# Odds Ratios Estimation of Rare Event in Binomial Distribution

Authors: Kobkun Raweesawat; Yupaporn Areepong; Katechan Jampachaisri; Saowanit Sukparungsee
Journal: Journal of Probability and Statistics
 (2016)

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## Abstract

We introduce the new estimator of odds ratios in rare events using Empirical Bayes method in two independent binomial distributions. We compare the proposed estimates of odds ratios with two estimators, modified maximum likelihood estimator (MMLE) and modified median unbiased estimator (MMUE), using the Estimated Relative Error (ERE) as a criterion of comparison. It is found that the new estimator is more efficient when compared to the other methods.


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## Body

## 1. Introduction

The odds ratio is a measure of association between two independent groups on a categorical response with two possible outcomes, success and failure. The two independent groups can be two treatment groups or treatment and control groups. The odds ratio is widely used in many fields of medical and social science research. It is most commonly used in epidemiology to express the results of some clinical trials, such as in case-control studies.A number of subjects in each group falling in each category can be summarized in a two-way contingency table. Total numbers of subjects in group 1 and group 2 aren1 and n2, which are assumed to be fixed. Numbers of successes in group 1 and group 2 are Y1 and Y2, which are considered as independent binomial random variables. Let π1 and π2 be probabilities of success in group 1 and group 2, respectively. The odds of success in group 1 are defined to be odds1=π1/1-π1, similar to group 2. The usual maximum likelihood estimator of odds ratio is defined as(1)OR^MLE=odds1odds2=π1/1-π1π2/1-π2=Y1/n1-Y1Y2/n2-Y2=Y1n2-Y2Y2n1-Y1.Odds ratio is nonnegative real value. When successes are similar in both groups, the odds ratio is equal to 1, meaning that groups are independent of response. When the odds of a positive response are higher in group 1 than in group 2, the odds ratio is greater than 1 and vice versa for the value less than 1. The father of odds ratio from 1 in a given direction represents stronger association. In addition, its sampling distribution is highly skewed. Sample natural logarithm of odds ratio, which is less skewed, is often utilized for inference. However, odds ratio can be zero (if zero cell count appears in numerator of (1)) or infinity (if zero cell count is in denominator of (1)) or undefined (if there are zero cell counts in both the numerator and denominator of (1)). Haldane [1] and Gart and Zweifel [2] suggested to add a correction term 0.5 to each cell, when having zero cell count, which gives the modified maximum likelihood estimator (MMLE) as(2)OR^MMLE=Y1+0.5n2-Y2+0.5Y2+0.5n1-Y1+0.5.Even though OR^MMLE still laid between 0 and infinity, some investigators discouraged adding 0.5 to each cell because of the appearance of adding “fake data”; see Bishop et al. [3] and Agresti and Yang [4]. Among controversy, several similar alternatives to this modified maximum likelihood estimator have been proposed. Hirji et al. [5] proposed the median unbiased estimator (MUE) of the odds ratio, obtained from the conditional noncentral hypergeometric distribution. However, the median unbiased estimator of the odds ratio still caused a problem when Y1=n1 and Y2=n2 or Y1=Y2=0, and then the MUE was undefined. Parzen et al. [6] proposed an estimator of the odds ratio based on MUE called the modified median unbiased estimator (MMUE) of which the estimated probability of success was always in the interval 0,1, even if there were 0 or n successes in each group. Consequently, the estimated odds ratio always laid between 0 and infinity. Additionally, this method performed well with respect to bias in small sample and was an alternative to adding “fake data.”In this paper, we focus on “rare events” which occasionally observed zero or small counts of interesting events which happened within a given time period or a given sample such as natural disasters or some diseases. As aforementioned, rare events caused difficulty in estimation of odds ratio due to the occurrence of zeros or small observed counts in numerator or in denominator or in both, resulting in the large standard error and therefore less precise confidence interval. Only a rough estimate of the odds ratio is thus obtained. Researches involving association between categorical variables in contingency table have long been studied, using both classical and Bayesian approaches. Good [7] studied association factor, at early stage, in large contingency table with small entries, assuming log-normal and Pearson type III distribution. The author also mentioned that these assumptions may be less accurate but easy to handle. Fisher [8] estimated the odds ratio based on hypergeometric distribution utilizing exact method in a 2×2 table. Thomas and Gart [9] constructed a table for 95% confidence limits of differences and ratio of two proportions, including odds ratio and one-tailed p value for Fisher-Irwin Exact test in various types of 2×2 table. Altham [10] studied association and exact p value in a 2×2 contingency table based on the cumulative posterior probabilities which was not easy to extract. Nurminen and Mutanen [11] proposed Bayesian approach for the estimation of difference between two proportions, risk ratio and odds ratio, using independent beta prior and provided integral expressions for the cumulative posterior distribution. They also applied the proposed method to real data regarding malignant lymphoma and colon cancer cases exposed to phenoxy acids and chlorophenols in agriculture. Nouri et al. [12] presented the estimation of the odds ratio in 2×2×J tables when exposure was misclassified. They compared the matrix and inverse matrix methods to the MLE method using simulation study and found that the inverse matrix method having a closed form was more efficient than the matrix method.As previously mentioned, the estimates of association measure in two-way contingency table can be carried out based on classical and Bayesian approaches. The exact distribution using classical approach is, however, rather difficult for mathematical tractability. In Bayesian approach, where prior belief is incorporated into derivation of posterior density, the hyperparameters, characterizing the prior density, are often unknown to researchers and need to be assessed irrespective of current data. However, controversy still exists. Alternatively, the estimation of hyperparameters is plausibly carried out with the notion of Empirical Bayes method using current data to estimate the unknown hyperparameters, contrary to Bayesian approach. As a consequence, we focus on the utilization of Empirical Bayes method to estimate the odds ratio in a two-way contingency table, focusing on small proportions of success. Our purposed estimation tends to outperform the traditional estimator, MMLE, and MMUE without interference in the original data.The rest of this paper is organized in the following sequence. In the next section, we discuss the median unbiased estimator. The third section describes the odds ratio estimation using EB method. The forth section illustrates simulated results, and the efficiency of EB is compared with MMLE and MUE. The fifth section displays the application of our method to real data. Our conclusion is drawn in the final section.


## 2. The Modified Median Unbiased Estimator of Odds Ratio

Parzen et al. [6] suggested the modified median unbiased estimator (MMUE) in two independent binomial distributions. Let p~ be the estimator of success probability which satisfies(3)Pp~≤p≥0.5,Pp~≥p≥0.5.To obtain p~t, they use the binomial distribution, Yt~Bnt,pt, where Yt denotes random variable representing success in the tth group t=1,2. Let yt be the observed value of Yt.(4)PYt=yt∣pt=ntytptyt1-ptnt-yt.The MMUE can be computed from the distribution of sufficient statistics for binomial data.Compute the values ofptL and ptU to be those value of pt for which (5)PYt≥yt∣pt=p~tL≥0.5,PYt≤yt∣pt=p~tU≥0.5,where ptL and ptU are the smallest and largest values of pt, respectively. Then, the MMUE is defined as(6)p~t=p~tL+p~tU2.When 0<Yt≤nt, we can find values of  p~tL and p~tU which satisfy(7)PYt≥yt∣pt=p~tL=PYt≤yt∣pt=p~tU=0.5.Then, solve from(8)0.5=PYt≥yt∣pt=p~tL=∑i=ytntntip~tLi1-p~tLnt-i,and solve p~tU from(9)0.5=PYt≤yt∣pt=p~tU=∑i=0ytntip~tUi1-p~tUnt-i.The values of  p~tL and p~tU can then actually be obtained by using the relationship between the cumulative beta distribution and the cumulative binomial distribution function as follows (Daly [13] and Johnson et al. [14]).LetU~Betaα,β:(10)Fpt,α,β=∫0ptΓα+βΓαΓβUα-11-Uβ-1dU=∑i=αα+β-1α+β-1ipti1-ptα+β-1-i=∑i=αntntipti1-ptnt-i.We need to find p~tL and p~tU such that(11)Fp~tL∣α=yt,β=nt-yt+1=0.5,Fp~tU∣α=yt+1,β=nt-yt+2=0.5.In particular,(12)p~tL=F-10.5∣α=yt,β=nt-yt+1,p~tU=F-10.5∣α=yt+1,β=nt-yt+2,where F-1Q∣α,β is the Qth quantile of the beta-distribution with parameters α and β.Now supposeyt=0, and then (13)PYt≥yt∣pt=p~tL=PYt≥0∣pt=p~tL=1,∑i=0ntntip~tLi1-p~tLnt-i=1.Any value of p~tL in the interval 0,1 satisfies(14)PYt≥0∣pt=p~tL≥0.5,where p~tL=0 is the smallest possible value of p~tL.Similarly, whenyt=0,  p~tU satisfies(15)PYt≤yt∣pt=p~tU=PYt=0∣pt=p~tU=0.5,nt0p~tU01-p~tUnt-0=0.5,p~tU=1-0.51/nt.Consequently, yt=0;  p~t equals(16)p~t=p~tL+p~tU2=1-0.51/n2.Similarly, when p~tU=1 is the largest possible value of p~tU, then p~tL satisfies(17)nnp~tLn1-p~tLn-n=0.5,p~tL=0.51/nwhen yt=n and p~t=p~tL+p~tU/2.Then, the MMUE of odds ratio estimation is defined as(18)OR^MMUE=p~1/1-p~1p~2/1-p~2,where p~1 and p~2 denote success probability estimators in groups 1 and 2, respectively.


## 3. Proposed Estimation of Odds Ratio

In this section, we proposed a new method for odds ratio estimation using Empirical Bayes method in two independent binomial distributions. LetY1 and Y2 be random variables, distributed as binomial with equal and unequal sample sizes and unknown probability, Y1~Binn1,p1 and Y2~Binn2,p2, where n1,n2 and p1,p2 denote two sample sizes and two unknown success probabilities. Adopt information priors on pi,  pi~betaαi,βi,  i=1,2, where αi and βi denote unknown hyperparameters. The estimation of hyperparameters can be obtained from the posterior marginal distribution function as follows:(19)my∣α,β=∫01fy∣pπpdp=nyΓα+βΓαΓβΓy+αΓn-y+βΓn+α+β.Consequently, the posterior marginal distribution function of y is the beta-binomial distribution (BBD).Then, both hyperparameters in each group can be estimated using maximum likelihood method. The likelihood function of posterior marginal distribution function is then written as(20)Ly∣α,β=∏i=1nnyiΓα+βΓαΓβΓyi+αΓn-yi+βΓn+α+β=∏i=1nnyiα+yi-1α+yi-2⋯αβ+n-yi-1β+n-yi-2⋯βα+β+n-1α+β+n-2⋯α+β.Applying Newton-Raphson method to solve a nonlinear equation, the r+1th maximum likelihood estimator of hyperparameters r=1,2,… can be obtained from(21)α^r+1β^r+1=α^rβ^r-∂2Lr∂α2∂2Lr∂α∂β∂2Lr∂β∂α∂2Lr∂β2-1∂Lr∂α∂Lr∂β,where(22)∂2Lr∂α2=-∑i=1n1αr+yi-12+1αr+yi-22+⋯+1αr2+∑i=1n1αr+βr+n-12+1αr+βr+n-22+⋯+1αr+βr2,∂2Lr∂α∂β=∑i=1n1αr+βr+n-12+1αr+βr+n-22+⋯+1αr+βr2,∂2Lr∂β2=∑i=1n1βr+n+yi-12+1βr+n+yi-22+⋯+1βr2+∑i=1n1αr+βr+n-12+1αr+βr+n-22+⋯+1αr+βr2,∂Lr∂α=∑i=1n1αr+yi-1+1αr+yi-2+⋯+1αr-∑i=1n1αr+βr+n-1+1αr+βr+n-2+⋯+1αr+βr,∂Lr∂β=∑i=1n1βr+n+yi-1+1βr+n+yi-2+⋯+1βr-∑i=1n1αr+βr+n-1+1αr+βr+n-2+⋯+1αr+βr,where the moment estimators of hyperparameters in beta-binomial distribution are used as initial values; see Minka [15].The posterior distribution function ofp is thus calculated, yielding(23)πp∣y,α,β=py+α-11-pn-y+β-1Γn+α+βΓy+αΓn-y+β.Substituting the estimators of α and β, we obtain(24)p∣y,α,β~Betay+α^,n-y+β^.Let p1′ and p2′ be estimators of p1 and p2, respectively, where(25)p1′=y1+α^1n1+α^1+β^1,p2′=y2+α^2n2+α^2+β^2.Thus, the EB estimator of odds ratio can be obtained as follows:(26)OR^EB=p1′/1-p1′p2′/1-p2′,where p1′ and p2′ denote success probability estimators in groups 1 and 2, respectively.


## 4. Simulation Study for MMLE, MMUE, and EB Method

Simulation studies have been carried out using R program (version 3.2.0) [16] to assess the efficiency of the EB method in comparison with two existing methods. Binomial data are generated with equal and unequal sample sizes: n1,n2=10,10,10,30 with 10,50 probabilities of success in group 1: p1=0.01,0.03,0.05,0.1, and 0.15. For each value of p1,p2 is varied to 0.01, 0.03, 0.05, 0.1, and 0.15. Each situation is repeated 5,000 times after removing the first 1,000 iterations (1,000 burn-ins). The efficiency of proposed estimator is evaluated using Estimated Relative Error (ERE), defined as(27)ERE=OR-OR^iOR,where OR denotes the usual maximum likelihood estimator of odds ratio and OR^i denotes the estimate of odds ratio using EB, MMLE, and MMUE i=1,2,3,…, respectively.The simulation results with odds ratio estimates for sample sizesn1,n2=10,10,10,30 and 10,50 are given in Tables 1–3. The performance of estimation uses ERE given in Tables 4–6 and compares this result with graph in Figure 1; the other case provides similar results. It is found that the odds ratio estimation using EB method mostly yields smallest ERE with 78.67%, while those using MMLE and MMUE methods result in smallest ERE with only 6.67% and 14.66%, respectively.Table 1
The estimated values of odds ratio forn1,n2=10,10.
p 1 , p 2 O R O R ^ E B O R ^ M M L E O R ^ M M U E (0.01, 0.01) 1.0000 1.3665 1.1514 1.2043 (0.01, 0.03) 0.3266 0.3931 1.0029 1.0377 (0.01, 0.05) 0.1919 0.2248 0.8723 0.8935 (0.01, 0.10) 0.0909 0.1040 0.6219 0.6184 (0.01, 0.15) 0.0572 0.0650 0.4481 0.4307 (0.03, 0.01) 3.0619 3.8746 1.6008 1.7848 (0.03, 0.03) 1.0000 1.1119 1.3933 1.5383 (0.03, 0.05) 0.5876 0.6363 1.2128 1.3244 (0.03, 0.10) 0.2784 0.2942 0.8640 0.9156 (0.03, 0.15) 0.1753 0.1838 0.6227 0.6378 (0.05, 0.01) 5.2105 6.5657 2.0724 2.4094 (0.05, 0.03) 1.7018 1.8851 1.8036 2.0763 (0.05, 0.05) 1.0000 1.0787 1.5693 1.7868 (0.05, 0.10) 0.4737 0.4989 1.1181 1.2356 (0.05, 0.15) 0.2982 0.3116 0.8059 0.8609 (0.10, 0.01) 11.0000 13.7434 3.3472 4.1489 (0.10, 0.03) 3.5926 3.9471 2.9135 3.5759 (0.10, 0.05) 2.1111 2.2585 2.5352 3.0777 (0.10, 0.10) 1.0000 1.0445 1.8068 2.1288 (0.10, 0.15) 0.6296 0.6523 1.3027 1.4839 (0.15, 0.01) 17.4706 21.7299 4.7827 6.1533 (0.15, 0.03) 5.7059 6.2349 4.1625 5.3026 (0.15, 0.05) 3.3529 3.5678 3.6225 4.5648 (0.15, 0.10) 1.5882 1.6498 2.5812 3.1568 (0.15, 0.15) 1.0000 1.0303 1.8602 2.1990Table 2
The estimated values of odds ratio forn1,n2=10,30.
p 1 , p 2 O R O R ^ E B O R ^ M M L E O R ^ M M U E (0.01, 0.01) 1.0000 1.3656 2.9352 2.7925 (0.01, 0.03) 0.3266 0.4023 2.0063 1.8490 (0.01, 0.05) 0.1919 0.2279 1.4146 1.2576 (0.01, 0.10) 0.0909 0.4043 0.6802 0.5486 (0.01, 0.15) 0.0572 0.0652 0.3981 0.2971 (0.03, 0.01) 3.0619 3.8640 4.0816 4.1395 (0.03, 0.03) 1.0000 1.1385 2.7872 2.7376 (0.03, 0.05) 0.5876 0.6452 1.9663 1.8632 (0.03, 0.10) 0.2784 1.1466 0.9457 0.8131 (0.03, 0.15) 0.1753 0.1845 0.5536 0.4406 (0.05, 0.01) 5.2105 6.5508 5.2833 5.5870 (0.05, 0.03) 1.7018 1.9307 3.6077 3.6950 (0.05, 0.05) 1.0000 1.0940 2.5446 2.5144 (0.05, 0.10) 0.4737 1.9430 1.2236 1.0969 (0.05, 0.15) 0.2982 0.3128 0.7163 0.5944 (0.10, 0.01) 11.0000 13.7159 8.5346 9.6223 (0.10, 0.03) 3.5926 4.0427 5.8289 6.3651 (0.10, 0.05) 2.1111 2.2907 4.1132 4.3339 (0.10, 0.10) 1.0000 1.0327 1.6556 1.5221 (0.10, 0.15) 0.6296 0.6549 1.1578 1.0247 (0.15, 0.01) 17.4706 21.6662 12.1932 14.2687 (0.15, 0.03) 5.7059 6.3850 8.3278 9.4395 (0.15, 0.05) 3.3529 3.6181 5.8732 6.4225 (0.15, 0.10) 1.5882 6.4257 2.8237 2.8019 (0.15, 0.15) 1.0000 1.0345 1.6529 1.5181Table 3
The estimated values of odds ratio forn1,n2=10,50.
p 1 , p 2 O R O R ^ E B O R ^ M M L E O R ^ M M U E (0.01, 0.01) 1.0000 1.5096 4.2803 3.9580 (0.01, 0.03) 0.3266 0.4210 2.3790 2.0799 (0.01, 0.05) 0.1919 0.6975 1.4513 1.1915 (0.01, 0.10) 0.0909 0.1057 0.6167 0.4475 (0.01, 0.15) 0.0572 0.0656 0.3656 0.2515 (0.03, 0.01) 3.0619 4.2760 5.9515 5.8667 (0.03, 0.03) 1.0000 1.1926 3.3063 3.0810 (0.03, 0.05) 0.5876 1.9756 2.0173 1.7653 (0.03, 0.10) 0.2784 0.2992 0.8578 0.6636 (0.03, 0.15) 0.1753 0.1856 0.5083 0.3728 (0.05, 0.01) 5.2105 7.2486 7.7012 7.9149 (0.05, 0.03) 1.7018 2.0224 4.2796 4.1586 (0.05, 0.05) 1.0000 3.3494 2.6114 2.3833 (0.05, 0.10) 0.4737 0.5073 1.1100 0.8954 (0.05, 0.15) 0.2982 0.3147 0.6579 0.5031 (0.10, 0.01) 11.0000 15.1780 12.4415 13.6337 (0.10, 0.03) 3.5926 4.2347 6.9165 7.1666 (0.10, 0.05) 2.1111 7.0132 4.2213 4.1087 (0.10, 0.10) 1.0000 1.0621 1.7938 1.5433 (0.10, 0.15) 0.6296 0.6589 1.0631 0.8669 (0.15, 0.01) 17.4706 23.9776 17.7763 20.2195 (0.15, 0.03) 5.7059 6.6884 9.8777 10.6225 (0.15, 0.05) 3.3529 11.0773 6.0256 6.0860 (0.15, 0.10) 1.5882 1.6775 2.5614 2.2869 (0.15, 0.15) 1.0000 1.0408 1.5181 1.2848Table 4
The percentages of the Estimated Relative Error of odds ratio estimation forn1,n2=10,10.
p 1 , p 2 E R E E B E R E M M L E E R E M M U E (0.01, 0.01) 36.6535 15.1385 20.4316 (0.01, 0.03) 20.3505 206.7910 217.7388 (0.01, 0.05) 17.1281 354.5204 365.6116 (0.01, 0.10) 14.3630 584.0771 580.2372 (0.01, 0.15) 13.4914 682.8616 652.4367 (0.03, 0.01) 26.5436 47.7187 41.7082 (0.03, 0.03) 11.1895 39.3332 53.8303 (0.03, 0.05) 8.2767 106.3838 125.3783 (0.03, 0.10) 5.6857 210.3971 228.9552 (0.03, 0.15) 4.8556 255.2977 263.9451 (0.05, 0.01) 26.0092 60.2273 53.7592 (0.05, 0.03) 10.7763 5.9852 22.0098 (0.05, 0.05) 7.8661 56.9330 78.6812 (0.05, 0.10) 5.3167 136.0498 160.8427 (0.05, 0.15) 4.4721 170.2257 188.6564 (0.10, 0.01) 24.9404 69.5705 62.2831 (0.10, 0.03) 9.8685 18.9032 0.4659 (0.10, 0.05) 6.9818 20.0893 45.7877 (0.10, 0.10) 4.4467 80.6791 112.8820 (0.10, 0.15) 3.6016 106.9073 135.6854 (0.15, 0.01) 24.3801 72.6244 64.7792 (0.15, 0.03) 9.2720 27.0495 7.0675 (0.15, 0.05) 6.4085 8.0392 36.1422 (0.15, 0.10) 3.8773 62.5224 98.7588 (0.15, 0.15) 3.0338 86.0169 119.9015Table 5
The percentages of the Estimated Relative Error of odds ratio estimation forn1,n2=10,30.
p 1 , p 2 E R E E B E R E M M L E E R E M M U E (0.01, 0.01) 36.5558 193.5166 179.2467 (0.01, 0.03) 23.1721 514.2925 466.1248 (0.01, 0.05) 18.7270 637.0581 555.2608 (0.01, 0.10) 344.7753 648.1943 503.4379 (0.01, 0.15) 13.8933 595.4561 419.0998 (0.03, 0.01) 26.1981 33.3063 35.1969 (0.03, 0.03) 13.8541 178.7227 173.7624 (0.03, 0.05) 9.7978 234.6089 217.0772 (0.03, 0.10) 311.9138 239.7364 192.0982 (0.03, 0.15) 5.2578 215.8678 151.3923 (0.05, 0.01) 25.7219 1.3963 7.2250 (0.05, 0.03) 13.4545 112.0004 117.1259 (0.05, 0.05) 9.3998 154.4641 151.4381 (0.05, 0.10) 310.1875 158.3124 131.5756 (0.05, 0.15) 4.8844 140.1677 99.2831 (0.10, 0.01) 24.6897 22.4126 12.5246 (0.10, 0.03) 12.5282 62.2482 77.1730 (0.10, 0.05) 8.5055 94.8364 105.2921 (0.10, 0.10) 3.2693 65.5584 52.2104 (0.10, 0.15) 4.0198 83.8863 62.7515 (0.15, 0.01) 24.0150 30.2071 18.3274 (0.15, 0.03) 11.9013 45.9518 65.4338 (0.15, 0.05) 7.9089 75.1670 91.5471 (0.15, 0.10) 304.5808 77.7868 76.4163 (0.15, 0.15) 3.4464 65.2920 51.8093Table 6
The percentages of the Estimated Relative Error of odds ratio estimation forn1,n2=10,50.
p 1 , p 2 E R E E B E R E M M L E E R E M M U E (0.01, 0.01) 50.9628 328.0343 295.7966 (0.01, 0.03) 28.9076 628.4030 536.8268 (0.01, 0.05) 263.4419 656.1777 520.8249 (0.01, 0.10) 16.2619 578.3828 392.2232 (0.01, 0.15) 14.5797 538.7736 339.4512 (0.03, 0.01) 39.6535 94.3767 91.6069 (0.03, 0.03) 19.2635 230.6324 208.1003 (0.03, 0.05) 236.1914 243.2889 200.4045 (0.03, 0.10) 7.4747 208.1561 138.4178 (0.03, 0.15) 5.8990 190.0357 112.7091 (0.05, 0.01) 39.1153 47.8002 51.9027 (0.05, 0.03) 18.8401 151.4810 144.3693 (0.05, 0.05) 234.9386 161.1356 138.3292 (0.05, 0.10) 7.0955 134.3384 89.0341 (0.05, 0.15) 5.5239 120.5896 68.6865 (0.10, 0.01) 37.9815 13.1041 23.9423 (0.10, 0.03) 17.8718 92.5224 99.4826 (0.10, 0.05) 232.2048 99.9578 94.6234 (0.10, 0.10) 6.2116 79.3838 54.3286 (0.10, 0.15) 4.6546 68.8420 37.6836 (0.15, 0.01) 37.2457 1.7496 15.7345 (0.15, 0.03) 17.2201 73.1152 86.1673 (0.15, 0.05) 230.3755 79.7100 81.5120 (0.15, 0.10) 5.6227 61.2754 43.9902 (0.15, 0.15) 4.0763 51.8150 28.4826Figure 1
The percentages of ERE for odds ratio estimation using EB, MMLE, and MMUE whenn1,n2=10,10.


## 5. Illustrative Examples Using Real Data

Our first example is taken from the studies of Good [7] and Hardell [17]. As shown in Table 7, subjects with malignant lymphoma and colon cancer cases and controls who are shortly exposed to phenoxy acids in agriculture and forestry were observed, including the true odds ratios and their estimates using EB, MMLE, and MMUE. For outcome in which Y1,Y2=8,9 out of n1,n2=24,53 for cases and control, respectively, the estimate of the odds ratio using EB method yields the least ERE with 0.5523, while those using MMLE and MMUE methods result in ERE with 1.2805 and 4.1483, respectively.Table 7
True odds ratios and their estimates using EB, MMLE, and MMUE, with corresponding percentages of ERE.
Methods True EB MMLE MMUE 1st example O R 2.4444 2.4309 2.4131 2.3430 ERE — 0.5523 1.2805 4.1483  2nd example O R 0.1169 0.1230 0.1642 0.1350 ERE — 5.2097 40.4643 15.5305The second example is taken from the study of Perondi et al. [18], as shown in Table 7, which compared high-dose epinephrine and standard-dose epinephrine in children with cardiac arrest with 34 children in each treatment, including the true odds ratios and their estimates using EB, MMLE, and MMUE. For outcome measure was survival at 24 hours in which Y1,Y2=1,7 out of n1,n2=34,34 for high dose and standard dose, respectively. The estimate of the odds ratio using EB method yields the least ERE with 5.2097, while those using MMUE and MMLE methods result in ERE with 15.5305 and 40.4643, respectively.


## 6. Conclusion

Based on simulated study for odds ratio estimation in rare events with two independent binomial data, the result indicates that the proposed method performs rather well. The EB estimator of odds ratio is also more efficient than the other two estimators, MMLE and MMUE. In addition, our purposed estimator is an alternative method for odds ratio estimation to the MMLE method without disturbing the original data.



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*Source: 3642941-2016-10-19.xml*