TESTING CATEGORIZED BIVARIATE NORMALITY WITH TWO-STAGE POLYCHORIC CORRELATION ESTIMATES

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1 IE Working Pper WP 0 / 03 08/05/003 TESTING CATEGORIZED BIVARIATE NORMALITY WITH TWO-STAGE POLYCHORIC CORRELATION ESTIMATES Alberto Mydeu Olivres Instituto de Empres Mrketing Dept. C/ Mri de Molin, , Mdrid Spin Alberto.Mydeu@ie.edu Abstrct We show tht when the thresholds nd the polychoric correltion re estimted in two stges, neither Person's X nor the likelihood rtio G goodness of fit test sttistics re symptoticlly chi-squre. We propose new test sttistic, M n, tht is symptoticlly chi-squre in this sitution. M n, my hve wide rnge of pplictions beyond the one considered here s it is symptoticlly chi-squre for brod clss of consistent nd symptoticlly norml estimtors. M n equls X with n djustment to tke into ccount tht the estimtor is not symptoticlly efficient. Also, M n X where M n = X in the cse of the one-stge mximum likelihood estimtor. Keywords LISREL, MPLUS, ctegoricl dt nlysis, pseudo-mximum likelihood estimtion, multinomil models * This reserch ws supported by grnt BSO of the Spnish Ministry of Science nd Technology, by Distinguished Young Investigtor Awrd of the Ctln Government, nd by grnt from the Fundción Instituto de Empres. I m indebted to Krl Jöreskog nd Hrry Joe for their comments to previous drft. 0

2 IE Working Pper WP 0 / 03 08/05/003. Introduction Consider bivrite stndrd norml density ctegorized ccording to (I - ) nd (J - ) thresholds, respectively. Within mximum likelihood frmework, Olsson (979) considered one nd two-stge pproches to estimte the q = (I - ) + (J - ) + prmeters of this model from the observed I J contingency tble. In the one-stge pproch ll prmeters re estimted simultneously. In the two-stge pproch, the thresholds re estimted seprtely from ech univrite mrginl, then the polychoric correltion is estimted from the bivrite tble using the thresholds estimted in the first stge. Of course, fter estimting the prmeters one must test the model (Muthén, 993). To this end, one my employ the likelihood rtio sttistic G or Person's X test sttistic. From stndrd theory (e.g, Agresti, 990), when the one-stge pproch is employed both sttistics re symptoticlly distributed s chi-squre with r = IJ - q - = IJ - I - J degrees of freedom. However, the distribution of G nd X when the two-stge pproch is employed remins to be investigted. Yet, ever since Olsson (979) concluded tht very similr results re obtined with the computtionlly simpler two-stge pproch, this pproch hs become the stndrd procedure for estimting this model. As such, it is the procedure implemented in computer progrms such s PRELIS/LISREL (Jöreskog & Sörbom, 993) nd MPLUS (Muthén & Muthén, 998). To ssess the goodness of fit of the moel, G is used in PRELIS/LISREL (Jöreskog, 00, July 6, personl communiction). No goodness of fit test is currently implemented in MPLUS. The purpose of this pper is to investigte the symptotic distribution of G nd X when the two-stge estimtor is employed.. Asymptotic distribution of prmeter estimtes Consider I J contingency tble. Let π = ( π,, π J,, πi,, πij,, πi,, πij) denote its IJ vector of probbilities nd p its ssocited vector of smple proportions. Furthermore, let π = ( π,,,, πi πi) nd π = ( π,, πj,, π J) denote the vectors of univrite mrginl probbilities, nd p nd p the vectors of its ssocited smple proportions. We note tht, π = Tπ π = Tπ, ()

3 IE Working Pper WP 0 / 03 08/05/003 for certin (implicitly defined) mtrices T nd T. For instnce, for I = nd J = 3, T = 03 3, = ( 3 3) T I I, where 3 nd 0 3 denote three-dimensionl column vectors of 's nd 0's respectively. Now, ssume the following model for π ij, ij τ i τj = ( z ) dz π φ () τ τ i j * * where τ =, τ = 0, τ = 0, τ = I nd φ J n ( ) denotes n-vrite stndrd norml density function. Thus, z * hs men zero nd correltion ρ mtrix. In prticulr, ρ τ i * * πi φ ( z) dz ( ) = τ i τj * * πj = φ z dz (3) In the sequel, let τ = ( τ, τ ), where τ nd τ denote the (I ) nd (J ) dimensionl vectors of thresholds implied by the model. Finlly, let κ = ( τ, ρ). We now provide the symptotic distribution of the one nd twostge prmeter estimtes using stndrd results for mximum likelihood estimtors for multinomil models. Agresti (990) is good source for the relevnt theory. Let π nd p be C-dimensionl vectors of multinomil probbilities nd smple proportions, respectively, nd let N denote smple size. Consider π prmetric structure for π, π(ϑ), with Jcobin mtrix =, nd suppose ϑ we estimte ϑ by mximizing C τj L( ϑ) = N p ln π ( ϑ ). (4) c= Then, under typicl regulrity conditions, it follows tht d c c N ( pπ) N ( 0, Γ ) Γ = D ππ (5)

4 IE Working Pper WP 0 / 03 08/05/003 where = Dig( ) N N D π, ( ) B = D D ( ˆ ) = B N ( p ) ϑ ϑ π (6) ( ˆ ) d N ( 0, ( D ) ) distribution, nd = denotes symptotic equlity. ϑ ϑ (7), d denotes convergence in. One-stge estimtion d Akin to (5) we write, N ( p π ) N ( 0, Γ ), where Γ = D ππ nd D = Dig( π ). Then, when ll the prmeters re estimted I J L τ, τ, ρ N p ln π τ, τ, ρ, by simultneously by mximizing ( i j ) = ij ij ( i j ) direct ppliction of (7) N i= j= ( ) ( ˆ ) d N 0, ( D ) κ κ (8) π π π = = κ τ ρ where Olsson (979). nd ll necessry derivtives cn be found in. Two-stge estimtion Consider now the following sequentil estimtor for κ (Olsson, 979): First stge: Estimte the thresholds for ech vrible seprtely by mximizing I J τ i i τ i L( ) = N pj ln πj ( j) i= j= L ( ) = N p ln π ( ) Second stge: Estimte the polychoric correltion by mximizing τ τ (9) L ( ρ, ˆ τ) N p ln π ( ρ, ˆ τ ) (0) I J = i= j= We shll now provide n lterntive derivtion of Olsson's results for this estimtor closely following Jöreskog's (994). We first notice tht ˆ τ nd ˆ τ re mximum likelihood estimtes, s (9) is the kernel of the log-likelihood function for estimting the thresholds from the univrite mrginls of the contingency tble. Similrly, (0) is the kernel of the log-likelihood function for estimting the polychoric correltion from the bivrite contingency tble given the estimted thresholds. Tht is, ˆρ is pseudo-mximum likelihood estimte in the ij ij i j 3

5 IE Working Pper WP 0 / 03 08/05/003 terminology of Gong nd Smniego (98). To obtin the symptotic distribution of the two-stge estimtes we first pply (6) to the first stge estimtes to obtin N ( τ τ ) = B N ( p π ) N ( τ τ ) = B N ( p π ) ˆ () where ( ) B = D D, = Dig( ) π D π, = τ derivtives cn lso be found in Olsson (979). Now, letting () nd () we get ˆ B, nd so on. These BT = BT, by N ( ˆ τ τ) = B N ( p π ). () Similrly, direct ppliction of (6) to the second stge estimtes yields N ( ρˆ ρ) = B N ( p π ( ρ, ˆ τ )) (3) π B D D, nd = ρ where = ( ). Note tht B nd re row nd column vector, respectively, despite the nottion. Now, we need the symptotic distribution of N p π ( ρ, τ ) to proceed. In Appendix, we show tht ( ) ˆ N ( p π ( ρ, ˆ τ) ) = ( I B ) N ( p π ) (4) π where =. Putting together (3) nd (4) we obtin τ Finlly, putting together () nd (5) we obtin N ( ρˆ ρ) = B ( I B ) N ( p π ). (5) N ( ˆ κ κ) = G N ( p π ) B G = ( ). (6) B I B nd since s shown in the Appendix, Gπ = 0, (7) ( ˆ ) d ( 0GD, G) N κ κ N (8) 4

6 IE Working Pper WP 0 / 03 08/05/003 where G nd D re to be evluted t the true popultion vlues. 3. Goodness of fit testing We shll first obtin the symptotic distribution of the unstndrdized residuls Nˆ e: = N ( p π ( κ )) when ˆκ re two- stge prmeter estimtes. ˆ In the Appendix it is shown tht π where = = ( ) κ Nˆe= ( I G) N ( p π ) (9). Thus, by (9) nd (7), d Nˆ e N ( 0, Ω ) Ω = ( I G) Γ ( I G). (0) We wish to investigte the symptotic distribution of Person's X sttistic, nd of nd the likelihood rtio sttistic G, I J ( p ( ) ij πij ˆ κ ) ˆ X = N = NˆeD ˆe π ( ˆ κ) i= j= ij () I J p ij G = N pij ln π ( ˆ κ ), () i= j= ij where Dˆ = Dig( π ( ˆ κ )), nd by convention when p ij = 0, pij p ij ln = 0. π ( ˆ κ) From stndrd theory (e.g., Agresti, 990), when the model prmeters re estimted simultneously, G = X χ IJ I J. When they re estimted in two stges, it lso holds tht G d = X (see Agresti, 990: p. 434). Therefore, we only consider here the symptotic distribution of X. Now, gin using stndrd results (Agresti, 990: p. 43), ˆ ˆ X = NeD ˆe = NˆeD ˆe. A necessry nd sufficient condition for X to be symptoticlly chi-squre distributed is (e.g., Schott, 997: Theorem 9.0) ij ΩD ΩD Ω = ΩD Ω (3) In the Appendix we show tht when the two-stge estimtor is employed ( ) ( )( )( ) ( )( ) ΩD = IK IJK IJ C Ω D = IK IJ C (4) where K = G, J = D K D nd C = π π D. Thus, (3) is not stisfied. Neither X nor G re symptoticlly chi-squred. Rther, these sttistics 5

7 IE Working Pper WP 0 / 03 08/05/003 converge in distribution to mixture of r = IJ - I - J independent chi-squre vribles with one degree of freedom (Box, 954: Theorem.). Thus, n lterntive test sttistic must be sought tht it is symptoticlly chi-squre under more generl conditions thn X nd G to test the goodness of fit of the model when the two-stge estimtor is employed. Let ϑ be consistent estimtor stisfying N ( ) = G N ( p ) ϑ ϑ π (5) for some q C mtrix G stisfying G = I. (6) Now, let e = ( p ( ) ) π ϑ nd consider the test sttistic M = N eue n U = D D D D (7) ( ) where U denotes U evluted t ϑ. We show in the Appendix tht under these conditions M d n C q χ. (8) We note tht M n cn be written s ( ˆ) M X N ˆ = ed D D e. (9) n Thus, M n X. M n equls X with n djustment to tke into ccount tht ϑ is not symptoticlly efficient. Note tht the second term in (9) becomes zero when ˆ ˆeD ˆ = 0. Since this is the grdient vector tht mximum likelihood estimtes stisfy, in the cse of one-stge mximum likelihood estimtion Mn = X. The two-stge estimtor under considertion is consistent nd with G given by (6) it stisfies (6) (see Appendix). Thus, with two-stge prmeter estimtes M d n IJIJ χ. 4. Numericl results Agresti (99) sked 6 respondents to compre the tste of Coke, Clssic Coke nd Pepsi using five point preference scle in pired comprison design {Coke vs. Clssic Coke, Coke vs. Pepsi, Clssic Coke vs. Pepsi}. The ctegories 6

8 IE Working Pper WP 0 / 03 08/05/003 were {"Strong preference for i", "Mild preference for i", "Indiference", "Mild preference for i ", nd "Strong preference for i ",}. For ech pir of vribles, we shll test the ssumption tht the observed 5 5 tble rises by ctegorizing stndrd bivrite norml density. Tht is, we re interested in testing substntive hypothesis of normlly distributed continuous preferences for the soft drinks in the popultion. In Tble we provide the thresholds nd polychoric correltion for ech pir of vribles estimted in two-stges nd the symptotic stndrd errors of these prmeters. The stndrd errors were obtined s the squre root of the digonl of (8) which ws consistently estimted by evluting ll derivtive mtrices nd probbilities t the estimted prmeter vlues Insert Tbles nd bout here In Tble we lso provide goodness of fit results for the two-stge estimtes using G, X nd M n. Inspecting the goodness of fit tests, we first notice tht for ll three bivrite tbles ll test sttistics suggest tht the ssumption of ctegorized bivrite normlity is resonble. We lso notice tht the estimted G sttistics re lrger thn the M n nd X sttistics. This is becuse we purposely chose numericl exmple with very smll smple size to highlight the differences between the sttistics. The estimted G sttistics re lrger becuse there re some empty cells in the observed bivrite tble nd these re not included in the computtion of G. On the other hnd, ll cells re used in the computtion of both M n nd X. The most surprising fct in Tble is tht the vlues for the symptoticlly correct M n nd the symptoticlly incorrect X re rther close. This is becuse s reflected in (9), the vlues of M n nd X will be very close if the estimtor used is highly efficient, yet not fully efficient. With these dt, the two stge estimtor is so highly efficient tht it is irrelevnt for prcticl purposes whether M n or X is used. To see this, in Tble we provide the results obtined when the model is estimted using one-stge mximum likelihood. Note tht in this cse, the thresholds estimted from different bivrite tbles need not be the sme cross tbles. We see in Tble tht the prmeter estimtes nd their stndrd errors for these tbles re indeed very similr to those obtined using the two stge pproch. As result, in this exmple the G nd X vlues obtined using the one nd two-stge estimtes re very close. To further illustrte the high symptotic reltive efficiency of the two stge estimtor we computed the popultion symptotic covrince mtrix of the 7

9 IE Working Pper WP 0 / 03 08/05/003 one-stge nd two-stge estimtors using (8) nd (8) t popultion vlues similr to those encountered in the exmple: τ = (, 0.5, 0.5, ), τ = (, 0.5, 0.5, ) nd ρ = 0.3. At these vlues, the determinnt of the symptotic covrince mtrix of the two-stge estimtes (8) is only.5% lrger thn the determinnt of the symptotic covrince mtrix of the one-stge estimtes. Also, the popultion symptotic vrinces of the two-stge prmeter estimtes re less thn % lrger thn for the one-stge prmeter estimtes. Yet, in our implementtion, the two-stge estimtes re on verge 7 times fster to compute thn the one-stge estimtes. To investigte the smll smple performnce of G, X nd M n we performed simultion study using the bove popultion vlues. The results for N = 50, N = 00, nd N = 000 cross 000 replictions re presented in Tble Insert Tble 3 bout here As cn be seen in this tble, in the criticl region {% to 0%} G tends to reject too often the null hypotheses when N = 50 nd N = 00. The behvior of X nd M n is cceptble even when N = 50 in these 5 5 contingency tbles. This is remrkble. Also, we see in this tble tht the empiricl distributions of X nd M n re very similr for ll smple sizes, with M n tking slightly smller vlues, in ccordnce to (9). Thus, the smll smple behvior of M n reltive to X mtches the symptotic efficiency results for the two-stge estimtes t these prmeter vlues. 5. Discussion The purpose of this reserch ws to investigte whether it ws theoreticlly justified the present use of G to test ctegorized bivrite normlity when the model prmeters re estimted in two stges. We hve shown tht with two-stge prmeter estimtes G is not symptoticlly chisqure distributed, nd neither is X. With two-stge prmeter estimtes, G nd X re symptoticlly equivlent, nd they re distributed s mixture of one-degree of freedom chi-squres. We hve proposed new test, M n, tht is symptoticlly distributed s chi-squre with two-stge prmeter estimtes. The expressions involved in computing M n re ctully side product of the computtions needed to obtin the two-stge estimtes nd their symptotic covrince mtrix (see Jöreskog, 8

10 IE Working Pper WP 0 / 03 08/05/ ). Our numericl results suggest tht G yields on verge smller p-vlues thn M n, prticulrly in smll smples. We hve lso shown tht M n reduces lgebriclly to X in the cse of one-stge prmeter estimtes. The more efficient the two stge estimtes, the closer M n will be to X. At the popultion vlues used in our simultion study, the two-stge estimtes re so efficient tht the empiricl distributions of M n nd X re very similr. However, t prmeter vlues where the two-stge estimtes re not so efficient, tht is, with lrge polychoric correltions, M n is not so close to X. Given the strightforwrd computtion of M n, we recommend tht this symptoticlly correct sttistic be used to ssess the goodness of fit of the model when two-stge prmeter estimtion is employed. 9

11 IE Working Pper WP 0 / 03 08/05/003 References Agresti, A. (990). Ctegoricl dt nlysis. New York: Wiley. Agresti, A. (99). Anlysis of ordinl pired comprison dt. Applied Sttistics, 4, Box, G.E.P. (954). Some theorems on qudrtic forms pplied in the study of nlysis of vrince problems: I. Effect of inequlity of vrince in the one-wy clssifiction. Annls of Mthemticl Sttistics, 6, Gong, G., & Smniego, F.J. (98). Pseudo mximum likelihood estimtion: Theory nd pplictions. Annls of Sttistics, 9, Jöreskog, K.G. (994). On the estimtion of polychoric correltions nd their symptotic covrince mtrix. Psychometrik, 59, Jöreskog, K.G. & Sörbom, D. (993). LISREL 8. User's reference guide. Chicgo, IL: Scientific Softwre. Khtri, (966). A note on Mnov model pplied to growth curves. Annls of the Institute of Mthemticl Sttistics, 8, Muthén, B. (987). LISCOMP: Anlysis of liner structurl equtions using comprehensive mesurement model. Mooresville, IN: Scientific Softwre. Muthén, B. (993). Goodness of fit with ctegoricl nd other non norml vribles. In K.A. Bollen & J.S. Long [Eds.] Testing structurl eqution models (pp ). Newbury Prk, CA: Sge. Muthén, L. & Muthén, B. (998). Mplus. Los Angeles, CA: Muthén & Muthén. Olsson, U. (979). Mximum likelihood estimtion of the polychoric correltion coefficient. Psychometrik, 44, Schott, J.R. (997). Mtrix nlysis for sttistics. New York: Wiley. 0

12 IE Working Pper WP 0 / 03 08/05/003 TABLE Two stge prmeter estimtes, estimted stndrd errors, nd goodness of fit tests for Agresti s soft drink dt Thresholds 3 4 vr (0.80) 0.06 (0.6) (0.69).50 (0.48) vr (0.87) (0.6) (0.66).0 (0.) vr (0.) (0.68) 0.03 (0.6) (0.84) Correltions nd Test Sttistics Vrs. Corr. M n p-vlue G p-vlue X p-vlue (,) 0.03 (0.40) (3,) (0.9) (3,) (0.4) Notes: N = 6; stndrd errors in prentheses; 5 d.f.

13 IE Working Pper WP 0 / 03 08/05/003 TABLE Joint prmeter estimtes, estimted stndrd errors nd goodness of fit tests for Agresti s soft drink dt Thresholds 3 4 vr (0.87) (0.6) (0.66).0 (0.) vr (0.80) 0.06 (0.6) (0.6).5 (0.49) vr (0.) (0.67) 0.04 (0.60) (0.84) vr (0.80) (0.60) (0.60).50 (0.50) vr (0.) (0.68) 0.03 (0.6) (0.84) vr (0.87) (0.6) (0.66).0 (0.) Correltions nd Test Sttistics Vrs. Corr. G p-vlue X p-vlue (,) 0.03 (0.44) (3,) (0.) (3,) (0.5) Notes: N = 6; stndrd errors in prentheses; 5 d.f.

14 IE Working Pper WP 0 / 03 08/05/003 TABLE 3 Simultion results Nominl rtes N Stt. Men Vr. % 5% 0% 0% 30% 40% 50% 60% 70% 80% 90% M n G X M n G X M n G X Notes: 000 replictions; 5 d.f.; τ = (, 0.5, 0.5, ), τ = (, 0.5, 0.5, ), ρ =

15 IE Working Pper WP 0 / 03 08/05/003 Appendix: Proofs of key results Proof of Eqution (4): A first order Tylor expnsion of π ( ρ, τ ) round τ = τ 0 yields ˆ π ( ρ, ˆ τ) = π ( ρ, τ) + ( ˆ τ τ ), π where = τ. Thus, ( ( ) ) ( ) N π ( ) ρ, ˆ τ π = ˆ τ τ = B N p π, where the lst symptotic equlity follows from (). Now, N ( p π ( ρ, ˆ τ) ) = N ( p π ) N ( π ( ρ, ˆ τ) π ) = ( I B ) N ( p π ) Proof of Eqution (7): B Tπ = 0 becuse D T π = D π = 0. B Tπ = 0 becuse D Tπ = D π = 0. Thus, Bπ = 0. Also, B π = 0 becuse D π = 0, so the proof is complete Proof of Eqution (9): A first order Tylor expnsion of π ( κ ) round κ = κ 0 yields ˆ π ( ˆ κ) = π ( κ) + ( ˆ κ κ ), π where = κ. Thus, ( ) ( ˆ ) N ˆ π ( ) π = κ κ = G N p π, where the lst symptotic equlity follows from (6). Now, N ( p ˆ π ) = N ( p π ) N ( ˆ π π ) = ( I G) N ( p π ) Proof of Eqution (4): We write Ω D = ( IK)( IJ) C = A C where K = G, C = D π π. Then, ( Ω ) J = D G D nd D = A AC CA+ C. Using the stndrd equlities, D π = = 0 π D = = 0 (30) nd (7) we first notice tht 4

16 IE Working Pper WP 0 / 03 08/05/003 JC = CJ = 0 KC = CK = 0 Thus, AC = C nd CA = C. Furthermore, using (30) nd π = ( sclr), C = C. Thus, ( Ω ) D = A C, but noting tht J = J K = K we find tht = (( )( )) = ( )( + )( ) A I K I J I K I JK I J. Therefore, ( D ) Ω Ω D. Proof of Eqution (8): First, using Tylor expnsion we find nlogously to the proof of (9) tht d N e N ( 0, Ω ) Ω = ( I G) Γ ( I G). (3) Then, by Lemm in Khtri (966), U in (7) cn be written s ( ) D U = (3) c c c c where c is (C - q) C mtrix stisfying c = 0. Now, we write Ω in (3) s Ω = YΓ Y, where Y = I G. Using (3), YU = U nd UY = Y. Thus, ( ) ΩUΩ = YΓUΓY = YΓY Y D Y, (33) where in (33) we hve used (7). When (6) holds, the second term is zero nd therefore ΩUΩ = Ω. The degrees of freedom vilble for testing re given by rnk( Ω U). Using the expression of U in (7) nd (6), ΩU = I Gπ. This mtrix is idempotent nd therefore its rnk equls its trce. The number of degrees of freedom vilble for testing is using (6) ( U) ( I) ( G) ( ) = tr Ω = tr tr tr π = C q. Eqution (6) cn be verified for the two-stge estimtor using T =, T, so tht B = I. Also, B = I. Finlly, T = 0, T = 0, so tht B = 0. 5