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DOI: 10.3102/1076998610381401
October 2010 published online 27JOURNAL OF EDUCATIONAL AND BEHAVIORAL STATISTICS
Xiaofeng Steven LiuSample Size for Confidence Interval of Covariate-Adjusted Mean Difference
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Sample Size for Confidence Interval of
Covariate-Adjusted Mean Difference
Xiaofeng Steven Liu
University of South Carolina
This article provides a way to determine adequate sample size for the
confidence interval of covariate-adjusted mean difference in randomized
experiments. The standard error of adjusted mean difference depends on covari-
ate variance and balance, which are two unknown quantities at the stage of
planning sample size. If covariate observations are viewed as randomly varying
from one study to another, the covariate variance and balance are related to a t
statistic in the standard error of adjusted mean difference. Using this t statistic
in the standard error, one can express the expected width of the confidence
interval as a function of the sample size. Alternatively, a sample size can be
found to achieve a desired probability of having the width of the confidence
interval smaller than a predetermined upper bound.
Keywords: sample size; confidence interval; experimental design
Adding a covariate can reduce error variance and adjust potential bias for the
estimated treatment effect in analysis of covariance (ANCOVA). Compared to
analysis of variance, ANCOVA removes the variability due to the covariate from
the within-group error variance (Cochran, 1957; Winer, Brown, & Michels,
1991). The smaller within-group error variance produces higher statistical power
for the overall test of any treatment effect. Following the overall test, the adjusted
mean difference, which takes into account the relation between the covariate and
the outcome, yields an unbiased estimate of the treatment difference, provided
that the assumptions of the ANCOVA model are met (i.e., linearity and additivity
of the covariate effect, perfect reliability of the covariate, cf. Elashoff, 1969).
The adjusted mean difference can be used to locate any difference in
average outcome performance among the treatment conditions. The most com-
mon comparison is a pairwise adjusted mean difference, which is routinely
presented in a confidence interval. The confidence interval of the adjusted
mean difference shows an estimated range of values that the true treatment dif-
ference may contain. A narrow width of the confidence interval pins down the
treatment difference to a small range of possible values, which suggests high
precision in the estimate.
Journal of Educational and Behavioral Statistics
Month 2010, Vol. 000, No. 00, pp. 1–12
DOI: 10.3102/1076998610381401
# 2010 AERA. http://jebs.aera.net
1
doi:10.3102/1076998610381401JOURNAL OF EDUCATIONAL AND BEHAVIORAL STATISTICS OnlineFirst, published on October 27, 2010 as
at UNIV OF TENNESSEE on February 28, 2013http://jebs.aera.netDownloaded from
Sample size plays an important role in achieving a narrow width in a
confidence interval. Although the confidence interval of the adjusted mean dif-
ference has been in wide use, there is not a comparable knowledge of planning
such a confidence interval. The extant literature provides ways to determine sam-
ple size for the confidence interval of unadjusted mean difference, but it does not
offer any solution in the case of adjusted mean difference. The difficulty with
adjusted mean difference lies in the fact that its standard error involves the suf-
ficient statistics of the covariate, which are unknown quantities in planning sam-
ple size because the covariate observations are treated as constants in the model.
One will not know the covariate observations or their sufficient statistics until
data are collected upon the completion of a study.
This article provides a way to determine adequate sample size for the confi-
dence interval of adjusted mean difference. The sufficient statistics of the covari-
ate are transformed into a t statistic in the standard error of the adjusted mean
difference. Using the t statistic in the standard error, one can express the expected
width of the confidence interval of the adjusted mean difference as a function of
the sample size. Alternatively, a sample size can be found to achieve a desired
probability of having the width of the confidence interval smaller than a pre-
determined upper bound, given the standard deviation of the outcome and the
correlation between the outcome and covariate.
Standard Error of Adjusted Mean Difference
Suppose the standard model for ANCOVA is
yij ¼ mþ tj þ b xij � �x� �
þ eij; eij � N 0;s2� �
; ð1Þ
where the subscript j represents the jth treatment conditions. The confidence
interval of the adjusted mean difference is
�y0j � �y0j0� �
� t0 s�y0j��y0
j0; ð2Þ
where �y0j and �y0j0 are the adjusted means for the jth and j0th group; t0 is
the 1� a=ð2pÞ percentile score of the t distribution with a degrees of freedom
J (n� 1)� 1; and p is the Bonferroni correction for the p number of simultaneous
confidence intervals. For simplicity of illustration, the p is set to one, but
the results are applicable to multiple comparisons with Bonferroni adjustment
(p > 1). The Bonferroni adjustments are known to be very conservative and are
appropriate for planning purpose. When a less conservative adjustment is
desired, the researcher can use the Holm method as an alternative. Compared
to the Bonferroni procedure, the Holm method uses the same adjustment for
the first test but less stringent adjustments for the subsequent tests (Kirk, 1995,
pp. 142–143).
The standard error of the adjusted mean difference is
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s�y0j��y0
j0¼ s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
nþ
�xj � �xj0� �2
Exx
sð3Þ
(Montgomery, 1997, p. 170), where Exx ¼P
j
Pi
xij � �xj
� �2:
The covariate affects the standard error of the adjusted mean difference in two
opposite ways. The covariate mean difference between two contrasted conditions
(i.e., covariate balance) increases the standard error. The more the two contrasted
conditions differ in covariate mean, the larger the standard error of the adjusted
mean difference becomes. On the contrary, the within-group covariate variance
Exx decreases the standard error of the adjusted mean difference. The ratio
between the covariate balance and the covariate variance determines the overall
effect of the covariate on the standard error.
A covariate is normally treated as fixed or non-stochastic in the general linear
model. The ratio of covariate balance and variance is assumed to be known in the esti-
mation process. This does not pose any problems during data analysis because the
data are presumably obtained at the conclusion of a study. However, a researcher can-
not know the covariate observations beforehand when planning such a study. Thus,
the covariate variance and balance are two unknown quantities at planning stage.
Although the covariate is viewed as fixed or non-stochastic in linear model
analysis, it can be viewed as random for the purpose of planning a study. The
covariate observations can be conceived of as randomly varying from one study
to another, if the same study is replicated again and again. Each replication will
generate a different set of covariate observations. When the covariate is treated as
random at planning stage, it is related to a t statistic in the standard error of the
pairwise adjusted mean difference, that is,
s�y0j��y0
j0¼ s 2=nð Þ1=2
1þ t2�
v� �1=2
; ð4Þ
where t ¼ �xj � �xj0ffiffiffiffiffiffiffiffiffiffiffiffi2n
Exx
v
� �q and v ¼ Jðn� 1Þ:
The numerator in the t is the covariate mean difference between the two con-
trasted treatment conditions, and the denominator in the t is the standard error of the
covariate mean difference. The pooled sum of squares of the covariate Exx is related
to a chi-square distribution with the degrees of freedom equal to v. Assume that
the treatment groups are formed by random assignment and the t ratio follows
an exact t distribution. Alternatively, the standard error can be written as
s�y0j��y0
j0¼ s 2=nð Þ1=2
hðtÞ; ð5Þ
where
hðtÞ ¼ 1þ t2�
v� �1=2
: ð6Þ
Sample Size for Confidence Interval
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The t statistic in effect tests the null hypothesis that the treatment groups have
equal population means on the covariate. Assume that the sample in the rando-
mized experiment has a population mean mx on the covariate. The random assign-
ment does not alter the population mean of the covariate observations among the
treatment groups (i.e., mxj¼ mxj0
¼ mx). The observed covariate mean difference
�xj � �xj0 can differ from zero by random chance, but the treatment groups should
not differ in their covariate attribute to the extent that they can be considered as
representing two different populations with unequal population means of the
covariate. In theory, the null hypothesis about the covariate should not be
rejected because the treatment groups formed by random assignment are sup-
posed to be equivalent prior to treatment intervention. Had the null hypothesis
about the covariate means been rejected, it would be possible that the treatment
effect is confounded with the nonrandom covariate mean difference between the
treatment groups. Thus, it would be prudent to stop the experiment, form the
groups by random assignment again, and then run the ANCOVA (Rubin,
2008). The most tolerable level of covariate imbalance that does not compromise
the ANCOVA analysis will correspond to the 97.5th percentile of the t statistic,
that is, tj j < t:975;v. This implies that at the 5% significance level, the treatment
groups are considered as having equal population means on the covariate. The
width wt.975 corresponding to the two-sided critical value t:975;v is a conservative
estimate of the covariate imbalance for planning purpose.
When planning sample size, the width w can be viewed as a random variable.
One can examine the stochastic property of the width through its expectation E(w)
and its cumulative distribution function P[w < U]. As the cumulative distribution
function may become very complicated, a useful variant can be used to simplify
the computation efforts. One can find the width (i.e., wt.975) corresponding to the
97.5th t percentile while holding s constant and then calculate the probability
P Wt:975 < U½ �. The probability thus computed will provide a conservative estimate
of the required sample size to achieve a narrow width in the confidence interval.
Expected Width of the Confidence Interval
Suppose that the half width of the confidence interval for the adjusted mean
difference is
w ¼ t0s 2=nð Þ1=2hðtÞ: ð7Þ
The estimated standard deviation s contains a chi distribution
s ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw2
v�1
�ðv� 1Þ
q¼ swv�1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=ðv� 1Þ
p:
ð8Þ
The expectation of the chi distribution wv�1 is
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E wv�1ð Þ ¼ffiffiffi2p
�ðv=2Þ� v� 1ð Þ=2ð Þ : ð9Þ
(Johnson, Kotz, & Balakrishnan, 1995a, p. 421).
Furthermore, the estimated standard deviation s is independent of the t statis-
tic (see the appendix). The expectation of the half width is therefore
EðwÞ ¼ t0s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
nðv� 1
sE wv�1ð ÞEðhðtÞÞ: ð10Þ
The expectation EðhðtÞÞ can be computed by expanding the function h(t) in a
Taylor series.
The function h(t) can be accurately approximated by the Taylor series expan-
sion around the mean of the t statistic
hðtÞ ¼ hðmtÞ þ h0ðmtÞ t � mtð Þ þ 1
2!h00ðmtÞ t � mtð Þ2
þ 1
3!h000ðmtÞ t � mtð Þ3þ 1
4!h0000ðmtÞ t � mtð Þ4þRðv; mtÞ:
ð11Þ
The second and fourth derivative of h(t) are
h00ðtÞ ¼ v2 v2 þ vt2� ��3=2
; ð12Þ
h0000ðtÞ ¼ v2 þ vt2� ��7=2
12v4t2 � 3v5� �
; ð13Þ
and the first central moment is mt ¼ 0. Taking the expectation of the Taylor series
yields
E hðtÞð Þ ¼ hðmtÞ þ1
2!h00ðmtÞE t � mtð Þ2þ 1
4!h0000ðmtÞE t � mtð Þ4þE Rðv; mtÞð Þ: ð14Þ
The even central moments remain in the expectation and the odd central
moments of t all become zero. For good approximation, the fourth central
moment is retained. Since the second and fourth central moments of the t statistic
are E t � mtð Þ2¼ v v� 2ð Þ�1and E t � mtð Þ4¼ 3v2 v� 4ð Þ�1
v� 2ð Þ�1, respec-
tively (Johnson, Kotz, & Balakrishnan, 1995b, p. 365), the expectation of h(t) is
E hðtÞð Þ � 1þ :125ð4v� 19Þðv� 4Þ�1ðv� 2Þ�1 ¼ gðvÞ: ð15Þ
It can be shown that the remainder Rðv; mtÞ in the Taylor expansion converges to
zero at the rate of 1/v4. The comparison shows that the approximation is very
close to the exact computation. Their differences are in the fourth or fifth decimal
place for small sample sizes (see Table 1).
Substituting g(v) into the formula for E(w) yields
Sample Size for Confidence Interval
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E wð Þ ¼ t0s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
nðv� 1Þ
s ffiffiffi2p
�ðv=2Þ� v� 1ð Þ=2ð Þ gðvÞ
¼ sfðvÞ;ð16Þ
where
fðvÞ ¼ t0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
nðv� 1Þ
s ffiffiffi2p
�ðv=2Þ� v� 1ð Þ=2ð Þ gðvÞ: ð17Þ
Alternatively, the expectation of the half width can be expressed in terms
of the standard deviation of the outcome sy and the correlation between the
outcome and covariate r, that is,
EðwÞ ¼ sy
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� r2
pfðvÞ
¼ sy �ðvÞ;ð18Þ
where �ðvÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� r2
pfðvÞ. Indeed, the expected width of the confidence inter-
val is a function of the sample size. Using the correlation between the outcome
and covariate, a researcher can tabulate the expected width of the confidence
interval against the sample size when planning the confidence interval. Table 2
lists sample size and expected width for J ¼ 2 and r ¼ .3. The expected width is
expressed in the multiple � of the standard deviation of the outcome. When the
sample size n is 28, the expected width is :514sy.
TABLE 1
Comparison Between E(h(t)) and its Approximation g(�)
v g(v) E(h(t)) g vð Þ � E h tð Þð Þ
15 1.035839 1.036328 �.000489
16 1.033482 1.033870 �.000388
17 1.031410 1.031723 �.000313
18 1.029576 1.029831 �.000255
19 1.027941 1.028153 �.000212
20 1.026476 1.026653 �.000177
21 1.025155 1.025304 �.000149
22 1.023958 1.024086 �.000128
23 1.022870 1.022980 �.000110
24 1.021875 1.021970 �.000095
25 1.020963 1.021046 �.000083
Note. The exact expectation E(h(t)) can be computed through numerical integration. The R codes for
the numerical integration are as follows:
integrand<-function(x, v¼15){sqrt(1þx*x/v)*dt(x, v, log¼FALSE)}
integrate(integrand, �Inf, þInf).
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Probability of Achieving a Certain Width
A researcher can also find a sample size to obtain a desired probability
of achieving a certain width in the confidence interval. Two random variables
(i.e., s and t) affect the width of the confidence interval. In particular, the
width of the confidence interval increases with covariate imbalance or t because
it is a monotonically increasing function of t2 tj j < t:975;v
� �. Given s, the half
width corresponding to the high level of covariate imbalance is evaluated at the
97.5th t percentile, t:975;v:wt:975 ¼ t0s 2=nð Þ1=2
h t:975;v
� �: ð19Þ
One can choose a sample size to achieve a high probability of having the width
wt:975 smaller than an upper bound U
� ¼ P wt:975 < U½ � > :8: ð20Þ
Since s2 ¼ s2w2v�1
�v� 1, it follows that
� ¼ P w2v�1 <
ðv� 1Þno2
2t20 h t:975;v
� �� �2
" #
¼ P w2v�1 < f ðn; JÞo2
� ð21Þ
where
o ¼ U=s and f ðn; JÞ ¼ ðv� 1Þn
2t20 1þ t2
:975;v
.v
� �2: ð22Þ
The function f n; Jð Þ depends on the sample size n and the number of treatment
groups J. The number of treatment groups is usually predetermined, and the sam-
ple size n can be found to attain a high probability �. As the function f n; Jð Þ is
monotonic, a few rounds of search can produce the sample size n satisfying the
probability equation.
The probability � is related to two conditional probabilities of achieving a cer-
tain width conditioning on whether the confidence interval includes the param-
eter of interest or not. The conditional probability �1 is the probability of
achieving a certain width given that the confidence interval includes the para-
meter, and the conditional probability �2 is the probability of achieving the same
width given that the confidence interval excludes the true parameter
� ¼ 1� að Þ�1 þ a�2 ð23Þ
(Beal, 1989). To calculate the conditional probability �1 for the mean compari-
son of two independent samples, one needs to know the joint probability g that
the confidence interval includes the true parameter and the half width of the
confidence interval is no larger than U. This joint probability conditioning on the
pooled sample variance estimate s2 is
Sample Size for Confidence Interval
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g s2 ¼ P Tj j � t0 s2
� ¼ P Tðs=sÞj j � t0ðs=sÞ s2
� ¼ P Zj j � t0ðs=sÞ s2
� ;
ð24Þ
where T is the regular t test corresponding to the confidence interval and Z is a
standard normal variable. The joint probability g can be obtained by integrating
P ¼ g s2�
over s2, whose density function is related to a chi-square. Thus, the
joint probability becomes
g ¼ 2v=2�1�v
2
� �h i�1ZU=ðt0ffiffiffiffiffiffi2=np
Þ
0
2F t0
s
s
� �� 1
h i 1
s
ffiffiffivp
s
s
� �v
exp � v
2
s2
s2
� �ds; ð25Þ
where F is the normal cumulative distribution function and v is the degrees of
freedom for two-sample independent t test. It follows from the theorem of con-
ditional probability that �1 ¼ g= 1� að Þ, which is named as the power of confi-
dence interval (Beal, 1989).
The conditional probability �1 and the unconditional probability � differ little
in value but greatly in computational complexity. When the probability �1 is
above .70, there is less than a percentage difference between �1 and �. However,
the latter is much easier to calculate than the former. The unconditional probabil-
ity � can be readily calculated with Excel, but the computation of �1 requires
numerical quadrature. For practical purposes, one can always compute � in lieu
of �1 (Grieve, 1991). If the conditional probability �1 is the criterion anyway, one
can work with the lower bound of �1 instead of computing it directly. It is obvi-
ous from the definition of � that the lower bound of �1 is related to �.
�� a1� a
< �1: ð26Þ
The lower bound �� að Þ= 1� að Þis approximately �� .01 for �1 larger than .70.
One can compute � and use � � .01 to achieve �1. If a chosen sample size yields
�, it means that the corresponding �1 will be at least as large as � � .01.
For example, suppose that a researcher wants to compare two types of
speed reading techniques (J ¼ 2) in terms of shortened reading time while
adjusting for the subjects’ prior reading ability. The covariate is the prior
reading ability, and it can be measured by a standardized reading test. Before
the researcher sets up the experiment, he or she needs to calculate the
required sample size to ensure the half width of the confidence interval accu-
rate up to 1 minute. Assume that it can take a person from 5 to 16 minutes to
finish reading the test materials in the experiment. The researcher estimates
that the standard deviation sy is (16–5)/6 minute, as the entire range of a dis-
tribution is about six standard deviations. Furthermore, the estimated standard
deviation 11/6 does not account for the correlation r between the covariate and
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the outcome. Adjusting by the correlation produces the standard deviation of the
residual
s ¼ sy 1� r2� �1=2
: ð27Þ
If prior research suggests that the correlation r is .3, the standard deviation s is
then 11=6ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� :09p
. The parameter o is then 6=�11
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� :09p �
. Table 3 lists
the sample size n and the probability �. To attain the probability � ¼ .85 or �1
¼ .84, the researcher needs to recruit 31 subjects into each group. This sample
size returns 85% chance of having the half width accurate to 1 minute for the con-
fidence interval of the adjusted mean difference (Table 3).
TABLE 2
Sample Size and Expected Width
n v � vð Þ
28 54 .514
29 56 .504
30 58 .495
31 60 .487
32 62 .479
33 64 .471
34 66 .464
35 68 .457
36 70 .450
37 72 .444
TABLE 3
Sample Size and Probability of Obtaining the Width
n v �
27 52 .56
28 54 .64
29 56 .72
30 58 .79
31 60 .85
32 62 .89
33 64 .93
34 66 .95
35 68 .97
36 70 .98
Sample Size for Confidence Interval
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Summary
Sample size plays an important role in planning the confidence interval of
adjusted mean difference. Although the ANCOVA offers the advantage of
reducing error variance, the lack of information on covariate at planning stage
makes it difficult to gauge the standard error of adjusted mean difference. If
covariate observations are viewed as randomly varying from one study to
another, the ratio between covariate variance and balance can be transformed
into a t statistic in the standard error of adjusted mean difference. Substituting
the average value of this ratio in the standard error, one can express the
expected width of the confidence interval as a function of the sample size.
Alternatively, one can find a sample size to attain a desired probability of hav-
ing the interval width corresponding to a high level of covariate imbalance
smaller than a predetermined upper bound.
Appendix
Independence Between t and sThe linear model for ANCOVA is
yij ¼ b0 þ b1aij þ b2 xij � �x� �
þ eij; i ¼ 1; 2; . . . ; 2n; j ¼ E or C;
where aij takes on .5 and �.5 for the observations in the experimental condition
(E) and the control condition (C), respectively. For simplicity, only two treatment
conditions are used in the proof, but the result applies to any pairwise mean com-
parison among more than two treatment conditions. The predicted outcome can
be presented in matrix notation
y ¼ b01þ b1aþ b2 x� �x1ð Þ:
The orthogonal basis of the design matrix ½1 a x� �x1ð Þ� can be found
through the Gram-Schmidt process. The orthogonal set is
u1 ¼ 1
u2 ¼ a
u3 ¼ x� �x1� aTx
aTaa:
The predicted outcome can be expressed in terms of the orthogonal set
y ¼ b01þ b1 þ b2
aTx
aTa
� �aþ b2 x� �x1� aTx
aTaa
� �
¼ b0u1 þ b1 þ b2
aTx
aTa
� �u2 þ b2u3:
(continued)
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Appendix (continued)
It can be proven that aTx aTað Þ�1u2
and u3k k areffiffiffiffiffiffiffiffin=2
p�xE � �xCð Þ and
ffiffiffiffiffiffiffiExx
p,
which correspond to the numerator and denominator of t=ffiffiffivp
. The two vectors
aTx aTað Þ�1u2 and u3 are orthogonal to the error vector e ¼ y� y because e is
orthogonal to y, which is the projection of y in the space spanned by the orthog-
onal set u1, u2, and u3. The length of e is related to s, that is, ek k ¼ sffiffiffiffiffiffiffiffiffiffiffiv� 1p
. In
short, the numerator and denominator of t=ffiffiffivp
and sffiffiffiffiffiffiffiffiffiffiffiv� 1p
are the respective
length of three orthogonal vectors.
The elements in the vector u3 are the residuals exijin the regression of the co-
variate xij on the treatment variable aij. Assume that exijand eij have a joint
normal distribution. The two residuals are independent because their correla-
tion is zero (u3 is orthogonal to e). Therefore, Exx and s are independent. A
similar argument can be made about �xE � �xC and s. Intuitively, the t statistic
measures the covariate imbalance, which can only result from random chance. Con-
sequently, the t statistic cannot be dependent on eij. Otherwise, the ANCOVA anal-
ysis is invalid because there is some systematic bias in the estimate of the
treatment effect.
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ments. Journal of American Statistical Association, 103, 1350–1353.
Winer, B. J., Brown, D. R., & Michels, K. M. (1991). Statistical principles in experimental
design. New York, NY: McGraw-Hill.
Author
XIAOFENG STEVEN LIU is Associate Professor in the Department of Educational Studies
at the University of South Carolina, Columbia, SC 29208; e-mail: xliu@mailbox.sc.edu.
Sample Size for Confidence Interval
11
at UNIV OF TENNESSEE on February 28, 2013http://jebs.aera.netDownloaded from
His areas of specialization are statistical power analysis, sample size determination, hier-
archical linear modeling, and educational policy study.
Manuscript received November 16, 2009
Revision received June 29, 2010
Accepted July 10, 2010
Liu
at UNIV OF TENNESSEE on February 28, 2013http://jebs.aera.netDownloaded from