Bayesian Sequential Estimation of Proportion of Orthopedic Surgery of Type 2 Diabetic Patients Among Different Age Groups A Case Study of Government

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1 Bayesia Sequetial Estimatio of Proportio of Orthopedic Surgery of Type Diabetic Patiets Amog Differet Age Groups A Case Study of Govermet Medical College, Jammu-Idia Roohi Gupta, Priyaka Aad ad *Rahul Gupta Departmet of Statistics, Uiversity of Jammu, Jammu *rahulgupta68@yahoo.com Abstract: A study to determie the proportio of diabetic cases hadled by the hospitals i Jammu EPH - Iteratioal Joural of Mathematics ad Statistics regardig orthopedic surgeries, especially amog differet age group, by employig a Bayesia model called Beta-biomial cougate model usig Bayesia sequetial estimatio method. From the hospital record of Govermet Medical College(GMC, Jammu. Data was collected ad collated over a period of 3 years from i terms of 4 quarters (Ja-March, April-Jue, July-Sep ad Oct- Dec. The study also compares the computed sample ad Empirical Bayes estimates over the three year period ad also the variaces of the computed estimates ad it has bee strogly advocated that Empirical Bayes estimators are better estimators o the basis of efficiecy ad cosistecy ad the proposed method ca be employed successfully i ay suitable locatio globally. Key Words: Bayesia, Empirical Bayes, Beta-Biomial cougate model, Estimatio of proportio. 1. INTRODUCTION AND BACKGROUND Bayesia solutios provide may istictive ad reasoable coclusios tha likelihood based ifereces, which is simply because Bayesia aalysis icludes prior iformatio as well as likelihood fuctio. I empirical Bayes(EB approach, data is used to help determiatio of the prior through estimatio of the so called hyperparameters. The cosiderable impact of Empirical Bayes (EB o statistical applicatios cotiues to obtai icreasig popularity sice its itroductio about four decades ago. The EB structure combies iformatio from several but similar sources. The primary iterest i EB aalysis is i the hyperparameters ( rather tha the parameters, from idividual studies (. More geerally, the hierarchical structure allows for appraisal of heterogeeity both withi ad betwee groups ( Carli ad Louis(1. Thus Bayesia approach to parameter estimatio, whe coditios of data allow, is to estimate the posterior distributio of the parameter(s i questio ( so that iferece o is the based o the posterior distributio. A prior distributio for is eeded to derive the posterior ad, i some cases, the prior may have its ow parameters, called hyperparameter(s (. Quite ofte, is ukow to the aalyst, i which case the prior is ot completely specified. Oe way of resolvig this problem is through empirical Bayes (EB aalysis. More importatly, EB ca lead to more precise estimates tha samplig theory approaches (Rubi(. EB aalysis also provides a more depedable rakig of parameters ad aids i the idetificatio of extreme values i the group(lik ad Hah(3. These properties of EB derive from the fact that it uses related supplemetary data which frequetist iferece igores(okafor(4. The EB cocept was first proposed by Robbis(5 i a o-parametric settig. Sigificat works i this regard has bee doe by Deely ad Lidley(6, Hui ad Berger(7, Morris(8, Wog(9, Smith(10, Raghuatha(11, Altma ad Casella(1, Efro(13 ad Carli ad Louis(14. Sigificat work o this field iclude Ogudei ad Okafor(15, Okafor,et al(16 ad Okafor ad Mbata(17.

2 By Type diabetes, the study refers to the brach of diabetes cocered with coditios ivolvig the edocrie system. Type diabetes is a progressive coditio, meaig that the loger someoe has it, the more help they will eed to maage blood glucose levels. This will require more medicatios & evetually, iected isuli will be eeded. People with type diabetes produce isuli, but their bodies do t use it correctly, this is referred to as beig isuli resistat. People with Type diabetes may also be uable to produce eough isuli to hadle the glucose i their body. I these istaces, isuli is eeded to allow the glucose to travel from the bloodstream ito our cells, where it s used to create eergy. I a attempt to EPH - Iteratioal determie Joural of Mathematics the ad Statistics proportio of diabetic cases hadled by the hospital regardig orthopedic surgeries, especially amog differet age group, we employed a Bayesia model called Beta-biomial cougate model usig Bayesia sequetial estimatio method, o the lies of Ogudei et.al(18. From the hospital record of Govermet Medical College(GMC, Jammu, data were collected ad collated over a period of 3 years from i terms of 4 quarters (Ja-March, April-Jue, July-Sep ad Oct-Dec. The study also compares the computed sample ad EB estimates over the three year period ad also the variaces of the computed estimates..the BAESIAN SEQUENTIAL PROCEDURE If a set of observatios x1, x, x3,.,., x geerates a posterior divisio ad, i a comparable situatio, supplemetary data are collected beyod these observatios, the the posterior distributio foud with earlier observatios becomes the ew prior distributio ad the additioal observatio give a ew posterior distributio ad coclusio ca be made from the secod posterior distributio. This process ca go o with eve more observatio. I other words, the secod posterior becomes the ew prior ad the ext set of observatios give the ext posterior from which the iferece ca be made. This is the priciple of Bayesia sequetial methodology that we applied to estimate the proportio of couted data obtaied from the hospital. Based o the Bayesia approach described above, data were collected mothly ad collated yearly for three years (014, 015 ad 016 from the hospital records of Govermet Medical College(GMC, Jammu. Tie populatio proportio of diabetic patiets admitted for orthopedic surgery i GMC is deoted by P0 while the proportio of diabetic patiets admitted for orthopedic surgery i age group is P ( = 1,,..., 5. X0 represets a radom outcome of patiet examied i age group. i 1,if ith diabetic Patiet is admitted for orthopedic surgery i age group ad i year k 0, otherwise ik i the total umber of diabetic patiets admitted for orthopedic surgeries i i1 age group i the year k, the total umber of diabetic patiets admitted for treatmets (both orthopedic ad o-

3 orthopedic surgeries i GMC Jammu i age group i year k. P the proportio of diabetic patiets admitted for orthopedic surgery i age group ad year k. For each year i each age group, we computed sample proportios P as follows: I 014 ad age group : P1 I 015 ad age group : P I 016 ad age group : P 3 Estimators of sample proportios: (1 Var( ad 3. THE BETA BINOMIAL CONJUGATE MODEL The Empirical Bayesia model to be realistic is a cougate beta biomial model where the biomial distributio represets the likelihood of the observed data likelihood ad the beta distributio take as the prior distributio of the biomial parameter. The posterior mea is P P f ( P, dp (1 A key compoet of this itegral is f ( P,. The posterior distributio of which is P. Uder the geeral Bayesia framework ad usig the beta cougate prior plus the biomial likelihood, the posterior distributio of P is: f ( P EPH - Iteratioal Joural of Mathematics ad Statistics 1 r 1 s1 P (1 P P (1 P B( r, s,, ( r. s ( 1 r 1 s1 P (1 P P (1 P dp B( r, s There is eed to estimate the hyper-parameters r ad s of the beta distributio i order to completely specify the prior. This ca be achieved easily through reparameterisatio of r f ( P, ad usig momet estimator. Lettig P0 ; M r s ad usig the prior r s

4 rs Po (1 P0 distributio of P; E( P P0 ad Var( P. These are ( r s 1( r s M 1 kow as prior mea ad variace respectively. Cosequetly, ˆ 1 a1 1 f ( P, ˆ, M P (1 P (3 B(, where, ˆ ˆ ˆ MP0 ; EPH - Iteratioal Joural of Mathematics ad Statistics ˆ Mˆ (1 i 0, 0 ad Mˆ d 0 (1 0 SP 1 0(1 0 SP N N ( where SP with M ad P0 estimated, the, ( N 1 i P ( /,, ˆ EB E P 0 M Mˆ P 0 ad M M ˆ ˆ (4 Var ( P EB. (5 ( 1( Mˆ Cosequetly, ˆ ad it ca be easily see that where Mˆ (the scale factor is large Mˆ relative to, is large ad ˆP 0 receives a larger weight tha. But large Mˆ implies small prior variace. Thus, the estimate which is associated with smaller variace receives larger weight i determiig the posterior mea P EB. O the other had, if Mˆ is small relative to, the sample mea receives more weight. Note that the posterior desity for the overall age group proportio P0 is obtaied by replacig ad i equatio (3 with ad N, respectively.

5 Uder cougacy, the EB estimator of a proportio if is a weighted mea of two estimators, the mea of the prior desity P0 ad the sample proportio estimator P ˆif. Thus, PEB P (1 0 if (6 P EB is the empirical Bayes Estimators with as the shrikage factor. is a fuctio of the prior ad sample estimator variaces such that, if variace of sample estimator is large, the weight of P ˆ0 ( i. e. will be large ad P EB will shrik towards P ˆ0. Two compoets of the above model ad ˆP EPH - Iteratioal Joural of 0 are Mathematics derived ad Statistics from the Empirical Bayesia process, proposed by Carli ad Louis( RESULTS The empirical results of the applicatio of Beta Biomial model ad Bayesia sequetial approach to the data of differet age group observatios(patiets for the three years (014, 015 ad 016 are preseted i Table 1 ad. The hyper parameters ad M are aticipated usig sample iformatio. These were cosequetly used to establish the statistical costats of the posterior distributios ad thereby etirely specifyig them. I this aalysis, comparisos are made o yearly basis estimated sample proportios ad Empirical Bayesia proportios as well as variaces of estimated sample proportios ad Empirical Bayesia proportios. Age group Table 1 (Comparative Aalysis of Estimated Sample proportios & EB Proportios Ja-March April-Jue July-Sept Oct-Dec Pi PEB P PEB P3 PEB P4 PEB

6 Age group Table (Comparative Aalysis of variaces of Estimated Sample proportios & EB Proportios Quarter 1 Quarter Quarter 3 Quarter 4 Var(Pi Var(PEB Var(P Var(PEB Var(P3 Var(PEB Var(P4 Var(PEB EPH - Iteratioal Joural of Mathematics ad Statistics Over the years uder deliberatio, the results show that the highest umber of patiets atteded to at the hospital is amog the age group 15 to 44 years but this age group also has the smallest proportio of diabetic surgeries. Similarly, the smallest umber of patiets ca be foud amog the two age groups 'less tha oe ear' ad 'greater tha 64'. Also the highest proportio of diabetic surgeries is amog patiets whose ages are greater tha 64 years followed by those who are less tha oe year old. The overall EB proportio of patiets admitted for diabetic surgeries i the hospital across the age groups elarged steadily (0.469, 0.58 ad over the" respective years of study. Furthermore, the computed variaces of the sample ad EB estimates are smallest amid the age groups '15-44 years' ad '44-64 years' while highest variaces are oted i age group 'less tha oe' ad 'greater tha 64 years' (see Table. 5. SUMMAR AND CONCLUSIONS This effort has bee able to demostrate how Bayesia sequetial estimatio of proportio ca be applied to statistical progressio cotrol for differet ages of the patiets ad years. The cosequece of the ivestigatio was compared both of yearly estimated sample proportios ad EB proportios plus the variaces. It was foud that overall EB proportio of diabetic patiets admitted for orthopedic surgeries i the hospital across the age groups icreased gradually. Similarly, the overall variaces of the proportios be iclied more to zero over the three years uder review. Thus, the results show that the Empirical Bayesia estimators are better estimators o the basis of efficiecy ad cosistecy properties of good estimators.

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