Open Journal of Statistics, 2012, 2, 48-59 doi:10.4236/ojs.2012.21006 Published Online January 2012 (http://www.SciRP.org/journal/ojs) Minimum MSE Weights of Adjusted Summary Estimator of Risk Difference in Multi-Center Studies Chukiat Viwatwongkasem1*, Jirawan Jitthavech2, Dankmar Böhning3, Vichit Lorchirachoonkul2
1Department of Biostatistics, Faculty of Public Health, Mahidol University, Bangkok, Thailand
2School of Applied Statistics, National Institute of Development Administration, Bangkok, Thailand
3Applied Statistics, School of Biological Sciences, University of Reading, Reading, UK
Received October 14,2011; received November 18, 2011; accepted November 30, 2011ABSTRACT
The simple adjusted estimator of risk difference in each center is easy constructed by adding a value c on the number of successes and on the number of failures in each arm of the proportion estimator. Assessing a treatment effect in multi-center studies, we propose minimum MSE (mean square error) weights of an adjusted summary estimate of risk difference under the assumption of a constant of common risk difference over all centers. To evaluate the performance of the proposed weights, we compare not only in terms of estimation based on bias, variance, and MSE with two other conventional weights, such as the Cochran-Mantel-Haenszel weights and the inverse variance (weighted least square) weights, but also we compare the potential tests based on the type I error probability and the power of test in a variety of situations. The results illustrate that the proposed weights in terms of point estimation and hypothesis testing perform well and should be recommended to use as an alternative choice. Finally, two applications are illustrated for the practi- cal use. Keywords: Minimum MSE Weights; Optimal Weights; Cochran-Mantel-Haenszel Weights; Inverse Variance Weights; 1. Introduction V p n p 1 p n 2c . They concluded that the
1 n 2 minimizes the Bayes risk (the
It is widely known that the conventional proportion esti-
average MSE of ˆp ) in the class of all estimators of the
mator, ˆp X n , is a maximum likelihood estimator
form X c n 2c with respect to uniform prior on
(MLE) and an uniformly minimum variance unbiased
[0,1] and Euclidean loss function; furthermore, the esti-
estimator (UMVUE) for the binomial parameter p
1 n 2 has smaller MSE than X n in
where the binomial random variable X is the number
the approximate interval 0.15, 0.85 of p . For another
of successes out of the number of patients n . However,
argumentation in the Bayesian approach, Casella and
Agresti and Coull [1], Agresti and Caffo [2], Ghosh [3],
Berger [7] showed that X n is a Bayes
and Newcombe [4,5] highlighted the point that ˆp might
estimator of p under the conditional binomial sampling
not be a good choice for p when the assumption of
n p and the prior beta distribution
np 5 and n 1 ˆp 5 was violated; this violation
p ~ beta , . Note that in case of 1 the beta
often occurs when the sample size n is small, or the
distribution has a special case as the uniform distribution
estimated probability ˆp is close to 0 or 1 (close to the
over [0,1]. Consequently, the estimator X c n 2c
boundaries of parameter space), leading to the problem
derived from the Bayesian approach and the Bayes risk
of the zero estimate of the variance of ˆp . The estimated
approach under the above mentioned criteria provides the
variance of ˆp , provided by V ( ˆp) ˆp 1 ˆp n , is zero
in the occurrence of any case: X 0 or X n . Böh-
With the idea of ˆp X c n 2c , the extension
ning and Viwatwongkasem [6] proposed the simple ad-
ˆp ˆp , the adjusted risk difference esti-
justed proportion estimator by adding a value c on the
mator between two independent binomial proportions,
number of successes and the number of failures; cones-
for estimating a common risk difference where
quently, ˆp X c n 2c is their proposed esti-
mate of p with the non-zero variance estimate
are proportion estimators for treatment and control arms.
In a multi-center study of size k , the parameter of in-
Copyright 2012 SciRes. OJS
terest is also a common risk difference that is as-
To obtain the minimum Q subject to a constraint
sumed to be a constant across centers. We concern about
f 1, we form the auxiliary function to seek
a combination of several adjusted risk difference estima-
ˆp ˆp from the thj center j 1, 2,, k
into the adjusted summary estimator of risk difference of
E f f 1
f where f are the weights
where is a Lagrange multiplier. The weights f
we would propose the optimal weights f as an alter-
and are derived by solving the following equations
native choice based on minimizing the MSE of ˆ
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-
details are presented in Appendix. The result of the
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
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 ˆ
j 1, 2,, k arrive at an adjusted summary estimator
f are non-random weights subject to the constraint
f 1. Please observe that for a single center ˆp2
In the particular case of c c 0 , our estimator
f 1 is a shrinkage estimator of a
ˆp ˆp . Our minimum
popular inverse-variance weighted estimator. Under a
MSE weights f of the adjusted summary estimator
common risk difference over all centers, the variance
were derived by following Lagrange’s method [8]
in the case of non-random weights f are ob-
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 observed 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
Suppose that a normal approximation is reliable, the
present the proposed weights minimizing the MSE of
E E f
Copyright 2012 SciRes. OJS
for testing H : we have the normal approximate test
der the null hypothesis of H : OR 1 0 . With
the null criterion, Mantel-Haenszel’s weight stated by
Sanchez-Meca and Marin-Martine [13] was equivalent to
n n 1 . Since the minor difference
between the conditional Mantel-Haenszel weight and the
We will reject H at level for two-sided test if
unconditional Cochran weight is in the denominators,
thus the two are often referred to interchangeably as the
percentile of the standard normal distribution. Alterna-
Cochran-Mantel-Haenszel weight. In this study, we use
tively, H is rejected when the p-value ( p ) is less than
or equal to p where p 2 1
and Z is the standard cumulative normal distribu-
3.2. Inverse Variance (INV) or Weighted Least Square (WLS) Weights 3. Other Well-Known Weights
Fleiss [14] and Lipsitz et al. [15] showed that the in- verse-variance weighted (INV) estimator or the weighted-
3.1. Cochran-Mantel-Haenszel (CMH) Weights
least-square (WLS) estimator for was in the summary
Cochran [10,11] proposed a weighted estimator of cen-
estimator of the weighted mean (linear, unbiased estima-
ter-specific sample sizes for a common risk difference
based on the unconditional binomial likelihood as
ˆp ˆp X n X
defined by the reciprocal of the variance as
The non-random and non-negative weights w yield
widely used as a standard non-random weight derived by
the minimum variance of the summary estimator ˆ
the harmonic means of the center-specific sample sizes.
w is also Cochran’s weight
However, the weights w cannot be used in practice
come common practice to replace them by their sample
p 1 p n p 1 p
suming that a normal approximation is reliable, the
Cochran’s Z-statistic for testing H : is provided
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
j j 1 j2
weighted test statistic for testing H : is
H ˆp 1 ˆp
The rejection rule of H follows the same as the previ-
Alternatively, Mantel and Haenszel [12] suggested the
test based on the conditional hypergeometric likelihood
V H . Also, the rule of H rejection
for a common odds ratio among the set of k tables un-
follows the same as the above standard normal test.
Copyright 2012 SciRes. OJS 4. Monte Carlo Simulation
bution over 0.1, 0.8 . Binomial random variables X
We perform simulations for estimating a common risk
are drawn with parameters n , p
n , p , respectively. All proposed test statistics are
difference and testing the null hypothesis H :
then computed. The procedure is replicated 5,000 times.
Parameters Setting:Let the common risk difference
From these replicates, the empirical power 1 of test
be some constants varying from 0 to 0.6, with incre-
mental steps of 0.1. Baseline proportion risks p
Number of rejections of H when H is true
j 1, 2,, k are generated from a uniform distribution
over 0, 0.95 . The correspondent proportion risks
5. Results
. For example, if 0.2 , then Since it is difficult to present all enormous results from
~ U 0, 0.75 and p p
the simulation study, we just have illustrated some in-
sample sizes n and n are varied as 4, 8, 16, 32,
stances. Nevertheless, the main results are concluded
100. The number of centers k takes values 1, 2, 4, 8, 16,
Statistics:Binomial random variables X and X5.1. Results for Estimating Risk Differences
in treatment and control arms are generated with pa-
Table 1 presents some results according to point estima-
tion of a common risk difference . However, we can
Estimation:All summary estimates of are com-
puted in a variety of different weights. The procedure is
The number of centers, k , can not change the order
replicated 5000 times. From these replicates, bias, vari-
of the MSE of all weighted estimators, even though
ance, and MSE (mean square error) are computed in the
an increase in k can decrease the variance and the
MSE of all estimators, leading to the increasing effi-
Type I Error:From the above parameter setting, we
ciency. Also, increasing n and n can decrease
assign under a null H : , so all tests are
the variance of all estimators while fixing k . The
computed. The replication is treated 5000 times. From
unbalanced cases of n and n for center j have
these replicates, the number of the null hypothesis reject-
a rare effect on the order of the MSE of all estimates.
tions is counted for the empirical type I error .
For most popular situations used under 0 ,
Number of rejections of H when H is true
cluding adjusted by c 2 is the best choice with the
The evaluation for two-sided tests in terms of the type
I probability is based on Cochran limits [18] as follow.
c 0.5 and the inverse-variance (INV) weighted es-
At 0.01 , the value is between 0.005, 0.015 .
timator c 0 are close together and are the second
At 0.05 , the value is between 0.04, 0.06 .
choice with smaller MSE. The Cochran-Mantel-
At 0.10 , the value is between 0.08, 0.12 .
Haenszel (CMH) weight performs the worst in this
simulation setting. This finding is very useful in gen-
Cochran limits, then the statistical test can control type I
eral situations of most clinical trials and most causal
relations between a disease and a suspected risk factor
Power of Tests:Before evaluating tests with their
since the risk difference is often less than 0.25 [19].
powers, all comparative tests should be calibrated to have
For 0.4 , the proposed estimator ˆ
the same type I error rate under H ; then any test whose
by c 1 performs best; for 0.5 , the proposed
power hits the maximum under H would be the best
adjusted by c 0.5 performs best;
test. To achieve the alternative hypothesis, we assume
for 0.6 , the INV weighted estimator ( c 0 )
0.1U 0.1 m2U
5.2. Results for Studying Type I Error
where U as an effect of between centers is assigned to
m m for a given m 0, 0.
Table 2 presents some results for controlling the empiri-
equivalently, U is an uniform variable over 0,
cal type I error. We can conclude the performance of
That is, E 0.1 and Var m
several tests according to the empirical alpha under H
we have p p where p be uniform distri-
Copyright 2012 SciRes. OJS Table 1. Mean, variance, MSE for estimating θ .
Mean: –0.001700 –0.000850 –0.001130 –0.000850 –0.000570
0.171245 0.042811 0.076109 0.042811 0.019027
MSE: 0.171250 0.042813 0.042813 0.019028
Mean: –0.000800 0.000400 –0.000640 –0.000530 –0.000400
0.088874 0.053058 0.056879 0.039499 0.022219
MSE: 0.088875 0.053058 0.056880 0.039500 0.022219
1 8 8 Mean: 0.002625 0.001965 0.002333 0.002100 0.001750
0.042575 0.035480 0.033641 0.027249 0.018923
MSE: 0.042584 0.035483 0.033647 0.027254 0.018926
1 16 16 Mean: –0.000050 0.000328 –0.000047 –0.000044 –0.000040
0.021759 0.020761 0.019275 0.017193 0.013926
MSE: 0.021759 0.020761 0.019275 0.017193 0.013926
Mean: –0.001900 –0.001950 –0.001840 –0.001790 -0.001690
0.010805 0.010674 0.010160 0.009572 0.008538
MSE: 0.010809 0.010678 0.010164 0.009575 0.008540
100 Mean: 0.000566 0.000572 0.000560 0.000555 0.000544
0.003482 0.003478 0.003413 0.003346 0.003219
MSE: 0.003482 0.003478 0.003413 0.003347 0.003219
16 2 2 Mean: 0.102200 0.051100 0.068133 0.051100 0.034067
0.178755 0.044689 0.079446 0.044689 0.019861
MSE: 0.178759 0.047080 0.047080 0.024210
16 4 4 Mean: 0.101900 0.071067 0.081520 0.067933 0.050950
0.093292 0.056358 0.059708 0.041462 0.023323
MSE: 0.093295 0.057194 0.060047 0.042490 0.025729
16 4 8 Mean: 0.091175 0.073915 0.078964 0.069820 0.056883
0.068527 0.048536 0.047903 0.036184 0.023445
MSE: 0.068605 0.049217 0.048345 0.037095 0.025305
16 4 16 Mean: 0.096425 0.086770 0.087330 0.080322 0.069865
0.057752 0.041273 0.040889 0.032469 0.024164
MSE: 0.057764 0.041448 0.041048 0.032856 0.025072
16 4 32 Mean: 0.103087 0.094537 0.095306 0.089488 0.080958
0.052651 0.037007 0.037127 0.030458 0.025400
MSE: 0.052662 0.037037 0.037149 0.030568 0.025763
16 8 8 Mean: 0.105625 0.091604 0.093890 0.084500 0.070417
0.047621 0.041375 0.037626 0.030478 0.021165
MSE: 0.047653 0.041446 0.037664 0.030718 0.022040
16 8 16 Mean: 0.100700 0.094838 0.093524 0.087382 0.077367
0.035620 0.031899 0.029404 0.024987 0.019128
MSE: 0.035620 0.031926 0.029445 0.025147 0.019641
16 8 32 Mean: 0.097381 0.093334 0.092488 0.088258 0.081217
0.028539 0.025407 0.023764 0.020808 0.017542
MSE: 0.028546 0.025452 0.023820 0.020945 0.017895
16 16 Mean: 0.099100 0.094834 0.093271 0.088089 0.079280
0.023792 0.023050 0.021075 0.018798 0.015227
MSE: 0.023793 0.023077 0.021120 0.018941 0.015656
32 32 Mean: 0.100794 0.099611 0.097741 0.094866 0.089594
0.011022 0.010951 0.010364 0.009764 0.008709
MSE: 0.011023 0.010951 0.010369 0.009790 0.008817
100 Mean: 0.100052 0.099934 0.099061 0.098092 0.096204
0.003728 0.003725 0.003654 0.003583 0.003446
MSE: 0.003728 0.003725 0.003655 0.003587 0.003461
Copyright 2012 SciRes. OJS Table 2. Empirical type I error for testing H : θ = θ at 5% significance level. 0 0 4.16 4.16
16 4.26 4.26 6.54 4.74 4.30
32 4.34 4.34 5.58 4.22 5.10
100 5.02 5.02 6.58
16 4.74 4.74 4.48 4.48 3.38
32 4.50 4.50 4.94 4.44 3.90
100 5.02 5.02 5.30 4.58 5.10
32 5.04 5.04 4.66 4.34 3.88
100 5.22 5.22 5.16 4.46 4.34
100 4.74 4.74 4.60 4.40 4.14 4.56 4.56
16 4.84 4.84 4.66 4.66 3.54
32 4.52 4.52 5.00 4.52 4.10
100 5.46 5.46 5.66 4.72 5.26
32 4.74 4.74 4.42 4.18 3.92
100 5.34 5.34 5.48 4.74 4.46
100 5.04 5.04 4.98 4.86 4.64
8 4.24 4.24 7.6 4.66 4.66
16 5.18 5.18 5.76 5.04 4.06
32 5.66 5.66 5.82 5.40 5.30
100 58.6 5.86 62.0 4.84 4.88
32 5.72 5.72 5.64 4.96 4.44
100 5.88 5.88 5.44 5.20 4.82
100 5.22 5.22 5.16 5.10 4.82 4.36 4.36 8.00
8 4.66 4.66 8.58 5.38 5.38 5.56 5.60 5.40 5.88
32 5.46 5.46 5.40 5.46 5.08
100 5.56 5.56 5.26 5.22 48.8
100 5.34 5.34 5.16 5.10 5.22
16 5.78 5.78 5.92 5.16 7.04
32 5.96 5.96 5.78 5.94 6.28
100 5.92 5.92 5.80 6.04 6.72
100 5.68 5.68 5.34 5.14 5.48
Bold values denote that the statistical tests can control the type I error.
Copyright 2012 SciRes. OJS
The increasing k cannot change the order of the
spectively. Also, the estimated standard errors of those of
empirical type I error rates of all tests. Also, the un-
overall differences are 0.014, 0.013, 0.014, respectively.
balanced cases of n and n for center j have a
slight effect on the order of the empirical type I error
tests at c 1 reject the null hypothesis at 5% level for
None of tests can control type I error rates when sam-
two-sided test and lead to the conclusion of a significant
ple size of treatment or control arm is very small
difference between the placebo and metoprolol mortality
4 ). There exists few tests that can
control type I error when sample size is small ( n 8
Turner et al. [21] presented data from clinical trials to
For 0 , almost all tests can control type I error
study the effect of selective decontamination of the di-
rates when the sample size is moderate to large
gestive tract on the risk of respiratory tract infection of
patients in intensive care units. See data and weights in
curs in practical use of H : 0 .
Table 5. The estimated overall differences and their es-
For 0.2 , 0.4 , and 0.6 , almost all tests
timated standard errors are 0.152 (0.012), 0.140 (0.011),
can control type I error rates when the sample size is
0.162 (0.012) for the CMH, the INV, and the proposed
large to very large ( n 32 or n
weights at c 1, respectively. All tests reject the null hypothesis with Z5.3. Results for Studying Power of Tests
13.719 and lead to the conclusion of a significant
Table 3 shows some more details of the powers. Fortu-
difference between treatment effect of selective decon-
nately, almost all tests under H : 0 can control type
tamination of the digestive tract on the risk of respiratory
I error rates when the sample size is moderate to large
parative tests when sample size is very small ( n 4 or
7. Conclusions and Discussion
4 ) since all of tests can not control type I error
In most general situations used by the risk difference
rates. The performance of several weighted tests accord-
lying on [0, 0.25], the results have confirmed that the
ing to the powers under H : 0.1U can be con-
minimum MSE weight of the proposed summary esti-
adjusted by c c c 1 (including
The empirical powers yield a similar pattern of results
c c c 2 ) is the best choice with the smallest MSE
like the MSE. An increase in the number of centers,
under a constant of common risk difference over all
k , can increase the power but it can not change the
k centers. The number of centers, k , cannot change
the order of the MSE of all weighted estimators, even
Overall, the proposed weights adjusted by c 1 in-
though an increase in k can decrease the variance and
cluding c 2 perform best with the highest power
the MSE of all weighted estimators. Also, increasing n
in a multi-center study of size k 2 when n 16
can decrease the variance of all estimators
while fixing k . The unbalanced cases of n and n
The INV weight and the CMH weight are achieved
for center j have a slight effect on the order of the
with the highest powers in one center study when
MSE of all estimates. The minimum MSE weight is de-
signed to yield more precise estimate relative to the
When the sample size is large to very large ( n 32
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
6. Numerical Examples
easy-to-compute formula, we have been solidly sug-
Two examples are presented to illustrate the implementa-
gested to use the proposed weight. In addition, the vari-
tion of the related methodology. Pocock [20] presented
ous choices for c have been considered again. The use
data from a randomized trial studying the effect of pla-
of c 0.5 as a conventional correction term [22] should
cebo and metoprolol on mortality after heart attack (AMI:
be revised. The better value of c in adding on the
Acute Myocardial Infarction) classified by three strata of
number of successes and the number of failures is sug-
age groups, namely, 40 - 64, 65 - 69, 70 - 74 years. Ta-
gested with at least for c 1 (including c 2 ). This
ble 4 shows the data and weights corresponding to the
result is supported by the ideas of Böhning and Viwat-
CMH, the INV, and the proposed strategies. The esti-
wongkasem [6], Agresti and Coull [1], and Agresti and
mated summary differences based on the CMH, the INV,
Caffol [2] that recommended to use the appropriate val-
and the proposed weights are 0.031, 0.024, 0.030, re-
ues of c greater than or equal to 1.
Copyright 2012 SciRes. OJS Table 3. Empirical power (percent) at m = 0.04 after controlling the estimated type I error at the nominal 5% level.
X X X X 11.2 11.2 10.6 10.6 16.4 16.4 15.4 14.8 14.6
X X X X 17.6 17.6 16.5 16.4 21.4 21.4 21.2 20.8 20.3 36.8 36.8 36.8 36.5 36.1 26.9 29.7 29.5 32.8 33.1 35.2 40.6 43.6 46.8 48.9 62.8 64.6 87.2 87.8 53.9 59.0 64.3 68.5 74.5 77.1 76.9 80.4 89.1 90.4 99.1 99.1 68.3 77.5 77.1 82.1 89.2 92.0 93.8 94.8 99.2 99.3 100.0 100.0 100.0 100.0 100.0
X X 81.8 83.2 92.7 95.6 95.1 96.7
94.5 95.0 97.5 99.9 99.9 99.9 99.9 99.9 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Copyright 2012 SciRes. OJS Table 4. Mortality data over three strata of age groups following Pocock. Table 5.Respiratory tract infections following Turner et al.
In terms of type I error estimates, when sample size is
when sample size is moderate to large ( n 16 or
type I error rates. In addition, there exists few tests that
In terms of power, we ignore to evaluate the power
can control type I error rates when sample size is small
when sample size is very small ( n 4 or n
8 ). This result is consonant with the
because all tests can not control type I error rates. The
comments of Lui [23] that none of conventional
results illustrate the same pattern like the MSE. The pro-
tests/weights under sparse data is appropriate. This inap-
posed weights adjusted by c 1 including c 2 per-
propriateness under sparse data can cope with the mini-
form best with the highest power in a multi-center study
mum MSE weights from this finding. The further work
of size k 2 when n 16 or n
to seek some appropriate tests/weights in sparse data
weight and the CMH weight are achieved with the high-
challenges for investigators to develop an innovation or
est powers in one center study when n 16 or
to improve much more reasonable tests/weights. In gen-
16 . When sample size is large to very large
eral results, almost all tests can control type I error rates
Copyright 2012 SciRes. OJS
strongly recommend to use the minimum MSE weight as
Common Chi-Square Test,” Biometrics, Vol. 10, No. 4,
an appropriate choice because of its highest power.
[12] N. Mantel and W. Haenszel, “Statistical Aspects of the
8. Acknowledgements
Analysis of Data from Retrospective Studies of Disease,” Journal of the National Cancer Institute, Vol. 22, 1959,
We would like to thank the editors and the referees for
comments which greatly improved this paper. This study
[13] J. Sanchez-Meca and F. Marin-Martinez, “Testing the
was partially supported for publication by the China
Significance of a Common Risk Difference in Meta-
Medical Board (CMB), Faculty of Public Health, Mahi-
Analysis,” Computational Statistics & Data Analysis, Vol.
[14] J. L. Fleiss, “Statistical Methods for Rates and Propor-
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Copyright 2012 SciRes. OJS Appendix
Under a true common risk difference over all k
centers ( j 1, 2,, k ), the mean square error of
Solving for by taking summation on f , it yields
To obtain the optimal weights ff 1 0 , we form the auxiliary
function by following Lagrange’s method to seek
E f
E f
a E f E 1 b f E
V f
a E f E aE 1
aV f f E a E b a E b
E E cj . The partial de-
rivatives with respect to and f yield
2 f V 2 f E E
aV f f E 1
aV f f E f E f E 1
Substitute each of the subscript j and rearrange
j 1 ; (aV E ) f f E f E f E 1
j 2 ; f E aV E
f E f E 1
j 3 ; f E f E aV E
f . . . f E 1
f E f E f E . . . (aV E ) f
It can be written in the matrix form as H f y
Copyright 2012 SciRes. OJS t f f f f y 1 1 1 1 e E E E
The matrix H can be illustrated as
The inverse of H is suggested in several textbooks
of linear model such as Rencher [24] and Sen and
H D t = D + t e = D
Therefore, f = D y f
In practice, we have to estimate the adjusted summary
estimator by replacing the sample estimates for the un-
known quantities: E , V , p , p , .
Copyright 2012 SciRes. OJS

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