TESTING EQUALITY OF CORRELATION COEFFICIENTS

FOR PAIRED BINARY DATA FROM MULTIPLE GROUPS

Outline• Background• Testing Methods• Simulation Study• Real Work Example• Conclusion

Background

Introduction• In clinical trials studying diseases at paired body parts, each

person contributes two measurements to the study.• Outcomes from the same patient can be highly correlated.• Taking eyes as an example, for a single patient, the probability that

one eye has disease often increases given the knowledge that the other one does.

• One of the models to deal with these binary correlated data is the equal correlation coefficients model.

• Before using this model, here we need to test if the correlation coefficients in each groups are actually equal.

Two Possible Method• Rosner’s modelAssume that for a single patient, the conditional probability of one eye having disease given disease

response at the other eye is R times the unconditional probability.

where Zijk=1 if the kth eye of jth individual in the ith group has a response at the end of the study, and 0 otherwise.Where constant R is the same in each of the g groups.

• Donner’s modelAssumes that the g groups share a common intra-class correlation coefficient.

Basically focus on this model in this paper.

Binary correlated data structure

N patients are randomly assigned to g group according to age.The number of patients in the each group are defined as , i=1,…g.The observed data are recorded as a vector:

Testing Method

Method(Equal correlation coefficients model)Donner (1988) proposed the model that assumed equal intra-class correlation in both of the subgroups.

• Let if there is response from the th eye(=1,2) of the th patient (=1,2,…, ) in the th group (1,2…,g ).

• The model implies that the non-improvement probabilities for none, one, or both sides of the paired body part are• ,

Pr() =Corr(, )=

• Our goal is to test the hypothesis that the two correlation coefficients from each treatment groups are equal using 3 test statistics.( are equal)

Likelihood ratio test Wald-type statistics Score test statistics

Log Likelihood Function• It is straightforward to show that the log-likelihood function for

is given by

• Under , we have and the log-likelihood function is

Likelihood Ratio Test• Setting differentiation of with respect to ’s and ’s equals to zero yields the MLEs of parameters.

• Under , let be the constrained maximum likelihood estimators. Similar approach that differentiation and set them to zero can be used.

• However no closed-form solution exist, we apply the fisher scoring algorithm.

• Since fisher scoring have problem of slow convergence, simplify the iteration by solving a third order polynomial.

• Obtain likelihood ratio test statistic which has asymptotically distribution under .

Wald-type test• Rewrite the null hypothesis in matrix form as here ,and

• The Wald-type test statistics has the form

• After some considerable algebra and plug in MLEs of the parameters, obtain test statistics, which has asymptotically distribution under .

where

Score test• Score test statistic uses MLEs of parameters under the null hypothesis.

• The score is a vector.

• The score test is given by

Where is the in formation matrix ,and it has asymptotically distribution under .

Simulation Study

• Type I error set up

; .(each configuration generate 10000 samples under the null)

• Power set up (Sample size=100)

g24

8

g2 =4 =8 =

Power plots with m=100,

Power plots with m=100, ,6.

Real Work Examples

• Example1:218 outpatients, aged from 20 to 29 with retinitis pigmentosa (RP),were assigned into 4 genetic type groups.

Table of the data(number of effected eyes for persons in each genetic type group)

Table of result(Statistic values and p-values of different test statistics)

• Example2:the extend and causes of blindness and visual impairment (VI) .

Table of the data prevalence of VI by age groups in the sample population.

Table of result(Statistic values and p-values of different test statistics)

• Example3:the MRSS case-control clinical trial which enrolled 168 patients with diffuse scleroderma. Patients are randomly given oral native collagen or placebo, and compare the MRSS(modified Rodnan Skin Scores) in treatment group and control group.

Conclusion

• Introduce three test statistics for testing the equality of correlation coefficients in paired binary data with different groups.

• Score test is recommended in practical use. Since simulation study showed that the Score test has not only robust empirical type I error for various number of groups and sample sizes, but also satisfactory power.

• With these asymptotic methods studied in this work, we consider developing exact tests for small samples as interesting future work.

Thank you!