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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

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doi:10.3102/1076998610381401JOURNAL OF EDUCATIONAL AND BEHAVIORAL STATISTICS OnlineFirst, published on October 27, 2010 as

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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