Spline-Based Semiparametric Projected Generalized Estimating Equation Method for Panel Count Data Lei Hua FXB 514, Center for Biostatistics in AIDS Research/HSPH 651 Huntington Avenue, Boston, MA 02466, U.S.A. [email protected]Ying Zhang Department of Biostatistics, The University of Iowa C22 GH, 200 Hawkins Drive, Iowa City, IA 52242, U.S.A. [email protected]Summary We propose to analyze panel count data using a spline-based semiparametric projected generalized estimating equation method with the semiparametric proportional mean model E(N(t)|Z )=Λ 0 (t) e β T 0 Z . The natural logarithm of the baseline mean function, log Λ 0 (t), is approximated by monotone cubic B-spline functions. The estimates of regression parameters and spline coefficients are obtained by projecting the generalized estimating equation esti- mates into the feasible domain using a weighted isotonic regression. The proposed method avoids assuming any parametric structure of the baseline mean function or the underlying counting process. Selection of the working-covariance matrix that represents the true corre- 1
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Spline-Based Semiparametric Projected Generalized
Estimating Equation Method for Panel Count Data
Lei Hua
FXB 514, Center for Biostatistics in AIDS Research/HSPH
Regression results with the three different working-covariance matrices are shown in Table
3. The tumor number at baseline is positively related to the recurrence of bladder tumor.
With one more tumor at baseline, the number of tumors at follow-ups increases by 15.5%,
23.1% and 39.1% on average using the working-covariance matrices V(i)1 , V
(i)2 and V
(i)3 , re-
spectively. Thiotepa instillation effectively decreases the number of recurrent tumors. The
number of recurrent tumors in patients with thiotepa instillation is 49.5%, 45.1% and 32.5%
of that in placebo group on average using V(i)1 , V
(i)2 and V
(i)3 , respectively. The tumor size
and pyridoxin pills are not significantly related to the number of recurrent tumors at follow-
up visits. The estimating results using the diagonal working-covariance matrix V(i)1 and the
working-covariance matrix based on Poisson process V(i)2 are consistent with the estimating
results based on the spline-based semiparametric pseudo-likelihood and the likelihood meth-
ods proposed by Lu et al. (2009). The proposed semiparametric projected GEE estimate
20
with the frailty Poisson covariance matrix V(i)3 provides an estimate of the over-dispersion
parameter as 1.32. It implies the over-dispersion of panel counts and possible positive corre-
lation among the tumor numbers in non-overlapping time intervals for the underlying tumor
progression. The effect of the tumor number at the study entrance and the treatment of
thiotepa are more significant when accounting for the correlation between cumulative tumor
numbers using the frailty variable. Figure 3 plots the estimated baseline mean function.
5. Final Remark
Modeling panel count data is a challenging task in general. The proposed spline-based semi-
parametric projected GEE method avoids assuming the underlying count process and bor-
rows the strength from discrete observations within subjects as well as those across subjects
to get a spline estimate of the mean function of the counting process. Choosing differ-
ent working-covariance matrices can accommodate different data structures. The proposed
spline-based projected GEE method with the working-covariance matrices V(i)3 accounts for
the over-dispersion and inter-correlation between non-overlapping counts. It improves the
estimating efficiency and provides a less biased standard error estimation using either the
“ad-hoc” parametric GEE sandwich formula or the bootstrap method when over-dispersion
is present in data. In our computing algorithm, over-dispersion parameter σ2 is fixed at
its estimate in the first stage and the parameters in the proportional mean function θ are
updated in the second stage. Our simulation results (not included in this paper) show that
update θ and σ2 alternately gives similar results.
21
The proposed model assumes that the observation times are noninformative to the un-
derlying counting process which may be violated in applications. Extension of the proposed
method to that scenario requires a further investigation.
22
References
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24
6. Appendix
In this section, we show that spline-based semiparametric GEE with V(i)1 , V
(i)2 and V
(i)3
coincide with the score equations under the different models. First, we define the followingnotations
B(i)Ki,j
=(B1
(T
(i)Ki,j
), · · · , Bqn
(T
(i)Ki,j
))T; B(i) =
(B
(i)Ki,1
, · · · , B(i)Ki,Ki
)Tµ(i)Ki,j
= exp(βTZi + αTB
(i)Ki,j
); µ(i) =
(µ(i)Ki,1
, · · · , µ(i)Ki,Ki
)T∆µ
(i)Ki,j
= µ(i)Ki,j
− µ(i)Ki,j−1; ∆µ(i) =
(∆µ
(i)Ki,1
, · · · ,∆µ(i)Ki,Ki
)T∆N(i)
Ki,j= N
(T
(i)Ki,j
)− N
(T
(i)Ki,j−1
); ∆N(i) =
(∆N(i)
Ki,1, · · · ,∆N(i)
Ki,Ki
)TAlso let 1Ki
= (1, 1, · · · , 1)TKi×1, then we have
∂µ(i)Ki,j
∂θ= exp
(βTZi + αTB
(i)Ki,j
)(ZT
i , B(i)T
Ki,j
)T;
∂µ(i)
∂θ=
(∂µ
(i)Ki,1
∂θ, · · · ,
∂µ(i)Ki,Ki
∂θ
)T
= diag(µ(i)Ki,1
, · · · , µ(i)Ki,Ki
) (1Ki
ZTi , B
(i))
6.1 Agreement between the GEE with V(i)1 and the score equation of the spline-based
pseudo-likelihood
Using V(i)1 as the working-covariance matrix, the U function of Equation (3) can be
rewritten as
U (θ) =n∑
i=1
(1Ki
ZTi , B
(i))T
diag(µ(i)Ki,1
, · · · , µ(i)Ki,Ki
)×(
diag(µ(i)Ki,1
, · · · , µ(i)Ki,Ki
))−1 (N (Ti)− µ(i)
)=
n∑i=1
(1Ki
ZTi , B
(i))T (N (Ti)− µ(i)
)This is exactly the score function of the spline-based pseudo-likelihood derived by Lu et al.(2009)
25
6.2 Agreement between the GEE with V(i)2 and the score equation of the spline-basedlikelihood
When using V(i)2 as the working-covariance matrix, the U function of Equation (3) can
be rewritten as
U (θ) =n∑
i=1
(1Ki
ZTi , B
(i))T
diag(µ(i)Ki,1
, · · · , µ(i)Ki,Ki
)V
(i)−1
2
(N(i) − µ(i)
).
Using the independence of the count increments based on the nonhomogeneous Poissonprocess assumption, the spline-based likelihood is given by
l̃n (θ;D) =n∑
i=1
Ki∑j=1
[∆N(i)
Ki,jlog∆Λ̃
(i)Ki,j
+∆N(i)Ki,j
βTZi − eβTZi∆Λ̃
(i)Ki,j
](6)
where
∆Λ̃(i)Ki,j
= exp
(qn∑l=1
αlBl
(T
(i)Ki,j
))− exp
(qn∑l=1
αlBl
(T
(i)Ki,j−1
))A careful examination of this likelihood shows that its score function can be rewritten in amatrix form,
∂
∂θl̃n (θ;D) =
n∑i=1
(∂∆µ(i)
∂θ
)T (diag
(∆µ
(i)Ki,1
, · · · ,∆µ(i)Ki,Ki
))−1 (∆N(i) −∆µ(i)
)Since
∂∆µ(i)Ki,j
∂θ= µ
(i)Ki,j
(ZTi , B
(i)T
Ki,j
)T− µ
(i)Ki,j−1
(ZTi , B
(i)T
Ki,j−1
)T=
{(−µ
(i)Ki,j−1, µ
(i)Ki,j
)(ZTi B
(i)T
Ki,j−1
ZTi B
(i)T
Ki,j
)}T
∂∆µ(i)
∂θ=
∂∆µ(i)Ki,1
∂θ, · · · ,
∂∆µ(i)Ki,Ki
∂θ
T
=
µ(i)Ki,1
0 · · · 0
−µ(i)Ki,1
µ(i)Ki,2
· · · 0...
......
...
0 0 −µ(i)Ki,Ki−1 µ
(i)Ki,Ki
(1kiZ
Ti , B
(i))
26
=
1 0 · · · 0−1 1 · · · 0...
......
...0 0 −1 1
diag(µ(i)Ki,1
, · · · , µ(i)Ki,Ki
)(1kiZ
Ti , B
(i))
The score function can be further written as
∂
∂θl̃n (θ;D) =
n∑i=1
(1Ki
ZTi , B
(i))T
diag(µ(i)Ki,1
, · · · , µ(i)Ki,Ki
)Σ(N(i) − µ(i)
),
where
Σ =
1 0 · · · 0−1 1 · · · 0...
......
...0 0 −1 1
T
diag(∆µ
(i)Ki,1
, · · · ,∆µ(i)Ki,Ki
)−1
1 0 · · · 0−1 1 · · · 0...
......
...0 0 −1 1
=
1
µ(i)Ki,1
− 1
µ(i)Ki,2
−µ(i)Ki,1
0 · · · 0
0 1
µ(i)Ki,2
−µ(i)Ki,1
− 1
µ(i)Ki,3
−µ(i)Ki,2
· · · 0
......
...... − 1
µ(i)Ki,Ki
−µ(i)Ki,Ki
0 0 0 · · · 1
µ(i)Ki,Ki
−µ(i)Ki,Ki
1 0 · · · 0−1 1 · · · 0...
......
...0 0 −1 1
=
1
µ(i)Ki,1
+ 1
µ(i)Ki,2
−µ(i)Ki,1
− 1
µ(i)Ki,2
−µ(i)Ki,1
· · · · · · 0
− 1
µ(i)Ki,2
−µ(i)Ki,1
1
µ(i)Ki,2
−µ(i)Ki,1
+ 1
µ(i)Ki,3
−µ(i)Ki,2
− 1
µ(i)Ki,3
−µ(i)Ki,2
· · · 0
......
......
...0 0 · · · · · · 1
µ(i)Ki,Ki
−µ(i)Ki,Ki−1
It is a straightforward algebra to verify that Σ =
(V
(i)2
)−1
, so the GEE with the working-
covariance matrix V(i)2 is the same as the score equation of the likelihood given in (6).
6.3 Agreement between the GEE with V(i)3 and the score equation of the likelihood of
Gamma-Frailty Poisson model
By the derivation of the equivalence between the GEE with V(i)2 and the score equation
of likelihood in (6), we have(∂µ(i)
∂θ
)T
V(i)−1
2
(N(i) − µ(i)
)=
Ki∑j=1
(∂∆µ
(i)Ki,j
∂θ
)(∆N(i)
Ki,j
∆µ(i)Ki,j
− 1
)(7)
27
This equality holds for any nonnegative and nondecreasing process N(i). Let N(i) = 2µ(i),then (
∂µ(i)
∂θ
)T
V(i)−1
2 µ(i) =
Ki∑j=1
∂∆µ(i)Ki,j
∂θ=∂µ
(i)Ki,Ki
∂θ=(ZT
i , B(i)T
Ki,Ki
)Tµ(i)Ki,Ki
(8)
Taking the β part of (7), we have(∂µ(i)
∂β
)T
V(i)−1
2
(N(i) − µ(i)
)=
Ki∑j=1
(∂∆µ
(i)Ki,j
∂β
)(∆N(i)
Ki,j
∆µ(i)Ki,j
− 1
)
The left hand side of (8) can be rewritten as
LHS = Zi1TKidiag
(µ(i)Ki,1
, · · · , µ(i)Ki,Ki
)V
(i)−1
2
(N(i) − µ(i)
)= Ziµ
(i)TV(i)−1
2
(N(i) − µ(i)
)and the right hand side of (8) can also be rewritten as
RHS =
Ki∑j=1
(µ(i)Ki,j
Zi − µ(i)Ki,j−1Zi
)(∆N(i)Ki,j
∆µ(i)Ki,j
− 1
)= Zi
(N(i)
Ki,Ki− µ
(i)Ki,Ki
).
This implies that
µ(i)TV(i)−1
2
(N(i) − µ(i)
)= N(i)
Ki,Ki− µ
(i)Ki,Ki
. (9)
Again letting N(i) = 2µ(i), we obtain
µ(i)V(i)−1
2 µ(i) = µ(i)Ki,Ki
(10)
The U function of Equation with V(i)3 as the working-covariance matrix can then be rewritten
as,
U (θ) =n∑
i=1
(∂µ(i)
∂θ
)T (V
(i)2 + σ2µ(i)µ(i)T
)−1 (N(i) − µ(i)
)
=n∑
i=1
(∂µ(i)
∂θ
)T
V (i)2 − σ2
1 + σ2µ(i)T(V
(i)2
)−1
µ(i)
(V
(i)2
)−1
µ(i)µ(i)TV −12
(N (Ti)− µ(i))
=n∑
i=1
{(∂µ(i)
∂θ
)T
V(i)−1
2
(N(i) − µ(i)
)− σ2
1 + σ2µ(i)TV −12 µ(i)
(∂µ(i)
∂θ
)T
V(i)−1
2 µ(i)
28
×µ(i)TV(i)−1
2
(N(i) − µ(i)
)}=
n∑i=1
{Ki∑j=1
(µ(i)Ki,j
(ZT
i , B(i)T
Ki,j
)T− µ
(i)Ki,j−1
(ZT
i , B(i)T
Ki,j−1
)T)(∆N(i)Ki,j
∆µ(i)Ki,j
− 1
)−
σ2
1 + σ2µ(i)Ki,Ki
(ZT
i , B(i)T
Ki,Ki
)Tµ(i)Ki,Ki
(N(i)
Ki,Ki− µ
(i)Ki,Ki
)}(by Equations (8)-(10))
=n∑
i=1
{Ki∑j=1
(µ(i)Ki,j
(ZT
i , B(i)T
Ki,j
)T− µ
(i)Ki,j−1
(ZT
i , B(i)T
Ki,j−1
)T) ∆N(i)Ki,j
∆µ(i)Ki,j
−1 + σ2N(i)
Ki,Ki
1 + σ2µ(i)Ki,Ki
(ZT
i , B(i)T
Ki,Ki
)Tµ(i)Ki,Ki
}
This is exactly the score function of the Gamma-frailty Poisson likelihood.
29
Table
1Sim
ulationsresu
ltsofthesp
lines-ba
sed
sievese
mipara
metric
GEE
estim
ators
with
threediff
erent
covariance
matricesfordata
from
scenario
1and
scenario
2
Scenario
1:Frailty
PoissonData
Scenario2:Mixture
PoissonData
N=50
N=100
N=50
N=100
V(i)
1V
(i)
2V
(i)
3V
(i)
1V
(i)
2V
(i)
3V
(i)
1V
(i)
2V
(i)
3V
(i)
1V
(i)
2V
(i)
3
β1
Bias
0.05
190.05
280.01
620.03
76
0.0370
-0.0001
0.0043
0.0049
0.0020
-0.0023
-0.0026
-0.0019
MC-sd
0.36
400.36
470.24
760.26
57
0.2639
0.1810
0.0810
0.0809
0.0640
0.0601
0.0606
0.0425
SSE
0.22
080.22
210.18
070.17
65
0.1772
0.1382
0.0564
0.0559
0.0546
0.0454
0.0452
0.0405
BSE
0.30
630.30
530.27
390.21
45
0.2137
0.1683
0.0732
0.0728
0.0649
0.0529
0.0528
0.0434
CP1
0.70
890.70
590.83
830.75
90
0.7500
0.8776
0.7990
0.7980
0.8980
0.8320
0.8299
0.9305
CP2
0.85
440.85
840.94
520.85
40
0.8480
0.9330
0.9050
0.9080
0.9370
0.9035
0.8994
0.9429
β2
Bias
-0.009
0-0.019
2-0.000
-0.0712
-0.0691
-0.0455
0.0017
0.0010
-0.0020
0.0016
-0.0001
0.0007
MC-sd
1.13
741.12
810.79
790.83
26
0.8196
0.5477
0.2540
0.2504
0.2015
0.1840
0.1812
0.1384
SSE
0.79
640.80
110.63
450.64
05
0.6407
0.4715
0.1999
0.1990
0.1800
0.1565
0.1556
0.1314
BSE
0.98
460.97
990.86
170.70
89
0.7048
0.5347
0.2312
0.2307
0.2045
0.1675
0.1668
0.1384
CP1
0.82
350.81
850.89
360.85
40
0.8660
0.9087
0.8640
0.8630
0.9150
0.8973
0.8973
0.9367
CP2
0.90
730.91
030.94
320.89
50
0.8960
0.9510
0.9190
0.9260
0.9390
0.9212
0.9315
0.9419
β3
Bias
0.03
910.03
750.02
36-0.0023
-0.0031
-0.0082
0.0012
0.0015
0.0029
0.0022
0.0027
0.0035
MC-sd
0.61
260.60
420.44
990.41
77
0.4145
0.2939
0.1443
0.1407
0.1232
0.1015
0.0981
0.0805
SSE
0.44
910.44
870.38
020.34
88
0.3479
0.2801
0.1179
0.1156
0.1098
0.0908
0.0892
0.0796
BSE
0.54
260.53
780.48
270.37
75
0.3751
0.3053
0.1321
0.1297
0.1208
0.0954
0.0939
0.0830
CP1
0.85
340.85
940.91
400.88
60
0.8900
0.9509
0.8750
0.8740
0.9070
0.9170
0.9232
0.9471
CP2
0.91
230.92
320.95
310.91
30
0.9170
0.9570
0.9150
0.9170
0.9340
0.9274
0.9336
0.9585
MC-sd:Monte-C
arlostandard
dev
iation;SSE:Ad-hocparametricsandwichstandard
errorestimation;BSE:Bootstrapstandard
errorestimation;CP1:95%
Coverage
basedonSSE;CP2:95%
CoveragebasedonBSE;
30
Table
2Sim
ulationsresu
ltsofthesp
lines-ba
sed
sievese
mipara
metric
GEE
estim
ators
with
threediff
erent
covariance
matricesfordata
from
scenario
3and
scenario
4
Scenario
3:PoissonData
Scenario4:NegativeBinomialCountData
N=50
N=100
N=50
N=100
V(i)
1V
(i)
2V
(i)
3V
(i)
1V
(i)
2V
(i)
3V
(i)
1V
(i)
2V
(i)
3V
(i)
1V
(i)
2V
(i)
3
β1
Bias
-0.000
10.00
030.00
020.00
12
0.0010
0.0009
-0.0004
-0.0004
-0.0008
0.0008
0.0009
0.0009
MC-sd
0.03
140.02
910.02
920.02
09
0.0195
0.0196
0.0357
0.0344
0.0350
0.0248
0.0231
0.0236
SSE
0.02
760.02
590.02
600.01
89
0.0176
0.0177
0.0312
0.0296
0.0302
0.0212
0.0200
0.0206
BSE
0.03
460.03
240.03
290.02
18
0.0203
0.0206
0.0390
0.0369
0.0383
0.0246
0.0232
0.0242
CP1
0.91
800.91
000.90
900.90
80
0.9220
0.9230
0.9070
0.9000
0.9040
0.9120
0.9080
0.9100
CP2
0.96
800.96
600.96
800.95
50
0.9630
0.9650
0.9630
0.9590
0.9630
0.9480
0.9490
0.9600
β2
Bias
0.00
230.00
160.00
150.00
00
0.0000
-0.0002
0.0014
0.0020
0.0017
-0.0019
-0.0024
-0.0026
MC-sd
0.10
160.09
690.09
720.06
87
0.0636
0.0641
0.1180
0.1100
0.1120
0.0791
0.0744
0.0752
SSE
0.08
790.08
260.08
280.06
25
0.0584
0.0584
0.0991
0.0947
0.0957
0.0707
0.0671
0.0679
BSE
0.10
450.09
760.09
860.06
83
0.0637
0.0643
0.1178
0.1122
0.1150
0.0771
0.0730
0.0747
CP1
0.91
300.89
600.89
600.91
90
0.9320
0.9300
0.8950
0.9080
0.9100
0.9200
0.9280
0.9270
CP2
0.95
000.94
100.94
200.94
00
0.9510
0.9540
0.9380
0.9480
0.9490
0.9360
0.9420
0.9490
β3
Bias
-0.000
2-0.000
8-0.000
80.00
14
0.0014
0.0015
-0.0038
-0.0038
-0.0037
0.0014
0.0016
0.0017
MC-sd
0.06
940.06
570.06
570.04
95
0.0449
0.0450
0.0799
0.0743
0.0749
0.0568
0.0538
0.0540
SSE
0.06
510.06
120.06
130.04
62
0.0431
0.0432
0.0738
0.0701
0.0706
0.0525
0.0496
0.0499
BSE
0.07
280.06
800.06
840.04
88
0.0454
0.0456
0.0822
0.0780
0.0791
0.0553
0.0522
0.0527
CP1
0.91
800.91
600.91
700.92
40
0.9290
0.9290
0.9190
0.9230
0.9220
0.9200
0.9090
0.9160
CP2
0.94
800.94
200.94
300.93
70
0.9430
0.9430
0.9480
0.9570
0.9570
0.9330
0.9290
0.9360
31
Gamma−Frailty Poisson Data
Observation Time
0
5
10
15
20
2 4 6 8
Bias^2: n=100 Variance: n=100
Bias^2: n=50
2 4 6 8
0
5
10
15
20
Variance: n=50V1V2V3
Mixture Poisson Data
Observation Time
0.0
0.2
0.4
0.6
0.8
1.0
1.2
2 4 6 8
Bias^2: n=100 Variance: n=100
Bias^2: n=50
2 4 6 8
0.0
0.2
0.4
0.6
0.8
1.0
1.2Variance: n=50
V1V2V3
Figure 1. Simulation results for estimations of the baseline mean function, Λ0 (t) = 2t1/232
Poisson Data
Observation Time
0.00
0.05
0.10
0.15
0.20
0.25
2 4 6 8
Bias^2: n=100 Variance: n=100
Bias^2: n=50
2 4 6 8
0.00
0.05
0.10
0.15
0.20
0.25
Variance: n=50
V1V2V3
Negative Binomial Count Data
Observation Time
0.0
0.1
0.2
0.3
2 4 6 8
Bias^2: n=100 Variance: n=100
Bias^2: n=50
2 4 6 8
0.0
0.1
0.2
0.3
Variance: n=50
V1V2V3
Figure 2. Simulation results for estimations of the baseline mean function, Λ0 (t) = 2t1/233
Table 3The spline-based sieve semiparametric inference for bladder tumor data