Psychologcal Test and Assessment Modelng, Volume 54, 2012 (3), 285-292 Bayesan predctve Confgural Frequency Analyss Eduardo Gutérrez-Peña 1 Abstract Confgural Frequency Analyss s a method for cell-wse nspecton of cross-classfcatons. CFA searches for patterns of varable categores that occur ether more often or less often than expected from a gven base model. In ths paper, we propose and dscuss an alternatve noton of types and anttypes that focuses on the lkely values of the cell frequences n future experments, as opposed to the average values of such frequences. The dea s developed from a Bayesan pont of vew. Key words: Bayesan methods, Confgural Frequency Analyss, predctve dstrbuton 1 Correspondence concernng ths artcle should be addressed to: Eduardo Gutérrez-Peña, PhD, Departamento de Probabldad y Estadístca, IIMAS-UAM, Apartado Postal 20-726, 01000 Méxco, D.F., Mexco; emal: eduardo@sgma.mas.unam.mx
286 E. Gutérrez-Peña Introducton Confgural Frequency Analyss (Lenert, 1969) s a method for cell-wse nspecton of cross-classfcatons. CFA searches for types (respectvely, anttypes ), that s, patterns of varable categores that occur more often (respectvely, less often) than expected from a gven base model. Bayesan CFA (Gutérrez-Peña and von Eye, 2000) defnes types and anttypes n terms of the true (unknown) values of the parameters, whch can be estmated but wll never be observed or fully known. On the other hand, the Bayesan predctve CFA ntroduced n ths paper focuses on the lkely values of the cell frequences n future experments, as opposed to the average values of such frequences. Both Bayesan CFA and Bayesan predctve CFA are capable of assgnng probabltes to patterns of types and anttypes, and thus allow for the comparson of such patterns by means of relatve probabltes. In the next secton we revew the Bayesan approach to CFA. We then ntroduce the Bayesan predctve CFA and llustrate the method usng a data set prevously analyzed n the lterature. Fnally, the last secton contans some concludng remarks. Bayesan CFA In ths secton we revew the work of Gutérrez-Peña and von Eye (2000) and ntroduce some notaton. Consder a cross classfcaton of d 2 categorcal varables. Let π denote the populaton probablty for cell (=1,2, K), and let π be the vector of such probabltes. For the sake of smplcty, here we shall only be concerned wth multnomal samplng, where M (the vector of observed frequences) can be regarded as an observaton from a ( K 1) -dmensonal multnomal dstrbuton wth ndex = m and unknown parameter vector π. Other samplng schemes can be dealt wth n a smlar fashon. From a Bayesan pont of vew, belefs concernng the value of π must be descrbed n terms of a pror dstrbuton. The usual conjugate pror for the multnomal parameter s the Drchlet dstrbuton. Ths dstrbuton s characterzed by a parameter vector β = ( β1,, β K ) such that E( π) = β/ β, where β = β. Small values of β mply vague pror nformaton; n partcular, the well-known Jeffreys rule corresponds to β = (1/ 2,,1/ 2). The posteror dstrbuton of π s also Drchlet, wth parameter β = ( m1 + 1/2,, m K + 1/2). Ths dstrbuton contans all the avalable nformaton about the populaton probabltes π, condtonal on the observed contngency table. Any base model mposes constrants on the possble values of π. In other words, under the base model the populaton probablty of cell s gven by π = f( π ) for some functons f. As smple example, consder a 2 2 cross classfcaton and a base model whch states that the two varables are ndependent. Then ( π ) ( ) ( ), π2 = π12 = f2( π ) = ( π1+ π2) ( π2 + π4), π π π π π π π and π = π = f ( π ) = ( π + π ) ( π + π ). π1 = π11 = f1 = π1+ π3 π1+ π2 3 = 21 = f3( ) = ( 3+ 4) ( 1+ 3) 4 22 4 3 4 2 4
Bayesan predctve Confgural Frequency Analyss 287 The base model can be tested on the bass of the posteror dstrbuton of π δ = log π π. Ths quantty s always nonnegatve and s zero f and only f the base model s correct. Thus, posteror dstrbutons of δ concentrated near zero support the base model, whereas posteror dstrbutons located away from zero lead to rejecton of the base model. Types and anttypes from a Bayesan perspectve. If we knew the actual value of π, then Cell could be regarded as a type f π > π, and as an anttype f π < π. However, even f π π, we would be unwllng to classfy Cell as a type (respectvely, anttype) unless π π was sgnfcantly greater than zero (respectvely, less than zero). Ths suggests the followng defnton of types and anttypes: Cell s regarded as a type f and only f u < π π, and as an anttype f and only f π π < l, where u and l are sutable threshold values (see von Eye and Gutérrez-Peña, 2004). From the posteror dstrbuton of π we can (for example) compute the posteror probablty of Cell beng a type, namely, Pr( u < π π ). Patterns of types and anttypes. An nterestng feature of the Bayesan approach s that t allows us to calculate the jont posteror probablty of several cells beng all types smultaneously. More generally, we can calculate the posteror probablty of any specfc pattern of types and anttypes n a cross-classfcaton. Gven a partcular base model, the posteror dstrbuton of π nduces a probablty dstrbuton on the set of all possble patterns. Consder, for example, a 2 2 cross classfcaton. Then such possble patterns nclude, T A T, A, A T T A, T, where T stands for type, A for anttype, and for nether. A Bayesan soluton to the Confgural Frequency Analyss problem would then be to report the most probable pattern. However, even for problems of moderate sze, the number of all possble patterns may be too large for a drect mplementaton of ths approach to be feasble. In practce, we can dramatcally reduce the numercal burden of the approach descrbed above f we only look at patterns n a neghbourhood of the partcular pattern suggested by an exploratory analyss whch looks at each cell ndvdually. In ths way, we can then compare two or more plausble patterns n terms of ther relatve posteror probablty.
288 E. Gutérrez-Peña The Bayesan predctve approach In ths secton, we ntroduce an alternatve noton of types and anttypes that focuses on the lkely values of the cell frequences n future experments, as opposed to the average values of such frequences. Ths approach may be useful n developmental or any other research area n whch repeated measurement desgns are employed. Types and anttypes. Let m denote the (as yet unobserved) count n Cell for a future experment, and m the correspondng count assumng the base model s true. The posteror predctve dstrbuton of the m s (.e., the condtonal dstrbuton of the m s gven the observed counts M) can be readly obtaned from the multnomal samplng model for M and the posteror dstrbuton of the parameters π. On the other hand, the m s are a functon of the m s just as the π s are a functon of π. Thus, Cell can be regarded as a type f E( m M ) < m, and as an anttype f m < E( m M ). As n the prevous case, however, even f m E( m M ) we would be unwllng to classfy Cell as a type (respectvely, anttype) unless m E( m M ) was sgnfcantly greater than zero (respectvely, less than zero). Ths suggests the followng predctve defnton of types and anttypes: Cell s regarded as a type f the observed count m falls on the rght tal of the posteror predctve dstrbuton of m, and as an anttype f the observed count m falls on the left tal of that dstrbuton. Specfcally, n what follows, Cell wll be labelled as a type f 0.95 ( ) q m M < m and as an anttype f m < q0.05 ( m M ), where q ( m α M ) denotes the α -quantle of the posteror dstrbuton of m. From the jont posteror predctve dstrbuton of the m s, we can now compute the correspondng posteror predctve probablty of any pattern of type and anttypes. An example (Alcohol abuse). Ths example concerns a sample of = 108 adult men who were dagnosed as alcohol abusers (Zucker, 1994). The dagnostc scale had the followng four levels: 1 = Alcohol user ; 2 = Mld abuser ; 3 = Severe abuser ; and 4 = Alcohol dependent Three years later the same ndvduals were dagnosed agan. Dagnostc categores were the same as at Tme 1. However, n addton, the dagnoss 0 = o user of alcohol was ncluded. Table 1 dsplays the 4 5 cross-classfcaton of the dagnoses at the two occasons. The base model s a log-lnear man effects model (frst-order CFA). Fgure 1 shows the posteror dstrbuton of δ. Ths dstrbuton s located away from zero, thus suggestng that the base model should be rejected (see the prevous secton). On the other hand, Table 2 compares the results of the Bayesan CFA and the classcal CFA. Accordng to the posteror dstrbuton of π, the pattern suggested by the Bayesan CFA s more than 30 tmes as lkely as the pattern suggested by the classcal CFA. See Gutérrez-Peña and von Eye (2000) for further detals.
Bayesan predctve Confgural Frequency Analyss 289 Table 1: Cross-Classfcaton of Alcohol Dagnoses at two Occasons ( = 108) Alcohol Abuse Dagnoses at Tme 2 Dagnoses, Categores 0 1 2 3 4 Dagnoses at Tme 1 1 10 8 1 0 0 2 8 2 11 3 3 3 4 2 7 10 3 4 9 3 4 6 14 Fgure 1: Posteror dstrbuton of δ for the alcohol abuse data Table 3 shows the results of the Bayesan predctve CFA and compares them wth those obtaned from the usual Bayesan CFA as dsplayed n Table 2. In ths case, accordng to the posteror predctve dstrbuton of m s, the pattern suggested by the Bayesan predctve CFA s about 10 tmes as lkely as the pattern suggested by the classcal CFA, and consderably more lkely than the pattern suggested by the usual Bayesan CFA. Perhaps more nterestngly, even under the orgnal Bayesan CFA (.e., wth respect to the posteror dstrbuton of π ), the pattern suggested by the Bayesan predctve CFA s about 10 tmes as lkely as the pattern suggested by the usual Bayesan CFA. We have also analysed a data set concernng sleep behavour and prevously dscussed n Gutérrez-Peña and von Eye (2000). In that case, all three approaches lead to the same pattern of types and anttypes.
290 E. Gutérrez-Peña Confguraton Obs. Freq. Table 2: Bayesan CFA for the alcohol abuse data Pr(Type) Pr(ether) Pr(Anttype) B-CFA Classcal CFA 1-0 10 0.5427 0.4573 0.0000 Type 1-1 8 0.7351 0.2649 0.0000 Type Type 1-2 1 0.0000 0.3585 0.6409 Anttype 1-3 0 0.0000 0.1017 0.8983 Anttype 1-4 0 0.0001 0.0799 0.9200 Anttype 2-0 8 0.0419 0.9439 0.0142 2-1 2 0.0026 0.7968 0.2006 2-2 11 0.6472 0.3528 0.0000 Type Type 2-3 3 0.0041 0.8062 0.1897 2-4 3 0.0027 0.7648 0.2325 3-0 4 0.0004 0.5099 0.4897 3-1 2 0.0053 0.8021 0.1926 3-2 7 0.1133 0.8854 0.0013 3-3 10 0.7047 0.2953 0.0000 Type Type 3-4 3 0.0041 0.8038 0.1921 4-0 9 0.0076 0.9065 0.0859 4-1 3 0.0020 0.7637 0.2343 4-2 4 0.0003 0.4925 0.5072 4-3 6 0.0202 0.9480 0.0318 4-4 14 0.9238 0.0762 0.0000 Type Type
Bayesan predctve Confgural Frequency Analyss 291 Table 3: Bayesan predctve CFA for the alcohol abuse data Confguraton Obs. Freq. Pr(Type) Pr(ether) Pr(Anttype) B-CFA BP-CFA 1-0 10 0.2560 0.7434 0.0006 Type 1-1 8 0.5085 0.4915 0.0000 Type Type 1-2 1 0.0009 0.9991 0.0000 Anttype 1-3 0 0.0000 1.0000 0.0000 Anttype 1-4 0 0.0000 1.0000 0.0000 Anttype 2-0 8 0.0421 0.9403 0.0176 2-1 2 0.0050 0.9950 0.0000 2-2 11 0.3360 0.6637 0.0003 Type 2-3 3 0.0118 0.8640 0.1242 2-4 3 0.0112 0.8674 0.1214 3-0 4 0.0035 0.8110 0.1855 3-1 2 0.0055 0.9945 0.0000 3-2 7 0.0706 0.9215 0.0079 3-3 10 0.4022 0.5971 0.0007 Type 3-4 3 0.0098 0.8712 0.1190 4-0 9 0.0213 0.9154 0.0633 4-1 3 0.0062 0.8724 0.1214 4-2 4 0.0018 0.8232 0.1750 4-3 6 0.0280 0.9574 0.0146 4-4 14 0.4157 0.5842 0.0001 Type
292 E. Gutérrez-Peña Dscusson Bayesan CFA s capable of assgnng probabltes to patterns of types and anttypes, and thus allows for the comparson of such patterns by means of relatve probabltes. However, t defnes types and anttypes n terms of the true (unknown) values of the parameters, whch can be estmated but wll never be observed or fully known. On the other hand, Bayesan predctve CFA focuses on the lkely values of the cell frequences n future experments. These can n prncple be observed and compared wth the values predcted by the model on the bass of prevous experments. Ths approach may be more approprate when sample szes are small. All n all, Bayesan predctve CFA seems to be more conservatve than the orgnal Bayesan CFA. Ths could be due to the fact that there s more uncertanty nvolved n the predcton of new observatons than n the estmaton of ther correspondng expected values (.e. the parameters). Author notes Eduardo Gutérrez-Peña, Department of Probablty and Statstcs, atonal Unversty, Mexco. Ths work was partally supported by Sstema aconal de Investgadores, Mexco. The author would lke to thank Prof. Alexander von Eye for several comments and suggestons that greatly mproved ths paper. References Gutérrez-Peña, E. and von Eye, A. (2000). A Bayesan approach to Confgural Frequency Analyss. Journal of Mathematcal Socology, 24, 151-174. Lenert, G.A. (1969). De Konfguratonsfrequenzanalyse als Klassfkatonsmethode n der klnschen Psychologe. In Irle, M. (Ed.), Bercht über den 26. Kongress der Deutschen Gesellschaft für Psychologe n Tübngen 1968, 244 253. Göttngen: Hogrefe. von Eye, A. and Gutérrez-Peña, E. (2004). Confgural Frequency Analyss: the search for extreme cells. Journal of Appled Statstcs, 31, 981-997. Zucker, R.A. (1994). Pathways to alcohol problems and alcoholsm: A developmental account of the evdence for multple alcoholsms and contextual contrbutons to rsk. In R.A. Zucker, J. Howard, & G. Boyd (Eds.), The development of alcohol problems: Explorng the bopsychosocal matrx of rsk. Rockvlle, MD: atonal Insttute on Alcohol Abuse and Alcoholsm.