Research Article Odds Ratios Estimation of Rare Event in ...downloads.hindawi.com/journals/jps/2016/3642941.pdfratio always laid between and innity. Additionally, this method performed
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Research ArticleOdds Ratios Estimation of Rare Event in Binomial Distribution
Kobkun Raweesawat1 Yupaporn Areepong1
Katechan Jampachaisri2 and Saowanit Sukparungsee1
1Department of Applied Statistics Faculty of Applied Science King Mongkutrsquos University of Technology North BangkokBangkok 10800 Thailand2Department of Mathematics Faculty of Science Naresuan University Phitsanulok 65000 Thailand
Correspondence should be addressed to Saowanit Sukparungsee saowanitsscikmutnbacth
Received 5 July 2016 Revised 12 September 2016 Accepted 15 September 2016
Academic Editor Shein-chung Chow
Copyright copy 2016 Kobkun Raweesawat et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We introduce the new estimator of odds ratios in rare events using Empirical Bayes method in two independent binomialdistributions We compare the proposed estimates of odds ratios with two estimators modified maximum likelihood estimator(MMLE) andmodifiedmedian unbiased estimator (MMUE) using the EstimatedRelative Error (ERE) as a criterion of comparisonIt is found that the new estimator is more efficient when compared to the other methods
1 Introduction
The odds ratio is a measure of association between twoindependent groups on a categorical response with twopossible outcomes success and failure The two independentgroups can be two treatment groups or treatment and controlgroups The odds ratio is widely used in many fields ofmedical and social science research It is most commonlyused in epidemiology to express the results of some clinicaltrials such as in case-control studies
A number of subjects in each group falling in eachcategory can be summarized in a two-way contingency tableTotal numbers of subjects in group 1 and group 2 are 1198991 and1198992 which are assumed to be fixed Numbers of successes ingroup 1 and group 2 are 1198841 and 1198842 which are considered asindependent binomial random variables Let 1205871 and 1205872 beprobabilities of success in group 1 and group 2 respectivelyThe odds of success in group 1 are defined to be odds1 =1205871(1minus1205871) similar to group 2Theusualmaximum likelihoodestimator of odds ratio is defined as
ORMLE = odds1odds2
= 1205871 (1 minus 1205871)1205872 (1 minus 1205872) =1198841 (1198991 minus 1198841)1198842 (1198992 minus 1198842)
= 1198841 (1198992 minus 1198842)1198842 (1198991 minus 1198841) (1)
Odds ratio is nonnegative real value When successes aresimilar in both groups the odds ratio is equal to 1 meaningthat groups are independent of response When the odds ofa positive response are higher in group 1 than in group 2 theodds ratio is greater than 1 and vice versa for the value lessthan 1 The father of odds ratio from 1 in a given directionrepresents stronger association In addition its samplingdistribution is highly skewed Sample natural logarithm ofodds ratio which is less skewed is often utilized for inferenceHowever odds ratio can be zero (if zero cell count appearsin numerator of (1)) or infinity (if zero cell count is indenominator of (1)) or undefined (if there are zero cell countsin both the numerator and denominator of (1)) Haldane [1]and Gart and Zweifel [2] suggested to add a correction term05 to each cell when having zero cell count which gives themodified maximum likelihood estimator (MMLE) as
ORMMLE = (1198841 + 05) (1198992 minus 1198842 + 05)(1198842 + 05) (1198991 minus 1198841 + 05) (2)
Even though ORMMLE still laid between 0 and infinity someinvestigators discouraged adding 05 to each cell becauseof the appearance of adding ldquofake datardquo see Bishop et al[3] and Agresti and Yang [4] Among controversy several
Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2016 Article ID 3642941 8 pageshttpdxdoiorg10115520163642941
2 Journal of Probability and Statistics
similar alternatives to this modified maximum likelihoodestimator have been proposed Hirji et al [5] proposedthe median unbiased estimator (MUE) of the odds ratioobtained from the conditional noncentral hypergeometricdistribution However the median unbiased estimator ofthe odds ratio still caused a problem when 1198841 = 1198991 and1198842 = 1198992 or 1198841 = 1198842 = 0 and then the MUE wasundefined Parzen et al [6] proposed an estimator of the oddsratio based on MUE called the modified median unbiasedestimator (MMUE) of which the estimated probability ofsuccess was always in the interval (0 1) even if therewere 0 or119899 successes in each group Consequently the estimated oddsratio always laid between 0 and infinity Additionally thismethod performed well with respect to bias in small sampleand was an alternative to adding ldquofake datardquo
In this paper we focus on ldquorare eventsrdquo which occasion-ally observed zero or small counts of interesting events whichhappened within a given time period or a given sample suchas natural disasters or some diseases As aforementioned rareevents caused difficulty in estimation of odds ratio due to theoccurrence of zeros or small observed counts in numeratoror in denominator or in both resulting in the large standarderror and therefore less precise confidence interval Only arough estimate of the odds ratio is thus obtained Researchesinvolving association between categorical variables in contin-gency table have long been studied using both classical andBayesian approaches Good [7] studied association factorat early stage in large contingency table with small entriesassuming log-normal and Pearson type III distribution Theauthor also mentioned that these assumptions may be lessaccurate but easy to handle Fisher [8] estimated the oddsratio based on hypergeometric distribution utilizing exactmethod in a 2 times 2 table Thomas and Gart [9] constructeda table for 95 confidence limits of differences and ratioof two proportions including odds ratio and one-tailed 119901value for Fisher-Irwin Exact test in various types of 2 times 2table Altham [10] studied association and exact 119901 value ina 2 times 2 contingency table based on the cumulative posteriorprobabilities which was not easy to extract Nurminen andMutanen [11] proposed Bayesian approach for the estimationof difference between two proportions risk ratio and oddsratio using independent beta prior and provided integralexpressions for the cumulative posterior distribution Theyalso applied the proposed method to real data regardingmalignant lymphoma and colon cancer cases exposed tophenoxy acids and chlorophenols in agriculture Nouri et al[12] presented the estimation of the odds ratio in 2 times 2 times 119869tables when exposure was misclassified They compared thematrix and inversematrixmethods to theMLEmethod usingsimulation study and found that the inverse matrix methodhaving a closed form was more efficient than the matrixmethod
As previously mentioned the estimates of associationmeasure in two-way contingency table can be carried outbased on classical and Bayesian approaches The exact distri-bution using classical approach is however rather difficult formathematical tractability In Bayesian approach where priorbelief is incorporated into derivation of posterior densitythe hyperparameters characterizing the prior density are
often unknown to researchers and need to be assessedirrespective of current data However controversy still existsAlternatively the estimation of hyperparameters is plausiblycarried out with the notion of Empirical Bayes method usingcurrent data to estimate the unknown hyperparameterscontrary to Bayesian approach As a consequence we focuson the utilization of Empirical Bayes method to estimate theodds ratio in a two-way contingency table focusing on smallproportions of success Our purposed estimation tends tooutperform the traditional estimator MMLE and MMUEwithout interference in the original data
The rest of this paper is organized in the followingsequence In the next section we discuss themedian unbiasedestimator The third section describes the odds ratio estima-tion using EB methodThe forth section illustrates simulatedresults and the efficiency of EB is compared withMMLE andMUEThefifth section displays the application of ourmethodto real data Our conclusion is drawn in the final section
2 The Modified Median Unbiased Estimatorof Odds Ratio
Parzen et al [6] suggested themodifiedmedian unbiased esti-mator (MMUE) in two independent binomial distributionsLet be the estimator of success probability which satisfies
119875 ( le 119901) ge 05119875 ( ge 119901) ge 05 (3)
To obtain 119905 they use the binomial distribution 119884119905 sim119861(119899119905 119901119905) where 119884119905 denotes random variable representingsuccess in the 119905th group (119905 = 1 2) Let 119910119905 be the observedvalue of 119884119905
(119899119905119894 ) (119871119905 )119894 (1 minus 119871119905 )119899119905minus119894 (8)
and solve 119880119905 from05 = 119875 (119884119905 le 119910119905 | 119901119905 = 119880119905 )= 119910119905sum119894=0
(119899119905119894 ) (119880119905 )119894 (1 minus 119880119905 )119899119905minus119894
(9)
The values of 119871119905 and 119880119905 can then actually be obtained byusing the relationship between the cumulative beta distri-bution and the cumulative binomial distribution function asfollows (Daly [13] and Johnson et al [14])
(119899119905119894 ) (119871119905 )119894 (1 minus 119871119905 )119899119905minus119894 = 1
(13)
Any value of 119871119905 in the interval [0 1] satisfies119875 (119884119905 ge 0 | 119901119905 = 119871119905 ) ge 05 (14)
where 119871119905 = 0 is the smallest possible value of 119871119905 Similarly when 119910119905 = 0 119880119905 satisfies119875 (119884119905 le 119910119905 | 119901119905 = 119880119905 ) = 119875 (119884119905 = 0 | 119901119905 = 119880119905 ) = 05(1198991199050) (119880119905 )
0 (1 minus 119880119905 )119899119905minus0 = 05119880119905 = 1 minus 05(1119899119905)
119871119905 + 119880119905 )2 = 1 minus 05(1119899)2 (16)
4 Journal of Probability and Statistics
Similarly when 119880119905 = 1 is the largest possible value of 119880119905 then 119871119905 satisfies
(119899119899) (119871119905 )119899 (1 minus 119871119905 )119899minus119899 = 05
119871119905 = (05)1119899(17)
when 119910119905 = 119899 and 119905 = (119871119905 + 119880119905 )2Then the MMUE of odds ratio estimation is defined as
ORMMUE = 1 (1 minus 1)2 (1 minus 2) (18)
where 1 and 2 denote success probability estimators ingroups 1 and 2 respectively
3 Proposed Estimation of Odds Ratio
In this section we proposed a newmethod for odds ratio esti-mation using Empirical Bayes method in two independent
binomial distributions Let 1198841 and 1198842 be random variablesdistributed as binomial with equal and unequal sample sizesand unknown probability 1198841 sim Bin(1198991 1199011) and 1198842 simBin(1198992 1199012) where 1198991 1198992 and 1199011 1199012 denote two sample sizesand two unknown success probabilities Adopt informationpriors on119901119894 119901119894 sim beta(120572119894 120573119894) 119894 = 1 2 where120572119894 and120573119894 denoteunknown hyperparameters The estimation of hyperparame-ters can be obtained from the posterior marginal distributionfunction as follows
Consequently the posteriormarginal distribution function of119910 is the beta-binomial distribution (BBD)Then both hyperparameters in each group can be esti-
mated using maximum likelihood method The likelihoodfunction of posterior marginal distribution function is thenwritten as
(119899119910119894)(120572 + 119910119894 minus 1) (120572 + 119910119894 minus 2) sdot sdot sdot 120572 (120573 + 119899 minus 119910119894 minus 1) (120573 + 119899 minus 119910119894 minus 2) sdot sdot sdot 120573(120572 + 120573 + 119899 minus 1) (120572 + 120573 + 119899 minus 2) sdot sdot sdot (120572 + 120573)
(20)
Applying Newton-Raphson method to solve a nonlinearequation the (119903 + 1)th maximum likelihood estimator ofhyperparameters (119903 = 1 2 ) can be obtained from
Let 11990110158401 and 11990110158402 be estimators of 1199011 and 1199012 respectively where11990110158401 = 1199101 + 11198991 + 1 + 1 11990110158402 = 1199102 + 21198992 + 2 + 2
(25)
Thus the EB estimator of odds ratio can be obtained asfollows
OREB = 11990110158401 (1 minus 11990110158401)11990110158402 (1 minus 11990110158402) (26)
where 11990110158401 and 11990110158402 denote success probability estimators ingroups 1 and 2 respectively
4 Simulation Study for MMLE MMUEand EB Method
Simulation studies have been carried out using R program(version 320) [16] to assess the efficiency of the EB methodin comparison with two existing methods Binomial data aregenerated with equal and unequal sample sizes (1198991 1198992) =(10 10) (10 30) with (10 50) probabilities of success ingroup 1 1199011 = 001 003 005 01 and 015 For each value
6 Journal of Probability and Statistics
Table 4 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 10)(1199011 1199012) EREEB EREMMLE EREMMUE
of 1199011 1199012 is varied to 001 003 005 01 and 015 Eachsituation is repeated 5000 times after removing the first1000 iterations (1000 burn-ins) The efficiency of proposedestimator is evaluated using Estimated Relative Error (ERE)defined as
ERE = [10038161003816100381610038161003816OR minus OR11989410038161003816100381610038161003816
OR] (27)
where OR denotes the usual maximum likelihood estimatorof odds ratio and OR119894 denotes the estimate of odds ratio usingEB MMLE and MMUE (119894 = 1 2 3 ) respectively
The simulation results with odds ratio estimates forsample sizes (1198991 1198992) = (10 10) (10 30) and (10 50) are givenin Tables 1ndash3 The performance of estimation uses ERE givenin Tables 4ndash6 and compares this result with graph in Figure 1the other case provides similar results It is found that theodds ratio estimation using EBmethodmostly yields smallestERE with 7867 while those using MMLE and MMUEmethods result in smallest ERE with only 667 and 1466respectively
5 Illustrative Examples Using Real Data
Our first example is taken from the studies of Good [7] andHardell [17] As shown in Table 7 subjects with malignant
Table 5 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 30)(1199011 1199012) EREEB EREMMLE EREMMUE
lymphoma and colon cancer cases and controls who areshortly exposed to phenoxy acids in agriculture and forestrywere observed including the true odds ratios and theirestimates using EB MMLE and MMUE For outcome inwhich (1198841 1198842) = (8 9) out of (1198991 1198992) = (24 53) for cases andcontrol respectively the estimate of the odds ratio using EBmethod yields the least ERE with 05523 while those usingMMLE and MMUE methods result in ERE with 12805 and41483 respectively
The second example is taken from the study of Perondiet al [18] as shown in Table 7 which compared high-doseepinephrine and standard-dose epinephrine in children withcardiac arrest with 34 children in each treatment includingthe true odds ratios and their estimates using EB MMLEand MMUE For outcome measure was survival at 24 hoursin which (1198841 1198842) = (1 7) out of (1198991 1198992) = (34 34) for highdose and standard dose respectivelyThe estimate of the oddsratio using EBmethod yields the least ERE with 52097 whilethose using MMUE and MMLE methods result in ERE with155305 and 404643 respectively
6 Conclusion
Based on simulated study for odds ratio estimation inrare events with two independent binomial data the resultindicates that the proposed method performs rather well
Journal of Probability and Statistics 7
002 004 006 008 010 012 014
0
50
100
150
200
250
300
350
ERE
p1
p2 = 005
002 004 006 008 010 012 014
0
100
200
300
400
500
600ER
E
p1
p2 = 01
002 004 006 008 010 012 014
20
30
40
50
60
70ER
E
p1
p2 = 001
002 004 006 008 010 012 014
0
50
100
150
200
ERE
p1
p2 = 003
002 004 006 008 010 012 014
0
100
200
300
400
500
600
700
ERE
p1
p2 = 015
EBMMLEMMUE
Figure 1 The percentages of ERE for odds ratio estimation using EB MMLE and MMUE when (1198991 1198992) = (10 10)
8 Journal of Probability and Statistics
Table 6 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 50)(1199011 1199012) EREEB EREMMLE EREMMUE
Table 7 True odds ratios and their estimates using EB MMLE andMMUE with corresponding percentages of ERE
MethodsTrue EB MMLE MMUE
1st example OR 24444 24309 24131 23430ERE mdash 05523 12805 41483
2nd example OR 01169 01230 01642 01350ERE mdash 52097 404643 155305
The EB estimator of odds ratio is also more efficient thanthe other two estimators MMLE and MMUE In additionour purposed estimator is an alternative method for oddsratio estimation to theMMLEmethodwithout disturbing theoriginal data
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors are grateful to the Graduate College KingMongkutrsquos University of Technology North Bangkok for thefinancial support
References
[1] J B S Haldane ldquoThe estimation and significance of thelogarithm of a ratio of frequenciesrdquo Annals of Human Geneticsvol 20 no 4 pp 309ndash311 1956
[2] J J Gart and J R Zweifel ldquoOn the bias of various estimators ofthe logit and its variance with application of quantal bioassayrdquoBiometrika vol 54 no 1-2 pp 181ndash187 1967
[3] Y M Bishop S E Fienberg and P W Holland DiscreteMultivariate Analysis Theory and Practice Springer New YorkNY USA 2007
[4] A Agresti andM-C Yang ldquoAn empirical investigation of someeffects of sparseness in contingency tablesrdquo ComputationalStatistics and Data Analysis vol 5 no 1 pp 9ndash21 1987
[5] K F Hirji A A Tsiatis and C R Mehta ldquoMedian unbiasedestimation for binary datardquo The American Statistician vol 43no 1 pp 7ndash11 1989
[6] M Parzen S Lipsitz J Ibrahim and N Klar ldquoAn estimate ofthe odds ratio that always existsrdquo Journal of Computational andGraphical Statistics vol 11 no 2 pp 420ndash436 2002
[7] I J Good ldquoOn the estimations of small frequencies in contin-gency tablesrdquo Journal of the Royal Statistical Society Series BMethodological vol 18 pp 113ndash124 1956
[8] R A Fisher ldquoThe logic of inductive inferencerdquo Journal of theRoyal Statistical Society vol 98 no 1 pp 39ndash82 1935
[9] D G Thomas and J J Gart ldquoA table of exact confidence limitsfor differences and ratios of two proportions and their oddsratiosrdquo Journal of the American Statistical Association vol 72no 357 pp 73ndash76 1977
[10] P M E Altham ldquoExact Bayesian analysis of a 2times2 contingencytable and Fisherrsquos lsquoexactrsquo significance testrdquo Journal of the RoyalStatistical Society Series B (Methodological) vol 31 pp 261ndash2691969
[11] M Nurminen and P Mutanen ldquoExact Bayesian analysis of twoproportionsrdquo Scandinavian Journal of Statistics vol 14 no 1 pp67ndash77 1987
[12] B Nouri N Zare and S M T Ayatollahi ldquoValidation studymethods for estimating odds ratio in 2 times 2 times 119869 tables whenexposure is misclassifiedrdquo Computational and MathematicalMethods in Medicine vol 2013 Article ID 170120 8 pages 2013
[13] L Daly ldquoSimple SAS macros for the calculation of exactbinomial and Poisson confidence limitsrdquo Computers in Biologyand Medicine vol 22 no 5 pp 351ndash361 1992
[14] N L Johnson A W Kemp and S Kotz Univariate DiscreteDistribution John Wiley amp Sons New Jersey NJ USA 2005
[15] T P Minka ldquoEstimating a Dirichlet distributionrdquo Tech RepThe MIT Press London UK 2000
[16] The R Development Core Team An introduction to R Vienna2010 httpR-projectorg
[17] L Hardell ldquoRelation of soft-tissue sarcoma malignant lym-phoma and colon cancer to phenoxy acids chlorophenols andother agentsrdquo Scandinavian Journal of Work Environment andHealth vol 7 no 2 pp 119ndash130 1981
[18] M B M Perondi A G Reis E F Paiva V M Nadkarniand R A Berg ldquoA comparison of high-dose and standard-doseepinephrine in children with cardiac arrestrdquo The New EnglandJournal of Medicine vol 350 no 17 pp 1722ndash1730 2004
similar alternatives to this modified maximum likelihoodestimator have been proposed Hirji et al [5] proposedthe median unbiased estimator (MUE) of the odds ratioobtained from the conditional noncentral hypergeometricdistribution However the median unbiased estimator ofthe odds ratio still caused a problem when 1198841 = 1198991 and1198842 = 1198992 or 1198841 = 1198842 = 0 and then the MUE wasundefined Parzen et al [6] proposed an estimator of the oddsratio based on MUE called the modified median unbiasedestimator (MMUE) of which the estimated probability ofsuccess was always in the interval (0 1) even if therewere 0 or119899 successes in each group Consequently the estimated oddsratio always laid between 0 and infinity Additionally thismethod performed well with respect to bias in small sampleand was an alternative to adding ldquofake datardquo
In this paper we focus on ldquorare eventsrdquo which occasion-ally observed zero or small counts of interesting events whichhappened within a given time period or a given sample suchas natural disasters or some diseases As aforementioned rareevents caused difficulty in estimation of odds ratio due to theoccurrence of zeros or small observed counts in numeratoror in denominator or in both resulting in the large standarderror and therefore less precise confidence interval Only arough estimate of the odds ratio is thus obtained Researchesinvolving association between categorical variables in contin-gency table have long been studied using both classical andBayesian approaches Good [7] studied association factorat early stage in large contingency table with small entriesassuming log-normal and Pearson type III distribution Theauthor also mentioned that these assumptions may be lessaccurate but easy to handle Fisher [8] estimated the oddsratio based on hypergeometric distribution utilizing exactmethod in a 2 times 2 table Thomas and Gart [9] constructeda table for 95 confidence limits of differences and ratioof two proportions including odds ratio and one-tailed 119901value for Fisher-Irwin Exact test in various types of 2 times 2table Altham [10] studied association and exact 119901 value ina 2 times 2 contingency table based on the cumulative posteriorprobabilities which was not easy to extract Nurminen andMutanen [11] proposed Bayesian approach for the estimationof difference between two proportions risk ratio and oddsratio using independent beta prior and provided integralexpressions for the cumulative posterior distribution Theyalso applied the proposed method to real data regardingmalignant lymphoma and colon cancer cases exposed tophenoxy acids and chlorophenols in agriculture Nouri et al[12] presented the estimation of the odds ratio in 2 times 2 times 119869tables when exposure was misclassified They compared thematrix and inversematrixmethods to theMLEmethod usingsimulation study and found that the inverse matrix methodhaving a closed form was more efficient than the matrixmethod
As previously mentioned the estimates of associationmeasure in two-way contingency table can be carried outbased on classical and Bayesian approaches The exact distri-bution using classical approach is however rather difficult formathematical tractability In Bayesian approach where priorbelief is incorporated into derivation of posterior densitythe hyperparameters characterizing the prior density are
often unknown to researchers and need to be assessedirrespective of current data However controversy still existsAlternatively the estimation of hyperparameters is plausiblycarried out with the notion of Empirical Bayes method usingcurrent data to estimate the unknown hyperparameterscontrary to Bayesian approach As a consequence we focuson the utilization of Empirical Bayes method to estimate theodds ratio in a two-way contingency table focusing on smallproportions of success Our purposed estimation tends tooutperform the traditional estimator MMLE and MMUEwithout interference in the original data
The rest of this paper is organized in the followingsequence In the next section we discuss themedian unbiasedestimator The third section describes the odds ratio estima-tion using EB methodThe forth section illustrates simulatedresults and the efficiency of EB is compared withMMLE andMUEThefifth section displays the application of ourmethodto real data Our conclusion is drawn in the final section
2 The Modified Median Unbiased Estimatorof Odds Ratio
Parzen et al [6] suggested themodifiedmedian unbiased esti-mator (MMUE) in two independent binomial distributionsLet be the estimator of success probability which satisfies
119875 ( le 119901) ge 05119875 ( ge 119901) ge 05 (3)
To obtain 119905 they use the binomial distribution 119884119905 sim119861(119899119905 119901119905) where 119884119905 denotes random variable representingsuccess in the 119905th group (119905 = 1 2) Let 119910119905 be the observedvalue of 119884119905
(119899119905119894 ) (119871119905 )119894 (1 minus 119871119905 )119899119905minus119894 (8)
and solve 119880119905 from05 = 119875 (119884119905 le 119910119905 | 119901119905 = 119880119905 )= 119910119905sum119894=0
(119899119905119894 ) (119880119905 )119894 (1 minus 119880119905 )119899119905minus119894
(9)
The values of 119871119905 and 119880119905 can then actually be obtained byusing the relationship between the cumulative beta distri-bution and the cumulative binomial distribution function asfollows (Daly [13] and Johnson et al [14])
(119899119905119894 ) (119871119905 )119894 (1 minus 119871119905 )119899119905minus119894 = 1
(13)
Any value of 119871119905 in the interval [0 1] satisfies119875 (119884119905 ge 0 | 119901119905 = 119871119905 ) ge 05 (14)
where 119871119905 = 0 is the smallest possible value of 119871119905 Similarly when 119910119905 = 0 119880119905 satisfies119875 (119884119905 le 119910119905 | 119901119905 = 119880119905 ) = 119875 (119884119905 = 0 | 119901119905 = 119880119905 ) = 05(1198991199050) (119880119905 )
0 (1 minus 119880119905 )119899119905minus0 = 05119880119905 = 1 minus 05(1119899119905)
119871119905 + 119880119905 )2 = 1 minus 05(1119899)2 (16)
4 Journal of Probability and Statistics
Similarly when 119880119905 = 1 is the largest possible value of 119880119905 then 119871119905 satisfies
(119899119899) (119871119905 )119899 (1 minus 119871119905 )119899minus119899 = 05
119871119905 = (05)1119899(17)
when 119910119905 = 119899 and 119905 = (119871119905 + 119880119905 )2Then the MMUE of odds ratio estimation is defined as
ORMMUE = 1 (1 minus 1)2 (1 minus 2) (18)
where 1 and 2 denote success probability estimators ingroups 1 and 2 respectively
3 Proposed Estimation of Odds Ratio
In this section we proposed a newmethod for odds ratio esti-mation using Empirical Bayes method in two independent
binomial distributions Let 1198841 and 1198842 be random variablesdistributed as binomial with equal and unequal sample sizesand unknown probability 1198841 sim Bin(1198991 1199011) and 1198842 simBin(1198992 1199012) where 1198991 1198992 and 1199011 1199012 denote two sample sizesand two unknown success probabilities Adopt informationpriors on119901119894 119901119894 sim beta(120572119894 120573119894) 119894 = 1 2 where120572119894 and120573119894 denoteunknown hyperparameters The estimation of hyperparame-ters can be obtained from the posterior marginal distributionfunction as follows
Consequently the posteriormarginal distribution function of119910 is the beta-binomial distribution (BBD)Then both hyperparameters in each group can be esti-
mated using maximum likelihood method The likelihoodfunction of posterior marginal distribution function is thenwritten as
(119899119910119894)(120572 + 119910119894 minus 1) (120572 + 119910119894 minus 2) sdot sdot sdot 120572 (120573 + 119899 minus 119910119894 minus 1) (120573 + 119899 minus 119910119894 minus 2) sdot sdot sdot 120573(120572 + 120573 + 119899 minus 1) (120572 + 120573 + 119899 minus 2) sdot sdot sdot (120572 + 120573)
(20)
Applying Newton-Raphson method to solve a nonlinearequation the (119903 + 1)th maximum likelihood estimator ofhyperparameters (119903 = 1 2 ) can be obtained from
Let 11990110158401 and 11990110158402 be estimators of 1199011 and 1199012 respectively where11990110158401 = 1199101 + 11198991 + 1 + 1 11990110158402 = 1199102 + 21198992 + 2 + 2
(25)
Thus the EB estimator of odds ratio can be obtained asfollows
OREB = 11990110158401 (1 minus 11990110158401)11990110158402 (1 minus 11990110158402) (26)
where 11990110158401 and 11990110158402 denote success probability estimators ingroups 1 and 2 respectively
4 Simulation Study for MMLE MMUEand EB Method
Simulation studies have been carried out using R program(version 320) [16] to assess the efficiency of the EB methodin comparison with two existing methods Binomial data aregenerated with equal and unequal sample sizes (1198991 1198992) =(10 10) (10 30) with (10 50) probabilities of success ingroup 1 1199011 = 001 003 005 01 and 015 For each value
6 Journal of Probability and Statistics
Table 4 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 10)(1199011 1199012) EREEB EREMMLE EREMMUE
of 1199011 1199012 is varied to 001 003 005 01 and 015 Eachsituation is repeated 5000 times after removing the first1000 iterations (1000 burn-ins) The efficiency of proposedestimator is evaluated using Estimated Relative Error (ERE)defined as
ERE = [10038161003816100381610038161003816OR minus OR11989410038161003816100381610038161003816
OR] (27)
where OR denotes the usual maximum likelihood estimatorof odds ratio and OR119894 denotes the estimate of odds ratio usingEB MMLE and MMUE (119894 = 1 2 3 ) respectively
The simulation results with odds ratio estimates forsample sizes (1198991 1198992) = (10 10) (10 30) and (10 50) are givenin Tables 1ndash3 The performance of estimation uses ERE givenin Tables 4ndash6 and compares this result with graph in Figure 1the other case provides similar results It is found that theodds ratio estimation using EBmethodmostly yields smallestERE with 7867 while those using MMLE and MMUEmethods result in smallest ERE with only 667 and 1466respectively
5 Illustrative Examples Using Real Data
Our first example is taken from the studies of Good [7] andHardell [17] As shown in Table 7 subjects with malignant
Table 5 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 30)(1199011 1199012) EREEB EREMMLE EREMMUE
lymphoma and colon cancer cases and controls who areshortly exposed to phenoxy acids in agriculture and forestrywere observed including the true odds ratios and theirestimates using EB MMLE and MMUE For outcome inwhich (1198841 1198842) = (8 9) out of (1198991 1198992) = (24 53) for cases andcontrol respectively the estimate of the odds ratio using EBmethod yields the least ERE with 05523 while those usingMMLE and MMUE methods result in ERE with 12805 and41483 respectively
The second example is taken from the study of Perondiet al [18] as shown in Table 7 which compared high-doseepinephrine and standard-dose epinephrine in children withcardiac arrest with 34 children in each treatment includingthe true odds ratios and their estimates using EB MMLEand MMUE For outcome measure was survival at 24 hoursin which (1198841 1198842) = (1 7) out of (1198991 1198992) = (34 34) for highdose and standard dose respectivelyThe estimate of the oddsratio using EBmethod yields the least ERE with 52097 whilethose using MMUE and MMLE methods result in ERE with155305 and 404643 respectively
6 Conclusion
Based on simulated study for odds ratio estimation inrare events with two independent binomial data the resultindicates that the proposed method performs rather well
Journal of Probability and Statistics 7
002 004 006 008 010 012 014
0
50
100
150
200
250
300
350
ERE
p1
p2 = 005
002 004 006 008 010 012 014
0
100
200
300
400
500
600ER
E
p1
p2 = 01
002 004 006 008 010 012 014
20
30
40
50
60
70ER
E
p1
p2 = 001
002 004 006 008 010 012 014
0
50
100
150
200
ERE
p1
p2 = 003
002 004 006 008 010 012 014
0
100
200
300
400
500
600
700
ERE
p1
p2 = 015
EBMMLEMMUE
Figure 1 The percentages of ERE for odds ratio estimation using EB MMLE and MMUE when (1198991 1198992) = (10 10)
8 Journal of Probability and Statistics
Table 6 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 50)(1199011 1199012) EREEB EREMMLE EREMMUE
Table 7 True odds ratios and their estimates using EB MMLE andMMUE with corresponding percentages of ERE
MethodsTrue EB MMLE MMUE
1st example OR 24444 24309 24131 23430ERE mdash 05523 12805 41483
2nd example OR 01169 01230 01642 01350ERE mdash 52097 404643 155305
The EB estimator of odds ratio is also more efficient thanthe other two estimators MMLE and MMUE In additionour purposed estimator is an alternative method for oddsratio estimation to theMMLEmethodwithout disturbing theoriginal data
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors are grateful to the Graduate College KingMongkutrsquos University of Technology North Bangkok for thefinancial support
References
[1] J B S Haldane ldquoThe estimation and significance of thelogarithm of a ratio of frequenciesrdquo Annals of Human Geneticsvol 20 no 4 pp 309ndash311 1956
[2] J J Gart and J R Zweifel ldquoOn the bias of various estimators ofthe logit and its variance with application of quantal bioassayrdquoBiometrika vol 54 no 1-2 pp 181ndash187 1967
[3] Y M Bishop S E Fienberg and P W Holland DiscreteMultivariate Analysis Theory and Practice Springer New YorkNY USA 2007
[4] A Agresti andM-C Yang ldquoAn empirical investigation of someeffects of sparseness in contingency tablesrdquo ComputationalStatistics and Data Analysis vol 5 no 1 pp 9ndash21 1987
[5] K F Hirji A A Tsiatis and C R Mehta ldquoMedian unbiasedestimation for binary datardquo The American Statistician vol 43no 1 pp 7ndash11 1989
[6] M Parzen S Lipsitz J Ibrahim and N Klar ldquoAn estimate ofthe odds ratio that always existsrdquo Journal of Computational andGraphical Statistics vol 11 no 2 pp 420ndash436 2002
[7] I J Good ldquoOn the estimations of small frequencies in contin-gency tablesrdquo Journal of the Royal Statistical Society Series BMethodological vol 18 pp 113ndash124 1956
[8] R A Fisher ldquoThe logic of inductive inferencerdquo Journal of theRoyal Statistical Society vol 98 no 1 pp 39ndash82 1935
[9] D G Thomas and J J Gart ldquoA table of exact confidence limitsfor differences and ratios of two proportions and their oddsratiosrdquo Journal of the American Statistical Association vol 72no 357 pp 73ndash76 1977
[10] P M E Altham ldquoExact Bayesian analysis of a 2times2 contingencytable and Fisherrsquos lsquoexactrsquo significance testrdquo Journal of the RoyalStatistical Society Series B (Methodological) vol 31 pp 261ndash2691969
[11] M Nurminen and P Mutanen ldquoExact Bayesian analysis of twoproportionsrdquo Scandinavian Journal of Statistics vol 14 no 1 pp67ndash77 1987
[12] B Nouri N Zare and S M T Ayatollahi ldquoValidation studymethods for estimating odds ratio in 2 times 2 times 119869 tables whenexposure is misclassifiedrdquo Computational and MathematicalMethods in Medicine vol 2013 Article ID 170120 8 pages 2013
[13] L Daly ldquoSimple SAS macros for the calculation of exactbinomial and Poisson confidence limitsrdquo Computers in Biologyand Medicine vol 22 no 5 pp 351ndash361 1992
[14] N L Johnson A W Kemp and S Kotz Univariate DiscreteDistribution John Wiley amp Sons New Jersey NJ USA 2005
[15] T P Minka ldquoEstimating a Dirichlet distributionrdquo Tech RepThe MIT Press London UK 2000
[16] The R Development Core Team An introduction to R Vienna2010 httpR-projectorg
[17] L Hardell ldquoRelation of soft-tissue sarcoma malignant lym-phoma and colon cancer to phenoxy acids chlorophenols andother agentsrdquo Scandinavian Journal of Work Environment andHealth vol 7 no 2 pp 119ndash130 1981
[18] M B M Perondi A G Reis E F Paiva V M Nadkarniand R A Berg ldquoA comparison of high-dose and standard-doseepinephrine in children with cardiac arrestrdquo The New EnglandJournal of Medicine vol 350 no 17 pp 1722ndash1730 2004
(119899119905119894 ) (119871119905 )119894 (1 minus 119871119905 )119899119905minus119894 (8)
and solve 119880119905 from05 = 119875 (119884119905 le 119910119905 | 119901119905 = 119880119905 )= 119910119905sum119894=0
(119899119905119894 ) (119880119905 )119894 (1 minus 119880119905 )119899119905minus119894
(9)
The values of 119871119905 and 119880119905 can then actually be obtained byusing the relationship between the cumulative beta distri-bution and the cumulative binomial distribution function asfollows (Daly [13] and Johnson et al [14])
(119899119905119894 ) (119871119905 )119894 (1 minus 119871119905 )119899119905minus119894 = 1
(13)
Any value of 119871119905 in the interval [0 1] satisfies119875 (119884119905 ge 0 | 119901119905 = 119871119905 ) ge 05 (14)
where 119871119905 = 0 is the smallest possible value of 119871119905 Similarly when 119910119905 = 0 119880119905 satisfies119875 (119884119905 le 119910119905 | 119901119905 = 119880119905 ) = 119875 (119884119905 = 0 | 119901119905 = 119880119905 ) = 05(1198991199050) (119880119905 )
0 (1 minus 119880119905 )119899119905minus0 = 05119880119905 = 1 minus 05(1119899119905)
119871119905 + 119880119905 )2 = 1 minus 05(1119899)2 (16)
4 Journal of Probability and Statistics
Similarly when 119880119905 = 1 is the largest possible value of 119880119905 then 119871119905 satisfies
(119899119899) (119871119905 )119899 (1 minus 119871119905 )119899minus119899 = 05
119871119905 = (05)1119899(17)
when 119910119905 = 119899 and 119905 = (119871119905 + 119880119905 )2Then the MMUE of odds ratio estimation is defined as
ORMMUE = 1 (1 minus 1)2 (1 minus 2) (18)
where 1 and 2 denote success probability estimators ingroups 1 and 2 respectively
3 Proposed Estimation of Odds Ratio
In this section we proposed a newmethod for odds ratio esti-mation using Empirical Bayes method in two independent
binomial distributions Let 1198841 and 1198842 be random variablesdistributed as binomial with equal and unequal sample sizesand unknown probability 1198841 sim Bin(1198991 1199011) and 1198842 simBin(1198992 1199012) where 1198991 1198992 and 1199011 1199012 denote two sample sizesand two unknown success probabilities Adopt informationpriors on119901119894 119901119894 sim beta(120572119894 120573119894) 119894 = 1 2 where120572119894 and120573119894 denoteunknown hyperparameters The estimation of hyperparame-ters can be obtained from the posterior marginal distributionfunction as follows
Consequently the posteriormarginal distribution function of119910 is the beta-binomial distribution (BBD)Then both hyperparameters in each group can be esti-
mated using maximum likelihood method The likelihoodfunction of posterior marginal distribution function is thenwritten as
(119899119910119894)(120572 + 119910119894 minus 1) (120572 + 119910119894 minus 2) sdot sdot sdot 120572 (120573 + 119899 minus 119910119894 minus 1) (120573 + 119899 minus 119910119894 minus 2) sdot sdot sdot 120573(120572 + 120573 + 119899 minus 1) (120572 + 120573 + 119899 minus 2) sdot sdot sdot (120572 + 120573)
(20)
Applying Newton-Raphson method to solve a nonlinearequation the (119903 + 1)th maximum likelihood estimator ofhyperparameters (119903 = 1 2 ) can be obtained from
Let 11990110158401 and 11990110158402 be estimators of 1199011 and 1199012 respectively where11990110158401 = 1199101 + 11198991 + 1 + 1 11990110158402 = 1199102 + 21198992 + 2 + 2
(25)
Thus the EB estimator of odds ratio can be obtained asfollows
OREB = 11990110158401 (1 minus 11990110158401)11990110158402 (1 minus 11990110158402) (26)
where 11990110158401 and 11990110158402 denote success probability estimators ingroups 1 and 2 respectively
4 Simulation Study for MMLE MMUEand EB Method
Simulation studies have been carried out using R program(version 320) [16] to assess the efficiency of the EB methodin comparison with two existing methods Binomial data aregenerated with equal and unequal sample sizes (1198991 1198992) =(10 10) (10 30) with (10 50) probabilities of success ingroup 1 1199011 = 001 003 005 01 and 015 For each value
6 Journal of Probability and Statistics
Table 4 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 10)(1199011 1199012) EREEB EREMMLE EREMMUE
of 1199011 1199012 is varied to 001 003 005 01 and 015 Eachsituation is repeated 5000 times after removing the first1000 iterations (1000 burn-ins) The efficiency of proposedestimator is evaluated using Estimated Relative Error (ERE)defined as
ERE = [10038161003816100381610038161003816OR minus OR11989410038161003816100381610038161003816
OR] (27)
where OR denotes the usual maximum likelihood estimatorof odds ratio and OR119894 denotes the estimate of odds ratio usingEB MMLE and MMUE (119894 = 1 2 3 ) respectively
The simulation results with odds ratio estimates forsample sizes (1198991 1198992) = (10 10) (10 30) and (10 50) are givenin Tables 1ndash3 The performance of estimation uses ERE givenin Tables 4ndash6 and compares this result with graph in Figure 1the other case provides similar results It is found that theodds ratio estimation using EBmethodmostly yields smallestERE with 7867 while those using MMLE and MMUEmethods result in smallest ERE with only 667 and 1466respectively
5 Illustrative Examples Using Real Data
Our first example is taken from the studies of Good [7] andHardell [17] As shown in Table 7 subjects with malignant
Table 5 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 30)(1199011 1199012) EREEB EREMMLE EREMMUE
lymphoma and colon cancer cases and controls who areshortly exposed to phenoxy acids in agriculture and forestrywere observed including the true odds ratios and theirestimates using EB MMLE and MMUE For outcome inwhich (1198841 1198842) = (8 9) out of (1198991 1198992) = (24 53) for cases andcontrol respectively the estimate of the odds ratio using EBmethod yields the least ERE with 05523 while those usingMMLE and MMUE methods result in ERE with 12805 and41483 respectively
The second example is taken from the study of Perondiet al [18] as shown in Table 7 which compared high-doseepinephrine and standard-dose epinephrine in children withcardiac arrest with 34 children in each treatment includingthe true odds ratios and their estimates using EB MMLEand MMUE For outcome measure was survival at 24 hoursin which (1198841 1198842) = (1 7) out of (1198991 1198992) = (34 34) for highdose and standard dose respectivelyThe estimate of the oddsratio using EBmethod yields the least ERE with 52097 whilethose using MMUE and MMLE methods result in ERE with155305 and 404643 respectively
6 Conclusion
Based on simulated study for odds ratio estimation inrare events with two independent binomial data the resultindicates that the proposed method performs rather well
Journal of Probability and Statistics 7
002 004 006 008 010 012 014
0
50
100
150
200
250
300
350
ERE
p1
p2 = 005
002 004 006 008 010 012 014
0
100
200
300
400
500
600ER
E
p1
p2 = 01
002 004 006 008 010 012 014
20
30
40
50
60
70ER
E
p1
p2 = 001
002 004 006 008 010 012 014
0
50
100
150
200
ERE
p1
p2 = 003
002 004 006 008 010 012 014
0
100
200
300
400
500
600
700
ERE
p1
p2 = 015
EBMMLEMMUE
Figure 1 The percentages of ERE for odds ratio estimation using EB MMLE and MMUE when (1198991 1198992) = (10 10)
8 Journal of Probability and Statistics
Table 6 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 50)(1199011 1199012) EREEB EREMMLE EREMMUE
Table 7 True odds ratios and their estimates using EB MMLE andMMUE with corresponding percentages of ERE
MethodsTrue EB MMLE MMUE
1st example OR 24444 24309 24131 23430ERE mdash 05523 12805 41483
2nd example OR 01169 01230 01642 01350ERE mdash 52097 404643 155305
The EB estimator of odds ratio is also more efficient thanthe other two estimators MMLE and MMUE In additionour purposed estimator is an alternative method for oddsratio estimation to theMMLEmethodwithout disturbing theoriginal data
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors are grateful to the Graduate College KingMongkutrsquos University of Technology North Bangkok for thefinancial support
References
[1] J B S Haldane ldquoThe estimation and significance of thelogarithm of a ratio of frequenciesrdquo Annals of Human Geneticsvol 20 no 4 pp 309ndash311 1956
[2] J J Gart and J R Zweifel ldquoOn the bias of various estimators ofthe logit and its variance with application of quantal bioassayrdquoBiometrika vol 54 no 1-2 pp 181ndash187 1967
[3] Y M Bishop S E Fienberg and P W Holland DiscreteMultivariate Analysis Theory and Practice Springer New YorkNY USA 2007
[4] A Agresti andM-C Yang ldquoAn empirical investigation of someeffects of sparseness in contingency tablesrdquo ComputationalStatistics and Data Analysis vol 5 no 1 pp 9ndash21 1987
[5] K F Hirji A A Tsiatis and C R Mehta ldquoMedian unbiasedestimation for binary datardquo The American Statistician vol 43no 1 pp 7ndash11 1989
[6] M Parzen S Lipsitz J Ibrahim and N Klar ldquoAn estimate ofthe odds ratio that always existsrdquo Journal of Computational andGraphical Statistics vol 11 no 2 pp 420ndash436 2002
[7] I J Good ldquoOn the estimations of small frequencies in contin-gency tablesrdquo Journal of the Royal Statistical Society Series BMethodological vol 18 pp 113ndash124 1956
[8] R A Fisher ldquoThe logic of inductive inferencerdquo Journal of theRoyal Statistical Society vol 98 no 1 pp 39ndash82 1935
[9] D G Thomas and J J Gart ldquoA table of exact confidence limitsfor differences and ratios of two proportions and their oddsratiosrdquo Journal of the American Statistical Association vol 72no 357 pp 73ndash76 1977
[10] P M E Altham ldquoExact Bayesian analysis of a 2times2 contingencytable and Fisherrsquos lsquoexactrsquo significance testrdquo Journal of the RoyalStatistical Society Series B (Methodological) vol 31 pp 261ndash2691969
[11] M Nurminen and P Mutanen ldquoExact Bayesian analysis of twoproportionsrdquo Scandinavian Journal of Statistics vol 14 no 1 pp67ndash77 1987
[12] B Nouri N Zare and S M T Ayatollahi ldquoValidation studymethods for estimating odds ratio in 2 times 2 times 119869 tables whenexposure is misclassifiedrdquo Computational and MathematicalMethods in Medicine vol 2013 Article ID 170120 8 pages 2013
[13] L Daly ldquoSimple SAS macros for the calculation of exactbinomial and Poisson confidence limitsrdquo Computers in Biologyand Medicine vol 22 no 5 pp 351ndash361 1992
[14] N L Johnson A W Kemp and S Kotz Univariate DiscreteDistribution John Wiley amp Sons New Jersey NJ USA 2005
[15] T P Minka ldquoEstimating a Dirichlet distributionrdquo Tech RepThe MIT Press London UK 2000
[16] The R Development Core Team An introduction to R Vienna2010 httpR-projectorg
[17] L Hardell ldquoRelation of soft-tissue sarcoma malignant lym-phoma and colon cancer to phenoxy acids chlorophenols andother agentsrdquo Scandinavian Journal of Work Environment andHealth vol 7 no 2 pp 119ndash130 1981
[18] M B M Perondi A G Reis E F Paiva V M Nadkarniand R A Berg ldquoA comparison of high-dose and standard-doseepinephrine in children with cardiac arrestrdquo The New EnglandJournal of Medicine vol 350 no 17 pp 1722ndash1730 2004
Similarly when 119880119905 = 1 is the largest possible value of 119880119905 then 119871119905 satisfies
(119899119899) (119871119905 )119899 (1 minus 119871119905 )119899minus119899 = 05
119871119905 = (05)1119899(17)
when 119910119905 = 119899 and 119905 = (119871119905 + 119880119905 )2Then the MMUE of odds ratio estimation is defined as
ORMMUE = 1 (1 minus 1)2 (1 minus 2) (18)
where 1 and 2 denote success probability estimators ingroups 1 and 2 respectively
3 Proposed Estimation of Odds Ratio
In this section we proposed a newmethod for odds ratio esti-mation using Empirical Bayes method in two independent
binomial distributions Let 1198841 and 1198842 be random variablesdistributed as binomial with equal and unequal sample sizesand unknown probability 1198841 sim Bin(1198991 1199011) and 1198842 simBin(1198992 1199012) where 1198991 1198992 and 1199011 1199012 denote two sample sizesand two unknown success probabilities Adopt informationpriors on119901119894 119901119894 sim beta(120572119894 120573119894) 119894 = 1 2 where120572119894 and120573119894 denoteunknown hyperparameters The estimation of hyperparame-ters can be obtained from the posterior marginal distributionfunction as follows
Consequently the posteriormarginal distribution function of119910 is the beta-binomial distribution (BBD)Then both hyperparameters in each group can be esti-
mated using maximum likelihood method The likelihoodfunction of posterior marginal distribution function is thenwritten as
(119899119910119894)(120572 + 119910119894 minus 1) (120572 + 119910119894 minus 2) sdot sdot sdot 120572 (120573 + 119899 minus 119910119894 minus 1) (120573 + 119899 minus 119910119894 minus 2) sdot sdot sdot 120573(120572 + 120573 + 119899 minus 1) (120572 + 120573 + 119899 minus 2) sdot sdot sdot (120572 + 120573)
(20)
Applying Newton-Raphson method to solve a nonlinearequation the (119903 + 1)th maximum likelihood estimator ofhyperparameters (119903 = 1 2 ) can be obtained from
Let 11990110158401 and 11990110158402 be estimators of 1199011 and 1199012 respectively where11990110158401 = 1199101 + 11198991 + 1 + 1 11990110158402 = 1199102 + 21198992 + 2 + 2
(25)
Thus the EB estimator of odds ratio can be obtained asfollows
OREB = 11990110158401 (1 minus 11990110158401)11990110158402 (1 minus 11990110158402) (26)
where 11990110158401 and 11990110158402 denote success probability estimators ingroups 1 and 2 respectively
4 Simulation Study for MMLE MMUEand EB Method
Simulation studies have been carried out using R program(version 320) [16] to assess the efficiency of the EB methodin comparison with two existing methods Binomial data aregenerated with equal and unequal sample sizes (1198991 1198992) =(10 10) (10 30) with (10 50) probabilities of success ingroup 1 1199011 = 001 003 005 01 and 015 For each value
6 Journal of Probability and Statistics
Table 4 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 10)(1199011 1199012) EREEB EREMMLE EREMMUE
of 1199011 1199012 is varied to 001 003 005 01 and 015 Eachsituation is repeated 5000 times after removing the first1000 iterations (1000 burn-ins) The efficiency of proposedestimator is evaluated using Estimated Relative Error (ERE)defined as
ERE = [10038161003816100381610038161003816OR minus OR11989410038161003816100381610038161003816
OR] (27)
where OR denotes the usual maximum likelihood estimatorof odds ratio and OR119894 denotes the estimate of odds ratio usingEB MMLE and MMUE (119894 = 1 2 3 ) respectively
The simulation results with odds ratio estimates forsample sizes (1198991 1198992) = (10 10) (10 30) and (10 50) are givenin Tables 1ndash3 The performance of estimation uses ERE givenin Tables 4ndash6 and compares this result with graph in Figure 1the other case provides similar results It is found that theodds ratio estimation using EBmethodmostly yields smallestERE with 7867 while those using MMLE and MMUEmethods result in smallest ERE with only 667 and 1466respectively
5 Illustrative Examples Using Real Data
Our first example is taken from the studies of Good [7] andHardell [17] As shown in Table 7 subjects with malignant
Table 5 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 30)(1199011 1199012) EREEB EREMMLE EREMMUE
lymphoma and colon cancer cases and controls who areshortly exposed to phenoxy acids in agriculture and forestrywere observed including the true odds ratios and theirestimates using EB MMLE and MMUE For outcome inwhich (1198841 1198842) = (8 9) out of (1198991 1198992) = (24 53) for cases andcontrol respectively the estimate of the odds ratio using EBmethod yields the least ERE with 05523 while those usingMMLE and MMUE methods result in ERE with 12805 and41483 respectively
The second example is taken from the study of Perondiet al [18] as shown in Table 7 which compared high-doseepinephrine and standard-dose epinephrine in children withcardiac arrest with 34 children in each treatment includingthe true odds ratios and their estimates using EB MMLEand MMUE For outcome measure was survival at 24 hoursin which (1198841 1198842) = (1 7) out of (1198991 1198992) = (34 34) for highdose and standard dose respectivelyThe estimate of the oddsratio using EBmethod yields the least ERE with 52097 whilethose using MMUE and MMLE methods result in ERE with155305 and 404643 respectively
6 Conclusion
Based on simulated study for odds ratio estimation inrare events with two independent binomial data the resultindicates that the proposed method performs rather well
Journal of Probability and Statistics 7
002 004 006 008 010 012 014
0
50
100
150
200
250
300
350
ERE
p1
p2 = 005
002 004 006 008 010 012 014
0
100
200
300
400
500
600ER
E
p1
p2 = 01
002 004 006 008 010 012 014
20
30
40
50
60
70ER
E
p1
p2 = 001
002 004 006 008 010 012 014
0
50
100
150
200
ERE
p1
p2 = 003
002 004 006 008 010 012 014
0
100
200
300
400
500
600
700
ERE
p1
p2 = 015
EBMMLEMMUE
Figure 1 The percentages of ERE for odds ratio estimation using EB MMLE and MMUE when (1198991 1198992) = (10 10)
8 Journal of Probability and Statistics
Table 6 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 50)(1199011 1199012) EREEB EREMMLE EREMMUE
Table 7 True odds ratios and their estimates using EB MMLE andMMUE with corresponding percentages of ERE
MethodsTrue EB MMLE MMUE
1st example OR 24444 24309 24131 23430ERE mdash 05523 12805 41483
2nd example OR 01169 01230 01642 01350ERE mdash 52097 404643 155305
The EB estimator of odds ratio is also more efficient thanthe other two estimators MMLE and MMUE In additionour purposed estimator is an alternative method for oddsratio estimation to theMMLEmethodwithout disturbing theoriginal data
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors are grateful to the Graduate College KingMongkutrsquos University of Technology North Bangkok for thefinancial support
References
[1] J B S Haldane ldquoThe estimation and significance of thelogarithm of a ratio of frequenciesrdquo Annals of Human Geneticsvol 20 no 4 pp 309ndash311 1956
[2] J J Gart and J R Zweifel ldquoOn the bias of various estimators ofthe logit and its variance with application of quantal bioassayrdquoBiometrika vol 54 no 1-2 pp 181ndash187 1967
[3] Y M Bishop S E Fienberg and P W Holland DiscreteMultivariate Analysis Theory and Practice Springer New YorkNY USA 2007
[4] A Agresti andM-C Yang ldquoAn empirical investigation of someeffects of sparseness in contingency tablesrdquo ComputationalStatistics and Data Analysis vol 5 no 1 pp 9ndash21 1987
[5] K F Hirji A A Tsiatis and C R Mehta ldquoMedian unbiasedestimation for binary datardquo The American Statistician vol 43no 1 pp 7ndash11 1989
[6] M Parzen S Lipsitz J Ibrahim and N Klar ldquoAn estimate ofthe odds ratio that always existsrdquo Journal of Computational andGraphical Statistics vol 11 no 2 pp 420ndash436 2002
[7] I J Good ldquoOn the estimations of small frequencies in contin-gency tablesrdquo Journal of the Royal Statistical Society Series BMethodological vol 18 pp 113ndash124 1956
[8] R A Fisher ldquoThe logic of inductive inferencerdquo Journal of theRoyal Statistical Society vol 98 no 1 pp 39ndash82 1935
[9] D G Thomas and J J Gart ldquoA table of exact confidence limitsfor differences and ratios of two proportions and their oddsratiosrdquo Journal of the American Statistical Association vol 72no 357 pp 73ndash76 1977
[10] P M E Altham ldquoExact Bayesian analysis of a 2times2 contingencytable and Fisherrsquos lsquoexactrsquo significance testrdquo Journal of the RoyalStatistical Society Series B (Methodological) vol 31 pp 261ndash2691969
[11] M Nurminen and P Mutanen ldquoExact Bayesian analysis of twoproportionsrdquo Scandinavian Journal of Statistics vol 14 no 1 pp67ndash77 1987
[12] B Nouri N Zare and S M T Ayatollahi ldquoValidation studymethods for estimating odds ratio in 2 times 2 times 119869 tables whenexposure is misclassifiedrdquo Computational and MathematicalMethods in Medicine vol 2013 Article ID 170120 8 pages 2013
[13] L Daly ldquoSimple SAS macros for the calculation of exactbinomial and Poisson confidence limitsrdquo Computers in Biologyand Medicine vol 22 no 5 pp 351ndash361 1992
[14] N L Johnson A W Kemp and S Kotz Univariate DiscreteDistribution John Wiley amp Sons New Jersey NJ USA 2005
[15] T P Minka ldquoEstimating a Dirichlet distributionrdquo Tech RepThe MIT Press London UK 2000
[16] The R Development Core Team An introduction to R Vienna2010 httpR-projectorg
[17] L Hardell ldquoRelation of soft-tissue sarcoma malignant lym-phoma and colon cancer to phenoxy acids chlorophenols andother agentsrdquo Scandinavian Journal of Work Environment andHealth vol 7 no 2 pp 119ndash130 1981
[18] M B M Perondi A G Reis E F Paiva V M Nadkarniand R A Berg ldquoA comparison of high-dose and standard-doseepinephrine in children with cardiac arrestrdquo The New EnglandJournal of Medicine vol 350 no 17 pp 1722ndash1730 2004
Let 11990110158401 and 11990110158402 be estimators of 1199011 and 1199012 respectively where11990110158401 = 1199101 + 11198991 + 1 + 1 11990110158402 = 1199102 + 21198992 + 2 + 2
(25)
Thus the EB estimator of odds ratio can be obtained asfollows
OREB = 11990110158401 (1 minus 11990110158401)11990110158402 (1 minus 11990110158402) (26)
where 11990110158401 and 11990110158402 denote success probability estimators ingroups 1 and 2 respectively
4 Simulation Study for MMLE MMUEand EB Method
Simulation studies have been carried out using R program(version 320) [16] to assess the efficiency of the EB methodin comparison with two existing methods Binomial data aregenerated with equal and unequal sample sizes (1198991 1198992) =(10 10) (10 30) with (10 50) probabilities of success ingroup 1 1199011 = 001 003 005 01 and 015 For each value
6 Journal of Probability and Statistics
Table 4 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 10)(1199011 1199012) EREEB EREMMLE EREMMUE
of 1199011 1199012 is varied to 001 003 005 01 and 015 Eachsituation is repeated 5000 times after removing the first1000 iterations (1000 burn-ins) The efficiency of proposedestimator is evaluated using Estimated Relative Error (ERE)defined as
ERE = [10038161003816100381610038161003816OR minus OR11989410038161003816100381610038161003816
OR] (27)
where OR denotes the usual maximum likelihood estimatorof odds ratio and OR119894 denotes the estimate of odds ratio usingEB MMLE and MMUE (119894 = 1 2 3 ) respectively
The simulation results with odds ratio estimates forsample sizes (1198991 1198992) = (10 10) (10 30) and (10 50) are givenin Tables 1ndash3 The performance of estimation uses ERE givenin Tables 4ndash6 and compares this result with graph in Figure 1the other case provides similar results It is found that theodds ratio estimation using EBmethodmostly yields smallestERE with 7867 while those using MMLE and MMUEmethods result in smallest ERE with only 667 and 1466respectively
5 Illustrative Examples Using Real Data
Our first example is taken from the studies of Good [7] andHardell [17] As shown in Table 7 subjects with malignant
Table 5 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 30)(1199011 1199012) EREEB EREMMLE EREMMUE
lymphoma and colon cancer cases and controls who areshortly exposed to phenoxy acids in agriculture and forestrywere observed including the true odds ratios and theirestimates using EB MMLE and MMUE For outcome inwhich (1198841 1198842) = (8 9) out of (1198991 1198992) = (24 53) for cases andcontrol respectively the estimate of the odds ratio using EBmethod yields the least ERE with 05523 while those usingMMLE and MMUE methods result in ERE with 12805 and41483 respectively
The second example is taken from the study of Perondiet al [18] as shown in Table 7 which compared high-doseepinephrine and standard-dose epinephrine in children withcardiac arrest with 34 children in each treatment includingthe true odds ratios and their estimates using EB MMLEand MMUE For outcome measure was survival at 24 hoursin which (1198841 1198842) = (1 7) out of (1198991 1198992) = (34 34) for highdose and standard dose respectivelyThe estimate of the oddsratio using EBmethod yields the least ERE with 52097 whilethose using MMUE and MMLE methods result in ERE with155305 and 404643 respectively
6 Conclusion
Based on simulated study for odds ratio estimation inrare events with two independent binomial data the resultindicates that the proposed method performs rather well
Journal of Probability and Statistics 7
002 004 006 008 010 012 014
0
50
100
150
200
250
300
350
ERE
p1
p2 = 005
002 004 006 008 010 012 014
0
100
200
300
400
500
600ER
E
p1
p2 = 01
002 004 006 008 010 012 014
20
30
40
50
60
70ER
E
p1
p2 = 001
002 004 006 008 010 012 014
0
50
100
150
200
ERE
p1
p2 = 003
002 004 006 008 010 012 014
0
100
200
300
400
500
600
700
ERE
p1
p2 = 015
EBMMLEMMUE
Figure 1 The percentages of ERE for odds ratio estimation using EB MMLE and MMUE when (1198991 1198992) = (10 10)
8 Journal of Probability and Statistics
Table 6 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 50)(1199011 1199012) EREEB EREMMLE EREMMUE
Table 7 True odds ratios and their estimates using EB MMLE andMMUE with corresponding percentages of ERE
MethodsTrue EB MMLE MMUE
1st example OR 24444 24309 24131 23430ERE mdash 05523 12805 41483
2nd example OR 01169 01230 01642 01350ERE mdash 52097 404643 155305
The EB estimator of odds ratio is also more efficient thanthe other two estimators MMLE and MMUE In additionour purposed estimator is an alternative method for oddsratio estimation to theMMLEmethodwithout disturbing theoriginal data
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors are grateful to the Graduate College KingMongkutrsquos University of Technology North Bangkok for thefinancial support
References
[1] J B S Haldane ldquoThe estimation and significance of thelogarithm of a ratio of frequenciesrdquo Annals of Human Geneticsvol 20 no 4 pp 309ndash311 1956
[2] J J Gart and J R Zweifel ldquoOn the bias of various estimators ofthe logit and its variance with application of quantal bioassayrdquoBiometrika vol 54 no 1-2 pp 181ndash187 1967
[3] Y M Bishop S E Fienberg and P W Holland DiscreteMultivariate Analysis Theory and Practice Springer New YorkNY USA 2007
[4] A Agresti andM-C Yang ldquoAn empirical investigation of someeffects of sparseness in contingency tablesrdquo ComputationalStatistics and Data Analysis vol 5 no 1 pp 9ndash21 1987
[5] K F Hirji A A Tsiatis and C R Mehta ldquoMedian unbiasedestimation for binary datardquo The American Statistician vol 43no 1 pp 7ndash11 1989
[6] M Parzen S Lipsitz J Ibrahim and N Klar ldquoAn estimate ofthe odds ratio that always existsrdquo Journal of Computational andGraphical Statistics vol 11 no 2 pp 420ndash436 2002
[7] I J Good ldquoOn the estimations of small frequencies in contin-gency tablesrdquo Journal of the Royal Statistical Society Series BMethodological vol 18 pp 113ndash124 1956
[8] R A Fisher ldquoThe logic of inductive inferencerdquo Journal of theRoyal Statistical Society vol 98 no 1 pp 39ndash82 1935
[9] D G Thomas and J J Gart ldquoA table of exact confidence limitsfor differences and ratios of two proportions and their oddsratiosrdquo Journal of the American Statistical Association vol 72no 357 pp 73ndash76 1977
[10] P M E Altham ldquoExact Bayesian analysis of a 2times2 contingencytable and Fisherrsquos lsquoexactrsquo significance testrdquo Journal of the RoyalStatistical Society Series B (Methodological) vol 31 pp 261ndash2691969
[11] M Nurminen and P Mutanen ldquoExact Bayesian analysis of twoproportionsrdquo Scandinavian Journal of Statistics vol 14 no 1 pp67ndash77 1987
[12] B Nouri N Zare and S M T Ayatollahi ldquoValidation studymethods for estimating odds ratio in 2 times 2 times 119869 tables whenexposure is misclassifiedrdquo Computational and MathematicalMethods in Medicine vol 2013 Article ID 170120 8 pages 2013
[13] L Daly ldquoSimple SAS macros for the calculation of exactbinomial and Poisson confidence limitsrdquo Computers in Biologyand Medicine vol 22 no 5 pp 351ndash361 1992
[14] N L Johnson A W Kemp and S Kotz Univariate DiscreteDistribution John Wiley amp Sons New Jersey NJ USA 2005
[15] T P Minka ldquoEstimating a Dirichlet distributionrdquo Tech RepThe MIT Press London UK 2000
[16] The R Development Core Team An introduction to R Vienna2010 httpR-projectorg
[17] L Hardell ldquoRelation of soft-tissue sarcoma malignant lym-phoma and colon cancer to phenoxy acids chlorophenols andother agentsrdquo Scandinavian Journal of Work Environment andHealth vol 7 no 2 pp 119ndash130 1981
[18] M B M Perondi A G Reis E F Paiva V M Nadkarniand R A Berg ldquoA comparison of high-dose and standard-doseepinephrine in children with cardiac arrestrdquo The New EnglandJournal of Medicine vol 350 no 17 pp 1722ndash1730 2004
of 1199011 1199012 is varied to 001 003 005 01 and 015 Eachsituation is repeated 5000 times after removing the first1000 iterations (1000 burn-ins) The efficiency of proposedestimator is evaluated using Estimated Relative Error (ERE)defined as
ERE = [10038161003816100381610038161003816OR minus OR11989410038161003816100381610038161003816
OR] (27)
where OR denotes the usual maximum likelihood estimatorof odds ratio and OR119894 denotes the estimate of odds ratio usingEB MMLE and MMUE (119894 = 1 2 3 ) respectively
The simulation results with odds ratio estimates forsample sizes (1198991 1198992) = (10 10) (10 30) and (10 50) are givenin Tables 1ndash3 The performance of estimation uses ERE givenin Tables 4ndash6 and compares this result with graph in Figure 1the other case provides similar results It is found that theodds ratio estimation using EBmethodmostly yields smallestERE with 7867 while those using MMLE and MMUEmethods result in smallest ERE with only 667 and 1466respectively
5 Illustrative Examples Using Real Data
Our first example is taken from the studies of Good [7] andHardell [17] As shown in Table 7 subjects with malignant
Table 5 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 30)(1199011 1199012) EREEB EREMMLE EREMMUE
lymphoma and colon cancer cases and controls who areshortly exposed to phenoxy acids in agriculture and forestrywere observed including the true odds ratios and theirestimates using EB MMLE and MMUE For outcome inwhich (1198841 1198842) = (8 9) out of (1198991 1198992) = (24 53) for cases andcontrol respectively the estimate of the odds ratio using EBmethod yields the least ERE with 05523 while those usingMMLE and MMUE methods result in ERE with 12805 and41483 respectively
The second example is taken from the study of Perondiet al [18] as shown in Table 7 which compared high-doseepinephrine and standard-dose epinephrine in children withcardiac arrest with 34 children in each treatment includingthe true odds ratios and their estimates using EB MMLEand MMUE For outcome measure was survival at 24 hoursin which (1198841 1198842) = (1 7) out of (1198991 1198992) = (34 34) for highdose and standard dose respectivelyThe estimate of the oddsratio using EBmethod yields the least ERE with 52097 whilethose using MMUE and MMLE methods result in ERE with155305 and 404643 respectively
6 Conclusion
Based on simulated study for odds ratio estimation inrare events with two independent binomial data the resultindicates that the proposed method performs rather well
Journal of Probability and Statistics 7
002 004 006 008 010 012 014
0
50
100
150
200
250
300
350
ERE
p1
p2 = 005
002 004 006 008 010 012 014
0
100
200
300
400
500
600ER
E
p1
p2 = 01
002 004 006 008 010 012 014
20
30
40
50
60
70ER
E
p1
p2 = 001
002 004 006 008 010 012 014
0
50
100
150
200
ERE
p1
p2 = 003
002 004 006 008 010 012 014
0
100
200
300
400
500
600
700
ERE
p1
p2 = 015
EBMMLEMMUE
Figure 1 The percentages of ERE for odds ratio estimation using EB MMLE and MMUE when (1198991 1198992) = (10 10)
8 Journal of Probability and Statistics
Table 6 The percentages of the Estimated Relative Error of oddsratio estimation for (1198991 1198992) = (10 50)(1199011 1199012) EREEB EREMMLE EREMMUE
Table 7 True odds ratios and their estimates using EB MMLE andMMUE with corresponding percentages of ERE
MethodsTrue EB MMLE MMUE
1st example OR 24444 24309 24131 23430ERE mdash 05523 12805 41483
2nd example OR 01169 01230 01642 01350ERE mdash 52097 404643 155305
The EB estimator of odds ratio is also more efficient thanthe other two estimators MMLE and MMUE In additionour purposed estimator is an alternative method for oddsratio estimation to theMMLEmethodwithout disturbing theoriginal data
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors are grateful to the Graduate College KingMongkutrsquos University of Technology North Bangkok for thefinancial support
References
[1] J B S Haldane ldquoThe estimation and significance of thelogarithm of a ratio of frequenciesrdquo Annals of Human Geneticsvol 20 no 4 pp 309ndash311 1956
[2] J J Gart and J R Zweifel ldquoOn the bias of various estimators ofthe logit and its variance with application of quantal bioassayrdquoBiometrika vol 54 no 1-2 pp 181ndash187 1967
[3] Y M Bishop S E Fienberg and P W Holland DiscreteMultivariate Analysis Theory and Practice Springer New YorkNY USA 2007
[4] A Agresti andM-C Yang ldquoAn empirical investigation of someeffects of sparseness in contingency tablesrdquo ComputationalStatistics and Data Analysis vol 5 no 1 pp 9ndash21 1987
[5] K F Hirji A A Tsiatis and C R Mehta ldquoMedian unbiasedestimation for binary datardquo The American Statistician vol 43no 1 pp 7ndash11 1989
[6] M Parzen S Lipsitz J Ibrahim and N Klar ldquoAn estimate ofthe odds ratio that always existsrdquo Journal of Computational andGraphical Statistics vol 11 no 2 pp 420ndash436 2002
[7] I J Good ldquoOn the estimations of small frequencies in contin-gency tablesrdquo Journal of the Royal Statistical Society Series BMethodological vol 18 pp 113ndash124 1956
[8] R A Fisher ldquoThe logic of inductive inferencerdquo Journal of theRoyal Statistical Society vol 98 no 1 pp 39ndash82 1935
[9] D G Thomas and J J Gart ldquoA table of exact confidence limitsfor differences and ratios of two proportions and their oddsratiosrdquo Journal of the American Statistical Association vol 72no 357 pp 73ndash76 1977
[10] P M E Altham ldquoExact Bayesian analysis of a 2times2 contingencytable and Fisherrsquos lsquoexactrsquo significance testrdquo Journal of the RoyalStatistical Society Series B (Methodological) vol 31 pp 261ndash2691969
[11] M Nurminen and P Mutanen ldquoExact Bayesian analysis of twoproportionsrdquo Scandinavian Journal of Statistics vol 14 no 1 pp67ndash77 1987
[12] B Nouri N Zare and S M T Ayatollahi ldquoValidation studymethods for estimating odds ratio in 2 times 2 times 119869 tables whenexposure is misclassifiedrdquo Computational and MathematicalMethods in Medicine vol 2013 Article ID 170120 8 pages 2013
[13] L Daly ldquoSimple SAS macros for the calculation of exactbinomial and Poisson confidence limitsrdquo Computers in Biologyand Medicine vol 22 no 5 pp 351ndash361 1992
[14] N L Johnson A W Kemp and S Kotz Univariate DiscreteDistribution John Wiley amp Sons New Jersey NJ USA 2005
[15] T P Minka ldquoEstimating a Dirichlet distributionrdquo Tech RepThe MIT Press London UK 2000
[16] The R Development Core Team An introduction to R Vienna2010 httpR-projectorg
[17] L Hardell ldquoRelation of soft-tissue sarcoma malignant lym-phoma and colon cancer to phenoxy acids chlorophenols andother agentsrdquo Scandinavian Journal of Work Environment andHealth vol 7 no 2 pp 119ndash130 1981
[18] M B M Perondi A G Reis E F Paiva V M Nadkarniand R A Berg ldquoA comparison of high-dose and standard-doseepinephrine in children with cardiac arrestrdquo The New EnglandJournal of Medicine vol 350 no 17 pp 1722ndash1730 2004
Table 7 True odds ratios and their estimates using EB MMLE andMMUE with corresponding percentages of ERE
MethodsTrue EB MMLE MMUE
1st example OR 24444 24309 24131 23430ERE mdash 05523 12805 41483
2nd example OR 01169 01230 01642 01350ERE mdash 52097 404643 155305
The EB estimator of odds ratio is also more efficient thanthe other two estimators MMLE and MMUE In additionour purposed estimator is an alternative method for oddsratio estimation to theMMLEmethodwithout disturbing theoriginal data
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors are grateful to the Graduate College KingMongkutrsquos University of Technology North Bangkok for thefinancial support
References
[1] J B S Haldane ldquoThe estimation and significance of thelogarithm of a ratio of frequenciesrdquo Annals of Human Geneticsvol 20 no 4 pp 309ndash311 1956
[2] J J Gart and J R Zweifel ldquoOn the bias of various estimators ofthe logit and its variance with application of quantal bioassayrdquoBiometrika vol 54 no 1-2 pp 181ndash187 1967
[3] Y M Bishop S E Fienberg and P W Holland DiscreteMultivariate Analysis Theory and Practice Springer New YorkNY USA 2007
[4] A Agresti andM-C Yang ldquoAn empirical investigation of someeffects of sparseness in contingency tablesrdquo ComputationalStatistics and Data Analysis vol 5 no 1 pp 9ndash21 1987
[5] K F Hirji A A Tsiatis and C R Mehta ldquoMedian unbiasedestimation for binary datardquo The American Statistician vol 43no 1 pp 7ndash11 1989
[6] M Parzen S Lipsitz J Ibrahim and N Klar ldquoAn estimate ofthe odds ratio that always existsrdquo Journal of Computational andGraphical Statistics vol 11 no 2 pp 420ndash436 2002
[7] I J Good ldquoOn the estimations of small frequencies in contin-gency tablesrdquo Journal of the Royal Statistical Society Series BMethodological vol 18 pp 113ndash124 1956
[8] R A Fisher ldquoThe logic of inductive inferencerdquo Journal of theRoyal Statistical Society vol 98 no 1 pp 39ndash82 1935
[9] D G Thomas and J J Gart ldquoA table of exact confidence limitsfor differences and ratios of two proportions and their oddsratiosrdquo Journal of the American Statistical Association vol 72no 357 pp 73ndash76 1977
[10] P M E Altham ldquoExact Bayesian analysis of a 2times2 contingencytable and Fisherrsquos lsquoexactrsquo significance testrdquo Journal of the RoyalStatistical Society Series B (Methodological) vol 31 pp 261ndash2691969
[11] M Nurminen and P Mutanen ldquoExact Bayesian analysis of twoproportionsrdquo Scandinavian Journal of Statistics vol 14 no 1 pp67ndash77 1987
[12] B Nouri N Zare and S M T Ayatollahi ldquoValidation studymethods for estimating odds ratio in 2 times 2 times 119869 tables whenexposure is misclassifiedrdquo Computational and MathematicalMethods in Medicine vol 2013 Article ID 170120 8 pages 2013
[13] L Daly ldquoSimple SAS macros for the calculation of exactbinomial and Poisson confidence limitsrdquo Computers in Biologyand Medicine vol 22 no 5 pp 351ndash361 1992
[14] N L Johnson A W Kemp and S Kotz Univariate DiscreteDistribution John Wiley amp Sons New Jersey NJ USA 2005
[15] T P Minka ldquoEstimating a Dirichlet distributionrdquo Tech RepThe MIT Press London UK 2000
[16] The R Development Core Team An introduction to R Vienna2010 httpR-projectorg
[17] L Hardell ldquoRelation of soft-tissue sarcoma malignant lym-phoma and colon cancer to phenoxy acids chlorophenols andother agentsrdquo Scandinavian Journal of Work Environment andHealth vol 7 no 2 pp 119ndash130 1981
[18] M B M Perondi A G Reis E F Paiva V M Nadkarniand R A Berg ldquoA comparison of high-dose and standard-doseepinephrine in children with cardiac arrestrdquo The New EnglandJournal of Medicine vol 350 no 17 pp 1722ndash1730 2004
Table 7 True odds ratios and their estimates using EB MMLE andMMUE with corresponding percentages of ERE
MethodsTrue EB MMLE MMUE
1st example OR 24444 24309 24131 23430ERE mdash 05523 12805 41483
2nd example OR 01169 01230 01642 01350ERE mdash 52097 404643 155305
The EB estimator of odds ratio is also more efficient thanthe other two estimators MMLE and MMUE In additionour purposed estimator is an alternative method for oddsratio estimation to theMMLEmethodwithout disturbing theoriginal data
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors are grateful to the Graduate College KingMongkutrsquos University of Technology North Bangkok for thefinancial support
References
[1] J B S Haldane ldquoThe estimation and significance of thelogarithm of a ratio of frequenciesrdquo Annals of Human Geneticsvol 20 no 4 pp 309ndash311 1956
[2] J J Gart and J R Zweifel ldquoOn the bias of various estimators ofthe logit and its variance with application of quantal bioassayrdquoBiometrika vol 54 no 1-2 pp 181ndash187 1967
[3] Y M Bishop S E Fienberg and P W Holland DiscreteMultivariate Analysis Theory and Practice Springer New YorkNY USA 2007
[4] A Agresti andM-C Yang ldquoAn empirical investigation of someeffects of sparseness in contingency tablesrdquo ComputationalStatistics and Data Analysis vol 5 no 1 pp 9ndash21 1987
[5] K F Hirji A A Tsiatis and C R Mehta ldquoMedian unbiasedestimation for binary datardquo The American Statistician vol 43no 1 pp 7ndash11 1989
[6] M Parzen S Lipsitz J Ibrahim and N Klar ldquoAn estimate ofthe odds ratio that always existsrdquo Journal of Computational andGraphical Statistics vol 11 no 2 pp 420ndash436 2002
[7] I J Good ldquoOn the estimations of small frequencies in contin-gency tablesrdquo Journal of the Royal Statistical Society Series BMethodological vol 18 pp 113ndash124 1956
[8] R A Fisher ldquoThe logic of inductive inferencerdquo Journal of theRoyal Statistical Society vol 98 no 1 pp 39ndash82 1935
[9] D G Thomas and J J Gart ldquoA table of exact confidence limitsfor differences and ratios of two proportions and their oddsratiosrdquo Journal of the American Statistical Association vol 72no 357 pp 73ndash76 1977
[10] P M E Altham ldquoExact Bayesian analysis of a 2times2 contingencytable and Fisherrsquos lsquoexactrsquo significance testrdquo Journal of the RoyalStatistical Society Series B (Methodological) vol 31 pp 261ndash2691969
[11] M Nurminen and P Mutanen ldquoExact Bayesian analysis of twoproportionsrdquo Scandinavian Journal of Statistics vol 14 no 1 pp67ndash77 1987
[12] B Nouri N Zare and S M T Ayatollahi ldquoValidation studymethods for estimating odds ratio in 2 times 2 times 119869 tables whenexposure is misclassifiedrdquo Computational and MathematicalMethods in Medicine vol 2013 Article ID 170120 8 pages 2013
[13] L Daly ldquoSimple SAS macros for the calculation of exactbinomial and Poisson confidence limitsrdquo Computers in Biologyand Medicine vol 22 no 5 pp 351ndash361 1992
[14] N L Johnson A W Kemp and S Kotz Univariate DiscreteDistribution John Wiley amp Sons New Jersey NJ USA 2005
[15] T P Minka ldquoEstimating a Dirichlet distributionrdquo Tech RepThe MIT Press London UK 2000
[16] The R Development Core Team An introduction to R Vienna2010 httpR-projectorg
[17] L Hardell ldquoRelation of soft-tissue sarcoma malignant lym-phoma and colon cancer to phenoxy acids chlorophenols andother agentsrdquo Scandinavian Journal of Work Environment andHealth vol 7 no 2 pp 119ndash130 1981
[18] M B M Perondi A G Reis E F Paiva V M Nadkarniand R A Berg ldquoA comparison of high-dose and standard-doseepinephrine in children with cardiac arrestrdquo The New EnglandJournal of Medicine vol 350 no 17 pp 1722ndash1730 2004