A Bayesian Alternative to the Chi-Squared test of Association in a Two-Way

Categorical Table Incorporating Intra-Class Correlation

Balgobin Nandram1 and Jai Won Choi

(bbn9@cdc.gov, jwc7@cdc.gov)

O±ce of research and Methodology

National Center for Health Statistics, CDC

3311 Toledo Road, Hyattsville, MD 20782

June 4, 2004

Abstract

It is straight forward to analyze data from a single multinomial table. Speci ̄cally, for the

analysis of a two-way categorical table, the common chi-squared test of independence between the

two variables and maximum likelihood estimators are readily available. When the counts in the

two-way categorical table are formed from familial data (clusters of correlated data), the common

chi-squared test no longer applies. We note that there are several approximate adjustments to

the common chi-squared test. For example, Choi and McHugh (1989, Biometrics 45, 979-996)

showed how to adjust the chi-squared statistic for clustered and weighted data. However, our main

contribution is the construction and analysis of a Bayesian model which removes all analytical

approximations. This is an extension of a standard multinomial-Dirichlet model to include the

intra-class correlation associated with the individuals within a cluster. We have used a key formula

described by Altham (1976, Biometrika 63, 263-269) to incorporate the intra-class correlation.

This intra-class correlation varies with the size of the cluster, but we assume that it is the same

for all clusters of the same size for the same variable. We use Markov chain Monte Carlo methods

to  ̄t our model, and to make posterior inference about the intra-class correlations and the cell

probabilities. Also, using Monte Carlo integration with a binomial importance function, we obtain

the Bayes factor for a test of no association. To demonstrate the performance of the alternative

test and estimation procedure, we have used data on activity limitation status and age from the

National Health Interview Survey and a simulation study.

Key Words: Bayes factor, Gibbs sampler, Monte Carlo integration, Multinomial-Dirichlet.

1Balgobin Nandram (balnan@wpi.edu) is on sabbatical leave from the Department of Mathematical Sciences,

Worcester Polytechnic Institute, 100 Institute road, Worcester, MA 01609.

1
1. Introduction

It is a common practice to use two-way categorical tables to present survey data. In this

situation it is assumed that the cell counts in the r £ c table follow a multinomial distribution.

However, because of strati ̄cation and clustering the joint distribution of the cell counts is no longer

multinomial. Thus, the standard chi-squared statistic no longer has a chi-squared distribution, and

therefore the test based on the multinomial distribution may be inadequate. It is standard practice

to make an adjustment to the standard chi-squared statistic, but in general the accuracy of this

adjustment is not well understood, and one can not estimate the cell probabilities based on this

adjustment. We propose a Bayesian alternative which is based on the Bayes factor to obtain a test

for association between the two categorical variables. Our Bayesian method also provides posterior

distributions for the cell probabilities.

Several authors have recognized inaccuracy in the analysis when the usual chi-squared test is

applied to correlated \multinomial" data. E®orts to correct for spurious in°ation in such tests

have been based on two approaches. The design-based approach provides inference with respect

to the asymptotic sampling distribution of estimates over repetitions of the sample design (Fellegi

1980, Holt, Scott and Ewings 1980, Rao and Scott 1981, 1984, Bedrick 1983, and Fay 1985).

For example, Rao and Scott (1981) investigate the e®ects of strati ̄cation and clustering on the

asymptotic distribution of Pearson's chi-squared statistic for goodness of  ̄t and independence.

They propose new measures called generalized design e®ects. See also Rao and Scott (1984) who

generalized the results of Rao and Scott (1981) to multi-way categorical tables. The model-based

approach postulates a probability distribution to model the sample data (Altham 1976, Cohen

1976, Brier 1980, Fienberg 1979, and Choi and McHugh 1989). For example, Choi and McHugh

(1989), applying the probabilistic development in Altham (1976), shows how to adjust the standard

chi-squared test statistic when there is an intra-class correlation.

The National Center for Health Statistics (NCHS) uses the National Health Interview Survey

(NHIS) to collect data on chronic and acute conditions, doctor visits, hospital episodes, disability,

2
household and personal information, and other special aspects of health of the U.S. population.

One of the variables we use in the NHIS is activity limitation status (ALS), a measure of longterm disability resulting from chronic conditions since 1957. ALS is de ̄ned as inability to carry

out the major activity for one's age-sex group such as working, keeping house or going to school;

restriction in the amount or kind of major activity; or restriction in relation to other activities

such as recreational, church and civic interests. ALS is typically classi ̄ed into three categories:

\unable to perform major activity", \limited in kind/amount major activity and in other activities"

and \not limited (includes unknowns)" ranging from severe individuals to individuals unnecessary

to classify. The relation between age and activity limitation status is of interest. In the health

interview survey, information (i.e., chronic disease and impairment) for each household member

about the major activity she/he usually performed during the 12 months prior to interview is

requested by the interviewer. There is possibly a positive association between ALS and age, and to

study this association three age groups (under 56 years, 56-70 years and more than 70 years) are

used. The analysis is complex because one can expect an intra-class correlation within households.

Let njk denote the number of individuals in the j

th row and kth column of the r £ c categorical

Pc P P P r r c table. Also let nj¢ = k=1 njk; j = 1;:::;r, n¢k = j=1 njk; k = 1;:::;c, n = j=1 k=1 njk

and ejk = nj¢n¢k=n; j = 1; : : : ; r; k = 1;:::;c. Then, Pearson's chi-squared statistic, under

independence of the row and column classi ̄cation, is

Xr cX Xu = (njk ¡ ejk)

2

=ejk:

j=1 k=1

If the responses from the individual members are independent and identically distributed, then

asymptotically (as n ! 1) Xu ! Â

2

(r¡1)(c¡1), a chi-squared random variable with (r ¡ 1)(c ¡ 1)

degrees of freedom. In practice, the validity of the chi-squared test depends on (a) the magnitude

of the expected values ejk and (b) whether the cell counts (njk; j = 1; : : : ; r; k = 1;:::;c) follow a

multinomial distribution given the sample size n (i.e., the individual responses are independent and

identically distributed). In (a) the test is valid if the ejk are larger than 5, and clearly the only way

to achieve this is to increase the sample size subject to cost. In (b) when there is correlation among

3
the members (e.g., familial correlation), the asymptotic distribution of Xu is no longer Â

2

(r¡1)(c¡1),

and the estimates of the cell proportions can be inaccurate. The problem about the asymptotic

distribution has received much attention, but the problem about the inaccuracy of the estimates

of the cell proportions has received virtually no attention. In this paper we address both problems

simultaneously within a Bayesian framework when there are familial count data.

We describe one solution that has been proposed for the problem about the asymptotic distribution. Let nt denote the number of members in all families of the same size t = 1;:::;T, and

let μt denote the intra-class correlation for clusters of size t (μ1  ́ 0). Motivated by Rao and Scott

(1981), Choi and McHugh (1989) derive the following adjusted chi-squared statistic

X

T

X ¡1 ^ ¡1

a = Xuf1 + n (t ¡ 1)ntμtg

t=1

where ^μ is the maximum likelihood estimator of μ under their model. The statistic Xa is an

improvement over Xu (i.e., more accurately Â

2

). The p-value corresponding to the adjusted (r¡1)(c¡1)

chi-squared statistic will be larger. For weighted data they further adjust Â

2 approximately (r¡1)(c¡1)

by the average weight.

We provide a Bayesian analysis of this problem. This is a direct extension of the probabilistic

development in Altham (1976) which is used to provide a likelihood function. Then proper but

noninformative priors are assigned to the parameters to provide a full Bayesian approach. The

model includes a nonnegative intra-class correlation which varies according to the number of individuals in a cluster (i.e., all clusters of the same size have the same intra-class correlation). In

this framework we can provide (a) the posterior densities of the cell probabilities and (b) a test of

association between the two categorical variables. In (b) we use the Bayes factor to quantify the

di®erence between a model with association and one without. This is the ratio of the prior odds of

one model to the other to their posterior odds (obtained through the use of Bayes' theorem), and

it is the same as the ratio of the marginal likelihoods of the data under two models, one without

association and the other with association. If two models, M0 and M1, are  ̄tted to data y, the

~

Bayes factor for comparing models M1 and M0 is de ̄ned as the ratio of the marginal likelihoods

4
of the data y as

~

Z

p(yjM1)

B10 = ~ with p(y j Mk) = p(yjμk; Mk)p(μk j Mk)dμk; k = 0; 1

p(yjM0) ~ ~ ~ ~ ~

~

where μk is the parameter vector under Mk, p(yjμk; Mk) is the probability density (or mass) function

~ ~ ~

and p(μk j Mk) is the prior density. For example, in application M0 is the model of no association

~

and M1 is the model of association. The Bayes factor summarizes the evidence provided by the

data in favor of one scienti ̄c hypothesis M1 relative to another M0. Kass and Raftery (1995)

gave a comprehensive description of Bayes factors including their interpretation. For example, if

0 · log(B10) < 1, the evidence against M0 is \not worth more than a bare mention"; if 1 ·

log(B10) < 3, the evidence against M0 is \positive"; if 3 · log(B10) < 5, the evidence against M0 is

\strong"; and if log(B10)  ̧ 5, the evidence against M0 is \very strong". There are several methods

to compute the marginal likelihood (e.g., see Section 1 of Chib and Jeliazkov 2001), and we note

that one standard method is Monte Carlo integration with an importance function.

In this paper, we introduce a Bayesian method to analyze data from an r £ c categorical table.

We consider the situation in which there are no missing data, but one in which the table is built up

by aggregating clustered multinomial data. In Section 2, we describe the methodology to obtain

estimates of the cell probabilities, and to obtain the Bayes factor for a test of no association between

the two categorical variables. We also show how to use Markov chain Monte Carlo methods to  ̄t

the models. We show how to use Monte Carlo integration with an importance function to compute

the marginal likelihoods under di®erent models. In Section 3 we illustrate our method using data

from the National Health Interview Survey. In Section 4, we perform several simulated examples to

compare inference using our model with another model which does not incorporate the intra-class

correlation. Finally, Section 5 has concluding remarks.

2. Bayesian Model and Computation

We describe the methodology to  ̄t \multinomial" data when there is an intra-class correlation.

We build our model based on the work of Altham (1976).

5
2.1 Model

Suppose there are siindividuals in the i

th cluster, i = 1;:::;`, and sijk individuals fall in the jth

P P r c row and k

th column in the r £ c table, j = 1; :::r; k = 1; :::; c. Here j=1 k=1 sijk = si

, sijk  ̧ 0.

Altham (1976) shows that the probability that all siindividuals fall in the j

th row and kth column

is

μsi1⁄4jk + (1 ¡ μsi

)1⁄4

si

; 0 · μsi · 1: (1) jk

There is exactly one sequence in (??). Note that (??) can be interpreted as a mixture of two

distributions. Let wsi be the latent variable such that wsi = 1 for perfect dependence and wsi =

0, for perfect independence, where dependence/independence refers to the intra-class correlation.

Then p(w = 1 j μsi)=1 ¡ p(w = 0 j μsi) = μsi s .

i si

Also, the probability that the individuals are in di®erent speci ̄ed cells is

Yr cY

(1 ¡ μsi

) 1⁄4

sijk (2) jk

j=1 k=1

where we allow the intraclass correlation μsi

, 0 · μsi · 1, to depend on the cluster size si

. Note

that there is at least one sequence in (??).

This model of clustering permits only positive association or independence among the individuals within a cluster, and this is typically the case for many demographic, social and economic

characteristics.

Note that μsi1⁄4jk + (1 ¡ μsi

)1⁄4

si

is strictly increasing in μsi

. When μsi = 0, the probability that jk

all individuals in the i

th cluster belong to cell (j; k) is 1⁄4

si

, and when μsi = 1, the probability that jk

all individuals in the i

th cluster belong to cell (j; k) is 1⁄4jk, which can be much larger. In addition,

Q Q r c sijk (1 ¡ μsi

) j=1 k=1 1⁄4 is a strictly decreasing function in μsi. When μsi = 0, the probability jk

that the individuals in the i

th cluster belong to di®erent speci ̄ed cells is 1⁄4

si

, and when μsi = 1, jk

the probability that the individuals in the i

th cluster belong to di®erent speci ̄ed cells is 0. Thus,

the intra-class correlation has an important role when inference is made about the 1⁄4jk and the

association between the two categorical variables. Henceforth, s1;:::;s` are assumed known.

6
Let C denote the set of clusters in which all individuals fall in a single cell of the r £ c table.

Then, letting si = (si11;:::;sirc); i = 1;:::;`,

~ 8

> μsi1⁄4jk + (1 ¡ μsi

)1⁄4

si <

; i 2 C jk

p(s ; 1⁄4) = (3)

~ ~ : r c sijk

i

j μsi >

(1 ¡ μsi

)si!

Q

j=1

Q

k=1 1⁄4jk =sijk! i =2 C:

Assuming independence over clusters and letting s = (s1;:::;s`), we have

~ ~ ~

Y Y Yr c Y Y Yr c

p(s j μ; 1⁄4) = 1⁄4jk + (1 ¡ μsi

)1⁄4

si f(1 ¡ μsi 1⁄4

sijk fμs : (4) i jkg )si

!

jk =sijk!g

~ ~ ~ i2C j=1 k=1 i =2C j=1 k=1

Observe that if μsi = 0; i = 1;:::;`,

Y

`

Yr Yc

sijk

p(s j μ; 1⁄4) = fsi

! 1⁄4jk =sijk!g; (5)

~ ~ ~ i=1 j=1 k=1

P` which is a product of multinomial probability functions. In (??) the statistics i=1 sijk = njk are

su±cient as in regular multinomial sampling (i.e., observations are from a simple random sample)

P P r c and each individual belongs to cell (j,k) with probability 1⁄4jk  ̧ 0, k=1 1⁄4jk = 1. j=1

Suppose that each cluster has size t; t = 1; :::; T; in applications T is 2 to 5 or so. Then letting

gtjk denote the number of clusters in C of size t with all individuals in cell (j; k) and g~t the number

~ of clusters of size t in C (i.e., outside C),

Y Y Y

T r c

jk)

gtjk p(s j μ; 1⁄4) = (μt1⁄4jk + (1 ¡ μt)1⁄4

t

(6)

~ ~ ~ t=1 j=1 k=1

Y

T r c

gt Y Y Y sijk £ f (1 ¡ μt)

~

g fsi! 1⁄4 =sijk!g:

jk

t=1 i =2C j=1 k=1

Finally for a full Bayesian approach, noting that μ1 = 0, we assume

iid μt » Uniform(0,1); t = 2;:::;T

and independently

1⁄4 » Dirichlet(1):

~ ~

These are noninformative but proper prior densities. Thus, the joint posterior density of (μ; 1⁄4) is

~ ~

proper.

7
The likelihood function is described in Appendix A where we introduce latent variables ztjk

(i.e., an augmented likelihood function used primarily to simplify the computations). Speci ̄cally,

in (??) we have described the joint probability mass function of (s; z) given (μ; 1⁄4). Then, letting

~ ~ ~ ~

z = fztjk; t = 2;:::;T; j = 1; : : : ; r; k = 1;:::;cg and using Bayes' theorem the joint posterior

~

density is

( ) Y

T

gt p(μ; 1⁄4; z j s) / (1 ¡ μt)~

~ ~ ~ ~ t=2

( Ã ! ) Y Y

r c T

g1jk+~sjk Y gtjk

jk) £ 1⁄4

gtjk¡ztjk (μt1⁄4jk)ztjk ((1 ¡ μt)1⁄4t

: (7)

j=1 k=1

jk

t=2 ztjk

The joint posterior density in (7) is complex, so we use the Gibbs sampler to draw samples which

are used to make inference about 1⁄4jk and μt

.

2.2 Computation

To run the Gibbs sampler we need starting values for μ and 1⁄4, and these are easy to obtain.

~ ~ P` P P r c Letting njk = i=1 sijk and n = j=1 k=1 njk, we take 1⁄4^jk = njk=n and μt = 1=t; t = 2; :::; T.

Note also that we estimate ztjk by ztjk = gtjk[μt1⁄4jk=fμt1⁄4jk + (1 ¡ μt)1⁄4

t

jkg].

The conditional posterior densities (cpd's) of each parameter given the others are needed to

implement the Gibbs sampler. Note that z1jk = μ1 = 0. Speci ̄cally, the cpd for μ is

8 9 ~

< XXr c r c XX = ind μt

j 1⁄4; z; s » Beta

:

1 + ztjk; 1+~gt + (gtjk ¡ ztjk)

;

; t = 2;:::;T;

~ ~ ~ j=1 k=1 j=1 k=1

the cpd for 1⁄4 is

~ ( ) X

T

1⁄4 j μ; z; s » Dirichlet 1 + g1jk + ~sjk + [ztjk + t(gtjk ¡ ztjk)]; j = 1; : : : ; r; k = 1;:::;c

~ ~ ~ ~ t=1

and the cpd for z is

~ ( ) ind μt1⁄4jk

ztjk j μ; 1⁄4; s » Binomial gtjk; ; t = 2;:::;T; j = 1; : : : ; r; k = 1; : : : ; c: μt1⁄4jk + (1 ¡ μt)1⁄4

t

~ ~ ~ jk

We \burn in" 1000 iterates, and took every tenth to get 1000 iterates which we use for inference.

These choices are very conservative, and the algorithm runs very quickly.

3. Alternative Test for Association and Computation

8
To test for association versus no association between the two categorical variables, we use

the Bayes factor, the ratio of the two marginal likelihoods. By no association we mean that

(1) (2) P P r (1) c (2)

1⁄4jk = q q where = = 1. A problem of the slightly less interest is to test j k j=1 qj k=1 qk

for no intra-class correlation. The model without intra-class correlation is given in Appendix B.

3.1 Bayes Factor

Consider our problem with intra-class correlation. Letting z = fzijk; t = 2; :::; T; j =

~

1; :::; r; k = 1; :::; cg, we de ̄ne Z = fz : 0 · ztjk · gtjk; t = 2; :::; T; j = 1; :::; r; k = 1; :::; cg.

~ Ã ! Q Q Q T r c gtjk For the model with association, taking μ1 = 0 and letting A(g; z) = k=1 , the t=1 j=1 ~ ~

ztjk

marginal likelihood is

X Z Z Y Y Y

T r c

f(μt1⁄4jk)

ztjk f(1 ¡ μt)1⁄4t gtjk¡ztjk pas(s)=(rc ¡ 1)! [A(g; z) jkg

~ z2Z ~ ~ t=1 j=1 k=1

~ 3

T r c

Y Y Y Y sijk

1⁄4

£ f (1 ¡ μ

gt

jk

5 t)

~

g fsi! gdμd1⁄4

t=1 i =2C j=1 k=1 sijk! ~ ~

and for the model without association the marginal likelihood is

X ZZZ Y Y Y

T r c

(1) (2))

ztjk f(1 ¡ μt)(q

(1) (2))

t gtjk¡ztjk pnas(s)=(r ¡ 1)!(c ¡ 1)! [A(g; z) (μtq q q g j k j k

~ z2Z ~ ~ t=1 j=1 k=1

~ 3

T r c (1) (2)) Y Y Y Y sijk (q q

g j k (1) (2) £f (1 ¡ μ

t 5 t)

~

g fsi! gdq dq dμ :

sijk! ~

j

~

k

t=1 ~ i =2C j=1 k=1

Q Q Q r c Then, letting d = (rc ¡ 1)!S and e = (r ¡ 1)!(c ¡ 1)!S with S = i =2Cfsi

!= j=1 k=1 sijk!g, it

is easy to show that

" # X Y

T

pas(s) = d A(g; z)f D(b1t + 1; b2t + 1)gD(a11 + 1; :::; arc + 1) (8)

~ z2Z ~ ~ t=2

~

and

" # X Y

T

pnas(s) = e A(g; z)f D (b1t + 1; b2t + 1)g D1(a) D2(a) (9)

~ z2Z ~ ~ t=2 ~ ~

P P r c r c ~ P P PT where b1t = j=1 k=1 ztjk, b2t = ~gt+ j=1 k=1(gtjk¡ztjk), ajk = g1jk+~sjk+ t=2fztjk+t(gtjk¡

P P c r ztjk)g, aj¢ = = j=1 ajk, j = 1;:::;r, k = 1;:::;c, D1(a) = D(a1¢ + 1;:::;ar¢ + 1) k=1 ajk, a¢k

~

and D2(a) = D(a¢1 + 1;:::;a¢k + 1). In (??) and (??) D(¢;:::; ¢) is the Dirichlet function, where

~

9
Q P · · for a ·-dimensional vector x; D(x) = s=1 ¡(xs)=¡( s=1 xs) (e.g., when · = 2, D(x1; x2) =

~ ~

¡(x1)¡(x2)=¡(x1 + x2) is the beta function).

3.2 Computation

To compute (??) and (??) we use Monte Carlo integration with the importance function,

ind

ztjk » Binomial(gtjk; qtjk); t = 2; :::; T; k = 1; :::; c: (10)

^ ^ · ^ In (??), qtjk = μt1⁄4^jk=fμ

^

t1⁄4^jk + (1 ¡ μt)^1⁄4jkg where · is a tuning constant and μt and 1⁄4^jk are

respectively the posterior means of μt and 1⁄4jk obtained from the Gibbs sampler. We choose the

tuning constant · = 2.

Then, simulation consistent estimators of pas(s) and pnas(s) are

~ ~

X

M QT (h) (h) (h) (h)

f t=2 D(b1t + 1; b2t + 1)gD(a11 + 1; :::; arc + 1) ) = dM¡1

pasd(s (h)

~ QT Q Q z (h)

h=1 r c tjk (1 ¡ qtjk)

gtjk¡z

tjk

t=2 j=1 k=1 qtjk

and

M QT (h) (h) d X f t=2 D(b1t + 1; b2t + 1)gD1(a

(h)

)D2(a

(h)

)

) = eM¡1

pnas(s (h) ~ ~

~ QT Qr Qc

z

tjk (1 ¡ qtjk)

gtjk¡z

(h)

t=2 j=1 k=1 qtjk

h=1 tjk

(h) Pr Pc (h) (h) Pr Pc (h) (h) PT (h) where b1t = j=1 k=1 ztjk, b2t = ~gt + j=1 k=1(gtjk ¡ ztjk), ajk = g1jk + ~sjk + t=2fztjk +

(h)

t(gtjk ¡ ztjk)g, and z

(h)

; h = 1; :::; M is a random sample from (??). We have chosen M=10,000.

~

4. An Illustrative Example

In the NHIS the households are poststrati ̄ed by states and there are data from all 51 states

(including the District of Columbia). For some states there are extremely small numbers of sampled

households (e.g., Iowa, Idaho, Wyoming) and for some states there are extremely large numbers of

sampled households (e.g., California, New York, Texas). We have studied these states individually

and to illustrate our procedure we use the data from Maryland (medium size state). In column 2 of

Table ?? we present the cell counts of the 3£3 table of age and ALS for Maryland. It is of general

interest to test the hypothesis that age and ALS are independent and to estimate the proportion of

10
individuals in each cell of the 3£3 table. We have compared our model with intra-class correlation

with the simple (without intra-class correlation) multinomial model; see Appendix B for a brief

discussion of the simple multinomial model.

Using the method of Rao (1965, p. 159) we have calculated the intra- class correlation coef-

 ̄cient for age and ALS separately for Maryland data. For age the estimates are: families of size

2 (:71); 3 (:45); 4 (:20) and for families of size 5 (-.03), and for ALS the estimates are: families of

size 2 (:29); 3 (:26); 4 (:20 and for families of size 5 (.02). Note that there are three families of

size 7 and 2 of size 8 that we have omitted from our data analysis, and the total number of individuals is 897 with 104 one-member families, 140 two-member families, 79 three-member families,

49 four-member families and 16  ̄ve-member families.

We have constructed 95% credible intervals for the intra-class correlations (μ2;:::; μ5) and they

are: μ2 (:39; :58); μ3 (:46; :70); μ4 (:57; :80); and μ5 (:28; :74). Thus, there is substantial intra-class

correlation especially among families of size 4.

We have also studied the tests (Bayes factor and adjusted chi-squared statistics). Working

with logarithm, for the model without intra-class correlation the Bayes factor is 15:6; the value

of the unadjusted chi-squared test statistic is 67:2 giving a p-value of virtually 0. For the model

with intra-class correlation the Bayes factor is 7:6 with a NSE of 3:3; the value of the adjusted

chi-squared test statistic is 45:8 giving a p-value of virtually 0 again. The Bayes factor gives very

strong evidence for an association between age and ALS with or without the intra-class correlation;

the same is true for the chi-squared test. Note that the count for cell (2; 1) is only 3, showing a

possible problem for the chi-squared test. We also note that in this example, because there is a

strong assosciation between age and ALS, the di®erence among these tests is small. However, in

cases where the association is not so large, there could be di®erences in these tests.

For the 1⁄4jk in Table ?? we present the posterior mean (PM), posterior standard deviation

(PSD), numerical standard error (NSE), and the 95% credible intervals. First, the NSE's are small

showing that the results can be reproduced. But note, as expected, the model without intra-class

correlation has PM's close to the 1⁄4^jk, but there are some cases where the PM's from the model with

11
intra-class correlation di®er (e.g., for cell (2,1) compare :067 with :027 and for cell (3, 1) compare

:706 with :675). Also the PSD's under the model with intra-class correlation are larger than those

under the model without intra-class correlation, as is expected. These di®erences are re°ected in

the 95% credible intervals (e.g., for cell (3,1) compare (:62; :73) with (:67; :73)). Except for cell (1,

2) the intervals under the model with intra-class correlation contain those under the model without

intra-class correlation.

It is interesting that inference about the 1⁄4jk can di®er under the model with intra-class correlation and the one without, re°ecting the presence of an intra-class correlation. The presence of a

substantial intra-class correlation has the e®ect of reducing the number of observations. Thus, we

can deduce that the absence of intra-class correlation in the simple multinomial model leads to an

under estimation of variability.

5. A Simulation Study

We have simulated data from our model to assess the quality of our methodology. Speci ̄cally,

we have studied how changes in the intra-class correlation a®ect inference about the 1⁄4jk and the

association between the variables in a r £ c categorical table.

We have chosen the 1⁄4jk to represent di®erent degrees of association between the categorical

variables in a 3 £ 3 table. Speci ̄cally, we have chosen three di®erent sets of 1⁄4jk: (a) low (or no)

assiciation (1⁄4jk = 1=9; j = 1; : : : r; k = 1;:::;c) (b) medium association (1⁄411 = :220, 1⁄412 = :150,

1⁄413 = :100, 1⁄421 = :075, 1⁄422 = :100, 1⁄423 = :075, 1⁄431 = :050, 1⁄432 = :100, and 1⁄433 = :130) and

(c) strong association (1⁄411 = :250, 1⁄412 = :050, 1⁄413 = :010, 1⁄421 = :030, 1⁄422 = :250, 1⁄423 = :030,

1⁄431 = :050, 1⁄432 = :080, and 1⁄433 = :250). We have taken μk = μ; k = 2;:::;T, and 5 values of μ

(:2; :4; :5; :6; :8). Thus, we study the e®ect of our choice of μ on inference about 1⁄4jk and the

association between the two categorical variables. We note that when μ is small (large), there is a

tendency for the simulated individuals to be in di®erent (same) cell(s) of the r £ c table. Letting

ck denote the number of clusters of size k; k = 1;:::;T = 5, we take c1 = 50, c2 = 70, c3 = 50,

c4 = 40 and c5 = 20 to get a total of 600 observations. These are held  ̄xed for all simulation

12
experiments. Thus, we study how the posterior distributions of the intraclass correlation and the

1⁄4jk are a®ected by choices of the intra-class correlation and the degree of association between the

two categorical variables.

We simulated the cell counts using the probabilities of allocation. Let si denote the counts for

~

the i

th cluster. Then, for t = 1, si

j 1⁄4 » Multinomial(1; 1⁄4). For t  ̧ 2 the probability that all t

~ ~ ~

individuals fall in the same cell is μ1⁄4jk + (1 ¡ μ)1⁄4

t and the probability that they fall in di®erent jk

Q Q P P r c sijk r c cells is (1 ¡ μ)(t!) j=1 k=1 1⁄4jk =sijk!;

j=1 k=1 sijk = t. Thus, it is straight forward to draw

the cell counts.

We have simulated 1000 datasets of size 600 for each value of μ and the degree of association

(i.e., there are 15000 datasets). We  ̄t our model incorporating the intra-class correlation to each

data set using the Gibbs sampler as described in Section 2.2. We compute (a) the posterior mean,

posterior standard deviation, and 95% credible intervals for the 1⁄4jk for each data set, (b) the Bayes

factor to test for association in the 3 £ 3 table and (c) we have also compared inference using the

model with intra-class correlation and the standrad multinomial-Dirichlet model, one that ignores

the intra-class correlation (see Appendix B).

We have presented results for our simulated examples in Tables ??, ??,?? and ??, taking

averages over the 1000 datasets for each quantity.

[Need to discuss!] In Tables ?? we have presented posterior mean (PM), posterior standard

deviation (PSD) and 95% credible interval for the μk and the 1⁄4jk. Apart from μ = :10 the posterior

summaries are concordant with the design values for μ2 to μ5.

[Need to discuss!] In Table ?? we have presented posterior mean (PM), posterior standard

deviation (PSD) and 95% credible interval for the and the 1⁄4jk. The posterior summaries indicate

that the model with the intra-class correlation is more in concordant with the design values than

the regular multinomial, and the regular multinomial degrades as μ increases.

[Need to discuss!] The log-Bayes factors in Table ?? for both the model with intra-class correlation and the regular multinomial are bigger than 5, indicating very strong evidence for no

13
association. But while for the model with intra-class correlation the Bayes factor decreases rapidly

as μ increases, the decrease under the regular multinomial model is not so steep.

We have studied the datasets generated from the design level of weak association (1⁄4jk = 1=9; j =

1; 2; 3; k = 1; 2; 3) to compare the adjusted ch-squared test and the Bayes factor from the model

with intra-class correlation; see Table ??. First, using the the model with intra-class correlation all

the datasets show some degree of evidence for no association; of the 5000 datasets 3812 show strong

evidence and 1188 show weak evidence for no association. Observe that the number of datasets

showing weak (strong) evidence increase (decrease) with the intra-class correlation. The situation

is di®erent when the adjusted chi-squared test is used; of the 5000 datasets 3948 do not reject

(accept in Table ??) and 1052 (much too large) reject no association at the 5% signi ̄cance level.

With aberrations the e®ect of increasing intra-class correlation is the same as for the Bayes factor.

Thus, inference using the adjusted chi-squared test can be incorrect.

6. Concluding Remarks

We have shown how to analyze multinomial data from r £ c categorical tables when there is an

intra-class correlation. We have also shown that by using the Bayes factor (ratio of the marginally

likelihoods of two models) we can test for association between the two categories.

We have analyzed 3 £ 3 categorical data of age and activity limitation status from the 1996

National Health Interview Survey. We have found moderately large intra-class correlations, and

these correlations have small e®ects on tests of hypothesis (both the standard chi-squared test and

our Bayesian alternative). While we have reported results for Maryland, we have found similar

results for many of the other states.

We have also performed a small simulation study to assess the impact of the intra-class correlations on the alternative to the chi-squared test and posterior inference of the cell probabilities.

It appears that the Bayes factor decreases as the intra-class correlation increases (further investigation is required), but for the examples we have not found much di®erence in inference between

14
the model with intra-class correlation and the one without. There are also small di®erences for

inference about the 1⁄4jk.

In future we can extend our methodology to accommodate (a) small areas (b) nonresponse and

(c) an intra-class correlation coe±cient corresponding to each categorical variable. In (a) we can

consider the states (including the District of Columbia) as small areas. There are very sparse data

from some of the states (e.g., Iowa, Idaho, Wyoming), and to make reliable inference about one

of these states, one needs to \borrow strength" across the states. In (b) there is a non-negligible

number of nonrespondents from each state, and one would need to construct a model that can

adjust for nonignorable nonresponse. Finally, in (c) for two categorical variables one intra-class

correlation is μsi

and the other is ·siμsi; 0 · · · μs

¡

i

1

. Then we can replace μsi s in Altham's i

formula by 1(·si + 1)μsi; 0 · μsi · 1; 0 < ·si · μ

¡1

; i = 1;:::;` with an appropriate joint prior 2 si

density on (μsi

; ·s ). i

APPENDIX A: Augmented Likelihood Function

We derive the likelihood function of μ and 1⁄4, augmented with latent variables. Let the cell

~ ~ XXr c

counts for the i

th cluster be sijk; i = 1; : : : ; `; j = 1; : : : ; c; k = 1;:::;c, and si = sijk. Also

j k

let si = (si11;:::;sirc). Assuming that the si are known, we have si given μand 1⁄4 are independent.

~ ~ ~ ~

We separate the derivation into two parts.

First, for each i2C (i.e., the  ̄rst part in which all members fall in the same cell),

X

1

)1⁄4

si 1⁄4jk)

!ijk f(1 ¡ μsi

)1⁄4

si 1¡!ijk p(s ; i

j μ; 1⁄4)=(μsi1⁄4jk) + (1 ¡ μsi jk = (μsi jkg

~ ~ ~ !ijk=0

j = 1; : : : ; r; k = 1;:::;c where !ijk is an indicator variable (i.e., !ijk = 0; 1). Then, by demarginalization over !ijk, we have

i

j μ; 1⁄4)=(μsi1⁄4jk)

!ijk f(1 ¡ μsi

)1⁄4

si 1¡!ijk ; !ijk = 0; 1:

~ ~ ~

p(!ijk; s jkg

μsi

1⁄4jk That is, !ijk j μ; 1⁄4 » Bernoullif si g; i 2 C: Note that for each i 2 C there is μsi

1⁄4jk+(1¡μsi)1⁄4

~ ~ jk

contribution from only one of the cells j = 1; : : : ; r; k = 1;:::;c (i.e., sijk  ́ si). Note also that

15
for i2C and si = 1, p(si

j μ; 1⁄4) = 1⁄4jk. Thus, for i 2 C letting ! denote the vector of all !ijk, by

~ ~ ~ ~

independence the joint probability mass function of (si

; i 2 C; !) is

~ ~ 2 3 2 3

Y Y p1(si

; i2C; ! j μ; 1⁄4) = 4 1⁄4

si 5 4 f(μsi1⁄4jk)

!ijk f(1 ¡ μsi

)1⁄4

si 1¡!ijk5 : (A.1) jk jkg

~ ~ ~ ~ i2C;si=1 i2C;si ̧2

Second, for i =2 C (i.e., the second part in which members fall in the di®erent cells),

Y Yr c

p(si

j μ; 1⁄4) = (1 ¡ μsi)fsi! 1⁄4

sijk =sijk!g jk ~ ~ ~ j=1 k=1

and the joint probability mass function of si

; i=2 C is

~

Y Y Y Yr c

p2(si

; i 2= C j μ; 1⁄4) = p(sij μ; 1⁄4) = (1 ¡ μsi)fsi! 1⁄4

sijk =sijk!g: (A.2) jk ~ ~ ~ ~ ~ ~ i =2C i =2C j=1 k=1

Thus, letting s = (s1;:::;s`), by independence the joint probability mass function of (s; !) is

~ ~ ~ ~ ~

p(s; ! j μ; 1⁄4) = p1(si

; i2C; ! j μ; 1⁄4)p2(si; i 2= C j μ; 1⁄4) (A.3)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

where p1(si

; !i; i2C j μ; 1⁄4) is given in (??) and p2(si; i 2= C j μ; 1⁄4) is given in (??).

~ ~ ~ ~ ~ ~ ~

Now, let T denote the largest cluster size and μ1 = 0. For the clusters in C, let gtjk denote the

number of clusters of size t with all their members in cell (j,k), and

X z = fztjk : ztjk = !ijk; t = 1;:::;T; j = 1; : : : ; r; k = 1;:::;cg: (A.4)

~ i2C;si=t

Note that in (??) for all i 2 C whenever sijk = 0, !ijk = 0. For the clusters outside C, let g~t denote

X the number of clusters of size t and s~jk = sijk, the total number of individuals in cell (j; k)

i =2C

outside C. Then, using the assumption that the intraclass correlation coe±cient depends only on

the cluster size with μsi = μt

, t = 1;:::;T and (??), the joint probability mass function of (s; z) is

8 9 ~ ~

Y < Y Yr c =

p(s; z j μ; 1⁄4) = si

!= sijk!

~ ~ ~ ~ : ; i =2C j=1 k=1

8 Ã ! 9 8 9 <Y Y Y

T r c

= <Y Y Y

T r c

=

gt

gtjk sjk (μt1⁄4jk)

zijk f(1 ¡ μt)1⁄4t gtjk¡ztjk ~ £

:

jkg

; :

(1 ¡ μt)

~ 1⁄4jk

;

: (A.5)

ztjk t=2 j=1 k=1 t=2 j=1 k=1

16
Note that the joint probability mass fucntion in (??) as a function of μ, 1⁄4 and z is the augmented

~ ~ ~

likelihood function (i.e., μ and 1⁄4 are augmented with z). Note also that from (??)

~ ~ ~

ind μt1⁄4jk

ztjk j μ; 1⁄4 » Binomialfgtjk; g;

μt1⁄4jk + (1 ¡ μt)1⁄4

t

~ ~ jk

t = 2;:::;T; j = 1; : : : ; r; k = 1;:::;c.

APPENDIX B: Product Multinomial-Dirichlet Model

XXr c

Letting 1⁄4jk; j = 1; : : : ; r; k = 1;:::;c denote the cell probabilities and si = sijk, the

j=1 k=1

multinomial-Dirichlet model for the cell counts sijk in a r £ c categorical table is

ind

si

j 1⁄4 » Multinomial(si; 1⁄4); i = 1;:::;` and 1⁄4 » Dirichlet(1;:::; 1); (B.1)

~ ~ ~ ~

where the si are assumed known.

P` Let njk = i=1 sijk denote the cell counts over all clusters, j = 1; : : : ; r; k = 1;:::;c. Then, a

posteriori

1⁄4 j s1;:::;s` » Dirichlet(n11 + 1;:::;nrc + 1):

~ ~ ~

Because the posterior density is in closed form, one can obtain inference about the 1⁄4jk in a straight

forward manner.

The corresponding marginal likelihoods (association: as, no association: nas) are

Q Q ( ) r c Y

`

(rc ¡ 1)! j=1 k=1 njk! si

!

pas(n) = Q Qc

(B.2) r

(n + rc ¡ 1)! k=1 ~ sijk!

i=1 j=1

and

Q Q r c (r ¡ 1)!(c ¡ 1)! (n + rc ¡ 1)! j=1 nj:! k=1 n:k!

pnas(n) = pas(n) Q Q (B.3) r c ~ ~ (rc ¡ 1)! (n + r ¡ 1)!(n + c ¡ 1)! k=1 njk!

j=1

Pc P P P r r c where nj: = k=1; njk; j = 1;:::;r, n:k = j=1; njk; k = 1;:::;c and j=1 k=1 njk = n.

References

17
Adams, P. F. and Marano, M. A. (1995). Current estimate from the National Health Interview

Survey, 1994. National Center for Health Statistics, CDC, Series 10, (193).

Altham, P. M. (1976). Discrete variable analysis for individuals grouped into families. Biometrika

63, 263-269.

Bedrick, E. J. (1983). Adjusted chi-squared tests for cross-classi ̄ed tables of survey data.

Biometrika 70, 591-595.

Brier, S. S. (1980). Analysis of contingency tables under cluster sampling. Biometrika 67, 591-596.

Chen, T. and Fienberg, S. E. (1974). Two-dimensional contingency tables with both completely

and partially cross-classi ̄ed data. Biometrics 30, 629-642.

Chib, S. and Jeliazkov, I. (2001). Marginal likelihood from the Metropolis-Hastings output. Journal

of the American Statistical Association 96, 270-281.

Choi, J. W. and McHugh, R. B. (1989). A reduction factor in goodness-of- ̄t and independence

tests for clustered and weighted observations," Biometrics 45, 979-996.

Cohen, J. E. (1976). The distribution of the chi-squared under cluster sampling from contingency

tables. Journal of the American Statistical Association 71, 591-596.

Fellegi, I. P. (1980). Approximate tests of independence and goodness of  ̄t based upon strati ̄ed

multistage samples. Journal of the American Statistical Association 75, 261-268.

Fay, R. E. (1985). Complex samples. Journal of the American Statistical Association 80, 148-157.

Fienberg, S. E.(1979). The use of chi-squared statistics for categorical data problems. Journal of

the Royal Statistical Society, Series B 41, 54-64.

Holt, D., Scott, A. J. and Ewings, P. O. (1980). Chi-squared tests with survey data. Journal of the

Royal Statistical Society, Series A 143, 302-320.

18
Kass, R. and Raftery, A. (1995). Bayes factors. Journal of the American Statistical Association 90,

773-795.

Rao, C. R. (1965). Linear Statistical Inference and Its Applications. New York: Wiley.

Rao, J. N. K. and Scott, A. J. (1981). The analysis of categorical data from complex sample

surveys: Chi-squared tests for goodness of  ̄t and independence in two-way tables. Journal of

the American Statistical Association 76, 221-230.

Rao, J. N. K. and Scott, A. J. (1984). On chi-squared tests for multiway contingency tables with

cell proportions estimated from survey data. The Annals of Statistics 12, 46-60.

19
Table 1: Comparison of the posterior means (PM), posterior standard deviations (PSD), numerical

standard deviation (NSE), and 95% credible intervals (CI) for 1⁄4jk with and without intra-class

correlation for Maryland

Intra-class No intra-class

Cell 1⁄4^ PM PSD NSE CI PM PSD NSE CI

(1, 1) .014 .024 .010 .002 (.008, .045) .015 .004 .001 (.008, .024)

(1, 2) .014 .027 .010 .002 (.011, .048) .016 .004 .001 (.008, .025)

(1, 3) .003 .008 .006 .001 (.001, .021) .004 .002 .000 (.001, .010)

(2, 1) .067 .027 .010 .002 (.012, .050) .067 .008 .002 (.053, .084)

(2, 2) .022 .028 .010 .002 (.011, .051) .023 .005 .001 (.015, .035)

(2, 3) .026 .024 .010 .002 (.009, .045) .027 .005 .001 (.017, .038)

(3, 1) .706 .675 .029 .005 (.616, .730) .699 .016 .004 (.669, .730)

(3, 2) .090 .099 .019 .004 (.067, .138) .091 .010 .002 (.071, .110)

(3, 3) .057 .088 .018 .003 (.056, .125) .057 .008 .001 (.043, .074)

NOTE: The total number of observations from Maryland is 897, and 1⁄4^jk is the observed

proportion of observations in cell (j; k). The row and column variables are age and activity

limitation status (ALS) respectively; Age (1: under 56 years; 2: 56-70 years; 3: over 70 years) and

ALS (1: unable to perform major activity; 2: limited in kind/amount major activity and limited

in other activities; 3: not limited (includes unknowns). The numerical standard errors are

obtained using the batch-means method with batches of 25 for the selected sample from the

Gibbs sampler.

20
Table 2: Ratio (R) and the probability content (C) of the 95% credible intervals for the intra-class

correlation (μ) over the 1000 simulated datasets by the degree of association (low, medium, high)

and  ̄ve values of the intra-class correlations (.2, .4, .5, .6, .8)

Low Medium High

μ R C R C R C

0.2

1.04

1.05

1.07

1.12

0.94

0.95

0.94

0.96

1.06

1.06

1.05

1.12

0.95

0.96

0.95

0.97

1.45

1.16

1.10

1.12

0.68

0.92

0.94

0.96

0.4

0.99

1.02

0.99

1.03

0.94

0.94

0.96

0.95

1.01

1.00

1.01

1.01

0.95

0.97

0.95

0.96

1.15

1.05

1.01

1.02

0.86

0.92

0.95

0.96

0.5

0.99

1.00

1.00

1.00

0.95

0.94

0.95

0.97

1.00

1.00

1.00

0.99

0.94

0.95

0.96

0.96

1.10

1.02

1.00

1.00

0.87

0.93

0.95

0.96

0.6

0.99

0.99

1.00

0.98

0.94

0.95

0.96

0.96

0.99

0.99

1.00

0.98

0.95

0.96

0.94

0.95

1.05

1.02

1.00

0.99

0.92

0.94

0.96

0.96

0.8

0.99

0.99

0.99

0.96

0.96

0.95

0.95

0.95

0.99

0.99

0.98

0.97

0.95

0.94

0.95

0.95

1.01

0.99

0.98

0.96

0.95

0.96

0.96

0.95

NOTE: For the i

th dataset Ri = PMi=DVi

, where PMiis the posterior maen of μ and DViis the

design value of μ; R is the average over the 1000 simulated datasets.

21
Table 3: Comparison of the two models via the ratio (R) and the probability content (C) of the

95% credible intervals for the 1⁄4jk over the 1000 simulated datasets by the degree of association

(low, medium, high) and three values of the intra-class correlation (.2, .5, .8)

Low Medium High

μ RI RN CI CN RI RN CI CN RI RN CI CN

1.00 1.00 0.95 0.91 0.99 0.99 0.94 0.89 1.00 1.00 0.96 0.88

1.01 1.01 0.96 0.90 1.00 1.00 0.95 0.90 1.04 1.02 0.96 0.91

1.00 1.00 0.96 0.91 1.00 0.99 0.95 0.91 1.19 1.13 0.96 0.91

1.00 0.99 0.95 0.91 1.01 1.01 0.96 0.91 1.06 1.04 0.95 0.89

0.2 1.00 1.01 0.95 0.90 1.00 1.00 0.96 0.90 0.97 0.99 0.95 0.88

1.00 1.00 0.96 0.92 1.01 1.01 0.95 0.90 1.07 1.05 0.95 0.90

1.00 1.01 0.94 0.88 1.03 1.02 0.93 0.88 1.05 1.02 0.95 0.91

0.99 0.99 0.96 0.89 1.00 1.00 0.95 0.90 1.02 1.00 0.95 0.89

1.00 1.00 0.95 0.90 0.99 1.00 0.96 0.89 0.98 0.99 0.94 0.90

1.01 1.00 0.94 0.81 0.99 1.00 0.95 0.93 1.00 1.00 0.96 0.83

1.00 1.00 0.96 0.82 0.99 1.00 0.95 0.83 1.02 1.00 0.96 0.82

1.00 1.00 0.95 0.82 1.00 1.00 0.94 0.81 1.25 1.15 0.94 0.82

0.99 0.99 0.94 0.81 1.02 1.02 0.95 0.84 1.07 1.03 0.95 0.81

0.5 1.00 1.00 0.95 0.82 1.00 1.00 0.94 0.82 0.97 0.99 0.96 0.84

1.00 1.00 0.95 0.83 1.01 1.01 0.96 0.84 1.08 1.04 0.94 0.83

1.01 1.01 0.94 0.82 1.02 1.01 0.94 0.83 1.05 1.02 0.95 0.82

1.00 1.00 0.94 0.81 1.01 1.00 0.96 0.82 1.02 1.01 0.97 0.84

1.00 1.00 0.95 0.83 0.99 0.99 0.95 0.82 0.98 0.99 0.95 0.83

0.99 1.00 0.95 0.74 0.99 0.99 0.94 0.75 0.99 0.99 0.94 0.75

1.00 1.00 0.96 0.76 0.99 0.99 0.96 0.75 1.06 1.04 0.95 0.75

0.99 0.99 0.95 0.75 1.00 1.00 0.95 0.77 1.30 1.15 0.96 0.75

1.01 1.00 0.96 0.78 1.00 0.99 0.94 0.75 1.09 1.05 0.95 0.77

0.8 1.00 1.00 0.95 0.77 1.00 1.00 0.95 0.75 0.98 1.00 0.95 0.77

1.01 1.01 0.94 0.76 1.02 1.02 0.94 0.75 1.10 1.05 0.95 0.76

1.00 1.01 0.96 0.77 1.04 1.02 0.95 0.78 1.03 1.01 0.95 0.75

1.00 1.00 0.95 0.77 1.01 1.01 0.95 0.75 1.01 1.00 0.97 0.79

1.01 1.00 0.94 0.76 1.00 1.00 0.94 0.75 0.97 0.99 0.94 0.77

NOTE: The two models are the model with intra-class (I) correlation and the model with no (N)

intra-class correlation (see Appendix B). For the i

th dataset Ri = PMi=DVi

, where PMiis the

posterior mean of 1⁄4jk and DVi

is the design value of 1⁄4jk; R is the average over the 1000 simulated

datasets.

22
Table 4: Comparison of the test based on the logarithm of Bayes factor and the test based on the

chi-squared statistic by the degree of association (low, medium, high) and  ̄ve values of intra-class

correlation (.2, .4, .5, .6, .8) averaged over the 1000 simulated datasets

Low Medium High

μ N Y 2 Âu

2 Âa N Y 2 Âu2 Âa N Y u2 Â a2 Â

0.2 -3.6 -4.2 5.8 5.2 21.8 17.8 56.5 49.9 226.3 194.3 489.0 426.6

0.4 -2.7 -3.9 7.8 6.2 22.7 14.8 58.0 46.5 227.0 167.6 490.4 389.7

0.5 -2.5 -3.8 8.1 6.2 22.3 13.1 57.0 43.7 228.6 153.6 493.0 374.5

0.6 -2.0 -3.6 9.2 6.7 23.3 12.1 59.2 43.3 228.5 138.5 492.0 358.1

0.8 -1.0 -3.2 11.2 7.5 24.7 9.5 62.0 41.8 227.3 110.6 489.1 329.2

NOTE: The Bayes factor is the ratio of the marginal likelihood for a model with (Y) association

P P r

j=1

c

k

(i.e., no restriction on 1⁄4jk; =1 1⁄4jk = 1) to the marginal likelihood for a model with no

(1) (2) P P (1)

k u a

(2) = 2 2 1). Also, Â and Â

r

j

c = 1⁄4 k=1 1⁄4

repectively the unadjusted and adjusted chi-squared statistic.

j

(N) association (i.e., 1⁄4jk 1⁄4 ; =1 1⁄4 = 1; are k j

Table 5: Classi ̄cation of the simulated data sets by statistical signi ̄cance using the adjusted chisquared statistic (reject: pvalue< :05; accept: pvalue ̧ :05) and the strength of evidence using the

logarithm of the Bayes factor (weak: log-Bayes factor < 3; strong: log-Bayes factor  ̧ 3) for  ̄ve

values of intra-class correlation (.2, .4, .5, .6, .8)

μ reject

weak

accept total reject

strong

accept total

.2

.4

.5

.6

.8

83

141

134

144

209

77

71

99

104

126

160

212

233

248

335

25

75

71

95

75

815

713

696

657

590

840

788

767

752

665

total 711 477 1188 341 3471 3812

NOTE: Inference using the model with intra-class correlation is compared with inference from the

adjusted chi-squared test.

23