QUASI-LIKELIHOOD APPROACH TO RATER AGREEMENT PLUS LINEAR BY LINEAR ASSOCIATION MODEL FOR ORDINAL CONTINGENCY TABLES

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1 Journal of Statstcs: Advances n Theory and Applcatons Volume 6, Number, 26, Pages -5 Avalable at DOI: QUASI-LIKELIHOOD APPROACH TO RATER AGREEMENT PLUS LINEAR BY LINEAR ASSOCIATION MODEL FOR ORDINAL CONTINGENCY TABLES Department of Statstcs Hacettepe Unversty Beytepe, 68 Ankara Turkey e-mal: spxl@hacettepe.edu.tr Abstract Quas-lkelhood estmaton methods, an alternatve to maxmum lkelhood estmaton of the regresson parameters n margnal models, are manly used to deal wth non-normal multvarate data. The goal of ths paper s to use quaslkelhood approach as a method for the analyss of rater agreement plus lnear by lnear assocaton model, whch s a type of -lnear models of rater agreement n R R contngency tables wth ordnal varables. We dscuss the dfferences between maxmum lkelhood and quas-lkelhood approaches and compared the standard errors.. Introducton The analyss of contngency tables plays an mportant role n many felds of appled statstcs. R R square contngency tables are the specal cases of two-way contngency tables (Bshop et al. [6]) and 2 Mathematcs Subject Classfcaton: 62J2. Keywords and phrases: contngency tables, quas-lkelhood, -lnear models, rater agreement. Receved July 2, Scentfc Advances Publshers

2 2 generally arse for dependent row and column varables. The dependency of the row and column varables n square contngency tables needs some specal solutons. Classcal -lnear models are extensvely used for modellng twoway contngency tables (Agrest [2]). In recent years, more attenton has been pad to the assessment of rater agreement n contngency tables, especally n medcal or behavoural scences (Osus [4]; Valet et al. [6]; Bagheban and Zayer [5]). Some types of -lnear models, such as dagonal parameter, lnear by lnear assocaton and agreement plus lnear by lnear assocaton models have been frequently suggested to descrbe the assocaton and structure of rater agreement (Agrest [3]). The classcal technque for estmatng the unknown parameters of a -lnear model s maxmum lkelhood (ML) method. In ths paper, quas-lkelhood (QL) estmaton method ntroduced by Wedderburn [8] s adopted to analyze rater agreement n R R square contngency tables, snce t s becomng very popular method as an alternatve to ML for statstcal modellng whch uses an approxmate lkelhood functon rather than a fully lkelhood (Lang and Zeger [2]; Zeger and Lang [2]; Lpstz et al. [3]; Ftzmaurce et al. [8]). Quas-lkelhood equatons are applcable to avod dstrbutonal assumptons for margnal models when the responses are count, dchotomous, categorcal, ordnal or contnuous data (Wedderburn [8]; Lang and Zeger [2]). Ths paper only focuses on the lnear by lnear assocaton plus agreement model R R for square contngency tables. Classcal approach s to ft lnear model to data and estmate the expected frequences and estmaton of model parameters. Ths paper consders quas-lkelhood equatons to analyze R R tables rather than ML and compares parameter estmates of quas-lkelhood method to those of the commonly used -lnear analyss wth ML method.

3 QUASI-LIKELIHOOD APPROACH TO RATER 3 In Secton 2, the agreement plus lnear by lnear assocaton model s descrbed. In Secton 3, quas-lkelhood approach for lnear by lnear assocaton plus agreement model n the analyss of R R tables s ntroduced. We provde a smulaton study n Secton 4 to gve detals on proposed methods wth small, moderate, and large sample szes. The numercal results are llustrated by an example n Secton 5. The conclusons are dscussed n Secton Assocaton Models Assocaton models for contngency tables are desgned to descrbe assocaton between row and column varables. In an R C contngency table, let denote the row varable wth categores =,, R and Y denote the column varable wth categores j =,, C. The cell frequences n j have expected frequences j. assocaton model as the form: Goodman [9] proposed the Y p j = λ + λ + λ j + βkuk ν jk, k = =,, R; j =,, C, () where λ s the grand mean, λ and λ Y j represent the row and the column margnal parameters and should satsfy the constrants R λ = = C Y λ j = j =. The Model () n Equaton () s referred as lnear by lnear assocaton model. The row and column scores { u } and { ν j } are assgned to reflect category orderngs. The ndependence model s the specal case β =. Snce the model has one more parameter ( β ) than the ndependence model, the resdual df of the model s equal to ( R ) ( C ). The parameter β n Model () specfes the drecton and strength of assocaton. When β >, tends to ncrease as Y ncreases (Agrest [3]). In ths model, the equal nterval scores are the common choce for the levels of the ordnal scales.

4 4 Agrest [] ntroduced a model havng the structure of unform assocaton plus an agreement parameter over the man dagonal for ordnal categorcal scales, Y j = λ + λ + λ + βu ν + δ, =,, R; j =,, C, (2) j j where β s the assocaton parameter between the row and the column varables, u and j ν j are the row and the column scores, respectvely, as ndcated n Model (). Scores are assgned as equal ntervals for the categores of the row varable, u u u, and for the categores 2 R of the column varable, ν ν ν. In Model (2) defned as 2 C Equaton (2), δ R = for all and δ Rj = for all j. The δ j n Equaton (2) denotes the agreement parameter as, δ, f = j, δj = (3), otherwse. The Model (2) has resdual degrees of freedom df = ( R ) 2 2 and can be also expressed n terms of the odds ratos θ j as follows: β + 2δ, = j, θj = β 2δ, j =, (4) β, j >. Agrest [] has referred to Model (2) as an agreement plus lnear by lnear assocaton model. The specal case β = s the Tanner and Young s [5] model for nomnal scale agreement; the specal case δ = s the lnear by lnear assocaton model, and the specal case β = δ = s the ndependence model. Ths model permts the lnear by lnear assocaton of the man dagonal.

5 QUASI-LIKELIHOOD APPROACH TO RATER 5 We can defne the 3 3 desgn matrx for Model (2) as the followng: δ β λ λ λ λ λ = Y Y The soluton of β parameter satsfes the Equaton (4). The -lnear model s one of the specalzed cases of generalzed lnear models for Posson-dstrbuted data. Therefore, the teratve proportonal fttng process generates ML estmates of the expected cell frequences (Agrest [2]). 3. Quas-lkelhood Approach Quas-lkelhood (QL) estmatng equatons ntroduced by Wedderburn [8] for generalzed lnear models are a useful framework for statstcal modellng wthout a fully specfed lkelhood for nonnormal multvarate data. Let Y be a response varable havng N ndependent observatons ( ) Y N Y Y,, 2, and let p,, 2, be p covarates. In generalzed lnear models, the mean response s lnearly related to the covarates va a lnk functon (). g as the followng form: ( ). p p l g β + + β + β =

6 6 The QL estmate of β = ( β, β,, ) T estmatng equaton n Equaton (5). N = β p s obtaned as a soluton of the T U ( β) = ( β) ( V ( Y )) ( Y ( β) ) =, (5) where the response Y, =,, N have mean and varance V ( Y ) = φν( ), ν( ) s a known varance functon and φ s a scale parameter for the overdsperson ( φ > ) (Wedderburn [8]). The QL estmator of β s consstent and asymptotcally normal even when the varance of the response has been msspecfed (Ftzmaurce et al. [8]). We used quas-lkelhood estmatng equaton n Equaton (5) to analyze the structure of assocaton and agreement n the -lnear Model (2). Usng the advantages of QL, we wll adopt t to the Model (2). Based on QL approach n the assessment of rater agreement n R R contngency tables, let Y be consdered as n j whch s the cell frequences n row and j column gven for both and Y raters and let j be the mean whch s lnearly related to the covarates va a lnk functon as Model (2): Y j = λ + λ + λ j + βuν j + δj,, j =,, R. (6) In R R contngency tables, n j are Posson responses, lnk functon can be defned as Posson. Soluton of equatons based on Model (2) gves QL estmates of parameters. We can obtan ML estmates for the cells under any -lnear model by teratve fttng algorthm. Parameter estmates and ther standard errors can be expressed as a lnear combnaton of the arthms of the observed cell frequences plus userspecfed covarates. But the man purpose of ths artcle s to compare between two compettve methods. In ths paper, we apply the QL to random varables N rs wth N rs the cell count for the ordnal outcomes ( Y = r, Y2 = s).

7 QUASI-LIKELIHOOD APPROACH TO RATER 7 4. Smulaton Study A smulaton study bascally has been carred out to compare the standard errors of agreement and assocaton parameters n Model (2). We used R-Project program to generate samples followng multnomal dstrbutons. Random samples wth gven fxed margnal totals are generated under multnomal samplng scheme. In the data generaton process, dmenson of tables and sample szes are set to R = 3, 4, 5, 6 and N = 5, 5, and 5, respectvely. Smulaton s replcated tmes. We often do not know the actual jont dstrbuton of the row and column varables but we can use the sample as the estmate of the jont dstrbuton. Detaled explanatons of how to assgn the margnal probabltes can be found n Yang [9] s work. Multnomal probabltes and the jont dstrbuton for 3 3 under multnomal dstrbuton are shown n Table 2. Margnal probabltes for 4 4, 5 5, and 6 6 as follows:.5 )..5,.9, For R = 4 : P ( x) = (.2,.,.2,.5) ; P ( y) = (.2,.,.2, For R = 5 : P ( x) = (.22,.6,.24,.25,.3) ; P ( y) = (.29,.2,.27,.8 ). For R = 6 : P ( x) = (.2,.4,.22,.23,.,.) ; P ( y) = (.27,.3,.25,.,.6 ). For all smulated tables, only the tables that hold the lnear by lnear plus assocaton model are selected. As the Kappa values greater than 8% represents very good agreement beyond chance (Altman [4]). All smulated tables are set to hgh agreement case by gven hgh probabltes to man dagonals. The sum of the probabltes on the man dagonals s approxmately 9%. Ths proporton generally corresponds to β and 2 δ 3.5. The δ values greater than represent hgh agreement beyond chance. An example of 3 3 random samples s dsplayed n Table. Y Y Y

8 8 Table. A smulated table for N = 5 /Y 2 3 Total Total Table 2. Margnal probabltes /Y 2 3 P ( x) P Y ( y) For data n Table, weghted Kappa coeffcent s calculated as 83.75% wth standard error.23. Table 3 dsplays the parameter estmates and ther standard errors of replcatons. ML estmates are calculated by the classcal -lnear model estmaton. All of the smulated tables are ftted to Model (2) at the level of % sgnfcance level. Due to the tables are generated under the multnomal model, parameter estmates are obtaned as small but statstcally sgnfcant at %. The coeffcents β are all postve, ths means the postve relaton between the row and column varables. Some of the δ coeffcents are close to zero.

9 QUASI-LIKELIHOOD APPROACH TO RATER 9 Table 3. Parameter estmates and ther standard errors under Model (2) Dmenson Sample Sze QL ML δ ˆ = (.22 ) δ ˆ = (.735) β ˆ =.725 (.48) β ˆ =.725 (.44) δ ˆ = 3.34 (.46 ) δ ˆ = 3.34 (.324) β ˆ =.37 (.64) β ˆ =.37 (.38) δ ˆ = 2.97 (.49 ) δ ˆ = 2.97 (.85) β ˆ =.222 (.6) β ˆ =.222 (.74) δ ˆ = (.8 ) δ ˆ = (.395) β ˆ =.274 (.46) β ˆ =.274 (.2) δ ˆ = (.3 ) δ ˆ = (.243) β ˆ =.2 (.39) β ˆ =.2 (.49) δ ˆ = 2.2 (.54 ) δ ˆ = 2.2 (.2) β ˆ =.95 (.28) β ˆ =.95 (.5) δ ˆ = (.24 ) δ ˆ = (.364) β ˆ =.556 (.58) β ˆ =.556 (.89) δ ˆ = 3.8 (.85 ) δ ˆ = 3.8 (.24) β ˆ =.8 (.27) β ˆ =.8 (.53) δ ˆ = (.8 ) δ ˆ = (.9) β ˆ =.3 (.68) β ˆ =.3 (.43) δ ˆ = 2.67 (.45 ) δ ˆ = 2.67 (.32) β ˆ =.886 (.235) β ˆ =.886 (.343) δ ˆ = 2.59 (.77 ) δ ˆ = 2.59 (.) β ˆ =.456 (.54) β ˆ =.456 (.253) δ ˆ =.928 (.75 ) δ ˆ =.928 (.98) β ˆ =.556 (.92) β ˆ =.556 (.34)

10 From the smulaton results, note that model parameter estmates are exactly the same but wth dfferent standard errors. Standard errors calculated by QL gve smaller values than for those calculated by ML. Standard errors of QL estmates are much smaller than ML estmates n each combnaton of smulaton parameters. When the sample sze s large, standard errors are expectedly becomng smaller. It can be seen that there s more dsagreement between margnal probabltes as well as hgh probabltes for off-dagonal probabltes. 5. Numercal Example The example n Table 4 gves frequences from the study on the dagnoss of multple scleross (MS) from Lands and Koch []. In ths example, 49 Wnnpeg patents were examned by two neurosts, one from New Orleans, and the other from Wnnpeg. The two neurosts classfed each patent nto one of the followng classes: ( = Certan MS, 2 = Probable MS, 3 = Possble MS, 4 = Doubtful, Unlkely, or Defntely not MS). Table 4. Dagnostc classfcaton regardng multple scleross for the Wnnpeg patents Wnnpeg neurost New Orleans neurost Total Total Model (2) addresses the agreement and assocaton between New Orleans neurost and Wnnpeg neurost. Classcal approach s to analyze data through general -lnear analyss. Lkelhood rato statstcs for goodness of ft test s 9.46 wth 7 df. Ths means that

11 QUASI-LIKELIHOOD APPROACH TO RATER mples that the model s statstcally sgnfcant at 5%. Model (2) fts the data well. Table 5 dsplays the parameter estmates under Model (2). Snce constants are not parameters under the multnomal assumpton, ther standard errors are not calculated. Table 5. Parameter estmates for Model (2) usng ML approach Parameter Estmate St. Error 95% CI Z Pr>Z Intercept.458 row = (3.5, 7.36) row = (2.587, 5.589) row = (.4, 3.458) row = 4 col = (5.488,.932) col = (3.564, 7.85) col = (.896, 3.62) col = 4 βˆ (.5,.8) 5.8. δˆ (.54,.448).5.99 As mentoned n earler sectons, QL methodoy s appled data for estmatng the parameters (.e., parameters n Table 5). Score statstcs (Table 6) for QL analyss show that only column varable and beta parameters are the statstcally sgnfcant model components. Table 6. Score statstcs based on Type 3 sums of squares n QL analyss Source df Ch-Square Pr>ChSq row column β δ.5.828

12 2 Table 7 gves the QL parameter estmates under Model (2). Agreement parameter δ ˆ =. 278 ndcates that there s no agreement between New Orleans neurost and Wnnpeg neurost. β ˆ =. 838 ndcates that there s a postve assocaton between two varables ( P <.). Table 7. Parameter estmates of Model (2) usng QL approach Parameter Estmate St. Error 95% CI Z Pr>Z Intercept ( 4.68, ) row = (3.66, 6.535) row = (2.962, 5.24) 7.. row = (.7, 3.59) row = 4 col = (5.64,.85) 6.8. col = (3.646, 7.769) col = (.26, 3.49) col = 4 βˆ (.536,.72) δˆ (.267,.2) The methods ML and QL are dffcult to compare (Hunger et al. []), however, t has been recognzed that the parameters n the QL are estmated wth more precson. Standard errors for model parameters are smaller than for those obtaned from ML method. Expected frequences under Model (2) from QL are presented n Table 8. Expected frequences reflect the assumptons concernng the relatonshps between the varables under study.

13 QUASI-LIKELIHOOD APPROACH TO RATER 3 Table 8. Expected frequences and 95% CI for mean response under Model (2) Wnnpeg neurost New Orleans neurost Total [33.32; 4.4] [4.43;.66] [.9;.25] [.2;.66] [27.93; 35.32] [9.87; 5.8] [.2; 3.42] [.39; 2.93] [9.53; 8.93] [9.2; 6.37] [2.93; 6.9] [3.36; 7.64] [.44; 4.8] [3.85; 7.49] [2.66; 6.78] [8.98; 2.88] Total Concluson In ths paper, we advocate the use of quas-lkelhood (QL) approach over the more common -lnear analyss by usng ML method n the analyss of R R contngency tables. The man purpose s merely to focus on the agreement plus lnear by lnear assocaton model whch allows researchers to consder the ordnal nature of contngency tables. We consder QL approach, snce t has several practcal advantages that can be summarzed as: () ts computatonal smplcty rather than maxmum lkelhood (ML) for categorcal data, () QL does not requre any multvarate dstrbutonal assumptons, () consstent parameter estmates can be obtaned even wth msspecfed varance. One can compare the emprcal estmates wth the model-based estmates. Whle constants are not parameters under the multnomal assumpton, therefore ther standard errors are not calculated, but QL treats constant term as a parameter and estmates t. There s no need for the model ft of the QL, because there s no lkelhood functon. The overall confdence

14 4 ntervals are much smaller n QL approach and parameter estmates from ML and QL methods are dentcal. Dfferences could be observed wth respect to standard errors, regardless of sample sze and dmenson of the table. References [] A. Agrest, A model for agreement between ratngs on an ordnal scale, Bometrcs 44 (988), [2] A. Agrest, Categorcal Data Analyss, 2nd Edton, New York: Wley, 22. [3] A. Agrest, Analyss of Ordnal Categorcal Data, 2nd Edton, New York: Wley, 2. [4] D. G. Altman, Practcal Statstcs for Medcal Research, London: Chapman and Hall, 99. [5] A. A. Bagheban and F. Zayer, A generalzaton of the unform assocaton model for assessng rater agreement n ordnal scales, Journal of Appled Statstcs 37(8) (2), [6] Y. M. M. Bshop, S. E. Fenberg and P. W. Holland, Dscrete Multvarate Analyss, Cambrdge, MA: MIT Press, 975. [7] P. J. Dggle, P. Heagerty, K. Y. Lang and S. L. Zeger, Longtudnal Data Analyss 2nd Edton, Oxford, UK: Oxford Unversty Press, 22. [8] G. Ftzmaurce, M. Davdan, G. Verbeke and G. Molenberghs, Longtudnal Data Analyss, Chapman & Hall. CRC Press, 28. [9] L. A. Goodman, The analyss of cross-classfed data havng ordered and or unordered categores-assocaton models, correlaton models, and asymmetry models for contngency tables wth or wthout mssng entres, Annals of Statstcs 3() (985), -69. [] M. Hunger, A. Dörng and R. Holle, Longtudnal beta regresson models for analyzng health-related qualty of lfe scores over tme, BMC Medcal Research Methodoy 2 (22), 44. [] J. R. Lands and G. G. Koch, The measurement of observer agreement for categorcal data, Bometrcs 33 (977), [2] K.-Y. Lang and S. L. Zeger, Longtudnal data analyss usng generalzed lnear models, Bometrka 73 (986), [3] S. R. Lpstz, K. Km and L. Zhao, Analyss of repeated categorcal data usng generalzed estmatng equatons, Statstcs n Medcne 3() (994), [4] G. Osus, Log-lnear models for assocaton and agreement n stratfed square contngency tables, Computatonal Statstcs 2 (997),

15 QUASI-LIKELIHOOD APPROACH TO RATER 5 [5] M. A. Tanner and M. A. Young, Modelng agreement among raters, Journal of Amercan Statstcal Assocaton 8 (985), [6] F. Valet, C. Gunot and J. V. Mary, Log-lnear non-unform assocaton models for agreement between two ratngs on an ordnal scale, Statstcs n Medcne 26(3) (26), [7] A. Von Eye and Y. E. Mun, Analyzng Rater Agreement: Manfest Varable Methods, Lawrence Erlbaum Assocates Inc., NJ, 25. [8] R. W. M. Wedderburn, Quas-lkelhood functons, generalzed lnear models, and the Gauss-Newton method, Bometrka 6(3) (974), [9] J. Yang, Measurement of Agreement for Categorcal Data, PhD Dssertaton, The Pennsylvana State Unversty The Graduate School Department of Statstcs, USA, 27. [2] S. L. Zeger and K.-Y. Lang, Longtudnal data analyss for dscrete and contnuous outcomes, Bometrcs 42 (986), 2-3. g