living related donor (LRD). The sample in this follow-up study is 1975 transplant operations.
Laird N. and D. Olivier (1981) Covariance analysis of censored survival data using log-linear analysis techniques, Journal of the American Statistical Association, Vol. 76, no. 374, pp The authors claim that they go beyond Holford (1980) ‘The analysis of rates and survivorship using log-linear models’, Biometrics, 36:
d:\s\data\laird\kidney\laird.doc

37. Life-table data on graft survival
Kidney Transplant Study
Life-table data on graft survival
Exposure (Exp) is calculated as follows:
Exp = [E - 0.5(W + D)]*# in days
where E = number entered
W = number withdrawn
D = number died
# = width of interval (the last open interval was taken as having 180 days)
608* *45
d:\s\data\laird\laird_lt.xls

38. Death rates (* 1000; per day)
Kidney Transplant Study
Death rates (* 1000; per day)

39. Kidney Transplant Study
CAD
LRD

40. SPSS Deaths 9-12 m: 107325 * exp[-8.7857+1.0087]=107325*0.0004193=45
Kidney Transplant Study
Model: Poisson
Design: Constant + TIME
SPSS
1 Constant
2 [TIME = 1]
3 [TIME = 2]
4 [TIME = 3]
5 [TIME = 4]
6 [TIME = 5]
7 [TIME = 6]
8 [TIME = 7]
9 [TIME = 8]
10 [TIME = 9]
11 [TIME = 10]
12 [TIME = 11]
13 [TIME = 12]
14 [TIME = 13]
15 [TIME = 14]
16 [TIME = 15]
17x[TIME = 16]
Parameter Estimate SE
Deaths 9-12 m: * exp[ ]=107325* =45

41. Kidney Transplant Study
Model: Poisson
Design: Constant + DONOR TYPE + TIME (unsaturated model)
Estimate SE
Constant
[CAD = 1.00]
x [LRD = 2.00]
[P1 = 1]
[P1 = 2]
[P1 = 3]
[P1 = 4]
[P1 = 5]
[P1 = 6]
[P1 = 7]
[P1 = 8]
[P1 = 9]
[P1 = 10]
[P1 = 11]
[P1 = 12]
[P1 = 13]
[P1 = 14]
[P1 = 15]
x [P1 = 16]
Deaths 9-12 m: * exp[ ]=53370* =31.49
Observed: 30