ANOVA models for Brownan moon Gordon Hazen Danel Apley Neehar Parkh Deparmen of Indusral Engneerng and Managemen Scences Norhwesern Unversy Transplan Oucomes Research Collaborave Norhwesern Unversy December 013 Correspondng Auhor: Gordon Hazen IEMS Deparmen, McCormck School of Engneerng Norhwesern Unversy Evanson IL 6008-3119 Runnng Head: ANOVA for Brownan Moon 1
Absrac We nvesgae longudnal models havng Brownan-moon covarance srucure. We show ha any such model can be vewed as arsng from a relaed meless classcal lnear model where sample szes correspond o longudnal observaon mes. Ths relaonshp s of praccal mpac when here are closed-form ANOVA ables for he relaed classcal model. Such ables can be drecly ransformed no he analogous ables for he orgnal longudnal model. We n parcular provde complee resuls for one-way fxed and random effecs ANOVA on he drf parameer n Brownan moon, and llusrae s use n esmang heerogeney n umor growh raes. KEY WORDS: LONGITUDINAL MODELS; BROWNIAN MOTION; ANALYSIS OF VARIANCE; ANOVA; RANDOM EFFECTS; STOCHASTIC HETEROGENEOUS TUMOR GROWTH. 1. Inroducon The sascal analyss of longudnal daa has a long hsory (e.g., (Dggle, 1988; Dggle, Heagery, Lang, & Zeger, 00; Snger & Wlle, 003; Verbeke & Molenberghs, 009)). Consder he suaon n whch repeaed measuremens x 0, x1,, x on ndvdual are aken a mes 0 0 1 J lnear model wh parameers,,, 1 : j 0 1 1 M M j j M J. We consder here a parcular x x ( u u ) j ~ normal(0, j ). (1) where u,, 1 u are specfed consans or observed covaraes for ndvdual. We allow he ndex o be M lexcographcally nesed, so ha (1) ncludes, among many possble examples, he addvely separable model wh = (g,h): x x 0 ( ) ghj gh g h ghj ghj n whch ( 1,, M ) = (, 1,, G, 1,, H). In (1) we resrc ourselves o he specfc correlaon srucure Corr( j, j) = j j f j j () ha arses when observaons come from a Brownan moon process (e.g. (Karln & Taylor, 1975; Oksendal, 1998)) wh drf u 11 umm and volaly.
Random effecs versons of model (1) are mporan when measuremens are avalable from mulple ndvduals (e.g., cancer growh daa from mulple paens see secon 5). In hs case, one mgh wonder wheher he populaon of ndvduals s homogeneous all sharng he same growh rae coeffcens,..., 1 M or heerogeneous. In he laer case, one mgh suspec ha ndvdual growh rae coeffcens are sampled from a hypohecal populaon (e.g., slow- versus fas-growng umors). Such models are aracve because hey allow an analys o forecas adapvely based on an ndvdual s hsory. For nsance, n a conrbuon no nvolvng repeaed measuremens, Ayer e al (Ayer, Alagoz, & Sou, 01) examne personalzng proocols for breas cancer screenng, n whch healhy screenng oucomes for an ndvdual yeld an adapve forecas of reduced cancer ncdence. Ths allows avodance of unnecessary screenng, whch would reduce coss and false-posve morbdes whou mpacng survval. In hs paper, we show ha any model of he form (1)-() can be vewed as arsng from a relaed meless classcal lnear model n whch sample szes correspond o observaon me ncremens j =,j+1 j. Moreover, nferences from he relaed classcal model apply drecly o model (1)-(), afer modfyng he degrees of freedom n he error sum of squares. When he classcal lnear model possesses a closed-form sum-of-squares able, hs able convers drecly no a sum-of-squares able for he desred model (1)-(). In parcular, we show ha he specal case of one way Brownan ANOVA, n whch he parameers are he paen drfs hemselves, arses from a classcal one-way ANOVA ha s smple enough o solve by hand or on a spreadshee. Recasng he analyss n a famlar ANOVA forma can provde greaer nsgh han he black box resuls from sascal sofware. Any nference procedure for Brownan moon would also apply o processes such as geomerc Brownan moon obaned by ransformaon (see secon 5), snce nference could occur on ransformed daa. We beleve he praccal mpor of hs paper wll be for analyss examnng longudnal daa, bu wh lmed access o or experence wh sascal sofware or general lnear models, who wsh o quckly oban ANOVA resuls n a famlar forma whou acqurng sofware or sudyng general lnear models. We srucure he paper as follows. In Secon we ransform o a general lnear model wh consan error varances, and derve closed-form esmaes and ess. In Secon 3 we specalze hese general resuls o one-way ANOVA on a Brownan drf parameer, and also consder esmaon and esng for a random-effecs versons of hs model. In Secon 4, we examne he connecon descrbed above beween model (1)-() and relaed meless classcal lnear models. Secon 5 provdes a one-way Brownan ANOVA on umor growh daa. Secon 6 concludes. 3
. ANOVA for he general longudnal model In model (1)-(), le x j = x j x,j1 j = j,j1 j = 1,,J. Then we have x j = j + j j ~ ndependen normal(0, j ) (3) where u 11 umm and here are a oal of J = observaons. We rewre he equaon for he drf vecor = ( 1,..., I ) n erms of he parameer vecor = ( 1,, M ): J = U (4) Noaon We consder boh fxed effecs and random effecs models. We use he noaon x x x x x x j j J 1 j j j j J 1 j j The maeral below uses he followng marx-vecor noaon. The prme symbol denoes marx ranspose. I J wll denoe an deny marx of dmenson J. If A 1,,A I are marces, hen dag (A ) s he block dagonal marx wh blocks A. If he A have he same number of columns, hen sack (A ) s he marx A sackng A 1 ono A ono ono A I. The followng deny wll be useful: dag (A ) sack (B ) = sack (A B ) f A,B are compable under marx mulplcaon. The basc model 1 I A obaned by We rea boh fxed and random effecs, bu nally do no dsngush whch case we are consderng. Wh he ransformaon 1/ y j = x j / j he basc model (3) becomes 1/ y j = j + e j e j ~ ndependen normal(0, ) In vecor noaon, hs s 1/ y = dag ( ) + e e ~ normal(0, I J ) (5) where 4
y sack ( y ) y sack ( y ) j j 1/ 1/ sack j( j ) = sack ( ). The resulng model s: 1/ y = dag ( )U + e. (6) Ths s a specal case of he lnear model y = X + e n whch 1/ X = dag ( )U. (7) In he followng, we explore wha happens o ANOVA for hs lnear model. Error sum of squares The leas-squares esmaes ˆα are he unresrced values of ha mnmze he sum of squares S() subjec o = U, where 1/ 1/ 1/ S( α) yxα ydag ( ) Uα ydag ( ) μ sack y x 1/ j jyj j j j j (8) Le ˆμ =Uα ˆ The resulng error sum of squares s gven by SS E = x j yxα ˆ ˆ j j (9) j SS E has J r degrees of freedom, where r s he rank of X = dag ( because f u s he h row of U, hen X = dag ( )U = sack ( elemen 1/ 1/ 1/ )U. The laer s equal o he rank of U u ). Assumng each u s nonzero, hen each 1/ u of hs sack X s an array of rank 1 wh rows proporonal o u. If r s he rank of U, hen here are r lnearly ndependen rows u, and X mus have rank r = r. Numeraor sum of squares Suppose he null hypohess o es s H: (agans an alernave hypoheses wh no resrcons on ) and consss of q ndependen lnear resrcons on. Le α ˆ be he leas squares esmae under H, and le μ ˆ = The numeraor sum of squares for an F-es of H s SS 1/ 1/ 1/ ˆ ˆ ˆ ˆ ˆ ˆ dag ( ˆ ˆ ) dag ( ˆ ˆ Xα Xα U αα μμ) sack ( ). SS has q degrees of freedom.. (10) Uα ˆ. 5
For he fxed-effecs scenaro, he expecaon of he mean square MS = SS/q can be obaned from Scheffe s Rule ((Scheffe, 1959), p.39), and s gven by E[MS] = 1 + q ( ) (11) where s obaned from he formula for ˆ by replacng each erm y j by s mean replacng each erm x j by j. F Tes 1/ j. Ths s equvalen o SS E and SS are ndependen, wh SS E / havng a ch-square dsrbuon wh he approprae degrees of freedom, and SS/ beng noncenral ch-square n general, and ch-square under. So he usual F-es of he hypohess H apples. The noncenraly parameer for SS/ may be obaned from Scheffe s Rule 1 ((Scheffe, 1959), p.39): If n SS, each observaon y j s replaced by s expeced value (or x j by j ), he resul s. 1/ j Random effecs In a random effecs model, we would have ~ normal( 0,). (If only some erms of are random, hs would be refleced by zeroes n.) The dsrbuonal and mean clams above reman condonally rue gven. In parcular, gven, he quany SS E / s condonally (Jr) a dsrbuon ha does no depend on. Therefore, SS E / s uncondonally (Jr). The random-effecs counerpar o he fxed-effecs null hypohess H: s a null hypoheses H ha places ceran resrcons on. Namely, H resrcs he varances of he q ndependen lnear conrass of correspondng o o be zero. Then under H, SS E / and SS/ are condonally ndependen gven wh (Jr) and (q) dsrbuons no dependng on, so are herefore ndependen (Jr) and (q) varables under H. The F sasc herefore sll has an F-dsrbuon under H, and he same F-es apples. 3. Specal Case of One way Brownan ANOVA As we have noed, our framework encompasses one-way or mul-way ANOVA models, nesed or crossed, wh or whou covaraes, ec. In hs secon, we specalze o he smples verson of Brownan ANOVA, a one-way model, whch we defne as he general model (1)-() wh he I drfs,, 1 I he parameers of neres. In he framework of he general model, for he one-way ANOVA specal case we have = and U = I I n (4), yeldng X 6
= dag ( 1/ ). The general resuls from he pror secon lead drecly o many of he asserons summarzed n Table 1 below. The fxed-effecs asserons n parcular fall ou n a sraghforward manner when he leas-squares esmaes x ˆ ˆ x are subsued. Hence, n he nex subsecon, we focus aenon on dervng he random-effecs asserons n Table 1. Table 1. One-way fxed and random effecs Brownan ANOVA Fxed effecs Random effecs Model,, 1 I 1 parameers I ndependen normal(, ) Hypohess H: 1 I H: = 0 Error sum of xj squares x SS E = j j wh J I degrees of freedom j SSE MS E = J I Hypohess x x sum of SS = squares wh I1 degrees of freedom SS MS = I 1 Expeced mean squares E[MS E ] = E[MS] = 1 + ( I 1) ( ) E[MS E ] = E[MS]= +. F-es Parameer esmaes 1 Rejec H a sgnfcance level f ˆ ˆ = MSE x ˆ seˆ ˆ 1/ MS MS E > F, I 1, J I ˆ 1 1. ( I 1) MSE 1 ˆ ( MS MS E ) x ˆ ˆ ˆ ˆ ˆ seˆ ˆ In case hs quany s negave, s common pracce o replace he esmae wh zero. ˆ 1/ 7
A key ake-away s ha Table 1 s nearly dencal o he analogous able for a meless classcal one-way ANOVA f one subsues observaon me ncremens j n he Brownan model for whn-cell sample szes n he classcal model. Ths neresng connecon connues o hold n he general model (1)-(). We elaborae n secon 4. Dervaon of Random effecs Resuls n Table 1 In he random effecs verson, we assume he effecs are dsrbued as { : = 1,,..., I} ~ ndependen normal(, ). Regardng he fxed-effecs expecaon of he mean square MS, we have n equaon (11) from he pror secon E x 1 1 1, j j whch we denoe by he quany n Table 1, as does no depend on. In he random effecs model, we have from (11) ha E[MS] = 1 +. ( I 1) E ( ) Usng Lemma A, hs becomes E[MS]= +, where ( I 1) 1 1, as shown n he able. Here, n parallel o classcal random effecs ANOVA, when he oal observaon mes are dencal across paens, he quany wll be her common value. Combnng he las equaon wh he fac ha MS E s an unbased esmae of, we oban n he convenonal ANOVA momen esmaor for : 1 ˆ ( MS MS E ). As n classcal random effecs ANOVA, s possble ha hs esmae can be negave, n whch case s common pracce o repor a zero esmae. A leas-squares esmae of he random effecs populaon mean can be obaned as follows when we rea and as known. Subsung ~ normal(1 I, I I ) no he random effecs model y = dag ( 1/ )+ e gves y = X1 I + g g ~ normal(0, IJ+ XX) 8
Defnng W = ( I J + XX) 1/ X1 I, he weghed leas-squares esmae of, reang and as known, s Wz 1X( I XX) ( I XX) y 1X( I XX) y ˆ WW 1X I XX I XX X1 1X I XX X1 1/ 1/ 1 I J J I J 1/ 1/ 1 I ( J ) ( J ) I I ( J ) I The marx o be nvered here smplfes o ( I XX) dag ( I ) 1 1/ 1/ 1 J J Use Lemma A3 o wre ( ) ( ) ( ) 1/ 1/ 1 1 1/ 1/ 1 IJ IJ I 1/ 1 1/ J IJ 1 ( IJ ) 4 1/ 1/ IJ 1 1/ 1/ IJ Subsue hs no he denomnaor of he las equaon for ˆ o ge 1 and no he numeraor o ge 1/ 1/ 1/ 1 1/ 1/ 1/ 1/ 1/ ( J ) I IJ. 9
10 1/ 1/ 1/ 1 1/ 1/ 1/ ( ) 1 J J y y x x x x x I I The resulng quoen s he esmae ˆ lsed n Table 1. To oban an esmaed sandard error for ˆ, calculae he numeraor varance usng ndependence of he x : 1 Var Var 1 Var E[ ] E Var[ ] 1 Var E 1. x x x x Then subsue o oban ˆ se lsed n he able. 4. Equvalence o classcal ANOVA models We noed n he pror secon he near equvalence beween one-way classcal and Brownan ANOVAs when observaon me s reaed as sample sze. Ths equvalence n fac exends o any longudnal model (1)-(), ha s, o any parameerzaon = U. Such parameerzaons nclude regresson and ANCOVA models. In hs secon, we explan how hs connecon arses. We resrc ourselves here o fxed-effecs models. For comparson, we rewre our longudnal model (3) here: x j = j + j j ~ NID(0, j ) = 1,,I; j = 1,,J, ()
Consder he relaed classcal ANOVA model y jk = + jk jk ~ NID(0, ) = 1,,I; j = 1,,J ; k = 1,,K j (*) havng he same parameers = ( 1,..., I ) and subjec o he same srucure = U. Le ˆα, α ˆ be he leassquares esmaes 1 of under models () and (*), respecvely; le α ˆ, α ˆ be he leas squares esmaes for hypohess H: under () and (*), respecvely; and le E SS, SS, SS, * SS E be he correspondng sums of squares. Le K K be he oal number of observaons. j j In he followng s assumed we can se K j = j for all,j. Of course hs requres he j o be negers, a condon ha can be acheved by suably rescalng me n model (). Proposon 1: The leas-squares esmaes α ˆ and α ˆ under model (*) depend on he varables y jk only hrough he sample sums y. Whenever yj j x and K j = j for all,j, he followng relaonshps hold: j ˆ α = α ˆ = * SS = ˆα ˆ α SS j k SS y y SS * E jk j E where y K y 1. j j j Proof: We begn by nroducng vecor noaon for (*): y = sack j (sack k (y = sack (y ) = sack (sack j (sack k ( ))) + = sack (sack j ( 1 K j ) + 1 We assume sde condons are mposed o nsure unqueness of he leas-squares esmaes, he same sde condons n (*) as n (). For model (), he esmaes are he ones dscussed n Secon, ha s, he weghed leas-squares esmaes obaned afer ransformng he daa o have homogeneous varances. Ths echncally may be no possble f some of he j are rraonal, bu n pracce daa would be expressed o only a fne number of decmal places, hence are effecvely raonal. 11
= sack ( 1K ) + = dag ( 1 )+. K Wh he parameerzaon = U, we have y = dag ( 1 )U+ = X * + K where X * = dag ( 1 )U. Under model (*), he leas-square esmaes α ˆ mnmze he sum of squares K subjec o = U, where: S * ( α) yx α sack ( y) dag ( 1K ) μ sack y 1 K yjk j k yjk yj j j k y j k yjk yj Kj yj S j k j S ( α) 1 So n effec, wha mus be mnmzed s subjec o = U. Of course he resulng α ˆ depends only on he sample sums va he sample means yj. Moreover, he quany s dencal o he sum of squares of (8) (we have added he superscrp o ndcae s for he model ()) under he assumed condons yj xj and K j = j for all,j. Therefore he resulng α ˆ mus be dencal o condons. The same argumen mples ha α ˆ = α ˆ. ˆα under hese Tha follows by nong ha, whch (by he argumens n he precedng paragraph) s he same as under he assumed condons yj xj and K j = j for all,j. Fnally, subsung α α, we oban, s ˆ. SS E under he assumed condons..,, (1) Noce n Proposon 1 ha f n addon y jk does no depend on k for all,j, hen we have no only,, ) = (,, ), bu also SS SS. Under he assumpon K j = j of he proposon, we have * E E 1
y x { j j, jk y does no depend on k} yjk xj j for all,j Proposon 1 herefore mples ha by performng a classcal ANOVA on model (*) whle makng he subsuons K j = j, yjk xj j, we can oban all relevan sascs for he longudnal model (). Bu does he (*)- ANOVA gve us he correc sascal nferences for model ()? I s apparen canno, because (Kr) dsrbuon, whereas whch * SS E / has a SS E / has a (Jr) dsrbuon. To see wha has happened, refer o (1) above, n * SS E s decomposed no he sum of wo ndependen quanes, and, wh K J and Jr respecve degrees of freedom. Assumng y jk does no depend on k, as n he subsuon above, resuls n, = 0 and =, ~ (Jr). In general, sascal nferences for he (*)-ANOVA depend on he jon dsrbuon of he quanes (,,,, jon because conclusons depend on probablsc ndependences ha may exs, such as ndependence of and needed for hypohess esng, and ndependence of and needed for confdence nerval consrucon. One mgh worry ha he resrcon ha y jk does no depend on k could do more han merely change he dsrbuon of / from (Kr) o (Jr) perhaps could furher aler he jon dsrbuon of (,,, n some way. Tha hs s no so s a consequence of he followng resul. Proposon : The jon dsrbuon of (,,,, s ndependen of he even A = {y jk does no depend on k for all,j}. Moreover, f K j = j for all,j, hen hs dsrbuon s dencal o he jon dsrbuon of (,,,. Proof: To derve he frs clam, observe he followng: () (,,,, ) depends on only va he whnsample sums. or, equvalenly, va he whn-sample averages.. ; (), s ndependen of. and, herefore, of (,,,, ), because, s a funcon of he whn-sample varances., whch are ndependen of he whn-sample averages for he normal samples n queson; and () he saed 13
even A s equvalen o, 0. Combnng hese hree resuls proves ha he jon dsrbuon of (,,,, s ndependen of he even A, as assered. To show he second clam noe frs ha by Proposon 1, when K j = j for all,j, we have (,,, ) = g({x j }) and (,,,, ) = g({ y j }) for some funcon g() he same funcon n boh cases. The second clam hen follows from he easly-verfed fac ha he jon dsrbuon of { y j } under model (*) s he same as he jon dsrbuon of {x j } under model (). Wha hen s he relevance o he longudnal model () of sascal nferences from classcal (*)-ANOVA when we make he subsuons K j = j, yjk xj j? Sep back for a momen and compare wo nferenal exercses: Sascal nference for parameers under model (*) and he condon A = {y jk does no depend on k for all,j}. The daa on whch such nferences are based may be aken, by Proposon 1, o be he sample sums { yj } and sample szes {K j }. Any such nferences depend by Proposon on he same jon dsrbuon F of (,,,, ha obans n he absence of condon A, ha s, he jon dsrbuon ha obans n he usual classcal ANOVA. The only dfference s ha under condon A, we have, Kr o Jr. 0, forcng he error sum of squares o equal,, and reducng s degrees of freedom from Sascal nference for parameers under model (). Ths s based on daa {x j } and observaon mes { j }, and depends on he same (by Proposon ) jon dsrbuon F of (,,,. Because of hese parallels, nference n hese wo suaons wll be dencal on dencal daa j = K j, xj yj. Bu as we have menoned, hese equales are equvalen o j = K j, yjk xj j. Ths leads us o he followng. Concluson: Sascal nferences on obaned by subsung yjk xj j and K j = j for all,j no he classcal (*)-ANOVA and changng he degrees of freedom for from Kr o Jr wll be vald for n he longudnal model (). 14
Alhough hs concluson s far-reachng, s praccal mplcaons are more lmed. If exreme scalng s needed o ge all-neger j, he resulng large sample szes K j and quanes of dencal daa y jk for he (*)-ANOVA wll be nconvenen for sofware packages, and he SS E degrees of freedom n sofware oupu would need o be correced from Kr o Jr. So hs s no n general an approach we recommend for sofware packages. However, none of hese dffcules arse f here are closed-form ables for ANOVA on model (*). Closed-form ables for unbalanced models lke (*) do exs n some cases, for example, for one-way ANOVA as we have already dscussed, as well as for nesed herarchcal models when effecs are weghed by sample sze (see (Scheffe, 1959), Ch. 5.3). We dscuss he laer n he example presened below. In general, o conver a closed-form classcal ANOVA able for model (*) o a correspondng ANOVA able for model (): 1. Replace he sample szes K j by observaon mes j everywhere.. In * SS E and * SS, replace y jk and yj by x j / j. 3. Change he degrees of freedom Kr for * SS E o Jr. I should be noed ha he replacemens n Sep also enal he followng replacemens: x y K K y x x 1 1 j 1 j j j j j j x y K K y x x 1 1 j 1 j j j j j j Noe n sep 1 ha one can om any scalng of he j o neger values. Ths s because any such scalng would cancel from numeraor and denomnaor of all F-sascs. Moreover, any scaled esmae of a scaled, hence he unscaled Example: A nesed model * MS E would be unbased for he unscaled. * MS E would be an unbased The followng llusraes how a specfc unbalanced nesed classcal ANOVA ranslaes o a nesed longudnal ANOVA usng he mehods jus dscussed. Scheffe ((Scheffe, 1959),Ch. 5.3) nroduces a procedure for consrucng classcal ANOVA ables for unbalanced nesed desgns. Hs procedure apples, for example, o he followng nesed desgn. 15
yhjk h hjk ~ NID(0, ) (13) hjk h h h h = 1,,H; = 1,,I h ; j = 1,,J h ; k = 1,,K hj. I h h I ; J h h J ; hj. To be concree, hs model reflecs, for example, he effec h j K K due o paens. The ndces par (h,) n (13) corresponds o he sngle ndex n (*). To elmnae parameer redundancy, Scheffe mposes he followng resrcons 3 : K h h 0 h and K h h 0 for each h. The leas-square esmaes and assocaed 1 confdence nervals are: h due o hospals h and whn hospal h he effec h ˆh y h y,/ K K 1 1 h (14) ˆh y h y h,/ K K (15) 1 1 h h where. For esng he wo dfferen null hypoheses H : h 0 for all h H : h 0 for all h, he ANOVA able s: Source SS d.f. SS Kh( yh y ) H 1 h h ( h h ) h I H SS K y y Error SS E = y h j k hjk yh K I Toal SS To = y hjk y K 1 h j k 3 I s o be noed ha hese specfc resrcons are needed n order ha he sums of squares n he ANOVA able below decompose properly. 16
The correspondng longudnal ANOVA resuls would be obaned by followng he nsrucons 1,,3 saed earler n hs secon. The ndex here corresponds o he par (h,) here, and he longudnal ANOVA model s he same as model (13) bu wh yhjk h hjk and NID(0, hj hjk ~ NID(0, ) replaced by xhj hhj hj and hj ~ ), respecvely. The leas-square esmaes and confdence nervals (14)-(15)ranslae o ˆh x h x,/ K K 1 1 h ˆh x h x h,/ K K 1 1 h h where and we have used JI nsead of KI as he degrees of freedom for. The classcal nesed ANOVA able above ranslaes o: Source SS d.f. SS h( xh x ) H 1 Error SS E = h h ( h h ) h I H xhj h j hj xh hj J I SS x x xhj h j hj x J 1 hj Toal SS To = These resuls arse by nvokng he gudelnes above o oban he subsuons K I J I Khj hj yhjk xhj hj yh xh Kh h Kh h h h y x y x under whch he nner sum over k n SS E smplfes snce s argumen no longer depends on k. 5. Example Analyss Ebara e al. (Ebara e al., 1986) publshed daa (see Fgure 1) on he growh of unreaed hepaocellular carcnoma umors. To llusrae he mehods presened here, we perform a one-way random effecs Brownan ANOVA (Table 1) o analyze growh-rae heerogeney n hs daa. 17
Fgure 1. Hepaocellular carcnoma umor growh n paens from Ebara e al. (1986). Sophscaed dfferenal equaon models of umor growh are avalable (see (Araujo & Mcelwan, 004)). Here for smplcy we focus on an exponenal growh model, n whch, f z s sze a me, and s growh rae a me, hen dz d z whch yelds exponenal growh z z0e when s a consan. A sochasc verson of hs model allows growh rae o vary n random ndependen ncremens over me: d d db where s mean growh rae, s growh rae volaly, and db s an ncremen of sandard Brownan moon. Ths assumpon forces z o be geomerc Brownan moon (e.g.,(oksendal, 1998)), ha s, x = ln(z ) s Brownan moon wh drf and volaly : dz z d(ln z ) d db. Cha e al. (Cha, P.Salzman, Plevrs, & Glynn, 004) employ hs model o explore smulaon-based parameer esmaon for breas cancer umor growh. If we allow mean growh rae o depend on paen, hen our model (3) s an mmedae resul. A random-effecs model allows growh raes for paens o be randomly sampled from a normal populaon wh mean and sandard devaon. Our one-way ANOVA procedures (Table 1) allow us o esmae and and 18
es he hypohess H: = 0. We performed such a one-way random-effecs ANOVA based on Table 1. The procedure s smple enough ha we carred ou on a spreadshee. A word abou uns: Because x = ln(z ), we have dx dz/ z, so ha dx (and herefore x self) s a unless quany. I follows ha he quanes,,, have uns of me 1, as do all sums of squares and mean squares. To make such quanes more meanngful n our resuls below, we conver hem o percens per 100 days. The resuls from applyng he ANOVA of Table 1 are: ˆ =1.4% /100 da. ˆ = 9.64% /100 da. ˆ = 13.% /100 da. se ˆ =.53% /100 da. Therefore, an esmaed 95% confdence nerval for he mean of he populaon of mean growh raes s 13.% ± z0.05.53% = 13. ± 4.96 per 100 days, and he sandard devaon of he populaon of mean growh raes s esmaed o be 9.64% per 100 days. The resuls of applyng he Table 1 ANOVA es of he null hypohess H: = 0 are Source SS df MS F p 118.7% 0 5.94% 4.174 1.45E 06 Error 16.6% 89 1.4% from whch we conclude ha he above esmae = 9.64% /100 days s sgnfcanly dfferen from 0 (p-value 1.45 10 6 ). We conclude ha growh-rae heerogeney s presen n he populaon from whch hese observaons arose. 6. Concluson The prmary heorecal resul from hs paper s ha any longudnal model (1)-() can be vewed as arsng from a correspondng meless classcal ANOVA model n whch sample szes are observaon mes; and ha sascal nferences for he former can be drecly exraced from he laer once error degrees of freedom are suably revsed. The models (1)-() nclude no only ANOVA models of all ypes, bu also regresson and ANCOVA models, for example. We concede, however, ha he praccal mpac of hs resul s lmed because: () as noed n Secon 4, 19
he requred sofware npus for he correspondng classcal model could be cumbersome, and he sofware oupus would need o be revsed n lgh of he alered degrees of freedom; and () s possble o ransform any longudnal model (1)-() no a general lnear model, and use avalable sascal sofware o numercally derve esmaes and hypohess es resuls whou any necessy of explong he connecon o classcal ANOVA. However, our procedure does have praccal usefulness when he desred longudnal model arses from a classcal ANOVA possessng a closed-form sum-of-squares able. In ha case, he classcal able convers drecly no a sum-of-squares able for he desred longudnal model. The eases verson of hs s lkely he one-way random effecs ANOVA (Table 1) n whch s desred o esmae he populaon of drfs from whch he daa have arsen, and o es for heerogeney n hs populaon. These esmaes and ess are smple enough o be derved on a spreadshee. We beleve he praccal mpor of hs paper wll be for analyss wh lmed access o or experence wh sascal sofware or general lnear models, who wsh o quckly oban famlar ANOVA resuls whou acqurng sofware or sudyng general lnear models. References Araujo, R. P., & Mcelwan, D. L. S. (004). A Hsory of he Sudy of Sold Tumour Growh: The Conrbuon of Mahemacal Modellng. Bullen of Mahemacal Bology, 66, 1039-1091. Ayer, T., Alagoz, O., & Sou, N. K. (01). OR Forum A POMDP approach o personalze mammography screenng decsons. Operaons Research, 60(5), 1039 1091. Cha, Y. L., P.Salzman, Plevrs, S. K., & Glynn, P. W. (004). Smulaon-based parameer esmaon for complex models: a breas cancer naural hsory modellng llusraon. Sascal Mehods n Medcal Research, 13, 507-54. Dggle, P. J. (1988). An approach o he analyss of repeaed measuremens. Bomercs, 44(4), 959-971. Dggle, P. J., Heagery, P., Lang, K., & Zeger, S. L. (00). Analyss of Longudnal Daa (second edon). New York: Oxford Unversy Press. Ebara, M., M.Oho, Shnagawa, T., Sugura, N., Kmura, K., Masuan, S., e al. (1986). Naural Hsory of Mnue Hepaocellular Carcnoma Smaller Than Three Cenmeers Complcang Crrhoss: A Sudy n Paens. Gasroenerology, 90, 89-98. 0
Graybll, F. A. (1983). Marces wh Applcaons n Sascs (Second Edon). Pacfc Grove, Calforna: Wadsworh & Brooks/Cole. Karln, S., & Taylor, H. M. (1975). A frs course n sochasc processes (d ed.). New York: Academc Press. Oksendal, B. (1998). Sochasc Dfferenal Equaons: An Inroducon wh Applcaons (Ffh edon ed.). Berln, Hedelberg, New York: Sprnger-Verlag. Scheffe, H. (1959). The Analyss of Varance. New York: John Wley and Sons. Snger, J. D., & Wlle, J. B. (003). Appled Longudnal Daa Analyss: Oxford Unversy Press. Verbeke, G., & Molenberghs, G. (009). Lnear Mxed Models for Longudnal Daa. New York: Sprnger Verlag. Appendx Suppose we have observaons y 1,,y J and weghs w 1,,w J. Defne he sum of squares SS = w j( y y j ) j 1 y w w j jyj where w wj. The followng resul s sandard and a consequence of elemenary algebra. j Lemma A1: SS = wy j j. j wy Lemma A: Suppose n addon ha he varables y are ndependen wh mean and varance. Then 1 E[SS] = w w w. Proof: We have E y j y has mean and varance w w, so E y w w. Then 1
E[SS] = E wy j j wy wj( ) w w w j j = w w w w w w 1 1 The followng s also a sandard resul (e.g., see Theorems 8.3.3 and 8.9.3 n (Graybll, 1983)). Lemma A3: Le he k k marx C be gven by CDab 1 where D s a nonsngular symmerc marx, a and b are each k 1 vecors, and s a scalar such ha 1aD b 0. Then C 1 1 1 1 = D 1 1 ad b D ab D.