Transcript Maximum likelihood - School of Geography, University of Leeds

1
17 July 2015
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):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’
3
Data: leaving parental home
Leaving parental home, 1961 cohort, micro-data
Survey Sept. 87 - Febr. 88
First 23 respondents. Three censored observations.
ID
Sex
Father
Month
Reason
1
2
2
268
2
2
1
3
268
2
3
1
2
202
1
4
2
2
320
4
5
1
1
237
1
6
1
1
295
2
7
1
1
272
2
8
2
1
231
1
9
2
1
312
3
10
1
2
289
2
11
1
1
316
2
12
2
1
321
4
13
2
1
260
1
14
2
2
281
2
15
2
1
273
2
16
1
2
251
3
17
2
2
212
1
18
2
2
320
2
19
1
2
221
3
20
2
2
322
4
21
2
1
221
2
22
2
3
308
2
23
1
2
233
1
TWO CATEGORIES leave home early (before age 20)
leave home late (at or after age 20)
Censored
Total
Sex
1 Female
2 Male
Father status
1 Low
2 Middle
3 High
Reason
1
2
3
4
Total
209
321
53
583
Educ/work
Marriage/cohabit
Freedom
Censored
at interview
%
35.85
55.06
9.09
100.00
Leaving home
4
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
Age
<20
>=20
Total
Censored
Total
Female
135
143
278
13
291
Sex
Male
74
178
252
40
292
Total
209
321
530
53
583
Exposures: number of months living at home (includes censored observations): PMij
Age
<20
>=20
Total
Female
15113
4876
19989
Sex
Male
16202
9114
25316
Total
31315
13990
45305
5
The log-rate model
λ
ij
ln
 u  u iA  u Bj  u ijAB
PMij
ln
ij = E[Nij]
PMij fixed
occurrences
counts
 ln
exposure
exposures
with A AGE [early (before age 10) = 0; late (at age 20 or later) =1] and B SEX [female = 0; male = 1]
A
B
AB


u



ui u j uij
ln λij ln PMij
offset
o
A
B
AB


u



ui u j uij
ln λij mij
The log-rate model is a log-linear model with OFFSET
(constant term)
6
The log-rate model
E[N]
λ
ERate  

 exp[η]
PM
PM
E[N]  λ  PM expη and ln λ  ln(PM)  η
Multiplicative form
Addititive form
: linear predictor
Ln(PM): offset
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: ln  ij  u  uiA  uBj
PM
ij
8
Leaving home
STEP 1: ITERATIVE PROPORTIONAL FITTING
liefbr\2_2\lograte\2_2.xls
OCCURRENCES
Sex
Age
Female
iter=2a
Male
Total
Age
Sex
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
Age
Female
<20
Age
Male
Total
Sex
Female
Male
Total
<20
127.1
82.1
209.2
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
ITERATIVE PROPORTIONAL FITTING
iter=1a
Age
Age
Male
Total
Female
82.0
209.0
>=20
151.0
170.0
321.0
Total
278.0
252.0
530.0
100.9
108.1
209.0
>=20
111.9
209.1
321.0
iter=3b
Total
212.7
317.3
530.0
Age
Age
Sex
Female
Male
Total
<20
131.8
85.9
217.7
>=20
146.2
166.1
312.3
Total
278.0
252.0
530.0
Total
127.0
<20
iter=1b
Male
<20
Sex
Female
Sex
Sex
Female
Male
Total
<20
127.0
82.0
209.0
>=20
151.0
170.0
321.0
Total
278.0
252.0
530.0
PREDICTIONS OF NUMBERS OF
PERSONS LEAVING HOME
9
Leaving home
STEP 2: PARAMETERS OF LOG-RATE MODEL (method of means)
Estimates/exposures * 1000
Logarithm of estimate - logarithm of exposure
Sex
Age
Female
Sex
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
Total
39.3715
23.7135
63.0850
Total
-8.2539
-9.2679 -17.5218
-7.4566
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)
A
B

exp[


u ui u j ]
λij PMij
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Leaving home
The log-rate model in SPSS: unsaturated model
Model and Design Information: unsaturated model
Model: Poisson
Design: Constant + SEX + TIMING
Parameter Aliased Term
1
2
3
4
5
Constant
[SEX = 1]
Ref. cat
x [SEX = 2]
[TIMING = 1]
x [TIMING = 2] Ref. cat
COUNTS PRED
Age
Female
< 20
127
> = 20
151
Total
278
Sex
Male
82
170
252
Total
209
321
530
EXPOSURE
Age
Female
< 20
15113
> = 20
4876
Total
19989
Sex
Male
16202
9114
25316
Total
31315
13990
45305
Parameter Estimates
Parameter Estimate
1
2
3
4
5
-3.9818
.5070
.0000
-1.3044
.0000
Asymptotic 95% CI
SE
Lower
Upper
.0694
.0878
.
.0897
.
-4.12
.33
.
-1.48
.
-3.85
.68
.
-1.13
.
ln 170/9114 (ref.cat)
[ln 151/4876]+3.9818
[ln 82/16202]+3.9818
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Leaving home
The log-rate model in SPSS: unsaturated model
PM *exp[
u  u u
A
B
i
j
]=
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+
0.03096
0.507] = 151.0
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The log-rate model in SPSS: unsaturated model
SEX
1
2
1
2
TIMING NUMBER
1
135
1
74
2
143
2
178
EXPOSURE
15113
16202
4876
9114
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 .
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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 = 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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Leaving home
The log-rate model in GLIM: unsaturated
model
Occ = Exp * exp[overall + sex + timing]
DATA:
O
i]
$fit
+t
[o]
scaled
[o]
[o]
[i]
$d
e$
[o]
[o]
1
[o]
2
[o]
3
[o]
sc
[o]
[i]
$d
r$
[o]
unit
[o]
1
[o]
2
[o]
3
[o]
4
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Leaving home
The log-rate model in GLIM: unsaturated model
Overall effect
Time(1)
Time(2)
Sex(1)
Sex(2)
Check:
Parameter
-4.779
0
1.304
0
-0.507
s.e.
0.0773
127.0
82.0
151.0
170.0
15113
16202
4876
9114
0.0897
0.0878
exp(-4.779)
exp(-4.779-0.507)
exp(-4.779+1.304)
exp(-4.779-0.507+1.304)
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The log-rate model in TDA
Leaving home
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
---------------------------------------------------------------------------1
0
0
53
53.00
128.47
0.00
144.00
1
0
1
530
530.00
72.63
0.00
140.00
Sum
583
583.00
Number of episodes: 583
Successfully created new episode data.
Idx SN Org Des MT Variable
Coeff
Error
C/Error Signif
------------------------------------------------------------------1 1
0
1 A Constant
-4.6098
0.0630
-73.1777 1.0000
2 1
0
1 A SEX1
0.3344
0.0870
3.8451 0.9999
Log likelihood (starting values): -2887.5967
Log likelihood (final estimates): -2880.1982
command file: ehd21.cf
data file: test.dat (micro data)
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Leaving home
LOG-RATE MODEL IN TDA: PROGRAMME
# ehd2.cf Basic exponential model with covariate SEX
nvar( dfile = test.dat,
ID = c1,
SN = c2,
TF = c3,
TF15 = TF-180,
SEX = c4,
SEX1 = SEX[1],
SEX2 = SEX[2],
DES
TFP
# data file
# identification number
# spell number
# TIME LEAVING HOME (=ENDING TIME)
# measured from age 0!!!!
# measured from age 15
# sex REASON = c5, # reason
# see boek p. 61 SEX1 = 1 for females and 0 for males
# MALES ref.cat
# = 1 for females
= if eq(REASON,4) then 0 else 1,
= TF15, # Blossfeld: TF+1 !!!!!!
# destination
);
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. 90-91)
rate(
xa (0,1) = SEX1,
pres = ehd21.res,
) = 2;
18
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
19
The unsaturated model
Similarity with log-rate model
20
Leaving home
The unsaturated log-linear model
• Assume: two-way classification; counts
unknown but marginal totals given
• Predict the expected counts (cell entries)
Number leaving parental home
by age and sex: data
Sex
Age
Female
Male
Total
<20
135
74
209
>=20
143
178
321
Total
278
252
530
21
Leaving home
ln λij  u  uiA  uBj
F
F
λ
F
i
j
ij

Number leaving parental home
by age and sex: predictions
Sex
Age
Female
Male
Total
<20
109.6
99.4
209
>=20
168.4
152.6
321
Total
278
252
530
22
Leaving home
The unsaturated log-linear model as a log-rate model
THE LOG-RATE MODEL IN TWO STEPS:
------------------------------------------------------
STEP 1: ITERATIVE PROPORTIONAL FITTING
ITERATIVE PROPORTIONAL FITTING
OCCURRENCES
iter=1a
Sex
Age
Female
Age
Male
Total
Sex
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
EXPOSURES
Age
Sex
Age
Female
Male
Total
<20
1
1
2
>=20
1
1
2
Total
2
2
4
Sex
Female
Male
Total
<20
109.6
99.4
209.0
>=20
168.4
152.6
321.0
Total
278.0
252.0
530.0
PREDICTIONS OF NUMBERS OF
PERSONS LEAVING HOME
Odds ratio = 1
23
Leaving home
STEP 2: PARAMETERS OF LOG-RATE MODEL (method of means)
Estimates/exposures
Logarithm of estimate - logarithm of exp.
Sex
Sex
Age
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 MODEL
Overall effect
4.8625 Mean of
Row effects
-0.2146 Mean of
0.2146 Mean of
Column effects
0.0491 Mean of
-0.0491 Mean of
check:
109.6
168.4
99.4
152.6
1
1
1
1
all 4 values
row 1 - overall mean
row 2 - overall mean
col. 1 - overall mean
col. 2 - overall mean
exp(-4.3805-0.6523+0.2535)
exp(-4.3805+0.6523+0.2535)
exp(-4.3805-0.6523-0.2535)
exp(-4.3805+0.6523-0.2535)
A
B

exp[


PM
ij
u ui u j ]
λij
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
STEP 1: ITERATIVE PROPORTIONAL FITTING
OCCURRENCES
Sex
Age
Female
iter=2a
Male
Total
Age
Sex
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
Age
Female
Age
Male
Total
Sex
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
ITERATIVE PROPORTIONAL FITTING
iter=1a
Age
Age
Sex
Female
Male
Total
209.0
>=20
143.1
177.9
321.0
Total
278.2
251.8
530.0
209.0
>=20
257.7
63.3
321.0
iter=3b
Total
446.3
83.7
530.0
Age
Female
Male
Total
<20
117.5
61.4
178.9
>=20
160.5
190.6
351.1
Total
278.0
252.0
530.0
Total
73.9
20.4
Age
Male
135.1
188.6
Sex
Female
<20
<20
iter=1b
Sex
Sex
Female
Male
Total
<20
135.0
74.0
209.0
>=20
143.0
178.0
321.0
Total
278.0
252.0
530.0
Odds ratio = 2.270837
26
Update table
Updating a table: THE LOG-RATE MODEL IN TWO STEPS
STEP 2: PARAMETERS OF LOG-RATE MODEL (method of means)
Estimates/exposures
Logarithm of estimate - logarithm of exposure
Sex
Sex
Age
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 MODEL
Overall effect
2.56102758 Mean of all 4 values
Row effects
0.23820507 Mean of row 1 - overall mean
-0.2382051 Mean of row 2 - overall mean
Column effects
-0.8113749 Mean of col. 1 - overall mean
0.81137494 Mean of col. 2 - overall mean
Interaction effects
0 F(11) - overall mean - row mean - col. mean
0 F(21) - overall mean - row mean - col. mean
0 F(12) - overall mean - row mean - col. mean
0 F(22) - overall mean - row mean - col. mean
27
Update table

Pr{N  n } 
exp[-  ]
nij
ij
ij
ij
n!
ij
ij
E[ Nij]  λ
ij
Var[ Nij]  λ
λij  exp[u  uiA  uBj  uijAB]
ln λij  u  uiA  uBj  uijAB
ij
28
Update table
Log-rate model: rate = events/exposure
N 

 exp [u  u  u  u ]
E  
m
m 
ij
ij
ij
ij
A
i
B
j
AB
ij
Gravity / spatial interaction model
   m
ij
i
j
ij
    expc 
ij
i
i and j are balancing factors
j
ij
29
Update table
IPF and biproportional adjustment
n

Pr{N  n } 
exp[-  ]
ij
ij
ij
ij
ij
n!
ij
   m
ij
i
j
ij
Log-likelihood function:


l( ,  ; n)  nij ln  i  j mij -  i  j mij - ln nij!
30
Update table
l ni 

 j  j mij  0
 i  i
ni 
ˆ i 
 ˆ m
l n j

 i  i mij  0
 j  j
n
j
ˆ
j 
 ˆ i mij
j
j
ij
i
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
Update table
Biproportional adjustment
   m
ij
Step 0: s (Step) = 0
Step 1


( 2 s 1)
j
i
1
(0)
i

ij
j
n j

(2s)
i
mij
i
Step 2

( 2 s2 )
i

ni 
( 2 s 1)
j
i
mij
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
Update table
Likelihood equations may be written as:
l ni   i 

0
 i
i
l n j   j

0
 j
j
Marginal totals are sufficient statistics
33
Update table
A different way of writing the spatial interaction model:
ij   i  j m ij
ni 
ˆ i 
 ˆ m
j
j
ij
ˆ j mij
ˆij


ˆij  ˆ i ˆ j mij 
ni   ni   ˆ ij ni 
ˆi 
 ˆ j mij

j
Link Poisson - Multinomial
34
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 ‘completedata’ 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. 231240. The authors claim that they go beyond Holford (1980) ‘The analysis of rates and
survivorship using log-linear models’, Biometrics, 36:299-306
d:\s\data\laird\kidney\laird.doc
37
Kidney Transplant Study
Life-table data on graft survival
CAD
LRD
Period
# days Entered Withdraw Deaths Exposure
Entered Withdraw Deaths Exposure
0-7d
7
1169
0
57
7984
806
0
21
5569
7-15d
8
1112
0
68
8624
785
0
13
6228
15-21d
6
1044
0
52
6108
772
0
11
4599
21d-1m
9
992
0
58
8667
761
0
14
6786
1-2m
30
934
0
115
26295
747
0
52
21630
2-3m
30
819
0
75
23445
695
0
27
20445
3-6m
90
744
0
106
62190
668
0
43
58185
6-9m
90
638
0
30
56070
625
0
18
55440
9-12m
90
608
0
30
53370
607
0
15
53955
1-1.5y
180
578
0
27
101610
592
0
11
105570
1.5-2y
180
551
18
27
95130
581
14
13
102150
2-2.5y
180
506
113
19
79200
554
99
10
89910
2.5-3y
180
374
90
14
57960
445
103
6
70290
3-3.5y
180
270
79
11
40500
336
106
4
50580
3.5-4y
180
180
90
3
24030
226
94
1
32130
>4y
180
87
84
3
7830
131
131
0
11790
Total
474
695
659013
547
259
695257
Source: Laird and Olivier, 1981, p. 238 (data aggregated from data on p. 237-238)
TOTAL
Entered Withdraw Deaths Exposure
1975
0
78
13552
1897
0
81
14852
1816
0
63
10707
1753
0
72
15453
1681
0
167
47925
1514
0
102
43890
1412
0
149
120375
1263
0
48
111510
1215
0
45
107325
1170
0
38
207180
1132
32
40
197280
1060
212
29
169110
819
193
20
128250
606
185
15
91080
406
184
4
56160
218
215
3
19620
1021
954 1354269
Exposure (Exp) is calculated as follows:
Exp = [E - 0.5(W + D)]*# in days
608*90 + 30*45
where
E = number entered
W = number withdrawn
D = number died
# = width of interval (the last open interval was taken as having 180 days)
d:\s\data\laird\laird_lt.xls
38
Kidney Transplant Study
Death rates (* 1000; per day)
Period
0-7d
7-15d
15-21d
21d-1m
1-2m
2-3m
3-6m
6-9m
9-12m
1-1.5y
1.5-2y
2-2.5y
2.5-3y
3-3.5y
3.5-4y
>4y
Total
CAD
7.14
7.88
8.51
6.69
4.37
3.20
1.70
0.54
0.56
0.27
0.28
0.24
0.24
0.27
0.12
0.38
1.05
LRD
3.77
2.09
2.39
2.06
2.40
1.32
0.74
0.32
0.28
0.10
0.13
0.11
0.09
0.08
0.03
0.00
0.37
TOTAL
5.76
5.45
5.88
4.66
3.48
2.32
1.24
0.43
0.42
0.18
0.20
0.17
0.16
0.16
0.07
0.15
0.70
39
Kidney Transplant Study
Kidney transplant survivors
1200
CAD
Number of survivors
1000
800
LRD
600
400
200
0
0
7
15
21
30
60
90
180
270
1.0 yr
1.5 yr
Duration since transplant (days/years)
2.0 yr
2.5 yr
3.0 yr
3.5 yr
4.0 yr
40
Kidney Transplant Study
Model: Poisson
Design: Constant + TIME
Parameter
1 Constant
1
2 [TIME = 1]
2
3 [TIME = 2]
3
4 [TIME = 3]
4
5 [TIME = 4]
5
6 [TIME = 5]
6
7 [TIME = 6]
7
8 [TIME = 7]
8
9 [TIME = 8]
9
10 [TIME = 9] 10
11 [TIME = 10] 11
12 [TIME = 11] 12
13 [TIME = 12] 13
14 [TIME = 13] 14
15 [TIME = 14] 15
16 [TIME = 15] 16
17x[TIME = 16] 17
Estimate
-8.7857
3.6281
3.5743
3.6502
3.4168
3.1263
2.7212
2.0913
1.0350
1.0087
.1819
.2822
.1147
.0197
.0742
-.7640
.0000
SPSS
SE
.5774
.5883
.5879
.5909
.5893
.5825
.5858
.5831
.5951
.5963
.5997
.5986
.6065
.6191
.6325
.7638
.
Deaths 9-12 m: 107325 * exp[-8.7857+1.0087]=107325*0.0004193=45
41
Kidney Transplant Study
Model: Poisson
Design: Constant + DONOR TYPE + TIME (unsaturated model)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
x
x
Estimate
Constant
-9.2184
[CAD = 1.00]
.8573
[LRD = 2.00]
.0000
[P1 = 1]
3.4734
[P1 = 2]
3.4260
[P1 = 3]
3.5097
[P1 = 4]
3.2837
[P1 = 5]
3.0026
[P1 = 6]
2.6089
[P1 = 7]
1.9928
[P1 = 8]
.9476
[P1 = 9]
.9258
[P1 = 10]
.1046
[P1 = 11]
.2116
[P1 = 12]
.0555
[P1 = 13]
-.0258
[P1 = 14]
.0349
[P1 = 15]
-.7890
[P1 = 16]
.0000
SE
.5791
.0730
.
.5885
.5880
.5910
.5893
.5826
.5858
.5832
.5952
.5963
.5997
.5986
.6065
.6191
.6325
.7638
.
Deaths 9-12 m: 53370 * exp[-9.2184+0.8573+0.9258]=53370*0.000590=31.49
Observed: 30