ASSESSING GOODNESS OF FIT OF GENERALIZED LINEAR MODELS TO SPARSE DATA USING HIGHER ORDER MOMENT CORRECTIONS

Transcription

1 ASSESSING GOODNESS OF FIT OF GENERALIZED LINEAR MODELS TO SPARSE DATA USING HIGHER ORDER MOMENT CORRECTIONS S. R. PAUL Department of Mathematcs & Statstcs, Unversty of Wndsor, Wndsor, ON N9B 3P4, Canada E-mal: D. DENG Department of Mathematcs & Statstcs, Unversty of Regna, Regna, SK S4S 0A, Canada E-mal: The purpose of ths paper s to assess goodness of ft propertes of the condtonal dstrbuton of the Pearson statstc, for non-canoncal generalzed lnear models wth data that are extensve but sparse, by Edgeworth approxmaton of p-values usng hgher order moment correctons. In ths paper we obtan approxmatons to the fourth moments of the uncondtonal and condtonal dstrbuton of the modfed Pearson statstc wth non-canoncal lns. Ths extends prevous results where approxmatons to the frst three moments are avalable and completes all usual hgher order moment calculatons of the modfed Pearson statstc. We consder the asymptotc lmt n whch the data are extensve but sparse and a supplementary estmatng equaton for the dsperson parameter. Specfc results for bnomal and Posson data are obtaned separately. The methods for assessng goodness of ft usng hgher order moments are dscussed. For testng goodness of ft of generalzed lnear models to sparse data some smulatons are conducted to compare, n terms of emprcal sze and power, the performance of the classcal Pearson ch-square statstc, a standardzed modfed Pearson ch-square statstc, a standardzed modfed devance Statstc, a modfed Pearson statstc based on Edgeworth approxmaton wth the frst three condtonal moments and a modfed Pearson statstc based on Edgeworth approxmaton wth the frst four condtonal moments. The hgher order moments corrected statstcs have defnte advantage over other statstcs dscussed here. The statstc based on Edgeworth approxmaton wth the frst four condtonal moments holds level most effectvely n most stuatons and has some power advantage. 1 Introducton The Pearson ch-square statstc and the lelhood rato ch-square statstc are generally used to test goodness of ft n contngency tables or to test goodness of ft of a generalzed lnear model to dscrete data n the form of counts or proportons. These statstcs usually perform poorly, n terms of level and power, when the cell frequences n the contngency table are small or when Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9, 011 1

2 the dscrete data dstrbuton departs from a dscrete generalzed lnear model. Improvements have been suggested by usng hgher order moment correctons. For example, Koehler and Larntz (1980) and Koehler (1986) obtan modfcatons of these statstcs for sparse contngency tables usng hgher-order moment approxmatons. These authors derve modfcatons usng moments of the uncondtonal dstrbuton of the Pearson ch-square statstc and the lelhood rato statstc. McCullagh (1986) argues that t s the condtonal dstrbuton of the statstc and not ts margnal dstrbuton that s relevant for assessng goodness of ft. He obtans condtonal dstrbutons of the Pearson ch-square statstc and the lelhood rato ch-square statstc for dscrete data for the case where the data are extensve but sparse. McCullagh (1985)obtans approxmatons to the frst three moments of the uncondtonal and condtonal dstrbutons of the Pearson ch-square statstc for canoncal exponental famly regresson models. Farrngton (1996) extends the results of McCullagh (1985) to models wth non-canoncal lns usng an estmatng-equatons approach followng Moore (1986). Paul and Deng (000) derve approxmatons to the frst three moments of the uncondtonal and condtonal dstrbuton of the modfed devance statstc, agan, usng an estmatng-equatons approach. Paul and Deng (000) further show through smulaton that the modfed devance statstc has some power advantage over the modfed Pearson statstc of Farrngton. They, however, report that both the modfed Pearson statstc of Farrngton (1996) and the modfed devance statstc of Paul and Deng (000) show nflated levels for small α (nomnal level) and small n (sample sze). The emprcal level and power propertes of the modfed Pearson statstc and the modfed devance statstc were studed by normal approxmaton usng the frst two condtonal moments. The purpose of ths paper s to assess propertes of these condtonal goodness of ft statstcs to sparse data by Edgeworth approxmaton of p-values usng hgher order moment correctons (up to the fourth moment). In ths paper we obtan approxmatons to the fourth moments of the uncondtonal and condtonal dstrbuton of the modfed Pearson statstc wth non-canoncal lns. Ths extends the results of Farrngton (1996) who obtans approxmatons to the frst three moments and completes all usual hgher order moment calculatons of the modfed Pearson statstc. As n Farrngton (1996) we consder the asymptotc lmt n whch the data are extensve but sparse and a supplementary estmatng equaton for the dsperson parameter. Specfc results for bnomal and Posson data are obtaned separately and specal cases are dscussed. All cumulants for both bnomal and Posson data have Smple and closed form expressons. The method for assessng goodness of Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9, 011

3 ft by Edgeworth approxmaton of p-values usng hgher order moments are dscussed. Some smulatons are conducted. Secton deals wth the modfed Pearson statstc and ts condtonal cumulants. Specfc results for bnomal and Posson data are also obtaned here. Secton 3 dscusses assessng goodness of ft by usng p-values. A smulaton study s conducted n Secton 4 and a dscusson follows n Secton 5. The modfed Pearson statstc and ts condtonal cumulants Let Y, = 1,..., n, denote ndependent random varables from an exponental famly dstrbuton wth mean µ and varance V. Consder a generalzed lnear regresson model wth the nverse ln functon h 1 (µ) = η = Xβ, where X s an n p model matrx and β s a vector of p regresson parameters. Maxmum lelhood estmates of the regresson parameters β 1,..., β p are obtaned as solutons of the p quas-lelhood estmatng equatons g r ( ˆβ) = 0, r = 1,..., p, where g r (β) = =1 y µ V µ β r. The usual Pearson test statstc for assessng the goodness of ft of the generalzed lnear model s X = n (y ˆµ ) =1. For over- or under-dspersed data ˆV a wder famly can be assumed wth varance φv. By usng a supplementary unbased estmatng equaton g q, { (y µ ) g q (β, φ) = a (µ )(y µ ) + φ, =1 Farrngton (1996) proposes a modfed Pearson statstc X = n ˆφ, where, ( ˆφ = 1 n ) (y ˆµ ) (y ˆµ )a (ˆµ ) +. n ˆV =1 The Pearson statstc X can be obtaned by puttng a = 0. Farrngton (1996) shows that the choce of the functon a = V /V maes ˆβ and ˆφ asymptotcally uncorrelated. Wth ths the modfed Pearson statstc s X = n φ (y ˆµ ) ˆV = (y ˆµ ). ˆV ˆV =1 =1 =1 =1 V Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9, 011 3

4 Farrngton (1996) shows that for generalzed lnear models both X and X are asymptotcally ndependent of ˆβ. Further, he proves that gven the maxmum lelhood estmate ˆβ, X depends only wealy on β. As he argues, ths supports the use of the condtonal dstrbuton of X gven β = ˆβ for assessng the goodness of ft even when the ln functon s not canoncal. Provdng detaled calculatons Farrngton (1996) obtans approxmatons to frst three uncondtonal and condtonal moments of X. In ths paper we follow smlar steps to obtan the approxmaton to the fourth uncondtonal and condtonal moments of X whch are presented n the Appendx. For convenence and completeness we gve all four condtonal cumulants of X gven β = ˆβ n what follows. Theorem.1 Let h = dh(η )/dη, h = d h(η )/dη, V = dv (µ )/dµ, = d V (µ )/dµ, W = dag(h /V ), and Q = (Q ) = X(X T W X) 1 X T. Then asymptotcally, the frst four condtonal cumulants of X gven β = ˆβ are V E(X ˆβ) = n p 1 =1 ˆV ĥ ˆV ˆQ, var(x ˆβ). = (n p)(ˆρ 4 ˆρ 3 + ), κ 3 (X ˆβ). = (n p)(ˆρ 6 3ˆρ ˆρ ˆρ 4 8ˆρ 3 ˆρ ), κ 4 (X ˆβ). = (n p)(ˆρ ˆρ 8 + 4ˆρ 6 16ˆρ 35 4ˆρ ˆρ ˆρ ˆρ 34 4ˆρ ˆρ ˆρ ) +3 ( ˆκ3 3 ˆκ 3ˆκ 4 + ˆκ 5 + ˆκ 3 )( ˆκ3 3 ˆκ 3 ˆκ 4 ˆV 5 ˆV 4 ˆV 3 ˆV ˆV 5 ˆV 4,=1 + ˆκ 5 ˆ + ˆκ 3 )ĥ ĥ ˆQ. ˆ κ t r κu where κ s s the s-order cumulant of y µ, ρ t ru = 1 n, s = V (rt+s)/ 3, 4, 5, 6, 7, 8; t = 1,, 3, 4; r = 3, 4, 6, 8 and u = 4, 5, 6, 7 and the crcumflex above the varates denotes evaluaton at β = ˆβ. Note that a degree of freedom correcton s ncluded n all the moments above. Now we consder applcaton to bnomal data. Let Y, = 1,..., n, denote ndependent random varables from a bnomal dstrbuton wth parameters m and π wth µ = m π = h(η ) where h s a ln functon, η = X β = X 1 β X p β p, X 1,..., X p are p explanatory varables, β 1,..., β p are p regresson parameters. Then, V = m π (1 π ), µ / β r = h X r and hence Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9, 011 4

5 the estmatng equatons to obtan ˆβ 1,..., ˆβ p are g r (β) = =1 and the approxmate cumulants are E(X ˆβ) = n p + (y µ ) m π (1 π ) h X r = 0 (r = 1,..., p) =1 ĥ m ˆπ (1 ˆπ ) ˆQ, var(x ˆβ) = (1 p n n ) (1 1 ), m κ 3 (X ˆβ) = 8(n p) ( =1 1 1 n =1 5m 4 m + 1 n =1 ) m 1 m ˆπ, (1 ˆπ ) κ 4 (X ˆβ) =. 48(1 p n ) (1 1 { m ) 17m + 31 m =1 + 1,=1 m + 3m 7 m ˆπ (1 ˆπ ) + 1 6m ˆπ (1 ˆπ ) (m )(m )(1 ˆπ )(1 ˆπ ) m m ˆπ ĥ ˆπ (1 ˆπ )(1 ˆπ ) ĥ ˆQ. For canoncal ln h(η ) = e η /(1 + e η ), h = m π (1 π ) and the frst and fourth cumulants smplfy to E(X ˆβ) = n p + ˆπ (1 ˆπ ) ˆQ, =1 κ 4 (X ˆβ) =. 48(1 p n ) (1 1 { m ) 17m + 31 m =1 + 1 m + 3m 7 m ˆπ (1 ˆπ ) + 1 6m ˆπ (1 ˆπ ) (1 ˆπ )(1 ˆπ )(1 )(1 ) m m ˆQ.,=1 For the stuaton wth no covarates, ˆπ = ˆπ = n =1 y / n =1 m, Q = Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9, 011 5

6 ( n =1 V ) 1 = [π(1 π) n =1 m ] 1 for, = 1,..., n and E(X ˆβ) = n(1 1 n + 1 n =1 m ), var(x ˆβ) = (n 1)(1 1 n ( κ 3 (X ˆβ) = 8(n 1) 1 1 n =1 =1 1 m ), 5m 4 m + 1 nˆπ(1 ˆπ) =1 ) m 1 m, κ 4 (X ˆβ) =. 48(n 1) (1 1 { m ) 17m + 31 m =1 m + 1n (1 ˆπ) (1 n n =1 1 m ) ˆπ(1 ˆπ) n. =1 m In partcular, for m = m, we have E(X ˆβ) = n (1 1 m ), + 3m 7 m ˆπ(1 ˆπ) + 1 6m ˆπ (1 ˆπ ) var(x ˆβ) = (n 1)(1 1 m ), κ 3 (X ˆβ) = 8(n 1)(1 1 ( m ) 1 4 ) m + 1, mˆπ(1 ˆπ) κ 4 (X ˆβ) =. 48(n 1)(1 1 m ) { m 17m + 31 m + 3m 7 m ˆπ(1 ˆπ) + 1 6m ˆπ (1 ˆπ ) + 1n(1 ˆπ) (1 m ). ˆπ(1 ˆπ)m In the extreme sparse stuaton, however, wth m = 1, E(X ˆβ) = n and all other moments become zero, mang the test useless. In ths stuaton the modfed devance statstc also degenerates. Ths s understandable, as n ths case each cluster contans only one response and therefore there s no ntra-cluster correlaton (over-dsperson). Note the goodness of ft test n ths paper, n Paul and Deng (000) and n Farrngton (1996) s of the generalzed lnear model aganst a general over-dspersed generalzed lnear model. Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9, 011 6

7 Next we consder applcaton to Posson data Y, = 1,..., n wth µ = h(η ) = h(x β). The estmatng equatons to obtan ˆβ 1,..., ˆβ p are g r (β) = =1 and the four cumulants are (y µ ) µ h X r = 0, r = 1,..., p. E(X ˆβ) = n p, var(x ˆβ) = (n p), ( κ 3 (X ˆβ) = 8(n p) n =1 ˆµ 1 ), κ 4 (X ˆβ) = 48(n p) n =1( 1ˆµ + 18 ˆµ ) + 1 4n,=1 ĥ ĥ ˆQ ˆµ ˆµ. For the canoncal ln h(η ) = exp(η ), the fourth cumulant smplfes further to κ 4 (X ˆβ) = 48(n p) ) + 1 ˆQ 6n ˆµ 4n. =1( 1ˆµ,=1 In stuatons where there s no covarate, ˆµ = ˆµ = n =1 y /n and Q = ( n =1 V ) 1 = (nµ) 1 and thus the thrd and fourth cumulants further smplfy to ( κ 3 (X ˆβ) = 8(n p) ), ˆµ { κ 4 (X ˆβ) = 48(n p) ˆµ ˆµ Note that n ths stuaton X = X snce (y ˆµ)/ˆµ = 0. 3 Assessng goodness of ft Goodness of ft s assessed by calculatng p-values. If we use the frst two moments the approxmate p-value s obtaned by calculatng the upper tal probablty P (Z z) of the standardzed quantty Z = {X Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9, 011 7

8 E(X ˆβ)/ var(x ˆβ) 1/. The p-values usng the hgher order moment correctons are calculated usng the Edgeworth approxmaton and P (Z z). = 1 Φ(z) + (z 1)ρ 3 (X ˆβ)φ(z)/3!. P (Z z). = 1 Φ(z) + φ(z){(z 1)ρ 3 (X ˆβ)/3! + (z 3 3z)ρ 4 (X ˆβ)/4! + (z 5 10z z)ρ 3(X ˆβ)/7. The quanttes ρ 3 (X ˆβ) and ρ 4 (X ˆβ) are the approxmate standardzed condtonal thrd and fourth cumulants, respectvely. 4 Smulaton A lmted smulaton study was conducted to compare the emprcal szes of the modfed Pearson statstc X after adustng for the frst three condtonal moments and the modfed Pearson statstc X after adustng for the frst four condtonal moments. We call these procedures Z 1 and Z respectvely. In the smulaton we also ncluded the usual Pearson ch-square statstc X, the standardzed modfed Pearson statstc X (nvestgated by and the standardzed modfed devance statstc D (see Paul and Deng, 000). For comparng emprcal szes smulatons have been conducted for the bnomal model wth p = and a sngle contnuous covarate chosen to nduce very strong regresson effects under both logstc and complementary log-log ln functons, for sample szes varyng from n = 10 to n = 100, bnomal denomnators m = 5 and 10 and nomnal α =0.01, 0.05, The results for the logstc and the complementary log-log ln functons are very smlar. In ths paper we present results for only the logstc model. Emprcal sze results are presented n Table 1. Results n Table 1 show that the statstc X, n general, under estmates the level. Both the statstcs X and D show lberal behavour for α = 0.01 (outsde tmes the standard error of ˆα, where ˆα s the emprcal sze). Otherwse, these two statstcs hold level except n the case of D for α = 0.05, m = 5 and n = 10 (emprcal level of s greater than se( ˆα)=0.054). These fndngs are smlar to those n Paul and Deng (000). Both the three and the four moments corrected statstcs Z 1 and Z hold level reasonably well, except for small α and small n. In these cases the statstc Z 1 s slghtly lberal. These smulatons have been extended for power comparson for the overdsperson parameter ψ=0.05, 0.10, 0.15, 0.0. For power comparson we smulated data from the beta-bnomal dstrbuton wth mean mπ and varance mπ(1 π){1 + (m 1)ψ where π = exp(xβ)/{1 + exp(xβ). Comparatve Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9, 011 8

9 Table 1. Emprcal szes of X, X, Z 1, Z and D for α =.01,.05,.10, m = 5, 10 and n = 10, 0, 50, 100 based on 10,000 replcatons. α m n X X D Z 1 Z power results are smlar for all values of ψ. So, we present results for only ψ = 0.1 n Table. Results n Table show that the statstc X s least powerful. Ths s nor surprsng as t under estmates the level. The comparatve power behavour of the statstcs X and D are smlar to that n Paul and Deng (000), namely that D s more powerful that X. For small α and small n the statstc Z 1 s slghtly more powerful than the statstc Z. Agan, ths s not surprsng as n these cases the statstc Z 1 s slghtly lberal. In other scenaros consdered here the statstc Z s more powerful, though margnally, than the statstc Z 1. 5 Dscusson We derved approxmate fourth cumulant of the condtonal dstrbuton of the Pearson statstc for non-canoncal generalzed lnear models wth data Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9, 011 9

10 Table. Emprcal powers of X, X, Z 1, Z and D for α =.01,.05,.10, ψ = 0.10, m = 5, 10 and n = 10, 0, 50, 100 based on 10,000 replcatons. α m n X X D Z 1 Z that are extensve but sparse. Ths completes all usual hgher order moment calculatons of the modfed Pearson statstc. Our smulatons show that, n general, the Z 1 procedure has better level property than the procedure based on the standardzed X. Ths s n lne wth Farrngton s (Farrngton, 1996, p358) fndng whch states that n general the Edgeworth approxmaton mproved the approxmatons to the tal probabltes n the regon of prmary nterest. Note that emprcal comparson of the sze and power propertes of the procedure based on the standardzed X and the Z 1 procedure was not done earler n the lterature. For small α and small n scenaros the statstc Z 1 s slghtly more powerful than the statstc Z. Ths s because n these cases the statstc Z 1 s slghtly lberal. In other scenaros consdered here the statstc Z s, n general, slghtly more powerful than the statstc Z 1. For both the bnomal and Posson dstrbuted data the formulas for all the condtonal cumulants of X have smple and easly computable closed form Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9,

11 expressons. The statstcs Z 1 and Z have defnte advantage over the other statstcs dscussed here. As for small α and small n scenaros the statstc Z 1 s slghtly lberal and n other scenaros Z s slghtly more powerful than Z 1 we recommend the use of Z n practcal applcatons. Note that the computatonal effort n usng Z as opposed to Z 1 s only slght as all the condtonal cumulants have smple closed forms. Paul and Deng (000) show that the modfed devance statstc D wth correcton for the frst two or three moments has power advantage over all other procedures except what we have developed here. Ths statstc, however, shows lberal behavour for small α or small n. It appears therefore that D based on the Edgeworth approxmaton wth correcton for the frst four cumulants mght show correct level property and be more powerful. However, dervaton of the fourth cumulant of D seems extremely complcated. Moreover, none of the cumulants of D for bnomal and Posson data have closed forms and therefore the method would be much more computatonally ntensve. Ths wll, however, be pursued n future research. Acnowledgments Ths research was partally supported by the Natural Scences and Engneerng Research Councl of Canada. Appendx Approxmate Fourth Uncondtonal and Condtonal Cumulants of X : For convenence and to save space we use the notaton of Farrngton (1996). Let θ denote the true parameter value (β 1,..., β p, 1) T and g = (g 1,..., g p, g q ) T. Then, by Taylor expanson of g r, r = 1,..., p, q, about θ and followng steps smlar to Farrngton (1996) we obtan κ 4 ( ˆφ) = n 4 E{( g1 ) 4 4( g1 ) 3 g q + 6( g1 ) (g q ) 4( g1 )(g q ) 3 + (g q ) 4 + O( 1 n 4 ). The terms 11, 1, g 1 are defned n Farrngton (000). Now, usng tensor notaton and omttng summaton, E[( g1 ) 4 ] = w w w w l h h h h lh 4 mq m Q m Q m Q lm (µ 4m 3Vm) m 4 + 3w w w w l h h h h lq Q l, Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9,

12 E[( g1 ) 3 g q ] = w w w w l h h h h l(q Q l + Q Q l + Q l Q ) + w w w (a l V 3 l µ 4l + V 4 l µ 3l V l V l 3V 1 l w l )h h h h 3 l Q l Q l Q l, and E[( g1 ) (g q ) ] = w w w w l h h h h l(q Q l + Q l Q ) + w w h h Q (w + µ 4 V 1 V + w w h h h Q Q (a V µ 4 + V 4 µ 6 + V 1 a V 1 V V 3 µ 4 3w 3 V 3 µ 4 + V V ), + a V 3 µ 5 E[( g1 )(g q ) 3 ] = 3w w h h Q (V w + µ 4 V 1 V 1 V + a h h Q (a 3 V 1 µ 4 + V 4 µ 7 + 3a V µ 5 3a V + 3a V 3 µ 6 3V 3 µ 5 + 3w 6a V µ 4 3V w 3 3V w µ 4 + 3w + 3V 1 V w ) E[(g q ) 4 ] = 3(w V + V µ 4 1 V 1 V ) + (a 4 µ 4 + V 4 µ 8 + 4a 3 V 1 µ 5 4a 3 V V + 4a V 3 µ 7 4V 3 µ a V µ 6 + 6a V + 6V µ 4 1a V 1 1a V µ 5 + 1a V 3 w 4 3V 4 µ 4 3 3V V 4 6V 1 w µ 4 + 6V w + 6w V + 6V µ 4 + 6V 3 V µ 4 µ 4 6V 1 ), V where µ = E(y µ ) and w = a + V /V. Note that µ 1 = κ 1 = 0, µ = κ = V, µ 3 = κ 3 = V V, µ 4 = κ 4 +3κ, µ 5 = κ 5 +10κ 3 κ, µ 6 = κ κ 4 κ + 10κ κ, µ 7 = κ 7 + 1κ 5 κ + 35κ 4 κ κ 3 κ and µ 8 = κ 8 +8κ 6 κ +56κ 5 κ 3 +35κ 4 +10κ 4κ +80κ 3 κ +105κ 4. Then, after some tedous calculaton and smplfcaton and notng that X = n ˆφ, t can be shown that κ 4 (X ) = n(ρ 8 + 4ρ ρ ρ ρ ρ ) + γ 4 κ 4 4 γ 3 ( κ 5 + 6κ 3 ) + 6 V =1 =1 4γ ( κ 7 = κ κ 3κ 4 =1 + 7 κ 3 V ). γ ( κ 6 V + 1 κ 4 V + 8 κ 3 V + 8V ) Now, snce the Pearson statstc and the regresson parameters are asymptotcally uncorrelated, then from McCullagh (1984) the asymptotc expanson Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9, 011 1

13 of the condtonal fourth cumulant of ˆφ gven ˆβ has the form, to the order O(n 4 ), κ 4 ( ˆφ ˆβ) = (ˆκ q,q,q,q + ˆκ r,q,qˆκ s,q,q [3]ĥr,s)/n, where κ q,q,q,q = n cum( ˆφ, ˆφ, ˆφ, ˆφ) = nκ 4 ( ˆφ), κ r,q,q = n cum( ˆβ r, ˆφ, ˆφ), h r,s = h rs h r h s = δ rs = 1 n n =1 1 µ µ V β r β s. Further, κ r,q,q = n cum( ˆβ r, ˆφ, ˆφ) = E[( ˆβ r β)( ˆφ φ) ] = n 1 E{[ n 1/ ( nb rr g r )][n 1/ µ (w b ss g s g q )] β s [n 1/ µ (w b tt g t g q )] β t = n 3/ {γ µ 3 V γ µ 4 + µ 5 where b rs = ( 1 11 ) rs wth 11 = (δ rs ) p p. Therefore, κ r,q,q κ s,q,q [3]h r,s = 3n 3/ {γ n 3/ {γ = 3n {γ 3 { γ = 3n {γ 3 { γ µ 3 V µ 3 V κ 3 V κ 3 V µ 3 V µ 3 V γ µ 4 γ µ 4 γ µ 4 γ µ 4 γ ( κ 4 γ ( κ 4 + µ 5 + µ 5 + µ 5 + µ 5 µ 3 V h X r b rr, µ 3 V h X r b rr µ 3 µ 3 µ 3 + 3) + κ 5 + 3) + κ 5 Thus, asymptotcally, to the order O(n 4 ) V + 8 κ κ 3 h X s b ss δ rs h h X r b rr X s b ss δ rs h h Q. κ 4 ( ˆφ ˆβ) = (ˆκ q,q,q,q + ˆκ r,q,qˆκ s,q,q [3]ĥr,s)/n = (nˆκ 4 (φ) + ˆκ r,q,qˆκ s,q,q [3]ĥr,s)/n = ˆκ 4 ( ˆφ) + ˆκ r,q,qˆκ s,q,q [3]ĥr,s)/n Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9,

14 and hence, = ˆκ 4 ( ˆφ) + 3n {ˆγ 4 ˆκ 3 ˆV { ˆγ ˆκ 3 ˆV ˆγ ( ˆκ 4 ˆ ˆγ ( ˆκ 4 ˆ + 3) + ˆκ 5 ˆ + 3) + ˆκ 5 ˆ + 8 ˆκ 3 ˆ + 8 ˆκ 3 ˆ ĥ ĥ ˆQ κ 4 (X ˆβ) = n(ˆρ 8 + 4ˆρ ˆρ ˆρ ˆρ ˆρ ) + ˆγ 4 ˆκ 4 4 ˆγ 3 ( ˆκ 5 + 6ˆκ 3 ) ˆV =1 =1 =1 ˆγ ( ˆκ ˆκ 4 ˆV ˆV 4ˆγ ( ˆκ 7 =1 [,=1 [ ˆ ˆγ ˆγ ˆκ 3 V ˆκ 3 V + 18 ˆκ 5 ˆ ˆγ ( ˆκ 4 ˆ ˆγ ( ˆκ 4 ˆ + 8 ˆκ ˆV ˆV ) + 3 ˆκ 3ˆκ 4 ˆ + 3) + ˆκ 5 ˆ + 3) + ˆκ 5 ˆ + 7 ˆκ 3 ˆV ) + 8 ˆκ ˆκ 3 ] ] ĥ ĥ ˆQ. For a = V /V, we have γ = a = κ 3 /V and thus, approxmately, to the order O(n 4 ), κ 4 (X ˆβ) = n(ˆρ ˆρ 8 + 4ˆρ 6 16ˆρ 35 4ˆρ ˆρ ˆρ ˆρ 34 4ˆρ ˆρ ˆρ ) + 3 ( ˆκ3 3 ˆκ 3ˆκ 4 + ˆκ 5 + ˆκ 3 ) ˆV 5 ˆV 4 ˆV 3 ˆV,=1 ( ˆκ3 3 ˆV 5 ˆκ 3 ˆκ 4 ˆV 4 + ˆκ 5 ˆ + ˆκ 3 )ĥ ĥ ˆQ. ˆ A better approxmaton s to ncorporate a degree of freedom correcton to the above formula whch replaces n by n p. For completeness we gve all four degree of freedom corrected condtonal cumulants n Theorem.1. Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9,

15 References 1. C. P. Farrngton (1996). On assessng goodness of ft of generalzed lnear models to sparse data. Journal of the Royal Statstcal Socety B, 58, K. J. Koehler (1986). Goodness-of-ft tests for log-lnear models n sparse contngency tables. Journal of the Amercan Statstcal Assocaton, 81, K. J. Koehler and K. Larntz (1980). An emprcal nvestgaton of goodness-of-ft statstcs for sparse multnomals. Journal of the Amercan Statstcal Assocaton, 75, P. McCullagh (1984). Tensor nataton and cumulants of polynomals. Bometra, 71, P. McCullagh (1985). On the asymptotc dstrbuton of Pearson s statstc n lnear exponental famly models.internatonal Statstcal Revew, 53, P. McCullagh (1986). The condtonal dstrbuton of goodness-of-ft statstcs for dscrete data. Journal of the Amercan Statstcal Assocaton, 81, D. F. Moore (1986). Asymptotc propertes of moments for overdspersed counts and proportons. Bometra, 73, S. R. Paul and D. Deng (000). Goodness of ft of generalzed lnear models to sparse data, Journal of the Royal Statstcal Socety B, 6, Submtted to Stat 011 Canada/IMST 011-FIM XX on Aprl 9,