Marginal Models for categorical data.

Transcription

1 Margnal Models for categorcal data Applcaton to condtonal ndependence and graphcal models Wcher Bergsma 1 Marcel Croon 2 Jacques Hagenaars 2 Tamas Rudas 3 1 London School of Economcs and Poltcal Scence 2 Tlburg Unversty 3 ELTE, Hungary Cambrdge Unversty, February 2012

2 Outlne 1 Introducton 2 Degrees of freedom CI models wth non d data 3 Smoothness of ntersectons of CI models

3 Margnal models: for what types of data? Interest les n populaton averaged quanttes, but through desgn data are dependent (clustered). For correct nference, margnal modelng s needed.

4 Margnal models: for what types of data? Interest les n populaton averaged quanttes, but through desgn data are dependent (clustered). For correct nference, margnal modelng s needed. Dependences n the data arse n many stuatons: Comparng margnal dstrbutons of two characterstcs measured on the same respondents, e.g., preference prme mnster and party preference.

5 Margnal models: for what types of data? Interest les n populaton averaged quanttes, but through desgn data are dependent (clustered). For correct nference, margnal modelng s needed. Dependences n the data arse n many stuatons: Comparng margnal dstrbutons of two characterstcs measured on the same respondents, e.g., preference prme mnster and party preference. Respondents are clustered, e.g., husbands and wves, but nterest les n overall populaton dfferences men and women.

6 Margnal models: for what types of data? Interest les n populaton averaged quanttes, but through desgn data are dependent (clustered). For correct nference, margnal modelng s needed. Dependences n the data arse n many stuatons: Comparng margnal dstrbutons of two characterstcs measured on the same respondents, e.g., preference prme mnster and party preference. Respondents are clustered, e.g., husbands and wves, but nterest les n overall populaton dfferences men and women. Panel studes (repeated measurements): are there overall changes n the populaton?

7 Margnal models: for what types of data? Interest les n populaton averaged quanttes, but through desgn data are dependent (clustered). For correct nference, margnal modelng s needed. Dependences n the data arse n many stuatons: Comparng margnal dstrbutons of two characterstcs measured on the same respondents, e.g., preference prme mnster and party preference. Respondents are clustered, e.g., husbands and wves, but nterest les n overall populaton dfferences men and women. Panel studes (repeated measurements): are there overall changes n the populaton? Trend studes: comparng changes n two varables over tme.

8 Ths talk: two types of margnal models 1 Condtonal ndependence models for certan non-d data.

9 Ths talk: two types of margnal models 1 Condtonal ndependence models for certan non-d data. 2 Intersectons of condtonal ndependences

10 Ths talk: two types of margnal models 1 Condtonal ndependence models for certan non-d data. 2 Intersectons of condtonal ndependences Problems: degrees of freedom and mnmal specfcaton, smoothness

11 Ths talk: two types of margnal models 1 Condtonal ndependence models for certan non-d data. 2 Intersectons of condtonal ndependences Problems: degrees of freedom and mnmal specfcaton, smoothness Focus on categorcal data

12 Example: longtudnal data Age (A) 13 % 14 % 15 % 16 % 17 % Boys Maruana use (B) 1. Never Once a month More than once a month Grls Maruana use (G) 1. Never Once a month More than once a month

13 Growth curves maruna use Mean Usage Boys Grls 1.1 r Varables of nterest: age (A), maruana use (M), gender (G). Longtudnal study,.e., same boys and grls at each pont n tme, so data n Table AMG not d.

14 Fttng and testng: maxmum lkelhood How to test a model such as M G A (at all ages, maruana use same for boys and grls)?

15 Fttng and testng: maxmum lkelhood How to test a model such as M G A (at all ages, maruana use same for boys and grls)? Model nduces constrants on multnomal probabltes n full table GM 1 M 2 M 3 M 4 M 5.

16 Fttng and testng: maxmum lkelhood How to test a model such as M G A (at all ages, maruana use same for boys and grls)? Model nduces constrants on multnomal probabltes n full table GM 1 M 2 M 3 M 4 M 5. Maxmze kernel of multnomal log-lkelhood subect to the constrant L = p log π π B log(a π) = 0 Use scorng type Lagrange multpler method, algorthm of B. (1997), based on Atchson and Slvey (1959) and Lang and Agrest (1994).

17 Problems For many models: no problems at all wth fttng and testng. For some models we encountered problems...

18 Outlne 1 Introducton 2 Degrees of freedom CI models wth non d data 3 Smoothness of ntersectons of CI models

19 How many degrees of freedom (df)? For = 1,..., K and = 1,..., K : π A BC + D + = π A + B CD + = π A + B + C D

20 How many degrees of freedom (df)? For = 1,..., K and = 1,..., K : π A BC + D + = π A + B CD + = π A + B + C D (K 2 1) restrctons per eq. (1)

21 How many degrees of freedom (df)? For = 1,..., K and = 1,..., K : π A BC + D + = π A + B CD + = π A + B + C Nave calculaton: df = 2(K 2 1) D (K 2 1) restrctons per eq. (1)

22 How many degrees of freedom (df)? For = 1,..., K and = 1,..., K : π A BC + D + = π A + B CD + = π A + B + C D (K 2 1) restrctons per eq. (1) Nave calculaton: df = 2(K 2 1) Wrong: K 1 of the restrctons are not needed.

23 How many degrees of freedom (df)? For = 1,..., K and = 1,..., K : π A BC + D + = π A + B CD + = π A + B + C D (K 2 1) restrctons per eq. (1) Nave calculaton: df = 2(K 2 1) Wrong: K 1 of the restrctons are not needed. Soluton usng margnal loglnear parameterzatons (B. & Rudas, 2002); (1) equvalent to λ A B = λ B C = λ C D (K 1) 2 restrctons per eq. λ A = λ B = λ C = λ D (K 1) restrctons per eq. These form restrctons on a parameterzaton, so ths s a mnmal specfcaton; hence df = 2(K 1) 2 + 3(K 1)

24 How many degrees of freedom (df)? Ex. 2 π B A = π C B = π D C

25 How many degrees of freedom (df)? Ex. 2 π B A = π C B = π D C K (K 1) restrctons per eq. (2)

26 How many degrees of freedom (df)? Ex. 2 π B A = π C B = π D C K (K 1) restrctons per eq. (2) Nave calculaton: df = 2K (K 1)

27 How many degrees of freedom (df)? Ex. 2 π B A = π C B = π D C K (K 1) restrctons per eq. (2) Nave calculaton: df = 2K (K 1) Correct. But how do we know?

28 How many degrees of freedom (df)? Ex. 2 π B A = π C B = π D C K (K 1) restrctons per eq. (2) Nave calculaton: df = 2K (K 1) Correct. But how do we know? Usng margnal loglnear parameters (2) equvalent to λ A B = λ B C = λ C D (K 1) 2 restrctons per eq. λ A B = λ B C = λ C D (K 1) restrctons per eq.

29 How many degrees of freedom (df)? Ex. 2 π B A = π C B = π D C K (K 1) restrctons per eq. (2) Nave calculaton: df = 2K (K 1) Correct. But how do we know? Usng margnal loglnear parameters (2) equvalent to λ A B = λ B C = λ C D (K 1) 2 restrctons per eq. λ A B = λ B C = λ C D (K 1) restrctons per eq. Agan, restrctons on a parameterzaton, so ths s mnmal specfcaton; df = 2(K 1) 2 + 2(K 1) = 2K (K 1)

30 Cause of problems Same loglnear effect s restrcted n two dfferent margnal tables. Frst example revsted: π A BC + D + = π A + B CD + = π A + B + C Nave margnal loglnear specfcaton: D λ A B = λ B C = λ C D λ A B = λ B C = λ C D λ A B = λ B C = λ C D The loglnear B-effect s restrcted n both tables AB and BC, and the C-effect n BC and CD problems!

31 A not-so-obvous example Panel study drugs use of youth wth 5 waves (ages 13 to 17) Response varables: alcohol and maruana use Margnal tables of nterest: transtons from tme t to t + 1 for both alcohol and maruana usage. Artfcal table IPRS: I - tem (maruana or alcohol) P - perod (age 13-14, 14-15, 15-16, or 16-17) R, S - Response at 1st and 2nd measurement Condtonal ndependence models for margnal table IPRS: IP RS: all turnover tables dentcal, whatever I, P P RS: for both alcohol and maruana (I), turnover tables same for all perods I RS P: for any perod, turnover table alcohol same as for maruana

32 Soluton to the not-so-obvous example Probabltes n table IPRS formed by sums of probabltes n the orgnal multnomal table M 1 M 2 M 3 M 4 M 5 A 1 A 2 A 3 A 4 A 5 (3 10 = 59, 049 cells). Soluton s to formulate models such as P RS n terms of restrctons on a margnal loglnear parameterzaton for orgnal table.

33 Specfcaton model Model IP RS T k : measurement at tme pont k. Model nvolves these restrctons on margnals of multnomal table: π I T 1 T 2 1 = π I 1 T 2 T 3 = π I 1 T 3 T 4 = π I 1 T 4 T 5 = π I 2 T 1 T 2 = π I 2 T 2 T 3 = π I 2 T 3 T 4 = π I 2 A mnmal specfcaton s obtaned by frst, mposng equalty of the condtonal margnal assocaton parameters: λ T 1T 2 1 = λ T 2 T 3 1 = λ T 3 T 4 1 = λ T 4 T 5 1 = λ T 1 T 2 2 = λ T 2 T 3 2 = λ T 3 and second, by constranng the followng unvarate margnals: λ T 1 T 4 T 4 T 5, 2 = λ T 4 1 = λ T 2 1 = λ T 3 1 = λ T 4 1 = λ T 5 1 = λ T 1 2 = λ T 2 2 = λ T 3 2 = λ T 4 2 = λ T The number of ndependent constrants n ths mnmal specfcaton s (K 1)(7K + 2). T

34 Specfcaton model Model P RS Model assumes that the turnover tables are dfferent for the two tems but are dentcal over tme for each tem. A mnmal specfcaton: and λ T 1T 2 λ T 1T 2 λ T 1 λ T 1 1 = λ T 2 2 = λ T 2 T 3 T 3 1 = λ T 3 2 = λ T 3 T 4 T 4 1 = λ T 4 2 = λ T 4 T 5 1, T 5 2, 1 = λ T 2 1 = λ T 3 1 = λ T 4 1 = λ T 5 2 = λ T 2 2 = λ T 3 2 = λ T 4 2 = λ T 5 1, 2. The number of ndependent constrants n ths mnmal specfcaton s (K 1)(6K + 2).

35 Specfcaton model Model I RS P Model assumes that the turnover tables change over perod, but are the same for both tems at each perod. Mnmal specfcaton: and: λ T 1T 2 λ T 2T 3 λ T 3T 4 λ T 4T 5 λ T 1 λ T 2 λ T 3 λ T 4 1 = λ T 1 1 = λ T 2 1 = λ T 3 1 = λ T 4 1 = λ T 1 1 = λ T 2 1 = λ T 3 T 2 2 T 3 2 T 4 2 T 5 2, = λ T 4

36 Outlne 1 Introducton 2 Degrees of freedom CI models wth non d data 3 Smoothness of ntersectons of CI models

37 Illustraton of problem Intersecton margnal and condtonal ndependence: A B A B C If C s bnary, then equvalent to unon A C B C so ntersecton nonsmooth at A B C. How can we know?

38 Soluton n ths case Agan same loglnear effect (of AB) restrcted n two dfferent margnal tables: A B λ A B = 0 ( ) A B C λ A BC k = 0 and λa BC = 0 No smplfcaton possble, so problems to be expected. Another example: A BC DE F BD C ABC (and FBDC) effects restrcted twce! AF BE DC But ntersecton smooth, how do we know? Next theorem needed.

39 General condtonal ndependence model Q = {P P : A B C (P)} where A, B, C V for a set of varables V, P the famly of postve probablty dstrbutons for V.

40 General condtonal ndependence model Q = {P P : A B C (P)} where A, B, C V for a set of varables V, P the famly of postve probablty dstrbutons for V. Any fnte set of axoms ncompletely descrbes relatons among models (Studeny, 2005).

41 General condtonal ndependence model Q = {P P : A B C (P)} where A, B, C V for a set of varables V, P the famly of postve probablty dstrbutons for V. Any fnte set of axoms ncompletely descrbes relatons among models (Studeny, 2005). But: we can dentfy well-behaved subsets of models (next theorem).

42 Identfcaton of smooth models Condtonal ndependence model: Q = {P P : A B Wth IP(.) denotng the power set, let C (P)} ID = ID (A, B, C ) = IP(A B C ) \ (IP(A C ) IP(B C )) (ID contans loglnear effects set to zero under th CI) Let M 1,..., M m = V be nondecreasng orderng of margnals. For E V, M(E) s frst of the M contanng E. Theorem Suppose C M(E) A B C for all and E ID. Then * Q s herarchcal margnal log-lnear and s hence smooth. * Smple formula can be gven for correct df

43 Chan graph whose Andersson Madgan Perlman nterpretaton s a smooth model by Theorem (famly contans nonsmooth models) A C E G B D F Smoothness not easly verfed wthout theorem

44 Further work By ncompleteness of axoms, condtonal ndependence theory as complex as number theory. Much to be dscovered! Other results by Mlan Studeny, Frantsek Matus.

45 Some references Bergsma, W. P. and T. Rudas (2002). Margnal models for categorcal data. Annals of Statstcs, Vol. 30, Bergsma, W. P., M. A. Croon and J. A. Hagenaars (2009). Margnal models for dependent, clustered and longtudnal categorcal data. Sprnger NY. Rudas, T. and W. P. Bergsma and R. Nemeth (2010). Margnal log-lnear parameterzaton of condtonal ndependence models. Bometrka, vol 97, ssue 4, pp