In section 2.3, we introduced methods which have been used to estimate the ICC for clustered ordinal data. In particular, we gave a detailed description of the ANOVA method.
In section 2.4, we proposed a kappa-type ICC estimator ρˆκ by extending Scott’s by Abraira and Vargas’s approach for clustered ordinal outcome data. Moreover, ρˆ was κ
shown to be asymptotically equal to the ANOVA ICC estimator ρˆA as the number of clusters becomes large. We further discussed ρˆ ’s properties, including its reduction to κ Scott’sπ, the minimum value, and options of imposed scores.
To summarize the ICC estimators discussed in this chapter, we list all ICC estimators Table 2.2. We will conduct simulation studies to evaluate ρˆA and ρˆκ and their relationships and properties in Chapter 6.
Table 2.2: Summary of the ICC estimators discussed in Chapter 2
Estimator Method General case
Special cases Minimum values
m mij = n ni = 2 = =m mij n ni = 2 = k Under HA Under H0 General case m mij = n ni = General case m mij = n ni = A
ρˆ ANOVA ICC estimator Equation (2.1)
and (2.7) Equation (2.9)
κ ρˆ
kappa-type ICC estimator from two populations
Equation (2.5) Equation (2.8) Equation (2.13) Equation (2.16) Equation (2.17) Equation (2.18) Equation (2.17) κ ρˆ' kappa-type ICC estimator from one single population
Equation (2.15)
π
ˆ Scott’sπ
ˆ Equation (2.2) Equation (2.14)overall
π
ˆ Scott’s from twoChapter 3
Adjusted Cochran-Armitage Tests for Clustered Ordinal
3
Outcomes
3.1 Introduction
In the previous chapter, we have presented methods for estimating the ICC for clustered ordinal outcome data. In the following chapters we discuss methods for analysis of clustered ordinal outcome data. We start with direct adjustment approaches which adapt simple corrections to the Cochran-Armitage test statistic for clustering effects.
The Cochran-Armitage trend test is a well-known approach for comparing binomial proportions among ordered groups. For independent ordinal outcome data, the Cochran- Armitage test statistic may equivalently be used to compare ordinal scores for two samples (Yates, 1948; Armitage, 1955). However, for clustered outcome data, the Cochran-Armitage trend test for comparing binary data can not be directly used to compare ordinal data.
She et al. (2010) extended the Cochran-Armitage test to genetic data from designs involving multistage cluster sampling. For each individual, they assigned the inverse of the product of the selection probabilities across all the stages of sampling as the weight. Then they adjusted all observed size in the Cochran-Armitage test statistic by the weights. However, its application to clustered ordinal outcomes was not discussed.
To extend the Cochran-Armitage test statistic for correlated ordinal data, Jung and Kang (2001) proposed a variance for the difference of scores between two groups that is obtained by standardizing the correlated scores. Although this approach takes into account the dependencies within clusters, the intraclass correlation coefficient (ICC) does not need to be specified.
Donner and Donald (1988) applied simple correction procedures to the Cochran- Armitage test to compare correlated binary outcomes on an ordinal cluster-level covariate. Their method, which utilizes an ICC for clustered binary data, offers such
advantages as simplicity and easy implementation. It does not necessarily require complicated computation and specified software. However, unlike the situation for independent outcome data, one cannot directly apply this adjusted test statistic to analyses of correlated ordinal data since the ICC for correlated binary outcome data is not equal to the ICC for correlated ordinal outcome data. Therefore a new ICC for correlated ordinal data must be used to obtain an adjusted version of the Cochran-Armitage trend test in this case.
In addition to Donner and Donald’s approach, we extend the Cochran-Armitage trend test to clustered data using a weighted least squares approach. The Cochran-Armitage test was originally derived from a simple linear probability model by using the ordinary least squares approach (OLS) (Cochran, 1954; Armitage, 1955). However, the underlying assumptions of the OLS procedure are violated in cluster randomization trials where clustering induces a correlation among observations. In this case a more efficient estimator obtained by the weighted least square (WLS) approach may be used instead as an extension of the OLS procedure although the bias of estimator is unaffected by the choice of using OLS or WLS approach. Thus we adjust the Cochran-Armitage test to clustered outcome data by extending the OLS approach to a WLS approach.
In this chapter, we develop three simple adjustments to the regular Cochran-Armitage chi-square statistics for clustered binary data and clustered ordinal data respectively. The first one is Donner and Donald (1988)’s adjustment which is obtained by modifying the observed sample sizes of both the point estimate and its variance estimate in the test statistic; the second one is distinct in that it adjusts only the variance estimator in the statistic; the third one derives the statistic using a WLS approach. We list all six statistics in Table 3.1. The subscript ‘CB’ denotes clustered binary and ‘CO’ denotes clustered ordinal. In addition, the subscript ‘(1)’ denotes the first adjustment method described above, ‘(2)’ the second adjustment method, and ‘(3)’ the third adjustment method.
The rest of the chapter is organized as follows. In section 3.2, we describe the Cochran- Armitage test for independent ordinal outcome data; in section 3.3, we present three
adjusted Cochran-Armitage tests for clustered binary outcome data; in section 3.4, we develop three adjusted Cochran-Armitage trend tests for clustered ordinal outcome data.