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Testing the Ratio of Two Poisson Rates Testing the Ratio of Two Poisson Rates Kangxia Gu1, Hon Keung Tony Ng∗1, Man Lai Tang2, and William R. Schucany1 1 Department of Statistical Science, Southern Methodist University, Dallas, Texas 75275-0332, USA 2 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Kowloon, Hong Kong Summary In this paper we compare the properties of four different general approaches for testing the ratio of two Poisson rates. Asymptotically normal tests, tests based on approximate p-values, exact conditional tests, and a likelihood ratio test are considered. The properties and power performance of these tests are studied by a Monte Carlo simulation experiment. Sample size calculation formulae are given for each of the test procedures and their validities are studied. Some recommendations favoring the likelihood ratio and certain asymptotic tests are based on these simulation results. Finally, all of the test procedures are illustrated with two real life medical examples. Key words: Asymptotic tests, Conditional test, Constrained maximum likelihood estimation, Level of sig- nificance, Mid-p; Monte Carlo simulation, Power. 1 Introduction The Poisson Distribution, named after the famous French mathematician Simeon Denis Poisson who first formulated it (Poisson, 1837), is a mathematical rule that assigns probabilities to the number of occurences. It is most commonly used to model the number of random occurrences of some phenomenon in a specified space or time interval. Sometimes the Poisson rate is of interest because it pertains to a unit of time or ∗ Corresponding author: e-mail: [email protected], Phone: +1-214-768-2465, Fax: +1-214-768-4035 2 K. Gu, H.K.T. Ng, M.L. Tang, and W.R. Schucany: Testing of Poisson Rates space. For example, the incidence rate of a disease (Rothman and Greenland, 1998) is defined as the number of events observed divided by the time at risk during the observation period. Biological, epidemiological, and medical research can produce data that follow a Poisson distribution. It is sometimes called the law of rare events, for it describes the distribution of counts of such events. For example, the number of cases of a rare disease or the frequency of single gene mutations may be modeled by a Poisson distribution. In two-sample situations, researchers are likely to be interested in testing the difference or ratio of the Poisson means. The comparison of the Poisson rates from two independent sam- ples is clearly of medical interest. For instance, Stampfer and Willett (1985) conducted a prospective study to examine the relationship of post-menopausal hormone use and coronary heart disease (CHD). With postmenopausal hormone use in 54308.7 person-years, there are 30 CHD cases; without postmenopausal hormone use in 51477.5 person-years, there are 60 CHD cases. The problem of interest is to test whether the post-menopausal hormone using group has less coronary heart disease. Another example in epidemi- ology is a breast cancer study reported in Rothman and Greenland (1998) and Graham, Mengersen, and Morton (2003). Two groups of women were compared to determine whether those who had been examined using x-ray fluoroscopy during treatment for tuberculosis had a higher rate of breast cancer than those who had not been examined using x-ray fluoroscopy. Forty-one cases of breast cancer in 28,010 person-years at risk are reported in the treatment group with women receiving x-ray fluoroscopy and 15 cases of breast cancer in 19,017 person-years at risk in the control group with women not receiving x-ray fluoroscopy. The problem of comparing two Poisson rates has been studied for a long time, however, most of the early studies focused on the equal time frame situation. Przyborowski and Wilenski (1939) first proposed a conditional test for the unity of the ratio of two Poisson rates based on a conditional binomial distribution. Subsequently, Chapman (1952), Birnbaum (1953), Brownlee (1967) and Gail (1974) studied the properties and some alternatives for testing the hypothesis or constructing confidence intervals based on the condi- tional approach. On the other hand, Hald (1960), Ractliffe (1964), Cox and Lewis (1966), Haight (1967) Testing of Poisson Rates 3 Detre and White (1970) and Sichel (1973) investigated asymptotic tests based on normal approximations for the equality of the two Poisson rates. Development for unequal sampling frames has received more attention recently. Shiue and Bain (1982) derived a uniformly most powerful unbiased (UMPU) test for equality of two Poisson rates and showed that a test based on the normal approximation of the binomial distribution is nearly as powerful as the UMPU test. Huffman (1984) proposed an improved asymptotic test statistic, which accelerated the rate of convergence to normality by a variance stabilizing transformation. Along the same line, Thode (1997) considered an alternative test statistic that is more powerful than Shiue and Bain statistic for large Poisson rates. Recently, Ng and Tang (2005) provided systematic comparisons among these tests and presented the associated sample size formulae. All these studies are devoted to the problem of testing the equality of the two Poisson rates (or the unity of the ratio) only. Krishnamoorthy and Thomson (2004) considered tests based on estimated p-values and showed that these have uniformly better power than the conditional test. Although many methods have been proposed to test the ratio of two Poisson rates, it is not clear when techniques are more appropriate, or how their performances might vary. Moreover, most of the works focus on testing the unity of the ratio. In the present article, we present an extensive and systematic comparative study of different test procedures for testing the non-unity of the rate ratio of two Poisson processes under unequal sampling frames. In Section 2, we review the problem of testing the general non-unity ratio of two Poisson rates over unequal-size sampling frames. Asymptotic tests based on normal approximations, tests based on approximate p-value methods, an exact conditional test, a mid-p conditional test, and a likelihood ratio test are considered. In Section 3, we present the formulae for sample size calculation for each of the test procedures. In Section 4, Monte Carlo simulations are used to study the properties of the different test procedures, as well as the validity of the sample size calculation formulae. Two real life examples are used to illustrate the test procedures in Section 5. Finally, a discussion of the results with some recommendations are presented in Section 6. 4 K. Gu, H.K.T. Ng, M.L. Tang, and W.R. Schucany: Testing of Poisson Rates 2 Test Procedures 2.1 Asymptotic tests based on normal approximations For fixed sampling frames t0 and t1, two independent Poisson processes (with parameters λ0 and λ1) are observed. Let X0 and X1 be the corresponding number of outcomes, i.e. Xi ∼ Poisson(λi) with λi = tiγi for i =0, 1. Here, γi, i =0, 1 are the Poisson rates. We denote the observed values of X0 and X1 by x0 and x1, respectively. We are interested in testing that the ratio of two Poisson rates is equal to a pre-specified positive number R versus greater than R, i.e. we are testing the following one-sided hypotheses H0 : γ0/γ1 = R against H1 : γ0/γ1 >R. (1) When R =1, it is equivalent to testing the equality of the two Poisson rates, i.e. γ0 = γ1. The properties of different test procedures for R =1under unequal sampling frames, i.e. t0 =6 t1, were investigated in Ng and Tang (2005). Since γi, i =0, 1 are the unknown parameters, two sample estimates for γi are considered as follows. • Unconstrained Maximum Likelihood Estimate (MLE) The maximum likelihood estimate of γi is given by Xi γˆi = ,i=0, 1. (2) ti • Constrained Maximum Likelihood Estimate (CMLE) Under the null hypothesis H0: γ0/γ1 = R, the CMLEs of γ0 and γ1 can be shown to be (see Appendix A) X0 + X1 X0 + X1 γ˜0 = and γ˜1 = , (3) t0(1 + 1/ρ) t1(1 + ρ) where d = t1/t0 and ρ = R/d. Note that the CMLEs of γ0 and γ1 are the same as the method of moment estimator under the null hypothesis H0 : γ0/γ1 = R. Specifically, E(X0+X1)=t0γ0+t1γ1 with γ0/γ1 = R yields the estimators in (3). Testing of Poisson Rates 5 Because we can re-express the null hypothesis in (1) as H0 : γ0 − Rγ1 =0, we can develop the test statistics based on the statistic W =ˆγ0 − Rγˆ1. The variance of the statistic W is given by 2 2 γ0 R γ1 σW = + t0 t1 and it can be estimated by ∗ 2 ∗ 2 γ0 R γ1 sW = + , t0 t1 ∗ where γi is any reasonable estimate of γi, i =0, 1. Hence, we consider the test statistic W/sW for H0. ∗ With MLEs γi =ˆγi, i =0, 1, we get X0/t0 − RX1/t1 X0 − X1ρ W1(X0,X1)= = . 2 2 2 2 pX0/t0 + X1R /t1 pX0 + X1ρ ∗ Similarly, with CMLEs γi =˜γi, i =0, 1, we obtain X0/t0 − RX1/t1 X0 − X1ρ W2(X0,X1)= = . 2 p(X0 + X1)(R /d + R)/(R + d) p(X0 + X1)ρ Note that when ρ =1, W1 is equivalent to W2 and is the test statistic studied by Shiue and Bain (1982) and Thode (1997). On the other hand, testing the null hypothesis in (1) is equivalent to testing ln(γ0/γ1) − ln(R)=0. Hence, we consider the statistic U = ln(ˆγ0/γˆ1) − ln(R) = ln(X0/t0) − ln(X1/t1) − ln(R). By the delta 2 method, the variance of U can be approximated by σU =1/(t0γ0)+1/(t1γ1) and it can be estimated by 2 ∗ ∗ ∗ sU =1/(t0γ0 )+1/(t1γ1 ), where γi is any reasonable estimate of γi, i =0, 1.
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