A WEIGHTED LEAST SQUARES METHOD FOR ESTIMATING THE SUCCESS RATE IN CLUSTERED BINARY DATA

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1 Jot Statstcal Meetgs - Bopharmaceutcal Secto A GHTED LEAST SQUAS METHOD FOR ESTMATNG THE SUCCESS RATE N CLUSTED BNARY DATA Mtchell J. Rose, Ph.D. Seor Bostatstca, Project Drector Departmet of Bostatstcs CON Clcal Research Church Rd. North Wales, PA 9454 ABSTRACT A weghted least squares (WLS method s proposed for estmatg the per ut agreemet rate clustered bary data. Three WLS estmators that are mathematcally equvalet to the rato, weghted, ad optmal estmators are proposed. Except the case of the weghted estmator, WLS estmators have dfferet stadard errors tha ther couterparts. WLS cofdece tervals are based o the t - dstrbuto stead of o the ormal dstrbuto. A Mote Carlo smulato was performed that employed a wde rage of parameter settgs. Tradtoal estmators were compared wth ther WLS ad bootstrap couterparts wth respect to stadard error ad emprcal cofdece coverage probablty (ECP. The success rate, tracluster correlato, sample sze, rage of cluster szes, ad dstrbuto of cluster szes were vared. The WLS verso of the optmal estmator performed best, but ECP was heavly depedet o the tracluster correlato. A sample sze of 50 was requred to guaratee adequate coverage whe the tracluster correlato was KEYWORDS Weghted least squares; Clustered bary data; Correlated bary data; Beta-Bomal dstrbuto; tracluster correlato NTRODUCTON Clustered bary data are characterzed by the assocato of two or more subuts wth a expermetal etty where the outcome varable s bary. Some examples of ths type of data clude developmetal toxcty expermets whch the presece or absece of a abormalty amog offsprg a ltter s uder study; clcal studes of male erectle dysfucto where the percet of successful attempts at tercourse s of terest; ad detal studes whch the umber of teeth exhbtg a abormalty s uder vestgato. The umber of subuts ofte vares across clusters. The ma statstcal problems the statstcal aalyss of clustered bary data are estmato of the success rate per subut, testg treatmet effects radomzed desgs, ad multvarate modelg of the bary outcomes. Ths paper s cocered wth the frst problem. t exteds work reported Rose (999. There are may approaches to the estmato problem. Let m ad y deote the umber of subuts ad umber of successes the th cluster, respectvely. Oe method s to estmate the mea of p y / m ad adjust the stadard error for the clusterg effect. The rato estmator (Cochra, 977 s the mea of y / m weghted by the umber of subuts per cluster. The weghted estmator (Lee ad Dub, 994 s the mea of y / m wth each cluster weghted equally. The optmal estmator (Jug ad Ah, 000 weghts each cluster by the verse of ts varace uder the assumpto of exchageablty (.e., equal correlato betwee all pars of subuts. Lee ad Dub also examed a bootstrap estmator that had smlar propertes to ther weghted estmator. Cofdece tervals for the rato, weghted, ad verse varace estmators are based o asymptotc ormal theory. Cofdece tervals for the bootstrap estmator ca be based o the percetle method (Lee ad Dub, 994. The rato, weghted, ad bootstrap estmators requre o dstrbutoal or correlatoal assumptos. Other approaches for estmatg the per ut success rate are ot smple fuctos of p (e.g., Lag ad Zeger, 986; Wllams, 975. Mote Carlo methods have typcally bee used to compare the stadard errors ad emprcal cofdece terval coverage probabltes of the rato, weghted, ad optmal estmators. Computer smulatos have employed ( a uform dstrbuto of cluster szes, ( a rage of cluster szes betwee ad 0, ad (3 a beta-bomal dstrbuto. The latter dstrbuto mples a exchageable correlatoal structure. The effects of varyg the dstrbuto ad rage of cluster szes ad the mportace of the assumpto of exchageablty have ot bee studed. ths paper the method of weghted least squares (WLS s proposed for estmatg the mea of y / m ad for obtag stadard errors ad cofdece tervals. WLS 948

2 Jot Statstcal Meetgs - Bopharmaceutcal Secto s frequetly used to model ratos (e.g., Rose et al., 998. A advatage of ths approach s that cofdece tervals are based o the t - dstrbuto stead of o the ormal dstrbuto, so the emprcal coverage probablty (ECP performace should be better small samples. METHODS Notato umber of clusters x j bary observato (0, for cluster subut j m umber of subuts cluster y m x j j y / m p prob(x j Kow Estmators umber of successes cluster The rato, weghted, ad optmal estmators are defed as follows: y ( m y m ( m [ + ( m m [ + ( m ρ ] ρ ] Ther varace estmators are, respectvely, (3 m ( ˆ ( ˆ m σ p (4 ( ( ˆ σ ( (5 ( ( σ ˆ ( (6 m [ + ( m ˆ ρ ] Weghted Least Squares Method Each of the above estmators ca be obtaed usg the followg zero-tercept weghted least squares model: E( y m βm (7 λ σ σ m Equato (7 s called the multplcatve heteroscedastcty model (Pdyck ad Rubfeld, 98. Settg λ, λ, ad λ + log[ + ρ ( m ]/ log m yelds (, (, ad (3, respectvely. The WLS varace estmators for ad are, respectvely, m ( ˆ p m ˆ σ ( ( WLS (8 ( ˆ σ ( ( WLS m [ + ( m ( ˆ] ρ m [ + ( m ( ˆ] ρ The WLS varace estmator for s the same as the Lee-Dub weghted varace estmator. WLS cofdece tervals are obtaed usg the t-dstrbuto wth - degrees of freedom. Bootstrap Estmators Lee ad Dub proposed the followg bootstrap estmator for p. Take a sample wth replacemet * * * * p, p,..., p from p, p,..., p. Compute p ad repeat the process a large umber of tmes. The mea of * * the p ( p s a bootstrap pot estmate of p. Ths bootstrap estmator s asymptotcally equal to the weghted estmator. A bootstrap pot estmate for the rato estmator ca be obtaed as follows. Take a radom sample * * * y, y,..., y from y, y,..., y. Compute (9 949

3 Jot Statstcal Meetgs - Bopharmaceutcal Secto r * j * y m j * (where m* correspods to y* ad repeat the process a large umber of tmes. The mea of * * the r ( r s the pot estmate of p. The percetle method ca be used to obta cofdece tervals. Mote Carlo Smulato A Mote Carlo smulato was coducted to compare the dfferet estmators. The parameters of the Mote Carlo smulato were as follows: Number of repettos 5000 Number of bootstrap samples p 0.50, tracluster correlato 0. (betabomal, 0.6 (beta-bomal, mxed 5 Cluster dstrbuto uform, egatve bomal 6 Rage of cluster szes -9, Sample szes 5, 30, 50 The estmators were compared as follows: Kow estmators,, ad (bas, stadard error, ad 95% ECP Weghted ad rato bootstrap estmators, (bas, stadard error, ad 95% ECP 3 WLS method,, ad (same bas ad stadard error as kow estmators; oly 95% ECP was computed Let ρ deote the commo tracluster correlato. The beta-bomal varate β was geerated as follows. Let β rab (seed,, z, where z x /( x + y, x ragam (seed, a, ad y ragam (seed, b. The parameters of the gamma dstrbuto are obtaed from a p( ρ / ρ ad b ( p( ρ / ρ. Ragam deotes the SAS radom umber geerator fucto for the gamma dstrbuto. The followg protectos agast overflow were utlzed: z > z z ad β - β 5 z < x0 z x0 5 Followg Jug ad Ah, ρ was estmated usg Doer ad Klar s (993 ANOVA estmator for Cohe s (960 κ wth κˆ trucated at zero. The egatve bomal dstrbuto was obtaed by settg the probablty of the frst successes to All smulatos were performed usg SAS Verso 8. o a Dell Optplex GX50 computer rug Mcrosoft Wdows 000 Verso 5.0. SULTS Results of the smulato are foud Tables 4.. The bas of every estmator was eglgble. o case dd the absolute value of the bas exceed These results are ot show.. Stadard errors were hgher ( for larger values of κ, ( for the egatve bomal (asymmetrc dstrbuto tha for the uform dstrbuto, (3 for smaller cluster szes ad (4 for smaller umbers of clusters. Stadard errors are show Table. 3. had cosstetly lower stadard errors, ad ths fdg carred over to the mxed correlatoal case. Other fdgs were: ( all the estmators had stadard errors as low as whe κ 0.; ( had stadard errors as low as for κ 0. ad 0.6; (3 the κ mxed case, the stadard errors of all estmators were smlar. See Table. 4. ECP for ad (o-wls was adequate whe > 30 ad p However, whe p 0.80 ECP deterorated; ths case ECP was acceptable oly whe 50, but stll adequate whe κ 0.6. ECP for all o-wls estmators the mxed case was ot adequate, eve whe 50. The percetle method for the bootstrap estmators was o more effectve tha the asymptotc approach. These results are show Table. 5. ECP usg the WLS approach was clearly better tha the other approaches. ECP for ad was adequate for all sample szes whe p 0.50; however, ECP for was oly adequate whe 50. ECP for ad was stll good whe p 0.80, but better at κ 0. tha at κ 0.6. ECP was adequate the mxed case whe 50. Results appear Tables 3 ad

4 Jot Statstcal Meetgs - Bopharmaceutcal Secto Table Stadard Errors (p 0.5. N Dstrbuto Type/Rage κ0. κ0.6 κmxed 5 U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ weghted estmator; rato estmator; verse varace weghted estmator; weghted bootstrap estmator; rato bootstrap estmator. U Uform; NB Negatve Bomal. The rage of the umber of subuts per cluster s approxmate the egatve bomal case. Whe κmxed half the maxmum umber of subuts have κ0. ad the other half have κ0.6. Correlatos betwee the subuts each half are zero. 95

5 Jot Statstcal Meetgs - Bopharmaceutcal Secto Table (cotued Stadard Errors (p 0.8 N Dstrbuto Type/Rage κ0. κ0.6 κmxed 5 U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ weghted estmator; rato estmator; verse varace weghted estmator; weghted bootstrap estmator; rato bootstrap estmator. U Uform; NB Negatve Bomal. The rage of the umber of subuts per cluster s approxmate the egatve bomal case. Whe κmxed half the maxmum umber of subuts have κ0. ad the other half have κ0.6. Correlatos betwee the subuts each half are zero. 95

6 Jot Statstcal Meetgs - Bopharmaceutcal Secto Table 95% Cofdece terval Coverage Probabltes (p 0.5 N Dstrbuto Type/Rage κ0. κ0.6 κmxed 5 U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ weghted estmator; rato estmator; verse varace weghted estmator; weghted bootstrap estmator; rato bootstrap estmator. U Uform; NB Negatve Bomal. The rage of the umber of subuts per cluster s approxmate the egatve bomal case. Whe κmxed half the maxmum umber of subuts have κ0. ad the other half have κ0.6. Correlatos betwee the subuts each half are zero. 953

7 Jot Statstcal Meetgs - Bopharmaceutcal Secto Table (cotued 95% Cofdece terval Coverage Probabltes (p 0.8 N Dstrbuto Type/Rage κ0. κ0.6 κmxed 5 U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ weghted estmator; rato estmator; verse varace weghted estmator; weghted bootstrap estmator; rato bootstrap estmator. U Uform; NB Negatve Bomal. The rage of the umber of subuts per cluster s approxmate the egatve bomal case. Whe κmxed half the maxmum umber of subuts have κ0. ad the other half have κ0.6. Correlatos betwee the subuts each half are zero. 954

8 Jot Statstcal Meetgs - Bopharmaceutcal Secto Table 3: 95% Cofdece terval Coverage Probabltes for Weghted Least Squares Estmators (p 0.5 Dstrbuto κ N Type/Rage 0. 5 U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ weghted estmator; rato estmator; verse varace weghted estmator. U Uform; NB Negatve Bomal. The rage of the umber of subuts per cluster s approxmate the egatve bomal case. 955

9 Jot Statstcal Meetgs - Bopharmaceutcal Secto Table 3 (cotued: 95% Cofdece terval Coverage Probabltes for Weghted Least Squares Estmators (p 0.8 Dstrbuto κ N Type/Rage 0. 5 U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ weghted estmator; rato estmator; verse varace weghted estmator. U Uform; NB Negatve Bomal. The rage of the umber of subuts per cluster s approxmate the egatve bomal case. 956

10 Jot Statstcal Meetgs - Bopharmaceutcal Secto Table 4: 95% Cofdece terval Coverage Probabltes for Weghted Least Squares Estmators (κ mxed Dstrbuto P N Type/Rage U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ U/ U/ NB/ NB/ weghted estmator; rato estmator; verse varace weghted estmator. U Uform; NB Negatve Bomal. The rage of the umber of subuts per cluster s approxmate the egatve bomal case. Whe κmxed half the maxmum umber of subuts have κ0. ad the other half have κ0.6. Correlatos betwee the subuts each half are zero. 957

11 Jot Statstcal Meetgs - Bopharmaceutcal Secto DSCUSSON Ths study demostrated that the weghted least squares method s more effectve tha asymptotc ormal methods for obtag cofdece tervals of the per ut success rate clustered bary data. The Mote Carlo smulato showed that WLS had uformly better emprcal cofdece coverage probablty the asymptotc ormal methods. WLS cofdece tervals are based o the t - dstrbuto stead of o the ormal dstrbuto. The results showed that had smaller stadard errors tha ad regardless of whether the exchageablty assumpto was satsfed. ECP for both ad was adequate as log as the WLS method was used. Fftee clusters were suffcet to guaratee good ECP whe p 0.50, but as may as ffty clusters were requred whe p Sce a wde rage of parameters was used the Mote Carlo smulato, the fdgs of ths study should be robust. The results suggest that vestgators should be careful whe plag studes usg clustered bary data, sce ECP dmshes the more p devates from Wth success rates of 0.80 or hgher, a sample sze of ffty clusters may be requred to guaratee good ECP whe the tracluster correlato s greater tha 0.5. Sample sze ad power estmato methods based o asymptotc ormal methods, such as those recommeded by Lee ad Dub (994, are lkely to be most useful the rage 0.3 p Whe plag studes whch the success rate s expected to be outsde of ths rage the use of Mote Carlo methods s advsable. 5. Jug S, Ah C. Estmato of Respose Probablty Correlated Bary Data: A New Approach. Drug formato Joural. 000; 34: Lee EW, Dub N. Estmato ad Sample Sze Cosderatos for Clustered Bary Resposes. Statstcs Medce. 994; 3: Lag KY, Zeger SL. Logtudal Data Aalyss usg Geeralzed Lear Models. Bometrka. 986; 73: Pdyck R, Rubfeld D. Ecoometrc Models ad Ecoomc Forecasts. New York, NY: McGraw-Hll, Rose M. A Weghted Least Squares Approach to Estmatg the Per Ut Agreemet Rate Clustered Bary Data. Proceedgs of the Bometrcs Secto of the Amerca Statstcal Assocato. 999; Rose MJ, Sork JD, Goldberg AP, Hagberg JM, Katzel L. Predctors of Age-Assocated Decle Maxmal Aerobc Capacty: A Comparso of Four Statstcal Models. Joural of Appled Physology. 998; 84: Wllams D. The Aalyss of Bary Resposes from Toxcologcal Expermets volvg Reproducto ad Teratogecty. Bometrcs. 975; 3: FENCES. Cochra WG. Survey Techques. New York, NY: Wley, Cohe J. A Coeffcet of Agreemet for Nomal Scales. Educatoal ad Psychologcal Measuremet. 960; 0: Doer A, Klar N. Cofdece terval Costructo for Effect Measures Arsg from Cluster Radomzato Trals. Joural of Clcal Epdemology. 993; 46: Hujoel P, Moulto, L, Loesche W. Estmato of Sestvty ad Specfcty of Ste-Specfc Dagostc Tests. Joural of Perodotal Research.990; 5:

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