Testing lack of fit in multiple regression

Transcription

1 Biometik (2), 87, 2, pp Biometik Tust Pinted in Get Bitin Testing lck of fit in multiple egession BY MARC AERTS Cente fo Sttistics, L imbugs Univesiti Centum, Univesitie Cmpus, B-359 Diepenbeek, Belgium mc.ets@luc.c.be GERDA CLAESKENS Deptment of Sttistics, Eindhoven Univesity of T echnology, Den Dolech 2, P.O. Box 513, NL -56 MB Eindhoven, T he Nethelnds g..m.cleskens@tue.nl AND JEFFREY D. HART Deptment of Sttistics, T exs A&M Univesity, College Sttion, T exs 77843, U.S.A. ht@stt.tmu.edu SUMMARY We study lck-of-fit tests bsed on othogonl seies estimtos. A common fetue of these tests is tht they e functions of scoe sttistics tht employ dt-diven model dimensions. The citei used to select the dimension e scoe-bsed vesions of AIC nd BIC. The tests cn be pplied in wide viety of settings, including both continuous nd discete dt. With two o moe covites, model sequence, i.e. pth in the ltentive models spce, hs to be chosen. Citicl points nd p-vlues of the lck-of-fit tests cn be obtined vi symptotic distibution theoy o by use of the bootstp. Dt exmples nd simultion study illustte the pplicbility of the tests. Some key wods: Additive model; Lck of fit; Multiple egession; Nonpmetic seies estimtion; Omnibus test; Penlised scoe; Scoe sttistic. 1. INTRODUCTION In ecent yes body of wok hs isen on omnibus lck-of-fit nd goodness-of-fit tests tht use dt-diven model selection citei; see Eubnk & Ht (1992), By ( 1993), Eubnk et l. (1995), Kllenbeg & Ledwin (1996), Kuchibhtl & Ht ( 1996), Aets, Cleskens & Ht (1999) nd Simonoff & Tsi (1999). In ll these woks, model selection citei ply n impotnt ole in tests of the null hypothesis tht function hs pescibed fom. These citei include cossvlidtion, Akike s infomtion citeion (AIC), the Byes infomtion citeion (BIC) nd estimtos of isk. The cuent ppe consides the poblem of testing whethe o not multiple egession function lies in pticul pmetic fmily. We popose tests tht utilise dt-diven model dimension. Rthe thn using penlised likelihood o some othe isk estimte, these tests employ penlised scoe sttistics, s does the goodness-of-fit test of Inglot, Kllenbeg & Ledwin ( 1997). The scoe-bsed tests e dvntgeous in t lest thee

2 46 M. AERTS, G. CLAESKENS AND J. D. HART wys. They e computtionlly simple, they cn be esily obustified to potect ginst model misspecifiction, nd they cn be pplied in logistic s well s continuous-esponse egession. The ppe hs two spects. Initilly we conside sevel tests in the simple egession setting, most of which e nlogues of peviously poposed tests. Ou im hee is to povide some insight into the powe of these tests. While no test is found to be eithe indmissible o unifomly most poweful, we e ble to chcteise the powe of ech test concisely, theeby mking esoned choice of tests possible. A second spect of the ppe is to conside ou scoe-bsed tests in the setting of multiple egession. In doing so, we ddess the issue of choosing n ppopite sequence of nested ltentive models. This issue is eltively tivil when testing the fit of function of one vible, but inceses quickly in complexity when the numbe of vibles inceses. We popose nd compe the powes of sevel schemes fo nesting models. The eminde of the ppe poceeds s follows. In 2 we popose sevel test sttistics in simple egession nd descibe thei lge-smple distibutions. Section 3 summises simultion study comping the powe of the tests fom 2. Section 4 poposes nd nlyses new lck-of-fit tests in multiple egession, 5 studies these tests vi simultion nd 6 povides some el-dt exmples. 2. TESTS IN SIMPLE REGRESSION Suppose we hve independent obsevtions (x,y ),...,(x,y ). The covite vlues 1 1 n n e ssumed to be fixed, which could coespond eithe to designed expeiment o to conditioning on the vlues of ndom covite. The density o pobbility mss function of ech Y hs the fom Y ~ f(y; c(x ), g), whee f is known up to the function of inteest i i i c(.) nd some k-dimensionl nuisnce pmete g (k<2). We wish to test null hypothesis of the fom H : c(.)µ{c(.; h):h=(h,...,h )µh}. (1) 1 p In this section we popose five nonpmetic tests of H tht ll hve two things in common: they e functions of scoe sttistics bsed on diffeent model dimensions, nd the model dimension is chosen by dt-bsed ule. The lge-smple distibution of ech test sttistic will be obtined unde genel conditions. Conside n ppoximto c(.; h,...,h )ofc(.) with the popety tht c(.; h, T) 1 p+ c(.; h), whee 1 nd denotes column vecto of zeos. The loglikelihood is denoted by L (g, h,...,h )= n log f{y; c(x ; h,...,h ), g} (=, 1,...). 1 p+ i i 1 p+ i=1 Fo ech, define U (g, h,...,h ) to be the column vecto of fist deivtives of L 1 p+ with espect to the pmetes g, h,...,h, nd A (g, h,...,h ) to be the expected 1 p+ 1 p+ Fishe infomtion. Define, fo 1, the scoe sttistic S=U (d@ )T{A (d@ )} 1U (d@ ), whee d@ =(d@,t ) nd d@ =(g@, h@ 1,...,h@ p ) is the mximum likelihood estimto of model pmetes ssuming tht H is tue. The null hypothesis is ejected in fvou of the ltentive H : c(x)=c(x; h 1,...,h p+ ) if the vlue of S is sufficiently lge. When H is tue nd ppopite egulity conditions hold (Ro, 1973), S conveges in disti-

3 T esting lck of fit in multiple egession 47 bution to x2 ndom vible s n2. One could lso use the obseved infomtion mtix in plce of A (d@ ), but thee is no genel consensus s to which is bette, nd often the expected infomtion leds to simple expessions. Ou inteest is in tests tht e sensitive to essentilly ny deptue fom H. The scoe test just descibed cn be expected to hve esonbly good powe when c is of the fom in H, but fo othe ltentives it need not even be consistent. Suppose we conside sequence {c(.; h,...,h ):=1, 2,...} of ppoximtos of c with the popety tht 1 p+ c(.; h,...,h ) c(.; h,...,h, ) fo ech =, 1,... nd ll llowble pmete 1 p+ 1 p+ vlues h,...,h. In othe wods, the models fo c e nested nd become incesingly 1 p+ complex s inceses. We desie futhemoe tht functions of the fom c(.; h,...,h ) 1 p+ come close nd close to spnning the spce of ll functions of inteest s 2. Fo exmple, c(.; h,...,h )=c(.; h,...,h )+ h u (.), 1 p+ 1 p p+j j j=1 whee {u (.), u (.),...}is complete fo the clss of functions tht e continuous on the 1 2 nge of the design points. In the simultions in 3 we will use othonomlised Legende polynomils nd cosine system. Associted with ech 1 is scoe sttistic S s defined bove. One expects test bsed on S, with 2, to be consistent ginst ny n n ltentive of inteest. Howeve, we now hve the poblem of multiplicity of tests. Which test should be used? One could bitily choose test, but thee is no ssunce tht the chosen test will be pticully good. This poblem is pecisely nlogous to tht of selecting the ode of Neymn (1937) smooth test. An ltentive to choosing one test is to pefom sequence of tests, with the citicl vlue of ech djusted so s to mintin n ovell level of significnce. Such n ppoch ws suggested in the goodness-of-fit context by Bickel & Ritov (1992). Hee we tke yet thid ppoch. The ide is to choose single vlue of, but to do so vi dt-bsed ule. This is in the sme spiit s the poposls of Eubnk & Ht (1992) nd Ledwin (1994). Ou method of choosing uses penlised scoe citeion, SIC(; C n )=S C n (=, 1,...), (2) whee S is defined to be nd C n is some constnt lge thn 1. We my choose to mximise SIC(; C n ) ove =,1,...,R n, with the uppe bound R n eithe fixed o tending to infinity with n. The initils SIC stnd fo scoe infomtion citeion. In contst, penlised loglikelihood citeion hs the fom 2L (g@, h@ 1,...,h@ p+ ) C n (=, 1,...), (3) whee (g@, h@ 1,...,h@ p+ ) is mximum likelihood estimto, i.e. the mximise of L. Tking C n equl to 2 nd log n in (3) yields the well-known AIC (Akike, 1974) nd BIC (Schwz, 1978) methods, espectively. When the null hypothesis is tue, the diffeence between SIC(; C n ) nd (3) is negligible. Fo this eson we shll efe to SIC(; 2) nd SIC(; log n) s scoe nlogues of AIC nd BIC. When the ltentive hypothesis is tue, the mximise of AIC is esonble estimto of the dimension of the model tht minimises Kullbck Leible isk. The sme cnnot be sid of SIC(; 2), but this is not necessily elevnt in testing context whee powe is the elevnt pefomnce mesue. The scoe citeion hs the dvntge of equiing mximum likelihood estimtos of model pmetes only unde the null hypothesis. In the goodness-of-fit context with independent nd identiclly

4 48 M. AERTS, G. CLAESKENS AND J. D. HART distibuted dt, Inglot et l. (1997) study citeion bsed only on U (.), nd not on the covince mtix A, nd they tke C =log n. n We conside the following test sttistics: S =g mx SIC(; 2); Rn S =S, =g mx SIC(; log n); b 1 Rn T ; T OS = mx S 1 R ; T mx =SIC(@ ;2). n Note mximises SIC(; log n) ove 1,..., R the thn,1,...,r. This definition b n n is used becuse of consistency popety of BIC-type ode selection citei; see Ledwin (1994). The sttistic T is stnddised vesion of S nd is nlogous to sttistic poposed by R. L. Eubnk, C.-S. Li nd S. Wng in the unpublished confeence ppe Testing lck of fit of pmetic egession models using nonpmetic egession techniques. It tuns out tht stnddising S s bove getly stbilises the null distibution of the sttistic, which hs decided effect on the powe fo T. One could similly stnddise S, but the null distibution of this sttistic is ledy quite stble nd the b stnddistion hs negligible effect. The ode selection test T is specil cse of OS one poposed by Aets et l. (1999), nd is nlogous to tht of Eubnk & Ht (1992). It is notewothy tht ejecting H fo lge vlues of T is equivlent to eject H when fo the ppopite choice of C. The null hypothesis is flse if nd only if the best n model dimension,, is t lest 1. It thus mkes sense to eject H >. Fo C = n 4 18, the esult is test with limiting size of 5. The sttistic T is the AIC-type scoe mx citeion evluted t its mximum. Simonoff & Tsi (1999) ecently poposed using the mximum of AIC s lck-of-fit sttistic. In fct, the test cn be tced to time seies poposl of Pzen (1977). Ht (1997, pp ) showed tht Pzen s test is equivlent to ejecting white-noise hypothesis when the mximum of n estimted isk citeion is sufficiently lge. We now conside the lge-smple null distibution of ech of the five sttistics poposed bove. To fcilitte the sttement of theoem, we intoduce the following nottion. Let Z,Z,... be sequence of independent nd identiclly distibuted stndd noml 1 2 ndom vibles, define V =, V =Z2+...+Z2, fo =1,2,...,nd A to be the vlue 1 of tht mximises V 2 ove =, 1,.... THEOREM 1. L et the pobbility model of 2 hold nd ssume tht A (g, h,...,h ) 1 1 p+1 is positive definite fo ech (g, h,...,h ) in the llowble set of pmete vlues. In 1 p+1 ddition we ssume the following conditions. (A) T he mtix A (d@ )/n conveges in pobbility to positive definite mtix A, whee A is the uppe left ( p+k+) (p+k+) submtix of A (=1, 2,...). +1 (B) T he mximum likelihood estimto d@ is solution of U (g, h,...,h )=. 1 p p+k (C) Fo ech 1, n DU (d@ ) conveges in distibution to (T, UT)T, whee U hs n p+k -vible noml distibution with men vecto nd covince mtix AB, whee AB 1 is the lowe ight submtix of A 1. (D) Given n incesing sequence of positive integes {R }, thee exist nonnegtive n function g, summble nonnegtive sequence {p :1} nd nonnegtive sequence {d } such tht n

5 T esting lck of fit in multiple egession 49 (i) fo ll n suyciently lge, p(s>) g()p +d, fo ll 1 5, =1,...,R ; n n (ii) lim g(y)=; nd y2 (iii) lim R d =. n2 n n It follows tht, if the null hypothesis (1) is tue nd n2, the sttistics S,S,T,T b OS nd T convege in distibution to V,V,(V A)/mx(1, A1/2), mx (V /) nd V 2A, mx A 1 A 1 A espectively. Poof. Fo the sttistics S,T,T OS nd T mx, we will pove the esult only fo T mx ; the othe thee poofs e vey simil. Define We hve nd hence c= mx 1 (V 2), c = mx (V 2), c = mx (S 2), R R,n 1 R 1 R b = mx (S 2). R,n R< R n p(t mx >x)=p(c R,n >x)+p(b R,n >x) p(c R,n >x ] b R,n >x), p(t mx >x) p(c>x) p(c R,n >x) p(c R >x) + p(c R >x) p(c>x) +p(b R,n >x). We must show tht, given ny e>, p(t mx >x) p(c>x) <e fo ll n sufficiently lge. By definition, p(c R >x) inceses to p(c>x) s R2, nd so thee exists R9 such tht p(c R >x) p(c>x) <e/3 fo ll RR9. Now, p(b >x)=p A pr n {S 2>x} B R n p(s>2+x). R,n =R+1 =R+1 If R2 x, the vey lst quntity is bounded by p{s R+1 >1 5(R+1)}+...+p(S Rn >1 5R n ), nd by ssumption (D) R n p(s>1 5) g(1 5) R n p +R d. n n =R+1 =R+1 Agin by (D), thee exists RB such tht the lst quntity is less thn e/3 fo ll RRB nd ll n sufficiently lge. Defining R =mx(r9,rb,2 x ) nd combining pevious steps, we mx hve p(c >x) p(c>x) +p(b >x)<2e/3 fo ll n sufficiently lge. Rmx Rmx,n The only emining tsk is to show tht p(c Rmx,n >x) p(c >x) <e/3 fo ll R n sufficiently lge. Fo ny R, c is continuous function of mx n DU (d@ ) nd the R,n R R elements of A (d@ )/n. This is consequence of the following fcts: R R (I) fo 1 R, A (d@ )/n is submtix of A (d@ )/n; R R (II) the vecto n DU (d@ ) is subvecto of n DU (d@ ), fo =1,...,R; R R (III) the quntity mx (s 2) is continuous function of s,...,s. 1 R 1 R By conditions (A), (B) nd (C), {n DU (d@ ), A (d@ )/n} conveges in distibution to R R R R {(T, UT)T, A }sn2. Since c is continuous function of {n DU (d@ ), A (d@ )/n}, p+k R R R,n R R R R it follows tht c conveges in distibution to the sme function of {(T, UT)T, A }, R,n p+k R R which is equl in distibution to c. R We now conside the distibution of S. If we use ssumptions (A) nd (B) nd the fct b

6 41 M. AERTS, G. CLAESKENS AND J. D. HART tht p(s >x)=p(s >x =1)+O{p(1<@ R 1 b b n the esult is poven by showing tht p(@ =1)1 s n2. We hve b p(@ >1) R n p{s S >log n( 1)} R n p{s>log n( 1)}, b 1 =2 =2 with the lst inequlity following fom the fct tht, by ssumption, S is lmost suely 1 nonnegtive. If we pply (D), the lst quntity is bounded by g{(log n)/2} R n p +R d, n n =2 which tends to by ssumption (D). % Some emks e in ode concening the conditions in Theoem 1. Conditions (A) (C) e stndd in the symptotic theoy of mximum likelihood estimtion. Condition (D) is sot of lge devition inequlity fo the scoe sttistic S. To motivte (D), conside testing the null hypothesis H* : c const, in model whee x =(i 1)/n, fo i=1,...,n. i 2 When we test H*, if seies ppoximtos of the fom c(x; h,...,h )=h +2 h cos(pjx) j+1 j=1 e used nd Y,...,Y e modelled s independent N(c(x ), s2) ndom vibles, then 1 n i S tkes the fom S= 2nh@2 j+1, s@2 j=1 in which h@ = 1 j+1 n n Y cos(pjx ) (j=, 1, 2,...), i i i=1 nd s@2=wn (Y i=1 i Y9 )2/n. Let s2 denote the vince Y unde the null hypothesis. Then, i fo 1 5, 2nh@2 j+1 > 5 s2 6 B +p AK s@2 1 K s2 1 6B. p(s>) p A j=1 When ndom vibles Y,...,Y e independent nd identiclly distibuted N(, s2), 1 n 2nh@2 /s2,..., 2nh@2 /s2 e independent nd identiclly distibuted x2 ndom vibles, 2 n 1 implying tht p A j=1 fo ny numbe tµ(, 1). Tking t=1, we hve 2 8 p A j=1 2nh@2 j+1 > 5 s2 2nh@2 j+1 > 5 s2 6 B (1 2t) /2 exp A 5t 6 B 6 B exp C 2q log A 3 4B +1/48 D =exp[ { 15 log(2/ 3)}] exp{ (5/48 15)} exp[ { 15 log(2/ 3)}] exp{ (5/48 15)}.

7 T esting lck of fit in multiple egession 411 In the nottion of condition (D), we my now tke p =exp[ { 15 log(2/ 3)}] nd g() =exp{ (5/48 15)}. Finlly, lge devition fomul fo s@ 2 yields p( s@2/s2 1 1) d =O(n 2). 6 n Note tht in the peceding exmple we my tke R =n nd still obtin R d. Hence, n n n the distibution theoy in Theoem 1 holds in this cse without plcing n bity uppe bound on the numbe of Fouie coefficients used. One cn similly check condition (D) in othe settings by using lge devition fomule fo d@ nd U (d )T{A (d )} 1U (d ). Note, howeve, tht in genel it my be necessy to tke R =o(n) in ode to obtin the limit distibutions of Theoem 1. n 3. SIMULATIONS IN SIMPLE REGRESSION MODELS To illustte the diffeent tests nd len bout thei powe chcteistics, we pefomed simultion study using two membes of the genelised line models fmily. In genel, the likelihood function fo genelised line model cn be witten s f(y ; c(x ), g)=exp([y c(x ) b{c(x )}]/(g)+c(y, g)), i i i i i i whee (.), b(.) nd c(.) e known functions, c(.) is the known ntul pmete of inteest nd g is n unknown dispesion pmete; see e.g. McCullgh & Nelde (1989, p. 28). As ppoximtos to c(.) we tke c(x; h,...,h )= p h c (x)+ h u (x) (=1, 2,...). 1 p+ j j p+j j j=1 j=1 It is undestood tht the functions u (.) e not line combintions of ny of the functions j c (.) ( j=1,...,p). j Let v nd w be ny two functions in B ={c,...,c,u,...,u}, nd suppose B is 1 p 1 othonoml in the sense tht 1 n n b {c@(x )}v(x )w(x )= (v w), i i i i=1 q1 (v / w). Then the scoe sttistic is given by whee S= n@2 j (g@ ), (4) = 1 j n n [Y b {c(x ; h@,...,h@ )}]u (x ). i i 1 p j i i=1 Expession (4) hs essentilly the sme fom s the sttistic of Neymn s clssicl smooth test; see Ht (1997, p. 14). To compe the five tests, we geneted continuous Y ~N{c(x ), g} nd biny i i Y ~Be(1/[1+exp{ c(x )}]), whee, in both cses, x =(i 1)/n, fo i=1,...,n.we focus i i i 2 on testing fo no effect, tht is c(x) is constnt, using the nomlised Legende polynomils u (.) L (.) on the intevl [1/(2n), 1 1/(2n)], o the cosine bsis u (.) 2 cos(pj.), fo j j j j=1,...,2.unless othewise stted, S-Plus is used fo the clcultions. Fom simultion bsed on 3 eplictions, we obtined citicl points, fo levels = 1, 5, 1, of the lge-smple distibution of ech test sttistic, except fo S, b

8 412 M. AERTS, G. CLAESKENS AND J. D. HART which is symptoticlly x2. Simulted type I eo pobbilities, not shown, fo simultion of size 5, with c(x), smple size 1 nd the Legende polynomil bsis, 1 show tht the esults fo Gussin nd binomil dt e quite simil to ech othe. In ech cse, the tue level of S is too high nd, t the 1% level, the level of S is too low. b The type I eo pobbilities e close to thei nominl vlues fo binomil dt, pesumbly s such dt equie no estimte of nuisnce pmete. The sme conclusions hold when cosine bsis is used. To exmine powe we conside two kinds of ltentive: c(x)=cos(pm x) nd c(x)=l (x). Fo the noml esponse Y, we took g=1 (esp. g= 1) fo the cosine m (esp. polynomil) ltentive. Fo m =1,2,...,1,these ltentive models e odeed fom low to high fequency. It is woth emking tht, when the ltentive hs combintion of low- nd high-fequency tems, the low-fequency tems will tend to dictte powe behviou. In othe wods, tests tht hve good powe fo low-fequency ltentives will pefom well in cses whee the ltentive is mixtue of low- nd high-fequency. In Fig. 1(), esults e shown fo 1 dtsets geneted fom Gussin model with polynomil ltentive. Figue 1( b) shows the simultion esults fo logistic egession model unde the cosine ltentives. In both cses, the Legende polynomil bsis is used, the smple size n equls 1 nd the level of significnce is equl to 5. Fo ll tests, citicl points wee clculted using 5 simulted dtsets unde the null hypothesis. In this wy, the powe esults fo the vious tests e diectly compble. () (b) S T mx T S b T OS m m Fig. 1. Simulted powe cuves fo () Gussin model nd ( b) logistic egession. The ode selection test T OS is vey good t the lowest fequency, but going to highe fequencies its powe deceses pidly. The BIC-bsed test S b shows bout the sme behviou. It hs good powe t low-fequency ltentives, but, since it hs lge penlty fo m lge, it hs lmost no powe t high fequency. At low-fequency ltentives, both T nd T mx impove upon the unstnddised S test, but, when m is t lest 4, S hs the lgest powe. On the bsis of the study summised by Fig. 1, we hve pefeence fo T mx nd T since both do vey well fo the low-fequency cses nd still hve esonble powe t high fequencies. Howeve, moe studies e equied befoe mking ny finl ecommendtions. In pticul, since low-fequency ltentives seem to pedominte in pctice, thoough study of diffeent types of low-fequency ltentive should be conducted. The conclusions dwn fom Fig. 1 hold fo ech of the 8 situtions, i.e. Gussin o

9 T esting lck of fit in multiple egession 413 logistic, cosine o polynomil bsis nd both types of ltentive. Fo the ltentives in this simultion study, thee is lmost no loss in powe when the wong bsis is used. In the simple egession setting, Yngimoto & Yngimoto (1987), By & Htign (199), Ht & Wehly (1992), Fn (1996) nd Lee & Ht (1998) hve poposed othe lck-of-fit tests utilising dt-diven model selectos. While it is of inteest to compe these tests with those consideed in this section, we do not do so hee since ou min inteest is in the multiple egession cse. The inteested ede is efeed to Lee & Ht (1998) fo modest compison of some tests of Fn (1996), Lee & Ht (1998) nd specil cse of S. 4. MULTIPLE REGRESSION 4 1. Omnibus tests in models with two covites To illustte how the methods of 2 my be genelised to multiple egession, we conside fist the eltively simple cse of two covites. Let c be n unknown function of the covites x nd x. We wish to test the null hypothesis 1 2 H : cµ{c(.,.; h):hµh}. (5) If we use cosine seies to epesent c, n ltentive model my be expessed s c(x,x )=c(x,x ; h)+ cos(pjx ) cos(pkx ). (6) jk 1 2 (j,k)µl The definition of the index set L will, in genel, depend on the specific model unde the null hypothesis. Fo exmple, if we wish to test the hypothesis tht c(x,x ) hs the fom 1 2 h +h cos(px )+h cos(px ), nd we use cosine bsis, then clely (1, ) nd (, 1) should not be included in L. Fo ese of nottion, we will now ssume tht the function c(x,x, h) is constnt, but genelistions e stightfowd. Unde the no-effect null 1 2 hypothesis, L is subset of {( j, k) : j, k<n, j+k>}. By nlogy with (2), we define the citeion SIC(L; C )=S C N(L), whee S is scoe sttistic nd N(L) denotes n L n L the numbe of elements in L. To cy out test s in 2 we mximise SIC(L; C ) ove some collection of subsets n L, L,...,L. It is impotnt tht this collection coespond to nested models, othewise 1 2 mn the distibutions of the esultnt test sttistics will, in genel, depend on pmetes of the null model, even when n2. We thus insist tht L 5L 5...5L, nd we cll 1 2 mn such collection of sets model sequence. The only poblem now is deciding on how to choose model sequence. One impotnt considetion is whethe o not given sequence will led to consistent test. To ensue consistency ginst vitully ny ltentive to H, we sk tht N(L )2 in such wy tht, fo ech (j,k)n(, ) ( j, k), mn (j,k)is in L fo ll n sufficiently lge. The choice of model sequence is futhe simplified mn if we conside only tests tht plce equl emphsis on the two covites. In othe wods, we could insist tht tems of the fom cos(pjx ) cos(pkx ) nd cos(pkx ) cos(pjx ) ente the model simultneously. We shll conside the fou choices of model sequence shown in Fig. 2. Appently, these sequences, nd slight vitions theeof, e the only possibilities if one wishes to tet the covites symmeticlly. In Fig. 2 these fou sequences e gphiclly epesented by the numbe of the step in which the bsis elements ente the model. The numbes on the xes e the indices of the bsis elements. One vition on these sequences is to dd ll min effects tems befoe intection tems e enteed. Fo the model sequence in Fig. 2() the numbe of model pmetes gows quickly,

10 414 M. AERTS, G. CLAESKENS AND J. D. HART Index in x 2 diection () Index in x 1 diection Index in x 2 diection (b) Index in x 1 diection Index in x 2 diection (c) Index in x 1 diection Index in x 2 diection (d) Index in x 1 diection Fig. 2. Fou exmples of model sequences in two dimensions. with 2j+1 tems dded to the pevious model t step j. This entils tht tests bsed on the sequence in Fig. 2() will, in genel, hve poo powe popeties. This poblem is lessened in the model sequence in Fig. 2(b), whee only j+1 tems e dded t the jth step. Even moe psimonious sequences e shown in Figs 2(c) nd (d). In the scheme in Fig. 2(c), min effects coesponding to fequency j ente the model t step 2j 1 nd j intection tems ente t step 2j. The sequence in Fig. 2(d) hs the ovell smllest step sizes, with t most two new tems dded t ech step. Intuitively, we hve slight pefeence fo the fouth sequence ove the othe thee since it dds t most two new tems to the model, whees the fist two dd n eve-incesing numbe t ech step. Othe omnibus tests e cetinly possible; Kllenbeg & Ledwin (1999) nd Bogdn (1999), fo exmple, popose model sequences fo tests of independence nd goodness of fit, espectively. Fo the ske of simplicity, though, we will estict ou ttention to sequences in Figs 2() (d). The lge-smple distibution of the sttistics s given in Theoem 1 cn be genelised to the multiple covite setting by chnging the definition of V nd A s follows. Fist define L =B, N j =N(L j ) N(L j 1 ), fo j=1,2,...,nd let Z jk, fo k=1,...,n j nd j=1, 2,..., be independent nd identiclly distibuted stndd noml ndom vibles. Now we define V =, V = N j Z2 (=1, 2,...), jk j=1 k=1 nd A=g mx{v 2N(L ):=, 1,...}. In the cse of thee o moe covites, it is woth pointing out the cost of n omnibus test s the numbe, d, of covites inceses. Regdless of wht d is, the mximum

11 T esting lck of fit in multiple egession 415 numbe of pmetes we should conside in model is O(n), nd fo simplicity let us just sy n. Fo n omnibus test tht plces the sme emphsis on ll d covites, this entils, oughly speking, tht R not exceed n1/d. Clely then, the bility of n omnibus test to n detect highe fequency ltentives quickly wnes s the dimension of the x-spce inceses T ests in dditive models Fo high-dimensionl covite vecto, the omnibus tests bsed on the fou model sequences of 4 1 become less ttctive. Howeve, if we cn ssume tht n dditive model fits the dt well, the cuse of dimensionlity cn be cicumvented. Unde this ssumption, n ltentive to the null model (5) cn, fo two covites nd the cosine bsis, be witten s c(x,x )=c(x,x ; h)+ k cos(pjx )+ l b cos(pjx ), j 1 j 2 j=1 j=1 whee k k nd l l fo =2, 3,.... Agin, thee e numbe of wys to constuct such sequence of nested models. If we insist tht k =l nd let k incese by t ech step, then L ={(1,),(,1),(2,),(,2),...,( j, ), (, j)}. We will efe to test j bsed on this model sequence s digonl test, since the pth {(k,l):1} coesponding to this test poceeds long the digonl {(k,k):1}. At ech step in digonl test two tems e dded to the pevious model. The only effect this hs on the symptotic distibution theoy of 2 is tht now V =(Z2+Z2)+...+(Z2 +Z2 ). This ppoch hs n obvious extension to the cse of moe thn two covites T he mx tests in models with ny numbe of covites Fist we explin the ide in two-covite models. Fo exmple, conside s n ltentive model c(x,x )=c(x,x ; h)+ cos(pjx ) (k=1 o 2), j k jµl whee only one of the covites is used to distinguish fom the null model. Of couse, othe bsis functions cn be used. Unless one hs pio belief tht only x would cuse k the lck of fit fom the null model, this ppoch is not ecommended. Insted we could tke s ou test sttistic the mximum of the vlues obtined by looking t ech covite diection septely; we efe to this s mx test. The level of such test cn be contolled by ppliction of Bonfeoni s inequlity o by bootstp method. The sme ide cn be used to extend the domin of ppliction to models with moe thn two covites. Fo model with d covites we conside, fo ech pi (, s) septely, ltentives c(x,...,x )=c(x,...,x ; h)+ cos(pjx ) cos(pkx ). 1 d 1 d jk s (j,k)µl Fo pticul choice of (, s), we my pefom ny of the tests fo two-covite models, e.g. one of the omnibus tests following the pth in Fig. 2(d). Next, we tke the mximum of ll d(d 1)/2 test sttistics. If the numbe of covites d is lge, using Bonfeoni s inequlity will esult in vey consevtive test, in which cse bootstp pocedue might be pefeble.

12 416 M. AERTS, G. CLAESKENS AND J. D. HART 4 4. T ests fo moe specific ltentives The test in 4 2 is not necessily consistent unless the ltentive is dditive. This dditivity ssumption cn be tested by modifiction of the omnibus tests. Conside, fo d=2, the null hypothesis H : c(x,x )=c (x ; h )+c (x ; h ). A genel dditive model cn be estimted by Fouie seies, but now the vecto h=(h, h ) is infinite dimensionl. 1 2 A possible ppoch to this poblem is fist to estimte, by use of model selection citeion, optiml odes k nd k fo seies estimtes of c (.; h ) nd c (.; h ). Then, the null model bsed on these estimted odes is extended with intection tems ccoding to one of the pths in Fig. 2. It is not immeditely cle how the dditionl model selection step will ffect the testing pocedue. This is n inteesting question tht will be ddessed in futue esech. Altentively, one could pefom tests of dditivity fo lge, fixed vlue of k=k = 1 k. A sensitivity nlysis on k might show to wht extent the choice of k nd k influences the finl conclusion. In 6 3 this method will be used on het-ttck dt tht wee used by Hstie & Tibshini (199) to illustte techniques on genelised dditive models. Fo moe thn two covites, the sttegy in 4 3 cn be followed. Finlly, we lso mention goodness-of-link o single-index test. In this cse the hypothesised model (5) is contsted with ltentive models of the fom c(x,x )=c(x,x ; h)+ u {c(x,x ; h)} j j 1 2 jµl Fo genelised line models, this povides wy of testing the dequcy of the link function. It is n ltentive to methods descibed by Collett (1991, 5 3) nd Bown (1982). 5. SIMULATIONS IN MULTIPLE REGRESSION MODELS In modest simultion study we comped the five tests S,S,T,T nd T.We b OS mx geneted independent noml esponse dt Y ~N{c(X,X ), g}, whee the covites i 1i 2i X nd X e independent nd identiclly distibuted fom unifom distibution on 1i 2i (, 1), fo i=1,..., 1, nd test the no-effect null hypothesis using the nomlised Legende polynomils u (.)=L (.). We ppoximted citicl points of the lge-smple j j tests by using 3 eplictions nd used Bonfeoni s inequlity fo the mx test. In Tble 1, bsed on 5 simulted dtsets unde the null hypothesis, type I eo pobbilities e estimted. As fo the one-covite models, the levels fo the S tests e b somewht too lge, nd t the nominl level of 1% the level of S is too smll. Thee is no substntil diffeence fo model sequences (c) nd (d). To study the powe chcteistics of the diffeent tests nd model sequences, we fist consideed the ltentive models c(x 1,x 2 )=2L m (x 1 )+2L n (x 2 ), g=1. (7) A lge vlue of m o n coesponds to highe fequency fo the coesponding covite. The vious tests wee pefomed fo sevel choices of m nd n. In the lefthnd gphs of Fig. 3, Figs 3(), (c), (e) nd (g), we used the dditive pth nd mx test fom 4 2, nd in the ight-hnd gphs, Figs 3(b), (d), (f ) nd (h), the intection pths shown in Figs 2(c) nd (d) wee used, lthough no intection ws pesent in the ltentive models. To obtin compble powe esults, citicl points wee obtined by simultion, bsed

13 T esting lck of fit in multiple egession Tble 1. Simulted type I eo pobbilities fothe nullhypothesis of no-evect in Gussin egession model with two covites using the nomlised L egende polynomils Digonl pth Mx test Sttistic = 1 = 5 = 1 = 1 = 5 = 1 S b S T T OS T mx Sequence in Fig. 2(c) Sequence in Fig. 2(d) Sttistic = 1 = 5 = 1 = 1 = 5 = 1 S b S T T OS T mx 417 on 5 dtsets geneted unde the null hypothesis. The level of significnce is equl to 5. The simulted powes wee bsed on 1 eplictions. In Figs 3(), (b), (c) nd (d), we chose m =n, in which cse both covites hve the sme fequency. The digonl pth is designed to be best fo this kind of ltentive. Remkbly, except fo the mx test, the cuves e odeed in the sme wy s in the simultion esults fo the univite cse. Whees the powe cuves fo S nd S b coss t m =n =2 fo the digonl nd omnibus test, this cossing is t bout m =n =4 fo the mx test. This is pobbly elted to the fct tht the mx test is bsed on two univite tests, fo which the ltentive model sequences incese by only one tem t time. A compison of the left- nd ighthnd gphs shows tht thee is some loss in powe when the intection pths e used. In Figs 3(e), (f ), (g) nd (h), the fequency of the fist covite is fixed t 1, nd the fequency of the second covite vies between 1 nd 6. In this setting, the loss in powe of the omnibus tests is usully negligible. Fo ll powe cuves thee seems to be chnge point t fequency n =2, t which point the powe deceses vey slowly. The explntion fo this phenomenon might be the dominnce of the lowest fequency tem, coesponding to m =1, in ll ltentive models (n =1,...,6).The S test hs emkbly low powe. Hee, stnddising hs beneficil effect on S fo ll ltentives. Finlly, fo ll cses, we obseve tht thee is no substntil diffeence between model sequences in Figs 2(c) nd (d). In Fig. 4 simultion esults e shown fo the intection type ltentive models c(x 1,x 2 )=4L m (x 1 )L n (x 2 ), g= 1. (8) A lge vlue of m o n coesponds to highe-ode intection. Note tht some of the cses consideed e quite exteme; fo exmple, m =n =3 coesponds to poduct of two thid-degee othogonlised polynomils. As expected, fo this kind of ltentive, the dditive type tests hve lmost no powe. Fo smll vlues of m nd n, the powe of the intection tests is eltively high; ecll tht we e testing fo no effect. As befoe, but now even moe ponounced, thee e two goups of tests, with the AIC-bsed tests being pefeed. Agin, the tests fo pths in Figs 2(c) nd (d) e lmost indistinguishble. We now illustte the tests of 4 3 using egession model with fou covites.

14 418 M. AERTS, G. CLAESKENS AND J. D. HART 8 6 () Digonl pth, m =n S T mx T S b T OS 8 6 (b) Sequence (c), m =n m =n m =n 8 (c) Mx test, m =n 8 (d) Sequence (d), m =n m =n m =n 8 (e) Digonl pth, m =1 8 (f) Sequence (c), m = n n 8 (g) Mx test, m =1 8 (h) Sequence (d), m = n n Fig. 3. Simulted powe cuves when the tue model is dditive s given in (7), fo sevel choices of m nd n ; () nd (e), digonl pth; (c) nd (g), mx test; (b) nd (f ), intection pth shown in Fig. 2(c); (d) nd (h), intection pth shown in Fig. 2(d).

15 T esting lck of fit in multiple egession () Digonl pth, m =n S T mx T S b T OS (b) Sequence (c), m =n m =n m =n 3 (c) Mx test, m =n 1 (d) Sequence (d), m =n m =n m =n (e) Digonl pth, m = (f) Sequence (c), m = n n 3 (g) Mx test, m =1 1 (h) Sequence (d), m = n n Fig. 4. Simulted powe cuves when the tue model hs intection stuctue s given in (8), fo sevel choices of m nd n ; () nd (e), digonl pth; (c) nd (g), mx test; (b) nd (f ), intection pth shown in Fig. 2(c); (d) nd ( h), intection pth shown in Fig. 2(d).

16 42 M. AERTS, G. CLAESKENS AND J. D. HART The model is s follows: the conditionl distibution of Y given the covite vlues x i ji (j=1,...,4)is N{c(x,...,x ), s2}, whee the covite vlues wee dwn independently fom Un(, 1) distibution nd kept fixed thoughout ll simulted dtsets. The 1i 4i smple size is 5. We will conside tests of the hypothesis H : c h +h x +...+h x Test sttistics using polynomil bsis e clculted using the softwe GAUSS. The pmetic bootstp, with 1 esmples, ws used to obtin p-vlues. The ltentives we consideed wee 2x +x 5x +x plus ech of the following functions, which live in spces of incesingly highe dimension: A, B 5x4 1, C 1 8x 1 x 2, C 2 8x 1 x 2 +5 log(x 2 +1), D 1 8x 1 x 2 +5 log(x 4 +1), D 2 5x 1 x 2 x 3, E 5x x 1 x x 1 x 2 x log(x 4 +1), nd we took s= 2. In Tble 2() we show the esults of the mx test, which looks in ll possible two-dimensionl diections, using the pth in Fig. 2(d) in ech diection. It is inteesting to compe the mx test with the one tht looks only in the diections of the vibles in the ltentive model, Tble 2( b). The ltte ones e efeed to s ocle tests, since they use infomtion tht nomlly only n ocle would know. Tble 2. Simulted powes of tests in Gussin egession model with fou covites () Mx test A B C 1 C 2 D 1 D 2 E = 5 S b T T OS = 1 S b T T OS (b) Ocle test B C 1 C 2 D 1 D 2 = 5 S b T T OS = 1 S b T T OS We only show the esults fo the sttistics S b,t nd T OS. Results fo T mx wee nely identicl to those of T, except fo ltentive B whee its powe ws 333 nd 157 t the 5% nd 1% levels, espectively. The simulted powe of S ws eveywhee the lowest, except fo B whee its powe ws 577 nd 161. Especilly t the 1% level, powes fo S wee extemely low, becoming s smll s 23 fo D 2. When the devition ws onedimensionl, B, the mx test, s expected, hd considebly lowe powe thn the ocle. The supeioity of the ocle dops the dmticlly fo the two-dimensionl ltentives C 1 nd C 2. At the 5% level, the loss in powe is fily smll fo test sttistics S b,t nd T mx, but, s befoe, the lge fo S. Fo the thee-dimensionl ltentives D 1 nd D 2, esults fo the mx tests e compble to those of the ocle tests.

17 T esting lck of fit in multiple egession EXAMPLES 6 1. T he poject on petem nd smll-fo-gesttionl ge dt This nd the thid exmple involve biny esponse dt with two covites, whees the second involves Gussin model nd thee covites. The poject on petem nd smll-fo-gesttionl ge infnts collected infomtion on 1338 infnts bon in the Nethelnds in 1983 nd hving gesttionl ge, x, less thn 1 32 weeks nd/o bithweight, x, less thn 15 gmmes; see Veloove & Vewey (1988) 2 fo moe detils. The outcome of inteest hee concens the sitution fte two yes. The biny vible Y is 1 if n infnt hs died within two yes fte bith o suvived with mjo hndicp, nd othewise. Afte deletion of obsevtions with missing dt, 131 infnts emin in the dtset. Le Cessie & vn Houwelingen (1991, 1993) exmined these dt to illustte lck-offit test bsed on weighted sum of kenel smoothed stnddised esiduls. Thei test filed to eject the null hypothesis of logistic model hving line nd qudtic tems in both covites x nd x. Likewise, likelihood tio test showed tht neithe one of 1 2 the thid-ode tems no fist-ode intection tem contibutes significntly to the model. Tble 3 shows the vlues of the omnibus, Fig. 2(d) sequence, test sttistics. The Legende polynomils L (x ) nd L (x ) wee used to epesent the models. With k 1 l 2 L (x) 1, the null hypothesis cn be witten s H : logit{e(y )}= 2 h L (x )+ 2 h L (x ). k k 1 l l 2 k= l=1 Fo the ltentive models we included min effects up to the sixth ode togethe with ll intection tems L (x )L (x ) such tht 2 k+l 6. k 1 l 2 Tble 3. Results of testing H : model is qudtic in x 1 nd x 2 fo the poject on petem nd smll-fo-gesttionl ge infnts in the Nethelnds, 1983 dt S b S T T OS b Vlue of sttistic P P B Tble 3 shows p-vlues bsed on the symptotic distibution, P, nd on the bootstp, 2 P. Using pmetic bootstp, we geneted 999 eplictions unde H, with the B pmetes tken to be the null mximum likelihood estimtes. No dditive o single index test, not shown, is significnt. All omnibus tests except S indicte some evidence b ginst H. The diffeent behviou of S is consequence of its lge penlty constnt, b C =log n j n The test esults suggest the ddition of cetin intections to the null model. The vlue =2 coesponds to the intection tems (Fig. 2(d)) L (x )L (x ), L (x )L (x ) nd/o L (x )L (x ). Inspied by these findings we investigted numeous new models nd fo ech one computed the clssicl model selection citeion (3) with C =2, tht is n AIC, nd C =log n, tht is BIC. The BIC citeion selected the null model, which gin is n consequence of the lge penlty. The model selected by AIC is logit{e(y )}= 5 h L (x )+ 2 h L (x )+h L (x )L (x )+h L (x )L (x ), k k 1 l l k= l=1 (9)

18 422 M. AERTS, G. CLAESKENS AND J. D. HART which cn be ewitten in tems of x,...,x5,x,x2,xx nd x x2. Both goups of highe-ode min effects x3,x4,x5 nd intections x x,x x2 e significnt t the 5% level T he penuts dt These dt come fom n expeiment concening device fo utomticlly shelling penuts ( Dickens & Mson, 1962). The dt consist of the esponse vible Y, gmmes of unshelled penuts, out of one kilogmme, nd the covites numbe of stokes pe minutes, x 1, nd length of stoke, x 2, nd b gid spcing, x 3, both in inches. Thee e n=67 obsevtions. The null hypothesis of inteest is qudtic polynomil esponse sufce; see Feund & Littell (1991, 5 6). Using the mx ppoch s descibed in 4 3, we dded 15 tems to this null model ccoding to the model sequence in Fig. 2(d) nd used polynomil bsis functions. Adding moe tems gve identicl conclusions. Bsed on 1 bootstp eplictes, the p-vlues of S b,t nd T OS wee, espectively, 34, 84 nd 21. Since thee e eplicted obsevtions, the clssicl F-test fo lck of fit my lso be pplied. Using SAS pocedue RSREG nd the option lckfit, one gets p-vlue of 1. This lso indictes tht thee is some evidence ginst the null hypothesis of qudtic esponse sufce T he het-ttck dt The tests fo dditivity fom 4 4 e illustted on het-ttck dt tht e descibed in Hstie & Tibshini (199). The biny esponse vible Y indictes the pesence of myocdil infction t the time of suvey fo 463 white mles between 15 nd 64 yes of ge. We will use the sme covites s in Hstie & Tibshini s dditive model, nmely x 1, the systolic blood pessue nd x 2, cholesteol tio. The hypothesised null model is logit{e(y )}=h + k 1 h u (x )+ k 2 h u (x ), (1) 1j 1j 1 2j 2j 2 j=1 j=1 whee we tke u (.)=u (.)=L (.), nd k nd k e unknown nd possibly equl 1j 2j j 1 2 to infinity. Insted of using dt-diven choice of the unknown odes k nd k, Tble 4 shows 1 2 the vlues of ll test sttistics fo k=k =k in {4, 5, 7, 8}. This shows how sensitive the 1 2 test sttistics e fo diffeent choices of k nd k. We dpted the model sequence in 1 2 Fig. 2(d) by excluding the tems on the xes. Fo exmple, when k=5, the sets of tems used, in thei ode of enty, wee {x x }, {x2x,x x2}, {x2x2}, {x3x,x x3}, {x3x2,x2x3}, {x4x,x x4}, Tble 4 gives the esults fo Legende nd cosine bsis functions. It lso shows, fo ech k, vlues nd p-vlues P obtined by the pmetic bootstp using 999 b B eplictions. Fo Legende polynomils, most p-vlues indicte tht the dditivity hypothesis cn be ejected. The vlues of the test sttistics nd thei coesponding p-vlues e compble fo diffeent vlues of k. At lest fo this exmple, the choice of the degee of the dditive null model is not elly cucil. A subject of ongoing esech is to exmine in moe genelity the impotnce of this choice. If we tun to the cosine bsis, T nd S fil to OS b eject dditivity. The othe tests e usully significnt except when k=7 o 8. Tble 4 indictes tht the choice of bsis cn be impotnt.

19 T esting lck of fit in multiple egession 423 Tble 4: Het-ttck dt. T est esults fo the dditivity hypothesis k S b S T T OS b Polynomil bsis Vlue P B Vlue P B Vlue P B Vlue P B Cosine bsis Vlue P B Vlue P B Vlue P B Vlue P B CONCLUDING REMARKS Diffeent sttistics fo testing lck of fit in multiple egession hve been poposed. Bsed on simultions, we pefe T nd T in combintion with the omnibus pth in mx Fig. 2(d). All tests e bsed on penlised scoe citeion, which only equies computtion of null pmete estimtes. Altentively, penlised loglikelihood citeion could be used. A detiled study of the pos nd cons of ech citeion seems to be wothwhile. Besides being computtionlly simple, the scoe citeion hs the dvntge tht it cn be obustified ginst likelihood misspecifiction. A obust vesion of T is studied in OS Aets et l. (1999). Similly, obust vesions of the othe sttistics poposed in the cuent ppe cn be constucted. Ou cuent esech is exmining ll the test sttistics in the even moe genel setting of estimting equtions. ACKNOWLEDGEMENT The esech of G. Cleskens ws suppoted by Resech Assistnt gnt of the Fund fo Scientific Resech Flndes, Belgium FWO nd conducted t the Cente fo Sttistics, Limbugs Univesiti Centum, J. D. Ht s esech ws conducted while visiting Limbugs Univesiti Centum. Pofesso Ht is most gteful to Limbugs Univesiti Centum fo thei geneous suppot duing his visit. The uthos gtefully cknowledge the helpful comments nd suggestions of the edito nd two efeees. REFERENCES AERTS, M., CLAESKENS, G. & HART, J. D. (1999). Testing the fit of pmetic function. J. Am. Sttist. Assoc. 94,

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