A Simulated Data Analysis on the Interval Estimation for the Binomial Proportion P

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1 Joural of Educatioal Policy ad Etrereeurial Research (JEPER) Vol., N0., October 0. P 77-8 A Simulated Data Aalysis o the Iterval Estimatio for the Biomial Proortio P Juge B. Guillea Advetist Medical Ceter College, Iliga City, Philiies ju0guillea@yahoo.com Abstract This study costructed a quadratic-based iterval estimator for biomial roortio. The method imosed a cotiuity correctio over the cofidece iterval. This quadratic-based iterval was comared to the differet existig alterative itervals through umerical aalysis usig the followig criteria: coverage, ad exected width for various values of, ad α = Simulated data results geerated the followig observatios: () the coverage of iterval is larger comared to that of the stadard ad o- itervals, for ay ad ; () the coverage of all the alterative methods aroaches to the omial 95% cofidece level as icreases for ay ;(3) the ad o- itervals have idistiguishable width differeces for ay as gets larger; () the exected width of the ad alterative itervals decreases as icreases for ad ay. Based o these observatios oe ca say that the method is a imrovemet of the stadard method. It is therefore recommeded to modify other existig alterative methods i such a way that there s a icrease i erformace i terms of coverage roerties, exected width, ad other measures. Keywords: Cofidece Iterval, Biomial Distributio, Stadard Iterval, Coverage Probability, Exected Width Itroductio Iferetial roblem like iterval estimatio arisig from biomial exerimets is oe of the classical roblems i statistics offerig may argumets ad disutes. Whe costructig a cofidece iterval, oe usually wishes the actual coverage to be close to the omial cofidece level, that is, it closely aroximates to. The uexected difficulties iheret to the choice of a cofidece iterval estimate of the biomial arameter, ad the relative iefficiecy (Marchad, E., Perro, F., ad Rokhaya, G., 00) f the stadard Wald cofidece iterval, has resurfaced recetly with the work of Brow, L. D., Cai, T.T., ad DasGuta, A. (999a ad 99b) ad Agresti ad Coull (998). Alog with this, several alterative iterval estimates have bee suggested. Some alterative itervals make use of a cotiuity correctio while others guaratee a miimum coverage for all values of the arameter. I lie with this, this study aims to develo a alterative method with slight modificatios of the method first develoed by Casella, et al., 990. As suggested, this modificatio imoses a cotiuity correctio factor. 77 htt:// Juge B. Guillea

2 Joural of Educatioal Policy ad Etrereeurial Research (JEPER) Vol., N0., October 0. P 77-8 Purose of the Study The objective of this study is to costruct a o-radomied cofidece iterval X Cx P C for, such that the coverage, where is some re-secified value betwee 0 ad (Casella ad Berger, 990) Secifically, the objective of this study is to comare umerically the erformace of the stadard, o ad itervals ad some alterative iterval estimators based o coverage ad exected width. BASIC CONCEPTS: Cofidece Iterval X,... f. Let l ( x) lx, x,... x ux, x x be two statistics satisfyig lx ux for which P l ( x) u( x) l ( x), u( x) is called a 00( )% cofidece iterval for ; Defiitio : Let u x),... ( the radom iterval cofidece coefficiet; ad (x), X X be a radom samle from the desity x ad. The is called the l ad u(x) are called the lower ad uer cofidece limits, resectively, for. Exected Width ad Coverage Probability: Some criteria for evaluatig iterval estimators are the iterval width ad coverage. Ideally, a iterval must have arrow width with large coverage, but such sets are usually difficult to costruct. Defiitio : The coverage of the cofidece set C x is defied as P CX I CxdFx where: is the samle sace of X ad I C(x) is a idicator fuctio for a oradomied set equal to if Cx, otherwise it is 0. Defiitio 3: The exected width is defied as: E f width of CX UX LX,, where x U X ad L X are the uer ad lower limits resectively of the cofidece set C x x0 Stadard Iterval Estimator: A stadard cofidece iterval for based o ormal aroximatio has gaied uiversal recommedatio i the itroductory statistics textbooks ad i statistical ractice. The iterval is kow to guaratee that for ay fixed, the coverage P C x as. To show this iterval estimator, let ad distributio, resectively. Let be the stadard ormal desity fuctio ad cumulative, x ˆ ad qˆ ˆ, where ˆ qˆ theory aroximatio of a cofidece iterval for biomial roortio is defied as: CX where is the / th quatile of the stadard ormal distributio. s ˆ.The ormal ˆ ˆ, 78 htt:// Juge B. Guillea

3 Joural of Educatioal Policy ad Etrereeurial Research (JEPER) Vol., N0., October 0. P htt:// Juge B. Guillea The Proosed Modified Iterval: Due to the discreteess of the biomial distributio ad as suggested by Casella, et al., 990, this roosed iterval imoses a cotiuity correctio, c, over the iterval. The factor is arbitrarily chose. Theorem : The aroximate cofidece iterval for with c is give by X C ˆ ˆ ˆ where the lower limit is give by, x L ˆ ˆ ˆ, ad uer limit is give by x U ˆ ˆ ˆ, Simulated Results ad Discussios: This sectio resets the comarative grahical ad umerical results ad comarisos of the differet alterative iterval estimators, i terms of its coverage behavior, ad exected width. I ivestigatig the erformace of the stadard iterval ad the alterative itervals, the usual = 0.05 is utilied. Simulatio of data values was doe through Male rogram. Comariso for Stadard, No-Modified ad Modified Itervals i terms of Coverage Probability: Figures resets the result of the coverage grahs of the stadard, the o- ad the itervals for = 0, 0, 70 ad 00 with omial 95%. It shows that both the o- ad the stadard itervals have sigificatly dowward sikes ear close to 0 or, while the iterval has a good coverage behavior for ay. The above aforemetioed results give evideces ad suorts to the followig claim: the coverage of the iterval has much better behavior over the stadard ad the o- itervals for ay ad.

4 Joural of Educatioal Policy ad Etrereeurial Research (JEPER) Vol., N0., October 0. P = 0, variable omial 95% level.0 = 0, variable omial 95% level coverage stadard o = 70, variable omial 95% level coverage stadard o- = 00, variable omial 95% level coverage stadard o- coverage stadard o Figure Comariso of coverage of the stadard, the o- ad the itervals for = 0, 0, 70 ad 00 with Comariso for Modified ad Alterative Itervals Figures shows the result of the coverage grahs of the Wilso, the Agresti Coull, the arcsie, the Wilso*, the Logit**, ad the itervals for = 70, 50, 300 ad 500 with variable for omial 95% cofidece level. It reveals that the Agresti-Coull iterval has coservative coverage ear = 0, which meas that most of the coverage is above the omial level. O the other had, the Wilso iterval has a fairly dowward sike ear 0 or, but has a good coverage away from the boudaries. The arcsie iterval has a erratic atter ear the boudaries, sice the coverage cuts off quickly at some values of 0.03, 0.05 or 0.96, with values below The iterval has some dowward sike ear the boudaries but gradually disaear as aroaches to 0.5 or away from 0 or. This iterval is comarable to other alterative itervals like the logit**, the Wilso, the arcsie but less comarable to the Agresti- Coull ad Wilso* itervals i terms of coverage behavior. Whe 0.0, 6 or 0.9, 0.99 the Agresti-Coull iterval aside from the Wilso* have coverage robabilities greater tha For larger values of, which i this case = 300 ad 500, the Wilso* has a cosistet coverage behavior that is greater tha or equal to 0.95 for all values of. The Wilso, arcsie, logit** ad itervals have some dowward sike ear = 0.0, but still the coverage of these itervals erform well i the middle arameter sace regio. These umerical fidigs show that the iterval has a comarable coverage behavior both i = 70, 50, 300 ad 500 for omial 95% cofidece level. These results give suort to the followig suggestio that the coverage behavior of all the methods aroaches to the omial 95% cofidece level as icreases for ay. 80 htt:// Juge B. Guillea

5 Joural of Educatioal Policy ad Etrereeurial Research (JEPER) Vol., N0., October 0. P = 70, v ariable omial 95% lev e.00 = 50, v ariable omial 95% lev el coverage w ils o agres ti-c oull arc s ie w ils o* logit** coverage w ils o agres ti-c oull arc s ie w ils o* logit** 0.86 = 300, v ariable omial 0. 95% lev0.6 el = 500, v ariable omial 95% lev el 0.99 coverage w ils o agres ti-c oull arc s ie w ils o* logit** coverage w ils o agres ti-c oull arc s ie w ils o* logit** Figure Comariso of coverage of the Wilso, the Agresti-Coull, the arcsie, the Wilso*, the logit** ad the itervals for = 70, 50, 300 ad 500 with Comariso for Stadard, No-Modified ad Modified Itervals i terms of Exected Width Figure 3 shows the comariso of exected width of the stadard, the o- ad the itervals for = 0, 0, 70 ad 00 with omial 95% level. Results show that at smaller ( 0 ), the iterval has larger width ear the boudaries 0 or, but as aroaches to 0.5, it has similar width with the o- iterval. The stadard iterval has wider width ear close to 0.5. But as icreases, they have comarable width erformace. The recedig results give validity to the cojecture that the o- ad the itervals have comarable exected width whe gets larger for ay. 8 htt:// Juge B. Guillea

6 Joural of Educatioal Policy ad Etrereeurial Research (JEPER) Vol., N0., October 0. P = 0, variable omial 95% level 0.35 = 0, variable omial 95% exected width stadard o- exected width stadard o = 70, variable omial 95% level = 00, variable omial 95% level exected width stadard o Figure 3 Comariso of Exected Width of the stadard, the o- ad the 0.0 itervals for = 0, 0, 70 ad 0 with exected width stadard o- 0.0 Comariso for Modified ad Alterative Itervals i terms of Exected Width Figure dislays the result for the grahs of the exected width of the Wilso iterval, the Agresti-Coull iterval, the arcsie iterval, the Wilso*, the logit** iterval ad the iterval for = 0, 80, 50 ad 300 with omial 95% cofidece level, resectively. Result shows that the iterval has the shortest width whe , the Wilso iterval ad Agresti-Coull iterval have a comarable width with the iterval whe aroaches 0.5, the Wilso* iterval is cosistet for havig the largest width whe 0.0 or , ad the logit** iterval is the largest at ear the boudaries or whe These umerical evaluatios show that the iterval has a better erformace i terms of exected width, the Wilso* has a larger width of what is exected sice this iterval is artly coservative i terms of coverage roerties esecially ear the boudaries. For = 50, the stadard iterval shows the shortest whe 0. or ; the iterval is the shortest whe 0. 5 or , ad still the Wilso (0.5) is the largest for most values of, ad the logit** iterval is the largest whe earer the boudaries. For = 300, the results show that the stadard iterval is the shortest whe 0. 0 or , the Wilso, Agresti-Coull, arcsie, logit** ad itervals have almost idistiguishable width differece whe 0.03 or , while the Wilso* is sigificatly larger. This suggests that the Wilso, Agresti-Coull, arcsie, Logit (-0.87) ad the itervals are all referable methods for larger values of i terms of exected width. But if the recisio of the estimate is referred for a icreased width, Wilso (0.5) iterval is referable esecially for larger values of. The aforemetioed results build u the followig evidece that the iterval that has a coverage closely aroximate to the omial 95% cofidece level, yields a arrower exected width. 8 htt:// Juge B. Guillea

7 Joural of Educatioal Policy ad Etrereeurial Research (JEPER) Vol., N0., October 0. P = 0, variable omial 95% level 0. = 80, variable omial 95% level exected width wilso agresti-coull arcsie wilso* logit* * exected width wilso agresti-coull arcsie wilso* logit* * = 50, variable omial 95% level = 300, variable omial 95% level exected width wilso agresti-coull arcsie wilso* logit* * exected width 0.0 wilso agresti-coull arcsie wilso* logit* * Figure Comariso of exected width of the Wilso, the Agresti-Coull, the arcsie, the Wilso*, the logit** ad the itervals for = 0, 80, 50 ad 300 with Coclusio ad Recommedatio The existig ad additioal results would suggest rejectio of the coditios made by several authors regardig the use of the stadard iterval, but istead utilie the alterative methods foud i the literature which erform better i terms of coverage roerties ad other criteria. The erformace of the alterative methods ad the roosed method by the researcher ad the results show that some of these itervals have very good coverage behavior ad smaller exected width. Give the varied otios, the best solutio will o doubt be iflueced by the user s ersoal refereces. A wise choice could be either oe of the Wilso, Agresti-Coull, Wilso*, logit**, arcsie ad itervals which show decisive imrovemet over the stadard iterval. Based o the aalysis ad results obtaied, the researcher s recommedatios to comare ad ivestigate the erformace (like coverage roerties) of the most robable classical ad Bayesia itervals ad examie the RMSE roerty of the iterval discussed i the curret study. Refereces Agresti, A., ad Caffo, B. (000). Simle ad Effective Cofidece Itervals for Proortios ad Differeces of Proortios Result from Addig Two Success ad Two Failures. The America Statisticia, 5, Boomsma, A. (005). Cofidece Itervals for a Biomial Proortio. Uiversity of Groige. Deartmet of Statistics & Measuremet Theory. 83 htt:// Juge B. Guillea

8 Joural of Educatioal Policy ad Etrereeurial Research (JEPER) Vol., N0., October 0. P 77-8 Brow, L. D., Cai, T. T., ad DasGuta, A. (999a). Iterval Estimatio of a Biomial Proortio. Uublished Techical Reort Brow, L. D., Cai, T. T., ad DasGuta A. (999b). Cofidece Itervals for a Biomial Proortio ad Edgeworth Exasio. Uublished Techical Reort. Brow, L. D., Cai, T. T., ad DasGuta A. (00). Iterval Estimatio for a Biomial Proortio (with discussio). Statistical Sciece, 6, Brow, L. D., Cai, T. T., ad DasGuta A. (00). Cofidece Itervals for a Biomial Proortio ad Asymtotic Exasios. The Aals of Statistics, 30, Casella, G., ad Berger, R. (990). Statistical Iferece. Pacific Coast, CA. Woodsworth ad Brooks/Cole. Dio, J. (00). Momet ad Cumulats i Stochastic Aroximatio. Mathematisches Istitut A, Uiversitat Stuttgart, Germay. Edwardes, M. D. (998). The Evaluatio of Cofidece Sets with Alicatio to Biomial Itervals. Statistica Siica, 8, Harte, D., (00). No Asymtotic Biomial Cofidece Itervals. Statistics Research Associates, PO Box 69, Welligto NZ. Marchad, E., Perro, F. & Rokhaya, G., (00). Miimax Esimatio of a Biomial Proortio whe [ ½] is bouded. Uiversite de Motreal 8 htt:// Juge B. Guillea