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over [0,1]. Consequently, the estimator

2

X

cnc

derived from the Bayesian approach and the Bayes risk

approach und er the ab ove men tion ed crite ria pr ovid es th e

same result at 1c

.

With the idea of

ˆ2

c

pXcnc , the extension

leads to 12

ˆˆˆ

ccc

pp

, the adjusted risk difference esti-

mator between two independent binomial proportions,

for estimating a common risk difference

where

11111

ˆ2

c

pXcnc and

22222

ˆ2

c

pXcnc

are proportion estimators for treatment and control arms.

In a multi-center study of size k, the parameter of in-

*Corresponding author.

C. VIWATWONGKASEM ET AL.

Copyright © 2012 SciRes. OJS

49

terest is also a common risk difference

that is as-

sumed to be a constant across centers. We concern about

a combination of several adjusted risk difference estima-

tors 12

ˆˆˆ

cjc jcj

pp

from the th

j center

1, 2,,jk

into the adjusted summary estimator of risk difference of

the form 1

ˆˆ

k

cwcj cj

jf

where cj

f

are the weights

subject to the condition that 11

k

cj

jf

. In this study,

we would propose the optimal weights cj

f

as an alter-

native choice based on minimizing the MSE of ˆcw

in

Section 2, then state the well-known candidates such as

the Cochran-Mantel-Haenszel (CMH) weights and the

inverse variance (INV) weights in Section 3. A simula-

tion plan for comparing the performance among weights

in terms of estimation and hypothesis testing is presented

in Section 4. The results of the comparison among the

potential estimators based on bias, variance, and MSE

and also the evaluations among tests related the men-

tioned weights through the type I error probability and

the power criteria lie on Section 5. Some numerical ex-

amples are applied in Section 6. Finally, conclusion and

discussion are presented in Section 7.

2. Deriving Minimum MSE Weights of

Adjusted Summary Estimator

Under the assumption of a constant of common risk dif-

ference

across k centers, we combine several ad-

justed risk difference estimators 12

ˆˆˆ

cjc jcj

pp

in

which

11111

ˆ2

cj jj

pXcnc and

22222

ˆ2

cj jj

pXcnc from the th

j center

1, 2,,jk arrive at an adjusted summary estimator

of risk difference of the form 1

ˆˆ

k

cwcj cj

jf

where

cj

f

are non-random weights subject to the constraint

that 11

k

cj

jf

. Please observe that for a single center

1k the adju sted summary estimator 1

ˆˆ

k

cwcj cj

jf

subject to 11

k

cj

jf

is a shrinkage estimator of a

simple adjusted estimator 12

ˆˆˆ

ccc

pp

. Our minimum

MSE weights cj

f

of the adjusted summary estimator

ˆcw

were derived by following Lagrange’s method [8]

under the assumption of a constant of common risk dif-

ference over all centers with the pooling point estimator

to estimate

. Lui and Chang [9] proposed the optimal

weights proportional to the reciprocal of the variance

with the Mantel-Haenszel point estimator under the as-

sumption of noncompliance. It was observ ed that both of

optimal weights provided the different formulae because

of different assumptions even though they were derived

from the same method of Lagrange. Now, we wish to

present the proposed weights minimizing the MSE of

ˆcw

as follows:

2

2

1

ˆˆ ˆ

k

cwcwcjc j

j

QMSEEE f

To obtain the minimum Q subject to a constraint

11

k

jcj

f

, we form the auxiliary function

to seek

cj

f

that minimize

2

11

ˆ1

kk

cj cjcj

jj

Ef f

where

is a Lagrange multiplier. The weights cj

f

and

are derived by solving the following equations

simultaneously: 0

, 0

cj

f

, 1,2,,jk. The

details are presented in Appendix. The result of the

weighted estimate for the th

j center yields

1

1

11

1

1

ˆ

ˆˆ

1

ˆˆ

ˆ

ˆˆ

ˆˆ

1

ˆ

ˆˆˆ

ˆˆ

jj pool

cj

k

mmmm pool

jj

k

mmmm

V

fa

VE

V

a

aEV

where 1112 22

112 2

ˆˆ

ˆ22

jcj jcj

jjj

npc npc

Enc nc

111222

22

112 2

ˆˆˆˆ

11

ˆ22

j

cj cjjcj cj

j

jj

npp npp

Vnc nc

, ˆ

ˆ

ˆˆ

jj

aE b

1

11

1ˆ

ˆˆ

kk

j

jj

j

aV

V

, 1

ˆ

ˆˆ

k

j

j

j

E

bV

, 12

ˆˆˆ

pool pp

11 1

11

1

11

11

ˆ

ˆ

kk

j

jj

jj

kk

j

j

jj

np X

pnn

,

22 2

11

2

22

11

ˆ

ˆ

kk

j

jj

jj

kk

j

j

jj

np X

pnn

In the particular case of 12

0cc, our estimator

1

ˆˆ

k

cwjcjcj

f

has a shrinkage estimator to be the

popular inverse-variance weighted estimator. Under a

common risk difference

over all centers, the variance

of ˆcw

in the case of non-random weights cj

f

are ob-

tained by

2

1

1112 22

222

1112 2

ˆˆ

11

22

k

cwcjcj

j

kjjjj jj

cj

jjj

VfV

npp npp

fncnc

Suppose that a normal approximation is reliable, the

asymptotic distribution is

1

2

1

ˆˆ

ˆ(0,1)

ˆ

ˆˆ

ˆˆ

k

cj cj

j

cw

k

cwcj cj

j

fN

VfV

C. VIWATWONGKASEM ET AL.

Copyright © 2012 SciRes. OJS

50

for testing 00

:H

we have the norm al approximate test

0

1

20

1

ˆˆ

ˆˆ

ˆ

k

cj cj

j

cw k

cjcj

j

f

Z

f

VH

We will reject 0

H

at

level for two-sided test if

2cw

Z

Z

where 2

Z

is the upper

100 2th

percentile of the standard normal distribution. Alterna-

tively, 0

H

is rejected when the p-value (p) is less than

or equal to

p

where

21 cw

pZ

and

Z

is the standard cumulative normal distribu-

tion function.

3. Other Well-Known Weights

3.1. Cochran-Mantel-Haenszel (CMH) Weights

Cochran [10,11] proposed a weighted estimator of cen-

ter-specific sample sizes for a common risk difference

based on the unconditional binomial likelihood as

1

1

ˆ

ˆk

j

j

j

CMH

j

k

j

w

w

where

1

121 2

12

11

j

jj jj

jj

wnnnn

nn

and

12

12 12

ˆˆˆ

j

j

jjj

j

j

X

X

pp nn

. Cochran’s weight

j

w is

widely used as a standard non-random weight derived by

the harmonic means of the center-specific sample sizes.

Note that 1

k

j

jj

j

fw w

is also Cochran’s weight

subject to the condition that 11

k

j

jf

. A straightfor-

ward derivation illustrates that ˆCMH

is an unbiased

estimate of

and the variance of ˆCMH

is readily

available as

2

1

2

1

ˆ

ˆ

k

j

j

j

CMH k

j

j

wV

Vw

where

1112 22

ˆ11

j

jjjjjj

Vppnppn

. As-

suming that a normal approximation is reliable, the

Cochran’s Z-statistic for testing 00

:H

is provided

as

0

2

20

11

11

ˆ

ˆ

ˆ

jj j

CMH

jj j

kk

jj

kk

jj

ww

Z

wV Hw

where

01 11222

ˆ

ˆˆˆˆˆ

11

j

jjjjjj

VHp pnppn

.

The rejection rule of 0

H

follows the same as the previ-

ous standard normal test.

Alternatively, Mantel and Haenszel [12] suggested the

test based on the conditional hypergeometric likelihood

for a common odds ratio among the set of k tables un-

der the null hypothesis of 0:1HOR

0

. With

the null criterion, Mantel-Haenszel’s weight stated by

Sanchez-Meca and Marin-Martine [13] was equivalent to

1212 1

jjjjj

wnn nn

. Since the minor difference

between the conditional Mantel-Haenszel weight and the

unconditional Cochran weight is in the denominators,

thus the two are often referred to interchangeably as the

Cochran-Mantel-Haenszel weight. In this study, we use

121 2

j

jj j j

wnnnn.

3.2. Inverse Variance (INV) or Weighted Least

Square (WLS) Weights

Fleiss [14] and Lipsitz et al. [15] showed that the in-

verse-variance weighted (INV) estimator or the weighted-

least-square (WLS) estimator for

was in the summa ry

estimator of the weighted mean (linear, unbiased estima-

tor) of th e form

11

ˆˆ

kk

I

NVj jj

jj

ww

where 1211 22

ˆˆˆ

j

jj jj jj

pp XnXn

and

j

w de-

fined by the reciprocal of the variance as

1

1122

12

11

1ˆ

jjj j

jjj

j

pppp

wnn

V

The non-random and non-negative weights

j

w yield

the minimum variance of the summary estimator ˆ

I

NV

for estimating

. The variance of ˆ

I

NV

is just given by

22

22

11

1

11

ˆ11

ˆjj jj

INV

j

jj

kk

jj

k

kk

j

jj

wV ww

Vw

ww

However, the weights

j

w cannot be used in practice

since 1

j

p and 2

j

p are unknown. Therefore, it has be-

come common practice to replace them by their sample

estimators. It yields

1122

1

12

ˆˆ

ˆˆ

11

ˆ

j

jj j

jjj

pppp

wnn

This weight was suggested in several textbooks of

epidemiology such as Kleinbaum et al. [16] or in text-

books of meta-analysis such as Petitt [17]. We assume

that a normal approximation is reliable; the inverse-variance

weighted test statistic for testing 00

:H

is

0

11

1

ˆ

1

jj

j

INV

j

kk

jj

k

j

ww

Zw

where

10

ˆ

ˆ

ˆjj

wV H

. Also, the rule of 0

H

rejection

follows the same as the above standard normal test.

C. VIWATWONGKASEM ET AL.

Copyright © 2012 SciRes. OJS

51

4. Monte Carlo Simulation

We perform simulations for estimating a common risk

difference

and testing the null hypothesis 00

:H

in the similar plans as follows:

Parameters Setting: Let the common risk difference

be some constants varying from 0 to 0.6, with incre-

mental steps of 0.1. Baseline proportion risks 2

j

p

21 222

,,,

k

pp p in the control arm for the th

j center

1, 2,,jk are generated from a uniform distribution

over

0, 0.95

. The correspondent proportion risks

1

j

p for the treatment arm in the th

j cen ter ar e ob ta in ed

as 12jj

pp

. For example, if 0.2

, then

2~0,0.75

j

pU and

12 ~ 0.2,0.95

jj

pp U

. The

sample sizes 1

j

n and 2

j

n are varied as 4, 8, 16, 32,

100. The number of centers k takes values 1, 2, 4, 8, 16,

32.

Statistics: Binomial random variables 1

j

X

and 2

j

X

in treatment and control arms are generated with pa-

rameters

11

,

j

j

np and

22

,

j

j

np for each center j.

Estimation: All summary estimates of

are com-

puted in a variety of different weights. The procedure is

replicated 5000 times. From these replicates, bias, vari-

ance, and MSE (mean square error) are computed in the

conventional way.

Type I Error: From the above parameter setting, we

assign 0

under a null 00

:H

, so all tests are

computed. The replication is treated 5000 times. From

these replicates, the number of the null hypothesis reject-

tions is counted for the empirical type I error

.

00

Number of rejections of when is true

Number of replications (5000 times)

HH

The evaluation for two-sided tests in terms of the type

I probability is based on Cochran limits [18] as follow.

At 0.01

, the

value is between

0.005, 0.015.

At 0.05

, the

value is between

0.04, 0.06.

At 0.10

, the

value is between

0.08, 0.12.

If the empirical type I error ˆ

lies within those of

Cochran limits, then the statistical test can control type I

error.

Power of Tests: Before evaluating tests with their

powers, all comparative tests should be calibrated to have

the same type I error rate under 0

H

; then any test whos e

power hits the maximum under 1

H

would be the best

test. To achieve the alternative hypothesis, we assume

the random effect model for

j

as

0.10.12 1

jm

UmU

where m

U as an effect of between centers is assigned to

be uniform

,mm for a given

0, 0.1m, or

equivalently, U is an uniform variable over

0,1.

That is,

0.1

j

E

and

2

212

j

Var m

. Also,

we have 12

j

jj

pp

where 2

j

p be uniform distri-

bution over

0.1, 0.8. Binomial random variables 1

j

X

and 2

j

X

are drawn with parameters

11

,

j

j

np, and

22

,

j

j

np, respectively. All proposed test statistics are

then computed. The procedure is replicated 5,000 times.

From these replicates, the empirical power

1

of test

is counted.

01

Number of rejections of when is true

1

Number of replications 5000 times

H

H

5. Results

Since it is difficult to present all enormous results from

the simulation study, we just have illustrated some in-

stances. Nevertheless, the main results are concluded

perfectly.

5.1. Results for Estimating Risk Differences

Table 1 presents some results according to point estima-

tion of a common risk difference

. However, we can

draw conclusions in the following.

The number of centers, k, can not change the order

of the MSE of all weighted estimators, even though

an increase in k can decrease the variance and the

MSE of all estimators, leading to the increasing effi-

ciency. Also, increasing 1

j

n and 2

j

n can decrease

the variance of all estimators while fixing k. The

unbalanced cases of 1

j

n and 2

j

n for center j have

a rare effect on the order of the MSE of all estim ates.

For most popular situations used under 0

,

0.1

, 0.2

, and 0.3

, the proposed sum-

mary estimator

cw

adjusted by 12

1ccc in-

cluding adjusted by 2c

is the best choice with the

smallest MSE. The estimator ˆcw

adjusted by

0.5c

and the inverse-variance (INV) weighted es-

timator

0c

are close tog ether and ar e the second

choice with smaller MSE. The Cochran-Mantel-

Haenszel (CMH) weight performs the worst in this

simulation setting. This finding is very useful in gen-

eral situations of most clinical trials and most causal

relations between a disease and a suspected risk factor

since the risk difference is often less than 0.25 [19].

For 0.4

, the proposed estimator ˆcw

adjusted

by 1c

performs best; for 0.5

, the proposed

estimator ˆcw

adjusted by 0.5c performs best;

for 0.6

, the INV weighted estimator (0c

)

performs best.

5.2. Results for Studying Type I Error

Table 2 presents some results for controlling the empiri-

cal type I error. We can conclude the performance of

several tests according to the empirical alpha under 0

H

as follows.

C. VIWATWONGKASEM ET AL.

Copyright © 2012 SciRes. OJS

52

Table 1. Mean , variance, M SE for estimating θ.

k 1j

n 2j

n Measure CMH INV

(0c

) 0.5c 1c 2c

0.0 1 2 2 Mean: –0.001700 –0.000850 –0.001130 –0.000850 –0.000570

Var: 0.171245 0.042811 0.076109 0.042811 0.019027

MSE: 0.171250 0.042813 0.076112 0.042813 0.019028

0.0 1 4 4 Mean: –0.000800 0.000400 –0.000640 –0.000530 –0.000400

Var: 0.088874 0.053058 0.056879 0.039499 0.022219

MSE: 0.088875 0.053058 0.056880 0.039500 0.022219

0.0 1 8 8 Mean: 0.002625 0.001965 0.002333 0.002100 0.001750

Var: 0.042575 0.035480 0.033641 0.027249 0.018923

MSE: 0.042584 0.035483 0.033647 0.027254 0.018926

0.0 1 16 16 Mean: –0.000050 0.000328 –0.000047 –0.000044 –0.000040

Var: 0.021759 0.020761 0.019275 0.017193 0.013926

MSE: 0.021759 0.020761 0.019275 0.017193 0.013926

0.0 1 32 32 Mean: –0.001900 –0.001950 –0.001840 –0.001790 -0.001690

Var: 0.010805 0.010674 0.010160 0.009572 0.008538

MSE: 0.010809 0.010678 0.010164 0.009575 0.008540

0.0 1 100 100 Mean: 0.000566 0.000572 0.000560 0.000555 0.000544

Var: 0.003482 0.003478 0.003413 0.003346 0.003219

MSE: 0.003482 0.003478 0.003413 0.003347 0.003219

0.1 16 2 2 Mean: 0.102200 0.051100 0.068133 0.051100 0.034067

Var: 0.178755 0.044689 0.079446 0.044689 0.019861

MSE: 0.178759 0.047080 0.080462 0.047080 0.024210

0.1 16 4 4 Mean: 0.101900 0.071067 0.081520 0.067933 0.050950

Var: 0.093292 0.056358 0.059708 0.041462 0.023323

MSE: 0.093295 0.057194 0.060047 0.042490 0.025729

0.1 16 4 8 Mean: 0.091175 0.073915 0.078964 0.069820 0.056883

Var: 0.068527 0.048536 0.047903 0.036184 0.023445

MSE: 0.068605 0.049217 0.048345 0.037095 0.025305

0.1 16 4 16 Mean: 0.096425 0.086770 0.087330 0.080322 0.069865

Var: 0.057752 0.041273 0.040889 0.032469 0.024164

MSE: 0.057764 0.041448 0.041048 0.032856 0.025072

0.1 16 4 32 Mean: 0.103087 0.094537 0.095306 0.089488 0.080958

Var: 0.052651 0.037007 0.037127 0.030458 0.025400

MSE: 0.052662 0.037037 0.037149 0.030568 0.025763

0.1 16 8 8 Mean: 0.105625 0.091604 0.093890 0.084500 0.070417

Var: 0.047621 0.041375 0.037626 0.030478 0.021165

MSE: 0.047653 0.041446 0.037664 0.030718 0.022040

0.1 16 8 16 Mean: 0.100700 0.094838 0.093524 0.087382 0.077367

Var: 0.035620 0.031899 0.029404 0.024987 0.019128

MSE: 0.035620 0.031926 0.029445 0.025147 0.019641

0.1 16 8 32 Mean: 0.097381 0.093334 0.092488 0.088258 0.081217

Var: 0.028539 0.025407 0.023764 0.020808 0.017542

MSE: 0.028546 0.025452 0.023820 0.020945 0.017895

0.1 16 16 16 Mean: 0.099100 0.094834 0.093271 0.088089 0.079280

Var: 0.023792 0.023050 0.021075 0.018798 0.015227

MSE: 0.023793 0.023077 0.021120 0.018941 0.015656

0.1 16 32 32 Mean: 0.100794 0.099611 0.097741 0.094866 0.089594

Var: 0.011022 0.010951 0.010364 0.009764 0.008709

MSE: 0.011023 0.010951 0.010369 0.009790 0.008817

0.1 16 100 100 Mean: 0.100052 0.099934 0.099061 0.098092 0.096204

Var: 0.003728 0.003725 0.003654 0.003583 0.003446

MSE: 0.003728 0.003725 0.003655 0.003587 0.003461

C. VIWATWONGKASEM ET AL.

Copyright © 2012 SciRes. OJS

53

Table 2. Empirical type I error for testing H0: θ = θ0 at 5% significance level.

0

k 1j

n 2j

n CMH INV (0c

) 0.5c

1c 2c

0.0 1 4 4 3.42 3.42 3.42 3.42 3.42

4 8 2.08 2.08 6.84 4.76 4.76

4 16 3.00 3.00 6.52 5.80 8.18

4 32 2.76 2.76 6.66 6.18 10.50

4 100 2.54 2.54 7.30 6.46 14.40

8 8 3.28 3.28 6.76 4.16 4.16

8 16 4.26 4.26 6.54 4.74 4.30

8 32 4.34 4.34 5.58 4.22 5.10

8 100 5.02 5.02 6.58 6.00 8.90

16 16 4.74 4.74 4.48 4.48 3.38

16 32 4.50 4.50 4.94 4.44 3.90

16 100 5.02 5.02 5.30 4.58 5.10

32 32 5.04 5.04 4.66 4.34 3.88

32 100 5.22 5.22 5.16 4.46 4.34

100 100 4.74 4.74 4.60 4.40 4.14

0.0 4 4 4 3.68 3.68 3.68 3.68 3.68

8 8 3.40 3.40 7.14 4.56 4.56

16 16 4.84 4.84 4.66 4.66 3.54

16 32 4.52 4.52 5.00 4.52 4.10

16 100 5.46 5.46 5.66 4.72 5.26

32 32 4.74 4.74 4.42 4.18 3.92

32 100 5.34 5.34 5.48 4.74 4.46

100 100 5.04 5.04 4.98 4.86 4.64

0.1 4 4 4 1.26 1.26 8.28 8.28 6.22

8 8 4.24 4.24 7.6 4.66 4.66

16 16 5.18 5.18 5.76 5.04 4.06

16 32 5.66 5.66 5.82 5.40 5.30

16 100 5.86 5.86 6.20 4.84 4.88

32 32 5.72 5.72 5.64 4.96 4.44

32 100 5.88 5.88 5.44 5.20 4.82

100 100 5.22 5.22 5.16 5.10 4.82

0.2 4 4 4 1.74 1.74 4.36 4.36 8.00

8 8 4.66 4.66 8.58 5.38 5.38

16 16 7.54 7.54 6.32 6.28 6.58

16 32 7.26 7.26 6.22 5.56 5.60

16 100 6.24 6.24 6.18 5.40 5.88

32 32 5.46 5.46 5.40 5.46 5.08

32 100 5.56 5.56 5.26 5.22 4.88

100 100 5.34 5.34 5.16 5.10 5.22

0.4 4 4 4 3.00 3.00 12.06 7.44 18.04

8 8 8.00 8.00 6.82 9.18 12.04

16 16 5.78 5.78 5.92 5.16 7.04

16 32 6.82 6.82 6.56 6.16 7.56

16 100 6.38 6.38 6.18 5.80 7.06

32 32 5.96 5.96 5.78 5.94 6.28

32 100 5.92 5.92 5.80 6.04 6.72

100 100 5.68 5.68 5.34 5.14 5.48

Bold values denote that the statistical tests can co ntrol the type I error.

C. VIWATWONGKASEM ET AL.

Copyright © 2012 SciRes. OJS

54

The increasing k cannot change the order of the

empirical type I error rates of all tests. Also, the un-

balanced cases of 1j

n and 2j

n for center j have a

slight effect on the order of the empirical type I error

rates of all tests.

None of tests can control type I error rates when sam-

ple size of treatment or control arm is very small

(14

j

n or 24

j

n). There exists few tests that can

control type I error when sample size is small (18

j

n

or 28

j

n).

For 0

, almost all tests can control type I error

rates when the sample size is moderate to large

(116

j

n or 216

j

n). This finding frequently oc-

curs in practical use of 0:0H

.

For 0.2

, 0.4

, and 0.6

, almost all tests

can control type I error rates when the sample size is

large to very large (132

j

n or 232

j

n).

5.3. Results for Studying Power of Tests

Table 3 shows some more details of the powers. Fortu-

nately, almost all tests under 0:0H

can control type

I error rates when the sample size is moderate to large

(116

j

n or 216

j

n). We ignore to consider the com-

parative tests when sample size is very small (14

j

n

or

24

j

n) since all of tests can not control type I error

rates. The performance of several weighted tests accord-

ing to the powers under 1:0.1

j

m

H

U

can be con-

cluded in the following:

The empirical powers yield a similar pattern of results

like the MSE. An increase in the number of centers,

k, can increase the power but it can not change the

order of power.

Overall, the proposed weights adjusted by 1c

in-

cluding 2c perform best with the highest power

in a multi-center study of size 2k when 116

j

n

or 216

j

n.

The INV weight and the CMH weight are achieved

with the highest powers in one center study when

116

j

n or 216

j

n.

When the sample size is large to very large (132

j

n

or 232

j

n), all weights perform well.

6. Numerical Examples

Two examples are presented to illustrate the implementa-

tion of the related methodology. Pocock [20] presented

data from a randomized trial studying the effect of pla-

cebo and metoprolol on mortality after heart attack (AMI:

Acute Myocardial Infarction) classified by three strata of

age groups, namely, 40 - 64, 65 - 69, 70 - 74 years. Ta-

ble 4 shows the data and weights corresponding to the

CMH, the INV, and the proposed strategies. The esti-

mated summary differences based on the CMH, the INV,

and the proposed weights are 0.031, 0.024, 0.030, re-

spectively. Also, the estimated standard errors of those of

overall differences are 0.014, 0.013, 0.014, respectively.

Since both of 2.237

CMH

Z

and 2.197

cw

Z are

greater than 21.96Z

, the CMH and the proposed

tests at 1c

reject the null hypothesis at 5% level for

two-sided test and lead to the conclusion of a significant

difference between the placebo and metoprolol mortality

rates whereas the INV test with 1.823

INV

Z fails to

reject the null hypothesis at 5% level.

Turner et al. [21] presented data from clinical trials to

study the effect of selective decontamination of the di-

gestive tract on the risk of respiratory tract infection of

patients in intensive care units. See data and weights in

Table 5. The estimated overall differences and their es-

timated standard errors are 0.152 (0.012), 0.140 (0.011),

0.162 (0.012) for the CMH, the INV, and the proposed

weights at 1c

, respectively. All tests reject the null

hypothesis with 12.584

CMH

Z

, 12.215

INV

Z,

13.719

cw

Z

and lead to the conclusion of a significant

difference between treatment effect of selective decon-

tamination of the digestive tract on the risk of respiratory

tract infection.

7. Conclusions and Discussion

In most general situations used by the risk difference

lying on [0, 0.25], the results have confirmed that the

minimum MSE weight of the proposed summary esti-

mator

cw

adjusted by 12

1cc c

(including

12

2cc c

) is the best choice with the smallest MSE

under a constant of common risk difference

over all

k centers. The number of centers, k, cannot change

the order of the MSE of all weighted estimators, even

though an increase in k can decrease the variance and

the MSE of all weighted estimators. Also, increasing 1

j

n

and 2

j

n can decrease the variance of all estimators

while fixing k. The unbalanced cases of 1

j

n and 2

j

n

for center j have a slight effect on the order of the

MSE of all estimates. The minimum MSE weight is de-

signed to yield more precise estimate relative to the

CMH and INV weights. Another benefit of the proposed

weight is easy to compute because of its closed-form

formula. With the basis of smallest MSE and the

easy-to-compute formula, we have been solidly sug-

gested to use the proposed weight. In addition, the vari-

ous choices for c have been considered again. The use

of 0.5c

as a conventional correction term [22] should

be revised. The better value of c in adding on the

number of successes and the number of failures is sug-

gested with at least for 1c (including 2c

). This

result is supported by the ideas of Böhning and Viwat-

wongkasem [6], Agresti and Coull [1], and Agresti and

Caffol [2] that recommended to use the appropriate val-

ues of c greater than or equal to 1.

C. VIWATWONGKASEM ET AL.

Copyright © 2012 SciRes. OJS

55

Table 3. Empirical power (percent) at m = 0.04 after controlling the estimated type I error at the nominal 5% level.

X = Controllable Type I error rates Empirical power rates

k 1j

n 2j

n CMH INV 0.5c1c

2c

CMH INV 0.5c 1c 2c

1 8 8 X X

6.8 6.8

8 16 X X X X 7.3 7.3 8.3 7.4

8 32 X X X X X 9.5 9.5

11.5 9.6 9.9

8 100 X X X 11.3 11.3

11.8

16 16 X X X X 11.2 11.2 10.6 10.6

16 32 X X X X 12.2 12.2

12.7 11.8

16 100 X X X X X

16.4 16.4 15.4 14.8 14.6

32 32 X X X X 17.6 17.6 16.5 16.4

32 100 X X X X X

21.4 21.4 21.2 20.8 20.3

100 100 X X X X X

36.8 36.8 36.8 36.5 36.1