Confidence Interval Estimation for a Difference ...

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Confidence Interval Estimation for a Difference Between Two Dependent Intraclass Correlation Coefficients With Variable Class Sizes

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Danuta Kowalik a , Yun-Hee Choi a , G Y Zou a,b,c,d , a Department

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b Robarts 8

of Epidemiology and Biostatistics, University of Western Ontario, London, Ontario, Canada N6A 5C1

Clinical Trials, Robarts Research Institute, University of Western Ontario, London, Ontario, Canada N6A 5K8

c Department 10

of Epidemiology and Biostatistics, School of Public Health, Southeast University, Nanjing, Jiangsu Province, PR China

d Corresponding 12

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to: GY Zou, Robarts Clinical Trials, Robarts Research Institute, P. O. Box 5015, 100 Perth Drive, London, Ontario, Canada N6A 5K8; Tel: 1-519-663-5777 Ext 24092; fax: 1-519-661-3766. E-mail: [email protected].

Abstract Four confidence intervals for a difference between two dependent intraclass correlation

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coefficients (ICCs) are developed, focusing on applications to family studies. The basic idea adopted here is that confidence intervals for a difference between two parameters can

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be obtained from confidence limits for each parameter. Among the four confidence procedures considered, the one based on the inverse hyperbolic tangent transformation for a

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single ICC performed best. Francis Galton’s data on human stature are used to illustrate the methodology.

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Key words: Family studies, Fisher’s z-transformation, Heritability, Interval estimation, Intraclass correlation coefficient, Sib-sib correlation.

Preprint submitted to Journal of Statistical Theory and Practice

26 March 2011

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1 Introduction

The intraclass correlation coefficient (ICC) is an important parameter in several fields of 26

research. In measurement studies, it is used to quantify reliability, where observations may be made on a sample of several assessors (Bartko, 1966). In genetic research, the ICC is

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commonly used to measure the degree of familial resemblance with respect to environmental or biological traits (Haggard, 1958). The ICC also plays a key role in adjusting for clustering

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effects in the design and analysis of clinical trials that randomize intact social units (Donner and Klar, 2000). See Donner (1986) and McGraw and Wong (1996) for extensive reviews on

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the topic. With few exceptions (Donner et al., 1984; Donner and Zou, 2002; Giraudeau et al., 2005),

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relatively little research has appeared on the inference involving dependent ICCs. Even if such a problem does arise, researchers tend to turn the question into comparing independent

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ICCs. For example, Naik and Helu (2007) recently analyzed Francis Galton’s familial data on human stature (Hanley, 2004) by randomly splitting the data into two parts. One part

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was used to compute ICC among boys and the other ICC among girls. The inference on gender difference in terms of similarity was then conducted using hypothesis procedures for

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independent ICCs. Furthermore, despite that confidence intervals are more informative than hypothesis testing,

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it is very rare to find that inference on a difference between two ICCs is conducted with confidence intervals. A confidence interval for the difference between the sister-sister and

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brother-brother ICCs would be more informative than a P-value resulted from hypothesis testing.

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Ramasundarahettige et al. (2009) recently applied a general confidence interval procedure for a difference to the case of dependent ICCs in the context of reliability studies having

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constant class sizes. Since the basic idea is to use confidence limits for single ICCs to recover variance estimates needed for a difference, we refer to the method as the MOVER, method 2

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of variance estimates recovery (Zou and Donner, 2008; Zou, 2008). The procedure based on exact confidence limits for a single ICC from the F-distribution performed best. However,

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the results may not hold for studies with variable class sizes in which Donner and Wells (1986) have found that the confidence interval for a single ICC based on F-distribution does

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not have superior performance. The purpose of this article is to adopt a similar approach to situations where the single ICCs

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are computed with variable class sizes, focusing on the case of family studies. We thus considered four procedures for a difference between two ICCs using confidence limits for single

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ICCs obtained from 1) variance formula by Smith (1957); 2) Fisher’s z-transformation; 3) the inverse hyperbolic tangent (arctanh) transformation; and 4) an approximate F-distribution

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(Thomas and Hultquist, 1978). Application of the MOVER resulted in four confidence interval procedures for a difference between two dependent ICCs. As in Ramasundarahettige

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et al. (2009), we take into account dependency between two ICCs using a result from Elston (1975).

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After setting up notations in Section 2, we summarize the four common procedures for a single ICC with variable class sizes in Section 3. Section 4 then applies the MOVER to ICCs.

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In Section 5, we report on a simulation study evaluating the performance of the procedures. An illustration of the procures is done using Galton’s data in Section 6. The article finishes

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with a discussion.

2 Notation and Terminology

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Let Y i = (Yi1 ,Yi2 , . . .,Yibi ,Yi,bi +1 ,Yi,bi +2 , . . .,Yi,bi +si ) denote measurements on the ith family, i = 1, 2, . . ., k, where Yi1 ,Yi2 , . . . ,Yibi are the scores on

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the bi brothers and Yi,bi +1 , . . . ,Yi,bi +si are the scores on the si sisters, respectively. We assume 3

the following model holds Y i ∼ N(µ i , Σi ), 74

where µ = (µb 1Tbi , µs 1Tsi ) and 



  [(1 − ρb)I bi + ρb J bi ]σb2 ρbs σb σs J bi ×si   Σi =     ρbs σb σs J Tbi ×si [(1 − ρs)I si + ρs J si ]σs2 In addition, here 1bi and 1si are the column vectors with all the bi and si elements equal to 1, 76

I bi and I si are bi × bi and si × si identity matrices, J bi and Jsi are bi × bi and si × si matrices with all the elements equal to 1 and, J bi ×si is a bi × si matrix for which all elements equal

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1. A necessary and sufficient condition for Σi to be positive definite, for all bi and si , is that 2 < ρ ρ . Under this model, it is assumed that the two intraclass correlation coefficients ρbs b s

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ρb and ρs , and the interclass correlation coefficient ρbs are constant among families in the population of studies, and in particular they are independent of sibling size bi + si . Our main

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interest is in developing and comparing methods for confidence intervals for ρb − ρs . In the case of constant class size, i.e., bi = b and si = s, for i = 1, 2, . . ., k, Elston (1975)

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derived the maximum likelihood estimators for ρb , ρs, and ρbs . In the case of variable class sizes, Donner and Koval (1980) found that analysis of variance (ANOVA) estimators for

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single ICCs are more stable. Let Yi j , for j = 1, . . ., bi , denote observation on jth member in the ith class, with class specific mean for brothers given by Y ib = ∑ j Yi j . The ANOVA

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estimator for ρb is given by

where

ρbb =

MSAb − MSEb , MSAb + (b0 − 1)MSEb

∑i bi (Y ib − ∑i ∑ j Yi j /N)2 MSAb = k−1 90

and ∑i ∑ j (Yi j −Y ib )2 MSEb = N −k 4

are the mean square errors between and within classes, respectively, b0 = 92

N − Σki=1 b2i /N , k−1

and N = ∑ bi is the total number of observations for brothers across families. The estimator for ρs is given analogously. Rosner (1982) suggested estimating ρbs by computing the Pear-

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son product-moment correlation over all possible brother-sister pairs. This estimator is given by b

b +s

i ∑ j′i=bii +1 (Yi j − Yeb )(Yi j′ − Yes ) ∑ki=1 ∑ j=1

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where

, ρbbs = q +si i 2] e ′ (Y − Y ) (Yi j − Yeb )2 ][∑ki=1 bi ∑bj′i=b [∑ki=1 si ∑bj=1 s ij i +1 ∑ bi siY ib ∑ bi siY is Yeb = i and Yes = i . ∑i bi si ∑i bi si

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3 Confidence intervals for a single intraclass correlation coefficient

As mentioned above, confidence interval for a difference between two ICCs, ρb − ρs , relies 100

on reliability of confidence interval for single ICCs. We therefore summarize four procedures for a single ICC, ρb . The results are also applicable to ρs .

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The most intuitive interval estimation approach is to evoke the central limit theorem in conjunction with Slutzky’s theorem, by which confidence limits are given by

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lSA , uSA = ρbb ± zα /2

q

c ρbb ), var(

where zα /2 is the upper α /2 quantile of the standard normal distribution, and  2(1 − ρb )2 {1 + ρb(b0 − 1)}2 var(ρbb ) = N −k b20

(k − 1)(1 − ρb){1 + ρb(2b0 − 1)} + ρb2 {∑ b2i − 2N −1 ∑ bi 3 + N −2 (∑ b2i )2 } + (k − 1)2 5

#

which was first derived by Smith (1957). Confidence interval for ρb is obtained by substitut106

ing ρbb for ρb in the variance formula. We refer to this method as the simple asymptotic (SA) method.

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It is well-known that the sampling distribution of the ICC is left-skewed. A transformation due to Fisher (1925) is commonly used to make inference for ICC with constant class size.

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The Fisher z-transformation is given by 1 1 + (b − 1)ρbb ln , 2 1 − ρbb

which is asymptotically distributed as  1 1 + (b − 1)ρb ln N , 2 1 − ρb 112

 1 o 1n 1 . + 2 k−1 N −k

In the case of variable class size, Weinber and Patel (1981) found replacing b by b0 performed well. The corresponding confidence interval for ρb is given by lz =

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exp(2lF ) − 1 , exp(2lF ) + b0 − 1

uz =

exp(2uF ) − 1 exp(2uF ) + b0 − 1

where

r n 1 1 + (b − 1)ρbb 1 1 1 o ± zα /2 lF , uF = ln . + 2 1 − ρbb 2 k−1 N −k An alternative approach to Fisher’s z-transformation is to apply the inverse hyperbolic tangent (arctanh) transformation, which Fisher (1925) suggested for the Pearson correlation, to

ρbb and derive its variance formula by the delta methods. By the central limit theorem, we have

1 1 + ρbb ln ∼N 2 1 − ρbb



1 1 + ρb var(ρbb) ln , 2 1 − ρb (1 − ρb2 )2



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where var(ρbb ) can be obtained using the Smith formula given above. The confidence interval

by arctanh transformation is given by la = where

exp(2lt ) − 1 , exp(2l) + 1

ua =

exp(2ut ) − 1 , exp(2ut ) + 1

p c ρbb ) 1 1 + ρbb zα /2 var( lt , ut = ln . ± 2 1 − ρbb 1 − ρbb2 6

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Finally, Thomas and Hultquist (1978) proposed an approximation to the exact confidence interval for ICC, given by le =

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F ∗ /Fu − 1 , Fu + bh − 1

ue =

F ∗ /Fl − 1 Fl + bh − 1

where F ∗ is the conventional F-statistic computed using the unweighted sum of square of class means, given by bh ∑i (Y i. − ∑i Y i. /k)2 /(k − 1) , ∑i ∑ j (Yi j −Y i. )2 /(N − k) and bh is the harmonic mean of numbers of brothers across families, Fl and Fu are α /2 and F∗ =

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1 − α /2 quantiles of the F-distribution with degrees of freedom k − 1 and N − k.

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4 Confidence intervals for a difference between two dependent intraclass correlation coefficients

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We first derive confidence interval for ρbb − ρbs assuming they are independent, and then extend to the case of dependent ICCs.

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By the central limit theorem, the 100(1 − α )% confidence limits for ρb − ρs are given by

and

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L = ρbb − ρbs − zα /2

p var(ρˆ b ) + var(ρˆ s)

U = ρbb − ρbs + zα /2

p

var(ρˆ b ) + var(ρˆ s)

where again zα /2 is the upper α /2 quantile of the standard normal distribution. The performance of (L,U ) depends on how the variances are estimated. Note that var(ρb) is a function of

ρ , indicating that variance of ρb depends on the parameter itself. Substituting ρb for ρ assumes

implicitly that point estimates are independent of variance estimates, resulting in symmetric 138

confidence intervals. In the spirits of score test (Rao, 1948; Bartlett, 1953, p. 15), we proceed by estimating variances needed for L and U in their respective neighborhoods.

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Now, given confidence limits for ρb and ρs as (lb , ub ) and (ls, us ), respectively, we can show 7

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that the distance between L and lb − us is given by q

q hq i

zα var(ρbb ) + var(ρbs) − var(ρbb ) + var(ρbs ) ,

which is shorter than that of L and ρbb − ρbs , given by

q

zα var(ρbb ) + var(ρbs) .

Thus, it is better to estimate variances needed for L under the assumption that ρb = lb and

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ρs = us . Similar arguments suggest variances for U be estimated at ρb = ub and ρs = ls. By again the central limit theorem, we can recover the variance estimates needed for L and

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U as follows. Since

q c ρbb ), lb ≈ ρbb − zα /2 var(

thus

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at ρb = lb . Similarly

c ρbb ) = var(

at ρb = ub . Furthermore, we have

c ρbb ) = var(

at ρs = ls , and

c ρbs) = var(

at ρs = us . Therefore

c ρbs) = var( L = ρbb − ρbs − zα /2 = ρbb − ρbs − zα /2

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Similar steps yield

= ρbb − ρbs −

U = ρbb − ρbs +

p

(ρbb − lb )2 zα2 /2

(ub − ρbb )2 zα2 /2 (ρbs − ls )2 zα2 /2 (us − ρbs )2 zα2 /2

p r

var(ρˆ b ) + var(ρˆ s) (ρbb −lb )2 zα2 /2

+

(us −ρbs )2 zα2 /2

(ρbb − lb )2 + (us − ρbs )2 .

q (ub − ρbb )2 + (ρbs − ls )2 . 8

Note that these confidence limits are in general asymmetric, as shown in Figure 1 (using the 154

Pythagorean theorem), unless confidence limits for ρb and ρs are symmetric. The approach can be easily extended in several ways. First, noting the confidence interval

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for −ρs is given by (−us , −ls ), we have a confidence interval for ρb + ρs given by  p  L = ρbb + ρbs − (ρbb − lb )2 + (ρbs − ls )2 p  U = ρb + ρbs + (u − ρb )2 + (us − ρbs )2 . b b b

This result is useful for constructing confidence intervals for linear contrasts of intraclass 158

correlation coefficients (Bhandary and Fujiwara, 2008). Second, when corr(ρbb , ρbs ) = r 6= 0, recalling that cov(ρbb, ρbs ) = r

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p

var(ρbb )var(ρbs ), we have

 p  L = ρbb − ρbs − (ρbb − ls )2 + (us − ρbs )2 − 2b r (ρbb − lb )(us − ρbs ) p  U = ρbb − ρbs − (ub − ρbb )2 + (ρbs − ls )2 − 2b r(ub − ρbb )(ρbs − ls )

where b r is the estimate of correlation between ρbb and ρbs , given by b r=

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p

bh (bh − 1)sh (sh − 1) ρbbs [1 + (bh − 1)ρbb ][1 + (sh − 1)ρbs]

(1)

(2)

with ρbbs denoting Rosner’s pairwise estimator of interclass correlation coefficient, and bh and

sh being the harmonic means of numbers of brothers and sisters across families, respectively. As the key step of the derivation is the recovery of variance estimates from confidence limits,

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we refer to this approach as the MOVER, method of variance estimates recovery (Zou and Donner, 2008; Zou, 2008).

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By the MOVER, we can now obtain a confidence interval for ρb − ρs as follows: 1) Obtain ρbb (lb , ub ) and ρbs (ls , us) using procedures described in Section 3; 2) Obtain b r using (2);

3) Apply the MOVER in (1) to obtain confidence limits, (L,U ) for ρb − ρs . 9

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5 Simulation Study

Since the MOVER is an asymptotic result, meaning that the finite sample properties are 172

intractable. We therefore assessed its performance using Monte Carlo simulation. We compared the performance of the four procedures, as developed from using the four

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procedures for single ICCs: 1) the simple asymptotic approach; 2) Fisher’s Z-transformation approach; 3) the arctanh transformation; and 4) the method based on an approximation of

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the F-distribution. The parameters for the simulation study included the number of families k, ρb , ρs , and ρbs .

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The total number of families was chosen to be k = 50 and 100 to reflect small and moderate sample sizes. The variable sibship sizes (s = bi + si ) were generated using the negative

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binomial distribution truncated below one given by Pr(s) =

(m + s − 1)!Q−1 (P/Q)s , (m − 1)!m!(1 − Q−m )

s = 1, 2, . . ., Q = 1 + P.

We chose the parameter values of m = 2.84 and P = 0.93, corresponding to the mean sibship 182

size of 3.12 and the variance of 4.52, reported by Brass (1958) for the United States in 1950. In order to avoid generating unreasonably large sibship sizes we truncated the extreme right

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tails of sibship sizes at the probability of 0.999. To classify each sibling as a male or a female, a random number generator based on the binomial distribution was used with the probability

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0.5. Parameter values for ρb and ρs were set to as (0.1, 0.1), (0.3, 0.1), (0.3, 0.3), (0.5, 0.1), (0.5, 0.3), (0.5, 0.5), (0.7, 0.3), (0.7, 0.5), (0.7, 0.7), (0.9, 0.5), (0.9, 0.7), (0.9, 0.9). For each

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combination of ρb and ρs, we set ρbs to range from 0.1 to min(ρb , ρs ) with an increment of 0.2 such that the variance-covariance matrix Σi to be positive definite.

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Without loss of generality µb = µs = 0 and σb2 = σs2 = 1, where µb , µs and σb2 , σs2 are the common mean and common variance of the observations obtained on brothers and sisters,

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respectively. For each parameter combination, observations for brothers and sisters were generated from a multivariate normal distribution. The number of runs was set to 10000 10

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such that the empirical coverage is expected (with 95% confidence interval) to vary between 94.6% to 95.4% for a nominal level of 95%. In the evaluation process, we considered three

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criteria: empirical coverage, balance of left and right tail errors, and interval width. Table 1 presents simulation study results for k = 50. The simple asymptotic method based

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on Smith’s variance formula tends to result in anti-conservative coverage for large difference between two ICCs, and conservative coverage for small difference of high ICC values. In

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addition, it also results in unbalanced tails errors for high values of ICCs. It is clear that the methods based on Fisher’s Z-transformation and the arctanh transformation gives compara-

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ble results when values of ICCs are small to moderate (≤ 0.5), providing good empirical coverage and balanced tail errors and competitive interval width. However, over the whole

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parameter space, the arctanh transformation provided better results in terms of coverage and the balance of tail errors. Moreover, this method tends to provide narrower intervals, com-

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pared to the simple asymptotic method. Confidence intervals based on the approximate Fdistribution appeared to have slightly conservative coverage with the largest interval width.

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This observation contrasts to the results by Ramasundarahettige et al. (2009) who reported that the method based on the exact limits for single ICCs performed the best. However, the

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observation here is expected, as Donner and Wells (1986) have concluded that the approximate F-distribution based method tends to be imprecise in the case of variable class sizes.

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Similar conclusions can be drawn for k = 100, as shown in Table 2. The problems associated with the simple asymptotic method are reduced somewhat but still overt. The method based

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on F-distribution is also slightly conservative.

6 Example: Galton’s data

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We now illustrate the four procedures using Galton’s data on siblings height, available at http://www.epi.mgill.ca/hanley/galton (Hanley, 2004). The data consist of 205 fam-

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ilies (486 sons, 476 daughters) with the sibship size ranging from 1 to 15. Among the 962 children, only 934 have have numeric values, which are the basis for this analysis. 11

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Based on the data we obtained ρbb = 0.385, ρbs = 0.421, ρbb − ρbs = −0.036, and ρbbs = 0.264, yielding b r = 0.061. The confidence interval estimates of ρb , ρs and ρb − ρs using the four

methods are summarized in Table 3. Based on the results there is no significant difference between the sex-specific ICCs at the α = 5% level regardless of the method used, as the

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resulting confidence intervals for ρb − ρs all contain zero. However, even for sample size

over 200, the differences among four confidence intervals for ρb − ρs are still distinguishable. 226

Consistent with the simulation, the method based on F-distribution provides wider intervals.

7 Discussion

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We have developed four confidence intervals for a difference between two correlated ICCs, focusing on family studies. The results can be seen as a direct extension of Ramasundara-

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hettige et al. (2009). Previous research has largely focused on hypothesis testing of equality (Donner et al., 1984), but confidence intervals are more informative as they indicate sta-

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tistical significance as well as the magnitude of the difference, providing basis for judging practical importance.

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The confidence intervals are simple to calculate, as only limits for single ICCs are required. The advantage of the MOVER is that it can take into account asymmetric sampling distri-

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bution for ρb. Another advantage of the MOVER is that it can be used when the raw data

are not available but the summary in terms of point estimates of ICCs and their respective

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confidence limits are given. It must be emphasized that our results are valid under several assumptions. First, the dis-

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tribution of class sizes was obtained from families typically seen in United States in 1950. In the cases of greater imbalance, the conclusions here may not be applicable. One way to

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reduce imbalance may be to exclude families with a single child in the analysis since Swiger et al. (1991) have suggested this can result in better estimates of ICC. The is also in accor-

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dance with the recommendation by Eliasziw and Donner (1991) that interclass correlation 12

coefficient be estimated by placing a restriction on the family size. Such a strategy is most 246

useful in the case of large sample size. Second, the validity of the MOVER depends on that of confidence limits for single ICCs. The confidence intervals for single ICCs relied on the

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assumption that the data are at least approximately normally distributed and the number of families is not too small. In situations when these assumptions may not hold, other proce-

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dures should be sought. One possible solution is to first transform the data using a power transformation and then obtain the desired confidence interval using the proposed method.

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However, due to the transformation the confidence intervals on the transformed scale may not be directly interpretable. Another possibility is to use the bootstrap. This procedure has

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been widely recommended for non-normal data. For example, Ukoumunne et al. (2003) implemented a non-parametric bootstrap confidence interval for ICC with constant class size

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in the context of cluster randomization trials. They showed that the application of bootstrapt method to the Fisher’s z-transformation of ICC performed well even with small sample

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sizes. Evaluation of the MOVER and the bootstrap for the difference of two correlated ICCs of small values arising from small numbers of large clusters, however, is left for further

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research.

Acknowledgments

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us ρb − ρs √Lower for 2 2

Upper for ρb + ρs √

(ρbb −lb ) +(us −ρbs )

(ub −ρbb )2 +(us −ρbb )2

ρbs ρbb

lb

Lower for ρb + ρs

√

ub

√

(ρbb −lb )2 +(ρbs −ls )2

Upper for ρb − ρs

(ub −ρbb )2 +(ρbs −ls )2

ls

Figure 1. Geometric illustration of margins of errors as obtained using the Method Of Variance Estimates Recovery (MOVER) for ρb − ρs and ρb + ρs , which are identical to that by the Pythagorean theorem. Note that a confidence interval is given by point estimate − lower margin of error, point estimate + upper margin of error

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Table 1 Performance (based on 10000 runs) of four procedures for constructing two-sided 95% confidence interval for a difference between two dependent intraclass correlation coefficients for k = 50. ρb ρs ρbs 0.1 0.1 0.0 0.1

Simple asymptotic CV(ML, MR)WD 94.19(2.87,2.94)0.91 93.96(2.97,3.07)0.90

Fisher’s z CV(ML, MR)WD 94.63(2.67,2.70)0.87 94.32(2.82,2.86)0.87

Arctanh CV(ML, MR)WD 95.16(2.41,2.43)0.88 95.09(2.41,2.50)0.87

F-distribution CV(ML, MR)WD 96.27(1.78,1.95)1.17 96.16(1.94,1.90)1.17

0.3 0.1 0.0 0.1 0.3 0.0 0.1 0.3

94.25(2.71,3.04)0.87 94.56(2.59,2.85)0.87 94.50(2.75,2.75)0.84 94.10(2.98,2.92)0.84 94.86(2.58,2.56)0.83

94.34(2.49,3.17)0.84 94.81(2.30,2.89)0.84 94.43(2.77,2.80)0.81 94.21(2.97,2.82)0.81 94.90(2.50,2.60)0.80

94.98(2.05,2.97)0.85 95.43(1.90,2.67)0.85 95.15(2.37,2.48)0.82 94.79(2.67,2.54)0.82 95.45(2.25,2.30)0.81

96.13(1.78,2.09)1.10 96.41(1.54,2.05)1.10 96.30(1.81,1.89)1.02 96.25(1.88,1.87)1.02 96.35(1.76,1.89)1.01

0.5 0.1 0.0 0.1 0.3 0.0 0.1 0.3 0.5 0.0 0.1 0.3 0.5

94.82(2.53,2.65)0.81 94.32(2.74,2.94)0.81 94.45(2.50,3.05)0.78 94.68(2.62,2.70)0.78 95.65(2.03,2.32)0.77 94.92(2.67,2.41)0.71 94.93(2.33,2.74)0.71 95.36(2.46,2.18)0.70 96.38(1.87,1.75)0.69

94.68(2.28,3.04)0.78 94.29(2.43,3.28)0.78 94.06(2.62,3.32)0.75 94.36(2.58,3.06)0.75 95.46(1.96,2.58)0.74 93.69(3.29,3.02)0.69 93.92(2.94,3.14)0.69 94.41(2.98,2.61)0.68 95.43(2.33,2.24)0.67

95.41(1.76,2.83)0.80 94.78(2.12,3.10)0.79 94.70(2.23,3.07)0.77 95.08(2.27,2.65)0.77 96.06(1.72,2.22)0.76 94.69(2.78,2.53)0.71 94.82(2.45,2.73)0.71 95.11(2.64,2.25)0.70 96.16(1.94,1.90)0.69

96.10(1.61,2.29)1.01 95.73(1.91,2.36)1.01 95.65(1.81,2.54)0.92 96.18(1.85,1.97)0.92 96.91(1.36,1.73)0.92 95.92(2.18,1.90)0.82 95.95(1.87,2.18)0.82 96.39(1.96,1.65)0.81 96.75(1.70,1.55)0.80

0.7 0.3 0.0 0.1 0.3 0.5 0.0 0.1 0.3 0.5 0.7 0.0 0.1 0.3 0.5 0.7

93.98(2.66,3.36)0.69 93.94(2.39,3.67)0.69 94.84(2.08,3.08)0.69 94.64(2.19,3.17)0.62 94.79(1.96,3.25)0.61 95.32(1.89,2.79)0.61 95.53(1.51,2.96)0.60 96.04(1.88,2.08)0.50 95.95(2.19,1.86)0.50 96.28(1.77,1.95)0.50 97.26(1.53,1.21)0.49 97.90(1.09,1.01)0.48

93.59(2.72,3.69)0.67 93.47(2.45,4.08)0.67 94.77(2.14,3.09)0.67 93.32(2.98,3.70)0.60 93.52(2.80,3.68)0.60 94.02(2.61,3.37)0.59 94.52(2.05,3.43)0.59 93.39(3.22,3.39)0.49 93.46(3.49,3.05)0.49 93.64(3.04,3.32)0.49 94.83(2.74,2.43)0.48 96.01(2.08,1.91)0.48

94.49(2.47,3.04)0.69 94.56(2.16,3.28)0.68 95.42(1.90,2.68)0.68 94.46(2.58,2.96)0.62 94.50(2.36,3.14)0.62 95.10(2.25,2.65)0.62 95.48(1.78,2.74)0.61 94.87(2.46,2.67)0.52 94.74(2.78,2.48)0.52 95.09(2.33,2.58)0.52 96.23(1.97,1.80)0.51 96.88(1.60,1.52)0.50

95.19(2.03,2.78)0.81 95.23(1.90,2.87)0.81 95.74(1.73,2.53)0.81 95.35(2.02,2.63)0.70 95.43(1.84,2.73)0.69 95.67(1.93,2.40)0.69 96.06(1.46,2.48)0.68 95.35(2.25,2.40)0.55 95.48(2.41,2.11)0.55 95.72(2.01,2.27)0.55 96.47(1.82,1.71)0.54 97.17(1.46,1.37)0.54

0.9 0.5 0.0 0.1 0.3 0.5 0.7 0.0 0.1 0.3 0.5 0.7 0.9 0.0 0.1 0.3 0.5 0.7 0.9

93.19(1.82,4.99)0.52 93.29(1.80,4.91)0.52 93.49(1.70,4.81)0.52 93.82(1.60,4.58)0.51 94.78(0.84,4.38)0.38 94.05(1.06,4.89)0.38 94.65(0.82,4.53)0.38 94.40(0.98,4.62)0.38 95.30(0.61,4.09)0.37 97.80(0.96,1.24)0.20 98.01(0.96,1.03)0.20 97.82(1.10,1.08)0.20 98.35(0.77,0.88)0.19 98.53(0.77,0.70)0.19 99.19(0.37,0.44)0.19

93.56(2.71,3.73)0.50 93.74(2.68,3.58)0.50 93.86(2.66,3.48)0.50 94.24(2.54,3.22)0.50 93.86(2.67,3.47)0.37 93.10(3.04,3.86)0.37 93.90(2.59,3.51)0.37 93.51(2.94,3.55)0.37 94.98(2.05,2.97)0.36 92.94(3.37,3.69)0.20 93.18(3.44,3.38)0.20 92.82(3.40,3.78)0.20 94.11(2.84,3.05)0.20 94.88(2.63,2.49)0.19 96.29(1.63,2.08)0.19

94.55(2.49,2.96)0.52 94.75(2.44,2.81)0.52 94.83(2.40,2.77)0.52 95.11(2.32,2.57)0.52 95.12(2.13,2.75)0.39 94.75(2.52,2.73)0.39 95.14(2.14,2.72)0.39 94.93(2.35,2.72)0.39 96.21(1.60,2.19)0.38 94.90(2.39,2.71)0.22 95.27(2.43,2.30)0.22 94.87(2.51,2.62)0.22 95.77(1.91,2.32)0.21 96.25(1.87,1.88)0.21 97.34(1.15,1.51)0.21

94.67(2.39,2.94)0.58 94.57(2.30,3.13)0.58 94.99(2.18,2.83)0.58 95.41(1.94,2.65)0.58 95.48(1.76,2.76)0.41 95.06(2.14,2.80)0.41 95.24(1.89,2.87)0.41 95.49(1.94,2.57)0.41 96.25(1.36,2.39)0.40 95.38(2.02,2.60)0.21 95.37(2.38,2.25)0.21 95.50(2.16,2.34)0.21 96.00(1.80,2.20)0.21 96.48(1.90,1.62)0.20 97.59(1.05,1.36)0.20

†CV: coverage %; ML: % of intervals missing the parameter value from the left; MR: % of intervals missing the parameter value from the right; WD: interval width.

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Table 2 Performance (based on 10000 runs) of four procedures for constructing two-sided 95% confidence interval for a difference between two dependent intraclass correlation coefficients for k = 100. ρb ρs ρbs 0.1 0.1 0.0 0.1

Simple asymptotic CV(ML, MR)WD 94.65(2.77,2.58)0.63 94.74(2.43,2.83)0.63

Fisher’s z CV(ML, MR)WD 94.85(2.70,2.45)0.62 94.95(2.31,2.74)0.62

Arctanh CV(ML, MR)WD 95.13(2.56,2.31)0.62 95.25(2.19,2.56)0.62

F-distribution CV(ML, MR)WD 96.41(1.91,1.68)0.82 96.43(1.61,1.96)0.82

0.3 0.1 0.0 0.1 0.0 0.3 0.1 0.3

94.70(2.67,2.63)0.61 94.98(2.52,2.50)0.61 94.93(2.57,2.50)0.59 94.70(2.72,2.58)0.59 95.55(2.43,2.02)0.58

94.73(2.49,2.78)0.60 94.93(2.41,2.66)0.60 94.73(2.70,2.57)0.57 94.41(2.88,2.71)0.57 95.30(2.55,2.15)0.57

95.09(2.24,2.67)0.60 95.40(2.06,2.54)0.60 95.19(2.40,2.41)0.58 94.99(2.54,2.47)0.58 95.72(2.35,1.93)0.58

96.26(1.70,2.04)0.76 96.18(1.77,2.05)0.76 96.26(1.87,1.87)0.70 96.08(2.00,1.92)0.70 96.51(1.85,1.64)0.70

0.5 0.1 0.0 0.1 0.3 0.0 0.1 0.3 0.5 0.0 0.1 0.3 0.5

94.65(2.76,2.59)0.57 94.61(2.68,2.71)0.57 94.65(2.59,2.76)0.54 95.19(2.28,2.53)0.54 95.45(2.12,2.43)0.54 95.13(2.35,2.52)0.50 94.91(2.67,2.42)0.50 95.55(2.13,2.32)0.49 96.50(1.64,1.86)0.49

94.48(2.56,2.96)0.55 94.44(2.41,3.15)0.55 94.15(2.67,3.18)0.53 94.72(2.45,2.83)0.53 94.99(2.23,2.78)0.52 94.13(2.88,2.99)0.48 93.79(3.23,2.98)0.48 94.50(2.60,2.90)0.47 95.62(2.03,2.35)0.47

94.96(2.29,2.75)0.56 94.91(2.16,2.93)0.56 94.82(2.43,2.75)0.54 95.28(2.12,2.60)0.54 95.57(1.97,2.46)0.54 95.10(2.38,2.52)0.50 94.89(2.65,2.46)0.50 95.44(2.18,2.38)0.49 96.43(1.68,1.89)0.49

95.87(1.80,2.33)0.70 95.55(2.01,2.44)0.70 95.92(1.91,2.17)0.63 96.12(1.75,2.13)0.63 96.32(1.60,2.08)0.63 96.20(1.86,1.94)0.55 95.90(2.12,1.98)0.55 96.28(1.91,1.81)0.55 96.90(1.44,1.66)0.54

0.7 0.3 0.0 0.1 0.3 0.5 0.0 0.1 0.3 0.5 0.7 0.0 0.1 0.3 0.5 0.7

95.23(2.15,2.62)0.48 94.81(2.06,3.13)0.48 94.71(2.33,2.96)0.48 95.02(2.17,2.81)0.43 95.25(1.98,2.77)0.43 95.35(1.90,2.75)0.43 96.07(1.64,2.29)0.42 95.71(2.06,2.23)0.35 95.13(2.45,2.42)0.35 95.94(2.05,2.01)0.35 96.62(1.69,1.69)0.34 97.45(1.32,1.23)0.34

94.68(2.36,2.96)0.47 94.34(2.25,3.41)0.47 94.13(2.58,3.29)0.46 93.93(2.76,3.31)0.41 93.89(2.81,3.30)0.41 94.05(2.74,3.21)0.41 95.13(2.22,2.65)0.41 93.41(3.10,3.49)0.34 92.94(3.57,3.49)0.34 94.04(2.91,3.05)0.33 94.77(2.76,2.47)0.33 95.79(2.17,2.04)0.32

95.34(2.15,2.51)0.48 95.07(2.04,2.89)0.48 94.87(2.32,2.81)0.48 94.98(2.41,2.61)0.43 95.11(2.34,2.55)0.43 95.20(2.18,2.62)0.43 96.07(1.85,2.08)0.42 94.97(2.43,2.60)0.36 94.40(2.82,2.78)0.36 95.41(2.32,2.27)0.35 96.12(2.00,1.88)0.35 96.98(1.57,1.45)0.34

95.75(1.99,2.26)0.56 95.52(1.83,2.65)0.56 95.55(2.07,2.38)0.55 95.41(1.95,2.64)0.47 95.89(1.85,2.26)0.47 95.66(1.99,2.35)0.47 96.41(1.64,1.95)0.46 95.63(2.11,2.26)0.37 95.05(2.51,2.44)0.37 95.93(2.02,2.05)0.36 96.20(2.06,1.74)0.36 97.07(1.45,1.48)0.36

0.9 0.5 0.0 0.1 0.3 0.5 0.7 0.0 0.1 0.3 0.5 0.7 0.9 0.0 0.1 0.3 0.5 0.7 0.9

94.08(1.83,4.09)0.36 94.27(1.83,3.90)0.36 94.29(1.71,4.00)0.36 94.60(1.58,3.82)0.36 94.64(1.50,3.86)0.26 94.35(1.23,4.42)0.26 94.64(1.24,4.12)0.26 95.41(0.99,3.60)0.26 95.83(0.88,3.29)0.26 96.50(1.79,1.71)0.13 96.55(1.80,1.65)0.14 96.87(1.50,1.63)0.13 97.06(1.46,1.48)0.13 97.52(1.13,1.35)0.13 98.49(0.81,0.70)0.13

93.65(3.05,3.30)0.35 94.17(2.72,3.11)0.35 93.98(2.69,3.33)0.35 94.35(2.55,3.10)0.35 93.58(2.93,3.49)0.25 93.19(2.93,3.88)0.25 93.74(2.78,3.48)0.25 94.26(2.55,3.19)0.25 94.75(2.38,2.87)0.25 93.08(3.53,3.39)0.13 92.98(3.35,3.67)0.13 93.60(3.28,3.12)0.13 93.97(3.04,2.99)0.13 94.73(2.54,2.73)0.13 96.39(1.86,1.75)0.12

94.74(2.70,2.56)0.36 95.06(2.37,2.57)0.36 94.95(2.42,2.63)0.36 95.46(2.14,2.40)0.36 95.15(2.25,2.60)0.27 94.82(2.31,2.87)0.27 95.24(2.23,2.53)0.27 95.90(1.81,2.29)0.26 95.86(1.96,2.18)0.26 95.03(2.55,2.42)0.14 94.97(2.53,2.50)0.14 95.42(2.23,2.35)0.14 95.72(2.11,2.17)0.14 96.34(1.74,1.92)0.14 97.59(1.31,1.10)0.13

95.06(2.31,2.63)0.40 95.06(2.22,2.72)0.40 95.25(1.99,2.76)0.39 95.19(2.10,2.71)0.39 95.05(2.29,2.66)0.27 95.10(1.98,2.92)0.27 95.44(1.93,2.63)0.27 95.88(1.72,2.40)0.27 96.04(1.73,2.23)0.27 95.16(2.53,2.31)0.13 95.14(2.39,2.47)0.14 95.56(2.23,2.21)0.13 95.87(2.04,2.09)0.13 96.67(1.66,1.67)0.13 97.75(1.21,1.04)0.13

†CV: coverage %; ML: % of intervals missing the parameter value from the left; MR: % of intervals missing the parameter value from the right; WD: interval width.

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Table 3 Two sided 95% confidence intervals for ρb − ρs using four different methods of confidence interval estimation for sex-specific ICCs Method ρbb = 0.385 ρbs = 0.421 ρbb − ρbs = −0.036 Simple asymptotic Fisher’s z arctanh F-distribution

ρbs = 0.264,b r = 0.061 (0.282, 0.488) (0.315, 0.527) (-0.179, 0.107) (0.284, 0.481) (0.317, 0.517) ( -0.171, 0.101) (0.278, 0.483) (0.310, 0.521) ( -0.178, 0.107) (0.274, 0.497) (0.301, 0.533) ( -0.189, 0.123)

Note: Mean (SD) for sons: 69.23(2.63) inch, N = 481. Daughters: 64.10(2.4) inch, N = 453

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