1 b. the log-rate model statistical analysis of occurrence-exposure rates
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B. The log-rate model
Statistical analysis of occurrence-exposure rates
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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):231-240
Holford, T.R. (1980) The analysis of rates and survivorship using log-linear models. Biometrics, 36:299-305
Yamaguchi, K. (1991) Event history analysis. Sage, Newbury Park, Chapter 4:’Log-rate models for piecewise constant rates’
References
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Leaving parental home, 1961 cohort, micro-dataSurvey Sept. 87 - Febr. 88
First 23 respondents. Three censored observations.ID Sex Father Month Reason Sex1 2 2 268 2 1 Female2 1 3 268 2 2 Male3 1 2 202 14 2 2 320 4 Father status5 1 1 237 1 1 Low6 1 1 295 2 2 Middle7 1 1 272 2 3 High8 2 1 231 19 2 1 312 3 Reason
10 1 2 289 2 1 Educ/work11 1 1 316 2 2 Marriage/cohabit12 2 1 321 4 3 Freedom13 2 1 260 1 4 Censored14 2 2 281 2 at interview15 2 1 273 216 1 2 251 317 2 2 212 118 2 2 320 219 1 2 221 320 2 2 322 421 2 1 221 222 2 3 308 223 1 2 233 1
Total %TWO CATEGORIES leave home early (before age 20) 209 35.85
leave home late (at or after age 20) 321 55.06Censored 53 9.09Total 583 100.00
Data: leaving parental home
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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
Age Female Male Total<20 135 74 209>=20 143 178 321Total 278 252 530Censored 13 40 53Total 291 292 583
Sex
Age Female Male Total<20 15113 16202 31315>=20 4876 9114 13990Total 19989 25316 45305
Sex
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The log-rate model
u u u u PMλln AB
ijBj
Ai
ij
ij
with A AGE [early (before age 10) = 0; late (at age 20 or later) =1] and B SEX [female = 0; male = 1]
u u u u ABij
Bj
Aiijij PMln λln
u u u u ABij
Bj
Ai
o
ijij mλln offset
exposures
countsln
exposure
soccurrenceln
The log-rate model is a log-linear model with OFFSET
(constant term)
ij = E[Nij]PMij fixed
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The log-rate model
]exp[η
PM
λ
PM
E[N] RateE
η ln(PM) λln and ηexp PM λ E[N]
Ln(PM): offset : linear predictor
The log-rate model is a log-linear model with OFFSET
(constant term)
Multiplicative form Addititive form
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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: uuPM
B
j
A
i
ij
ij u ln
8STEP 1: ITERATIVE PROPORTIONAL FITTING
OCCURRENCES
Sex iter=2a Sex
Age Female Male Total Age Female Male Total
<20 135 74 209 <20 126.5 82.5 209.0
>=20 143 178 321 >=20 150.3 170.7 321.0
Total 278 252 530 Total 276.8 253.2 530.0
EXPOSURES iter=2b Sex
Sex Age Female Male Total
Age Female Male Total <20 127.1 82.1 209.2
<20 15113 16202 31315 >=20 150.9 169.9 320.8
>=20 4876 9114 13990 Total 278.0 252.0 530.0
Total 19989 25316 45305
iter=3a Sex
ITERATIVE PROPORTIONAL FITTING Age Female Male Total
<20 127.0 82.0 209.0
iter=1a Sex >=20 151.0 170.0 321.0
Age Female Male Total Total 278.0 252.0 530.0
<20 100.9 108.1 209.0
>=20 111.9 209.1 321.0 iter=3b Sex
Total 212.7 317.3 530.0 Age Female Male Total
<20 127.0 82.0 209.0
iter=1b Sex >=20 151.0 170.0 321.0
Age Female Male Total Total 278.0 252.0 530.0
<20 131.8 85.9 217.7
>=20 146.2 166.1 312.3 PREDICTIONS OF NUMBERS OF Total 278.0 252.0 530.0 PERSONS LEAVING HOME
liefbr\2_2\lograte\2_2.xls
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STEP 2: PARAMETERS OF LOG-RATE MODEL (method of means)Estimates/exposures * 1000 Logarithm of estimate - logarithm of exposure
Sex Sex
Age Female Male Total Age Female Male Total
<20 8.4033 5.0613 13.4646 <20 -4.7791 -5.2861 -10.0653
>=20 30.9682 18.6522 49.6204 >=20 -3.4748 -3.9818 -7.4566
Total 39.3715 23.7135 63.0850 Total -8.2539 -9.2679 -17.5218
PARAMETERS OF LOG-RATE MODEL: contrast coding
Overall effect -4.3804603 Mean of all 4 values
Row effects -0.6521677 Mean of row 1 - overall mean
0.65216769 Mean of row 2 - overall mean
Column effects 0.25349767 Mean of col. 1 - overall mean
-0.2534977 Mean of col. 2 - overall mean
Interaction effects 0 F(11) - overall mean - row mean - col. mean
-4.441E-16 F(21) - overall mean - row mean - col. mean
-8.882E-16 F(12) - overall mean - row mean - col. mean
-4.441E-16 F(22) - overall mean - row mean - col. mean
check: 127.0 15113 exp(-4.3805-0.6523+0.2535)
151.0 4876 exp(-4.3805+0.6523+0.2535)
82.0 16202 exp(-4.3805-0.6523-0.2535)
170.0 9114 exp(-4.3805+0.6523-0.2535)
] exp[ PM uuuλ
B
j
A
iijij
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The log-rate model in SPSS: unsaturated model
Model and Design Information: unsaturated model
Model: PoissonDesign: Constant + SEX + TIMING
Parameter Aliased Term
1 Constant 2 [SEX = 1] 3 x [SEX = 2] 4 [TIMING = 1] 5 x [TIMING = 2]
Parameter Estimates
Asymptotic 95% CIParameter Estimate SE Lower Upper
1 -3.9818 .0694 -4.12 -3.85 2 .5070 .0878 .33 .68 3 .0000 . . . 4 -1.3044 .0897 -1.48 -1.13 5 .0000 . . .
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The log-rate model in SPSS: unsaturated model
PM *exp[ ] = RATE
9114*exp[-3.982 ] = 170.0 0.01865
16202*exp[-3.982-1.304 ] = 82.0 0.00506
15113*exp[-3.982-1.304+0.507] = 127.0 0.00840
4876*exp[-3.982+ 0.507] = 151.0 0.03096
uuuB
j
A
i
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The log-rate model in GLIM: unsaturated modelOcc = Exp * exp[overall + sex]
DATA: Occurrence matrix and exposure matrix (2*2)[i] $fit +sex$ [o] scaled deviance = 218.48 (change = -14.80) at cycle 4 [o] d.f. = 2 (change = -1 ) [o] [i] $d e$ [o] estimate s.e. parameter [o] 1 -4.275 0.05997 1 [o] 2 -0.3344 0.08697 SEX(2) [o] scale parameter taken as 1.000 Females 278 = 19989 * exp[-4.275] RATE = exp[-4.275] = 0.0139 Males 252 = 25316 * exp [-4.275 - 0.3344] RATE = exp [-4.275 - 0.3344] = 0.0100 [i] $d r$ [o] unit observed fitted residual [o] 1 135 210.19 -5.186 [o] 2 74 161.28 -6.873 [o] 3 143 67.81 9.130 [o] 4 178 90.72 9.163
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The log-rate model in GLIM: unsaturated model
Occ = Exp * exp[overall + sex + timing]
DATA: Occurrence matrix and exposure matrix (2*2)i] $fit +timing$ ! SEX + TIMING[o] scaled deviance = 2.1044 (change = -216.4) at cycle [o] d.f. = 1 (change = -1 )[o][i] $d e$[o] estimate s.e. parameter[o] 1 -4.779 0.07727 1[o] 2 -0.5070 0.08777 SEX(2)[o] 3 1.304 0.08969 TIMI(2)[o] scale parameter taken as 1.000[o][i] $d r$[o] unit observed fitted residual[o] 1 135 127.00 0.710[o] 2 74 82.00 -0.884[o] 3 143 151.00 -0.651[o] 4 178 170.00 0.614
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The log-rate model in GLIM: unsaturated model
Parameter s.e.Overall effect -4.779 0.0773Time(1) 0Time(2) 1.304 0.0897Sex(1) 0Sex(2) -0.507 0.0878
Check: 127.0 15113 exp(-4.779)82.0 16202 exp(-4.779-0.507)
151.0 4876 exp(-4.779+1.304)170.0 9114 exp(-4.779-0.507+1.304)
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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
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I. The unsaturated model Similarity with log-rate model
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The unsaturated log-linear model
• Assume: two-way classification; counts unknown but marginal totals given
• Predict the expected counts (cell entries)
Age Female Male Total<20 135 74 209>=20 143 178 321Total 278 252 530
Sexby age and sex: data
Number leaving parental home
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μ ln μμλB
j
A
iij
FFF
λji
ij
Age Female Male Total<20 109.6 99.4 209>=20 168.4 152.6 321Total 278 252 530
Sexby age and sex: predictions
Number leaving parental home
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STEP 1: ITERATIVE PROPORTIONAL FITTING ITERATIVE PROPORTIONAL FITTING
OCCURRENCES iter=1a Sex
Sex Age Female Male Total
Age Female Male Total <20 104.5 104.5 209.0
<20 135 74 209 >=20 160.5 160.5 321.0
>=20 143 178 321 Total 265.0 265.0 530.0
Total 278 252 530
iter=1b Sex
EXPOSURES Age Female Male Total
Sex <20 109.6 99.4 209.0
Age Female Male Total >=20 168.4 152.6 321.0
<20 1 1 2 Total 278.0 252.0 530.0
>=20 1 1 2
Total 2 2 4 PREDICTIONS OF NUMBERS OFPERSONS LEAVING HOME
THE LOG-RATE MODEL IN TWO STEPS:
The unsaturated log-linear model as a log-rate model
Odds ratio = 1
20
] exp[ PM uuuλB
j
A
iijij
STEP 2: PARAMETERS OF LOG-RATE MODEL (method of means)Estimates/exposures Logarithm of estimate - logarithm of exp.
Sex SexAge Female Male Total Age Female Male Total <20 109.63 99.37 209.00 <20 4.70 4.60 9.30 >=20 168.37 152.63 321.00 >=20 5.13 5.03 10.15 Total 278.00 252.00 530.00 Total 9.82 9.63 19.45
PARAMETERS OF LOG-RATE MODELOverall effect 4.8625Mean of all 4 valuesRow effects -0.2146Mean of row 1 - overall mean
0.2146Mean of row 2 - overall meanColumn effects 0.0491Mean of col. 1 - overall mean
-0.0491Mean of col. 2 - overall mean
check: 109.6 1exp(-4.3805-0.6523+0.2535)168.4 1exp(-4.3805+0.6523+0.2535)99.4 1exp(-4.3805-0.6523-0.2535)
152.6 1exp(-4.3805+0.6523-0.2535)
With PMij = 1
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II. Update a tableSimilarity with log-rate model
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Updating a table: THE LOG-RATE MODEL IN TWO STEPS
Odds ratio = 2.270837
STEP 1: ITERATIVE PROPORTIONAL FITTING
OCCURRENCES
Sex iter=2a Sex
Age Female Male Total Age Female Male Total
<20 135 74 209 <20 137.2 71.8 209.0
>=20 143 178 321 >=20 146.8 174.2 321.0
Total 278 252 530 Total 284.0 246.0 530.0
INITIAL GUESS iter=2b Sex
Sex Age Female Male Total
Age Female Male Total <20 134.3 73.5 207.8
<20 18.49 2.00 20.49 >=20 143.7 178.5 322.2
>=20 31.55 7.75 39.30 Total 278.0 252.0 530.0
Total 50.04 9.75 59.79
iter=3a Sex
ITERATIVE PROPORTIONAL FITTING Age Female Male Total
<20 135.1 73.9 209.0
iter=1a Sex >=20 143.1 177.9 321.0
Age Female Male Total Total 278.2 251.8 530.0
<20 188.6 20.4 209.0
>=20 257.7 63.3 321.0 iter=3b Sex
Total 446.3 83.7 530.0 Age Female Male Total
<20 135.0 74.0 209.0
iter=1b Sex >=20 143.0 178.0 321.0
Age Female Male Total Total 278.0 252.0 530.0
<20 117.5 61.4 178.9
>=20 160.5 190.6 351.1
Total 278.0 252.0 530.0
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STEP 2: PARAMETERS OF LOG-RATE MODEL (method of means)Estimates/exposures Logarithm of estimate - logarithm of exposure
Sex SexAge Female Male Total Age Female Male Total <20 7.2999 36.9885 44.2884 <20 1.9879 3.6106 5.5985 >=20 4.5333 22.9702 27.5035 >=20 1.5114 3.1342 4.6456 Total 11.8332 59.9587 71.7919 Total 3.4993 6.7448 10.2441
PARAMETERS OF LOG-RATE MODELOverall effect 2.56102758 Mean of all 4 valuesRow effects 0.23820507 Mean of row 1 - overall mean
-0.2382051 Mean of row 2 - overall meanColumn effects -0.8113749 Mean of col. 1 - overall mean
0.81137494 Mean of col. 2 - overall meanInteraction effects 0 F(11) - overall mean - row mean - col. mean
0 F(21) - overall mean - row mean - col. mean0 F(12) - overall mean - row mean - col. mean0 F(22) - overall mean - row mean - col. mean
Updating a table: THE LOG-RATE MODEL IN TWO STEPS