A Non-Parametric Predictive Model for Missing Data: A Case of Philippine Public Hospitals

Transcription

1 Proceedngs of the 2011 Internatonal Conference on Industral Engneerng and Operatons Management Kuala Lumpur, Malaysa, January 22 24, 2011 A Non-Parametrc Predctve Model for Mssng Data: A Case of Phlppne Publc Hosptals Vctor John M. Cantor, Rchard C. L, Martha Lauren L. Tan, Rachelle Joy S. Yu Department of Industral Engneerng De La Salle Unversty, Manla 1004, Phlppnes Abstract Organzatons have an abundance of data but actually have ncomplete nformaton. Incomplete nformaton happens when there s mssng or unrelable data whch can lead to wrong decsons. A predctve model was developed usng Hurwcz crteron by means of lnear programmng (LP). Ths predctve model estmates the organzaton s mssng or unrelable data usng external data from other smlar organzatons. The proposed predctve model was tested usng data from Phlppne publc hosptals under the Department of Health. The model was able to provde a range of data that can test the valdty of ncomplete nformaton encompassng the optmstc and pessmstc decsons made by the organzaton. Keywords Hurwcz crteron, lnear programmng, predctve model, performance measurement 1. Introducton Mssng data pertans to performance-related data that are unavalable due to several reasons that nclude: admnstratve fault n whch the staff faled to collect the data, malfunctonng of equpment that resulted to data corrupton and refusal of respondent to answer the questons from a survey among others. Performance measurement s data ntensve thus n any case wheren data s mssng, results may appear skewed and the relablty of the measurement s questonable [1]. Mssng data exsts and nevtable n all companes n dfferent sectors thus should be accounted for. Excluson or elmnaton of the cases or categores that contans mssng data produces a dstorted result of effcency as some neffcent systems may be consdered effcent n the absence of consderng a sgnfcant nput or output n evaluaton. Moreover, the mpact of mssng data s detrmental not only through ts potental hdden bases of the results but also n ts practcal mpact on the sample sze avalable for analyss. Therefore, estmaton of mssng data s left for selecton. In choosng so, several estmaton methodologes exst and must be further analyzed wth regards to ther assumptons and applcablty wth the study. There are two knds of estmaton methodologes, the parametrc whch assumes data come from a type of probablty dstrbuton and makes nferences about the parameters of the dstrbuton and the non-parametrc whch often referred to as dstrbuton free methods as they do not rely on assumptons that the data are drawn from a gven probablty dstrbuton. For the purpose of ths study, the non-parametrc technque was opted to not go nto several assumptons about the data to be gathered. 2. Revew of Related Lterature 2.1 Mssng Data Snce effcency measurements for most organzatons produce a wde range of outputs and use numerous nputs, data for these nputs and outputs must be well measured and collected n order to guarantee the accuracy of the measurement. However, more often than not t s not easy to get track of the records and nformaton that are needed as much as sometmes there s no systematzed recordng or documentaton system that would allow storage of records more accurately and tmely. 789

2 In ths premse, the problem of mssng data arses frequently n practce. Mssng data are performance related data that are unavalable durng the collecton of data to be used for measurng performance [2]. For example, wth the study of [3], consder a large survey of famles conducted n 1967 wth many socoeconomc varables recorded, and a follow-up survey of the same famles n Not only s t lkely that there wll be a few mssng values scattered throughout the data set, but also t s lkely that there wll be a large block of mssng values n the 1970 data because many famles studed n 1967 could not be located n Another wth [4], consderng the 160 studes that have been dentfed as havng mssng data, only 54 (33.75%) explctly acknowledged the problem. In contrast, 106 (66.25%) of the studes that have been dentfed as havng mssng data dd not menton the problem, and mssng values were nferred from degrees of freedom values that were nconsstent wth the stated sample sze and desgn characterstcs. Mssng data are a common problem n quanttatve research studes. Standard statstcal procedures were developed for data sets, so mssng values represent a consderable nusance to the analyst. Tradtonally, mssng data were dealt by means of varous ad hoc methods that attempted to fx that data before analyss. The blanket removal of cases wth mssng data (.e., lst-wse deleton) s one such strategy. Another method nvolves substtutng mssng values wth the varable mean. Unfortunately, these ad hoc tradtonal methods can serously bas sample statstcs and have been crtczed n the methodologcal lterature [5]. 2.2 Parametrc Estmaton Methodology Regresson analyss s used to predct the mssng values of a varable based on ts relatonshp to other varables n the data set. Although t has the appeal of usng relatonshps already exstng n the sample as the bass of predcton, ths method also has several dsadvantages. Frst, t renforces the relatonshps already n the data. As the use of ths method ncreases, the resultng data become more characterstc of the sample and less generalzable. Second, unless stochastc terms are added to the estmated values, the varance of the dstrbuton s understood. Thrd, ths method assumes that the varable wth mssng data has substantal correlatons wth other varables. If these correlatons are not suffcent to produce a meanngful estmate, then other methods, such as mean substtuton, are preferable. Fnally, the regresson procedure s not constraned n the estmates t makes. Thus, the predcted values may not fall n the vald ranges for varables, thereby requrng some form of addtonal adjustment. Even wth all of these potental problems, the regresson method of mputaton holds promse n those nstances for whch moderate levels of wdely scattered mssng data are present and for whch the relatonshps between varables are suffcently establshed so that the researcher s confdent that usng ths method wll not mpact the generalzablty of the results. However, for most cases the data are beng collected for analyss does not follow normal dstrbuton thus for the purpose of ths study a non-parametrc estmaton method s more preferred. 2.3 LP Predctve Estmaton A non-parametrc technque that fnds the best weghts to be used n the lnear equaton based on the data set and ts objectve s to mnmze the predcton values errors, that s, both negatve and postve resduals. Mathematcally, ts objectve functon s to mnmze the standard error of estmated usng the resduals as the varables. 2.4 Hurwcz Crteron The Hurwcz crteron attempts to fnd a mddle ground between the extremes posed by the optmst and pessmst crtera. Instead of assumng total optmsm and pessmsm, Hurwcz crteron ncorporates a measure of both by assgnng a certan percentage weght to optmsm and the balance to pessmsm. A weghted average can be computed for every acton alternatve wth an alpha-weght, α, called the coeffcent of realsm. Realsm here means that the unbrdled optmsm of Maxmax s replaced by an attenuated optmsm as denoted by the α. Note that 0α1. Thus, a better name for the coeffcent of realsm s coeffcent of optmsm. An α = 1 denotes absolute optmsm (Maxmax) whle an α = 0 ndcates absolute pessmsm (Maxmn). The α s selected subjectvely by the decson maker. Hurwcz crteron wll be ncorporated n the estmaton of mssng to consder the dfferng level of optmsm of dfferent decson makers. 790

3 3. Methodology The study follows a systematc methodology n developng an LP predctve estmaton model up untl the valdaton of the estmaton logc. Intally, the development of the LP predctve estmaton model wth the ncluson of Hurwcz crteron was done. The entre model was encoded n Mcrosoft Excel for data valdaton. Data were gathered from lterature and from dfferent database sources about healthcare performance n the Phlppnes wheren complete nformaton was avalable. The ntenton of collectng a complete set of data was to purposely have the freedom to delete some data whch wll represent the mssng data; and the deleted nformaton wll then become the bass of the desrablty of the estmated value of the LP predctve estmaton model. After the collecton of data was completed, the LP predctve estmaton model was run usng the gathered data and predctve results were obtaned. Several runs were done for each data set that were collected usng dfferent estmaton technques (.e., regresson analyss, mean substtuton, and case substtuton) n order to compare the predctve results of these estmaton technques wth that of the LP predctve estmaton model. Fnally, after the comparson of results wth the dfferent estmaton technques, analyss of model logc and extracton of nsghts from the model results were conducted to ensure that the LP predctve estmaton model gave a sound and logcal results. 4. LP Predctve Estmaton Methodology 4.1 LP Predctve Estmaton Model To adjust data sets where there are mssng data occurrences, an LP predctve estmaton model s suggested to estmate the mssng data. The LP predctve estmaton model s non-parametrc n nature; as such there s no assumpton of normalty of data. Whether the data s non-normal or normal the LP predctve estmaton model can be used to estmate the data. The model s as follows: Indces = data set Parameters C = actual value of parameter to be estmated n data set X = predctor parameter for data set Varables A 1 = postve slope coeffcent A 2 = negatve slope coeffcent B 1 = postve y slope coeffcent B 2 = negatve y slope coeffcent = estmate error for data set Objectve Functon The objectve of the model s to fnd the best combnaton of coeffcents to mnmze the sum of estmate error from what s estmated by the model, and the actual value of the parameter to be estmated (ths s denoted by the varable ). Mn n 1 (1) Constrants Estmate errors are obtaned and calculated through error constrants lmts the error should be the dfference of the predcted value s greater that the actual value (negatve error), and the other type would be that the predcted value s less than the actual value (postve error). To dfferentate postve and negatve errors two constrants are ntroduced for the two possble error types. o Negatve Error Constrant The nequalty constrans that f an estmated value s less than the actual value, the estmate error should be greater than zero. C A1 A2 * X B1 B2 0, (2) o Postve Error Constrant 791

4 Ths constrant lmts that f an estmated value s less than the actual value, the estmate error should be greater than zero. C A 1 A 2 * X B 1 B 2 0, (3) 4.2 The Predcton Methodology Usng the model, par wse comparson of Standard Error s done to select the most approprate predctor for the data to be predcted. Predctor wth least standard error s selected. Use functon to estmate data range The LP estmaton model generates a sngle pont estmate of the mssng data, however snce the values s just an estmate there s no guarantee that the calculated value wll represent the real value of the mssng data. An nterval estmate would better sut to represent the range of values n whch the mssng value can be expected. From each of the par wse combnaton comparsons the standard error of the lnear functon was calculated usng the followng formula: SSE S e (4) n 2 It s usually true that approxmately 68% of the estmated values of y wll be wthn Se, and approxmately 95% of estmated y wll be wthn 2Se [6]. In the three data set used t was found that approxmately 83% are wthn Se. For the purpose of the estmate ntervals, these are approxmated by Se. Dependng on the degree of optmsm/pessmsm set Hurwcz crteron Snce mssng data are expressed n terms of ntervals, the concept of Hurwcz crteron s suggested such that the degree of optmsm or pessmsm s ncorporated n obtanng a performance measure. The Hurwcz crteron attempts to fnd a mddle ground between the extremes posed by the optmst and pessmst crtera. Instead of assumng total optmsm or pessmsm, Hurwcz ncorporates a measure of both by assgnng a certan percentage weght to optmsm and the balance to pessmsm. Usng data range and Hurwcz crteron, predct mssng data 5. Results and Analyss 5.1 Data For the purpose of numercal valdaton, a case study s done on the urban health system of Local Government Unts (LGU) n Metro Manla. There are 17 LGUs to be consdered. The data set and nformaton used for ths case study s gathered from the offcal webste and resources of the respectve ctes and from the materals of the study conducted by [7]. Data used are populaton, health establshments, health personnel, treatments servced and cases examned. Health personnel refer to all the personnel nvolved n assstng the needs of the people n terms of ther health problems. Some of the personnel nvolved n adng the people are medcal doctor, regstered nurses, nutrtonsts or detcan, dentsts, mdwves, other techncal and non-techncal health worker. The value of the health personnel s solved usng weghted average method. The health personnel s equal to the sum product of the number of personnel per manpower type and weght per manpower type. Health establshments are the total number of health facltes n the LGU such as hosptals, health centers and clncs. Treatments servced refer to the total number of servce provded to the consttuents of each cty. It concerns treatments and vaccnes that each LGU provded to the ctzens of the cty. Ths ncludes the followng treatments and vaccnes: HEPA B mmunzaton, tetanus toxod, vtamn A, dewormng, dental, tuberculoss, rabes, and leprosy treatment. In addton, cases examned refers to the total number of patents examned by the health 792

5 personnel n each LGU namely tuberculoss cases, STD cases, measles cases, dengue fever cases, and pneumona cases. As can be seen n Table 1, there are cases of mssng data for health establshments and health personnel. Table 1: LGUs Health Data Set Used for Case Study Cty/Muncpalty Health Health Treatments Cases Populaton Establshments Personnel Servced Examned Malabon 324, , , Navotas 231, , , Valenzuela 555, , , Caloocan Cty 1,445, , , Markna 445, , , Pasg 557, , , Pateros 60, , , Tagug 630, , , Quezon Cty 2,468, ,008, , Makat 389, , , Mandaluyong 264, , , San Juan 107, , , Manla 1,468, , , Las Pñas 575, , , Muntnlupa 348, , , Parañaque 550, , , Pasay Cty 285, , , To valdate results of the LP estmaton technque, standard error of estmates of the resultng functon was used to compare the resultng data estmatons. Lnear functon coeffcents were obtaned through regresson analyss as well as the LP estmaton methodology. 5.2 Par-wse Regresson Results Usng regresson analyss for estmaton for number of health establshments, the best predctor would be the data for treatments servced wth a standard error of As for health personnel estmates, the best predctor would be the health establshment data set wth standard error of (See Table 2). Table 2: Par-wse Regresson Results n Predctng Establshment and Personnel Varables 5.3 Par-wse LP Estmaton Results Y X SSE Se Coeffcents X Intercept Establshment Populaton E Establshment Personnel Establshment Treatments Establshment Cases Personnel Populaton Personnel Establshment Personnel Treatments Personnel Cases Usng LP estmaton methodology for the estmaton of establshments, the best predctor would be treatments wth a standard error of As for health personnel estmates the best predctor would be the establshments wth standard error of (See Table 3). For both regresson and LP estmaton, the predctors chosen are the same, but the LP estmaton methodology resulted n lower standard error. Table 3: Par-wse Regresson Results n Predctng Establshment and Personnel Varables Y X SSE Se Coeffcents X Intercept Establshment Populaton E Establshment Personnel Establshment Treatments Establshment Cases Personnel Populaton Personnel Establshment Personnel Treatments Personnel Cases

6 5.4 Par-wse LP Estmaton Results Resultng estmates are logcal as the resultng estmates are qute proportonal to other nputs. Example for the cty of Markna, ts data for populaton, personnel, treatments and cases are small to md sze n magntude, t s only logcal that the estmate for health establshment s also small to md sze. Ths behavor s observed wth other ctes as well where estmates of data are obtaned. Table 4 and Table 5 show varous optons of results that can be obtaned usng the LP estmaton methodology. These varyng types of results can be used dependng on the needs and use of the estmated data. Pont estmate smply gves the resultng estmate of the predctng functon; whle a range of values (mnmum and maxmum) can be used to have the confdence that the real value or data falls between the predcted range. Hurwcz crteron on the other hand, gves the flexblty for the user on how optmstc he or she s on the LGU or cty n terms of nput usage or output producton. For example, n estmatng for the health establshment data for Markna (the user s a bt pessmstc n the effcency of Markna Health Care System); snce there s a degree of pessmsm, a lower alpha crteron can be used (0.4) to predct a hgher value but stll wthn the max and mn range. Hurwcz crteron can gve the user estmaton flexblty, dependng on the degree n whch they know the operatons of the LGU or cty n whch a data s beng estmated. 5.5 Concluson Table 4: Estmate Results for Health Establshments Estmates for Health Establshments Cty Pont estmate Mn value Max Value Wth Hurwcz alpha = 0.4 alpha = 0.6 Markna Pateros Quezon Cty Las Pñas Table 5: Estmate Results for Health Personnel Estmates for Health Personnel Cty Pont estmate Mn value Max Value Wth Hurwcz alpha = 0.4 alpha = 0.6 Pasg Manla The LP estmaton methodology had smlar results wth regresson analyss n terms of the best predctor for a lnear estmaton functon. However, usng the LP estmaton methodology resulted n lower standard error of estmate. Advantages of usng the LP estmaton methodology nclude: lower standard error of estmate, and that t s not lmted to the assumptons of data. LP estmaton can be used even when data sets are small (less than 30) or when data exhbts non-normalty. Also, resultng estmates were logcal n terms of the proportonalty wth other nput data set across cty/lgu. The estmaton methodology also allows for varyng types of data estmates, such as ranges and usng of Hurwcz crteron to allow for user flexblty for optmstc and pessmstc estmates. References 1. Kao, C., and Lu, S.T., 2000, Data Envelopment Analyss wth mssng data: An applcaton to Unversty Lbrares n Tawan, The Journal of the Operatonal Research Socety, 51(8), Lttle, R.J.A. and Rubn, D.B., 2002, Statstcal Analyss wth Mssng Data, 2 nd edton, New York: John Wley. 3. Rubn, D.B., 1976, Inference and mssng data, Bometrka, 63, Roth, P.L., 1994, Mssng data: A conceptual revew for appled psychologsts, Personnel Psychology, 47(3), Enders, C., and Peugh, J., 2004, Mssng Data n Educatonal Research: A Revew of Reportng Practces and Suggestons for Improvement, Revew of Educatonal Research, 74(4), Wnston, W.L., 2004, Operatons research: Applcatons and algorthms, 4th edton, Calforna: Thomson. 7. Manalang, A., 2009, A baselne comparatve of the urban health systems n the natonal captal regon, APIEMS conference proceedngs