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1 Appendx: Persn-rened mehds and hery Supplemenal Onlne Appendx Ths nlne appendx accmpanes Serba and Bauer (2010) Machng mehd wh hery n persn-rened develpmenal psychpahlgy research Develpmen & Psychpahlgy, 22, Here we presen mdel equans fr each persn-rened mehd dscussed n he manuscrp. Here we als prvde examples f hw es persn-rened prncples lsed n manuscrp Table 1 where pssble usng a gven mehd. In each case rejecng he null hyphess lsed ( H ) cnsues suppr fr he persn-rened prncple. Less-resrcve varable rened mehds Laen grwh mdel (LGM). Le all srucural equan mdels, he LGM cnsss f a measuremen mdel and a srucural mdel. Le y be a p x 1 vecr f repeaed measures fr persn. In Fgure 1 Panel A, p=5. The measuremen mdel s y = υ+ Λη + ε where fen ε ~ N(0, Θ ). (1) The srucural mdel s η = α+ ς where ς ~ N(0, Φ ). (2) υ s a p x 1 vecr f em nerceps fxed 0 (n shwn n Fgure 1 Panel A). Λ s a p x q marx f facr ladngs (fxed 1, 1, 1, 1, 1 fr he frs clumn and 0, 1, 2, 3, 4 fr he secnd clumn n Fgure 1 Panel A defne nercep and slpe grwh facrs). η s a q x 1 vecr f laen grwh facr scres, and n Fgure 1 Panel A q=2. ε s a p x 1 vecr f me-specfc resduals. Θ s a ypcally-dagnal p x p cvarance marx f ε. α s a q x 1 vecr f grwh facr means (n shwn n Fgure 1 Panel A). ς s a q x 1 vecr f ndvdual devans frm hse grwh facr means. Φ s a ypcally-unsrucured q x q cvarance marx f ς. Manuscrp Table 2 lss whch persn-rened prncples are esable wh LGM. Nex we gve examples f hw hese prncples culd be esed. (1) Inerndvdual dfferences/nrandvdual change prncple. Assumng paern summarzan and paern parsmny prncples are nvald, an example f esng he A1

2 Appendx: Persn-rened mehds and hery nerndvdual dfferences/nrandvdual change prncple s H0 : φ 11 = 0 n Fgure 1 Panel A (.e. n slpe varably). (2) Indvdual specfcy prncple. Under he same assumpn, an example f esng he ndvdual specfcy prncple s H : Φ = 0 0 (.e. n varance r cvarance n grwh facrs). (3) Cmplex-neracns prncple. Under he same assumpn, an example f esng he cmplex-neracns prncple n he LGM n Fgure 1 Panel A s expand he srucural mdel n Equan (2) regress grwh facrs n a vecr f persn-level predcrs. An example vecr f persn-level predcrs s x = [ x1, x2, x1 x 2], hugh mre predcrs and neracn erms culd ceranly be ncluded. Ths yelds: η = α+ Γx + ς (3) where η, ς, and α are 2 x 1, x s 3 x 1, and Γ s a 2 x 3 marx f regressn ceffcens. Then, we can, fr example, es: H0 : γ 23 = 0 (me by x1 by x2 neracn). (4) Hlsm prncple. Lmed esng f he hlsc prncple n he LGM s pssble by expandng he unvarae Equan (1)-(2) nclude ne r mre parallel grwh prcesses. Suppsng he rgnal grwh prcess (labeled (a)) and addnal grwh prcess (labeled (b)) each had p=5 and q=2, hs wuld enal sacng he vecrs f repeaed measures, nerceps, and resduals fr he w prcesses y y υ = ( b), υ = ( b), and ε y υ ε = ( b) such ha ε y, υ, and ε are nw each 10 x 1. We wuld als sac vecrs f grwh facr scres, grwh facr means and mean devans fr he w prcesses, η η α = ( b), α = ( b), and η α ς ς = ( b) such ς ha η, ς, and α are nw each 4 x 1. We wuld expand Λ 0 Λ = ( b) be 10 x 4 and 0 Λ blc dagnal wh 5 x 2 blcs and expand ( a, b) Θ Θ Θ = ( ab, ) ( b) be 10 x 10 where Θ Θ Θ, ( a, b) ( b) ( ab, ) Φ Φ Θ, and Θ are each 5 x 5 dagnal marces. Fnally, we wuld expand Φ = ( ab, ) ( b) Φ Φ ( b) ( ab, ) be 4 x 4 where Φ, Φ, and Φ are each 2 x 2 and unsrucured. Tesng he nerdependency aspec f he hlsm prncple culd nvlve seeng f grwh facrs rac A2

3 Appendx: Persn-rened mehds and hery Φ 0 geher ver me,.e. H : Φ = ( b). Tesng he recprcy aspec f he hlsm 0 Φ prncple culd nvlve nsead seeng f grwh facrs predc each her by reparameerzng he mdel s ha Φ s blc dagnal wh 2 x 2 blcs, and hen expandng he srucural mdel n Equan (4): η = α+ βη + ς (4) Here agan η, ς, and α are 4 x 1, and β s 4 x 4. Then, an example f esng he recprcy aspec f he hlsm prncple wuld be H : β = 0. 0 Classfcan mehds Laen class grwh mdel (LCGM). The measuremen mdel fr p repeaed measures n persn n laen class s: y = υ + Λη + ε where fen ε ~ N(0, Θ ) (5) The srucural mdels are η = α (6) π = exp( ν ) K = 1 exp( ν ) Here y s a p x 1 vecr f repeaed measures fr persn n class, where p=4 n Fgure 1 Panel B. There are a al f K classes. υ s a p x 1 vecr f class-specfc em nerceps fxed 0 (n shwn n Fgure 1 Panel B). Λ s a p x q marx f ladngs f repeaed measures n q grwh parameers n class. η s a q x 1 vecr f class-specfc laen grwh parameers. ε s a p x 1 vecr f me-specfc resduals fr class. Θ s a ypcally-dagnal p x p cvarance marx f ε. α s a q x 1 vecr f class-specfc means (n shwn n Fgure 1 Panel B). Fnally π s he prbably f membershp n class whch s calculaed frm a npredcr mulnmal lgsc regressn wh nercep ν. Manuscrp Table 2 lss whch persn-rened prncples are esable wh LCGM. Nex we gve examples f hw hese prncples culd be esed. A3

4 Appendx: Persn-rened mehds and hery (1) Paern parsmny. Assumng ha he paern summary prncple s vald, esng he paern parsmny prncple n he LCGM culd nvlve cmparng he f f K=2, 3 class mdels and asceranng wheher he pmally fng number f classes s < a predefned small number. (2) Cmplex neracns prncple. Under he same assumpn, esng he cmplex neracns prncple n he LCGM culd nvlve addng a vecr f persn-level predcr(s) f class membershp such as x = [ x1, x2, x1 x 2 ] n Equan (7) ( π ) = K exp( ν + δ x ) = 1 exp( ν + δ x ) (7) Where here δ s 1 x 3 and here x s 3 x 1. Then esng mplcly fr neracns n he predcn f grwh parameer values culd enal, H :. 0 δ = δ (3) Hlsm prncple. Lmed esng f he hlsm prncple s pssble by expandng he unvarae equan (5)-(6) als mdel, fr example, a secnd lngudnal behavr, havng j=1 J rajecry classes. Fr each f j classes n he secnd grwh prcess, a be specfed andθ and j Λ j wuld need α j wuld need be esmaed. Fnally, he w grwh prcesses wuld be lngudnally lned by esmang π j, he cndnal prbably f membershp n class f prcess 1 gven membershp n class j f prcess 2 (see Nagn & Tremblay, 2001). Gven ha π j was esmaed frm he frs prcess n Equan (6) and π j was esmaed frm he secnd prcess, bh f hese quanes can be used slve fr: π j, he cndnal prbably f membershp n class j f prcess 1 gven membershp n class f prcess 2, and π j,he jn prbably f membershp n class j and. Then, esng he nerdependency aspec f he hlsm prncple culd nvlve H0 :" π j n dfferen han chance r π chance r H : 0 j π j j n dfferen han n dfferen han chance and esng s recprcy aspec culd nvlve π n dfferen han chance and π j n dfferen han chance. Laen Marv mdel. The laen Marv mdel fr a respnse paern n ne bnary varable measured a 4 mepns (e.g. 1,0,1,1 r 0,0,0,1 r 1,1,0,0), as shwn n Fgure 2 Panel C, s A4

5 Appendx: Persn-rened mehds and hery K M N O Py ( ) = δ ρτ ρτ ρτ ρ (8) = 1 m= 1 n= 1 = 1 m m nm n n Here, δ and ρ s are scalar, measuremen mdel parameers and τ s are scalar, srucural mdel parameers. Here als here are K laen sauses a me 1, M a me 2, N a me 3 and O a me 4. δ are nal laen saus prbables, whch sum 1 acrss K. ρ s he prbably f em endrsemen a mepn 1 gven membershp n laen saus a mepn 1. ρm s he prbably f em endrsemen a mepn 2 gven membershp n laen saus m a mepn 2. ρn s he prbably f em endrsemen a mepn 3 gven membershp n laen saus n a mepn 3. ρ s he prbably f em endrsemen a mepn 4 gven membershp n laen saus a mepn 4. (Ne ha f here were n ne bu J measures per mepn, as n a laen ransn mdel, we wuld smply replace ρ, ρ, ρ, ρ wh J J J J ρj, ρ jm, ρ jn, ρ j n Equan (8). Ne als ha he laen Marv j= 1 j= 1 j= 1 j= 1 mdel requres ρ = ρ = ρ = ρ bu he laen ransn mdel des n.) ρ, ρ, ρ, ρ each m n sum 1 acrss her respecve bnary respnse caegres.τ s are scalar ransn prbables frm a parcular laen saus a a prr mepn a parcular laen saus a he curren mepn. Hence, τ m denes he prbably f ransnng membershp n saus m a mepn 2 gven membershp n saus a mepn 1 (here are a K x M such prbables). τ nm denes he prbably f ransnng membershp n saus n a mepn 3 gven membershp n saus m a mepn 2 (here are a M x N such prbables). Fnally, τ n denes he prbably f ransnng membershp n saus a mepn 4 gven membershp n saus n a mepn 3 (here are N x O such prbables). Manuscrp Table 2 lss whch persn-rened prncples are esable wh laen Marv mdel. Nex we gve examples f hw hese prncples culd be esed. (1) Paern parsmny prncple. Assumng ha he paern summary prncple s vald, esng he paern parsmny prncple n he laen Marv mdel culd nvlve cmparng he f f K=2, 3 sauses, M=2, 3 sauses, N=2, 3 sauses, O=2, 3 sauses and asceranng wheher he pmally fng number f sauses/mepn s < a predefned small number. m n m n A5

6 Appendx: Persn-rened mehds and hery (2) Cmplex neracns prncple. Under he same assumpn, esng he cmplex neracns prncple n he laen Marv mdel culd nvlve, fr example, addng a vecr, x, f persnlevel predcr(s) f laen ransn prbables. Ths wuld enal ncludng a mulnmal lgsc regressn predc laen ransn prbables: ( τ ) = m M exp( α + β + γ x ) m= 1 m m m exp( α + β + γ x ) m m m In Equan (9), βm denes he dfference n lg dds f beng n class m vs. he reference class a me 2 fr persns n class a me 1 cmpared he reference class. The γ m allws he effec f x n --m ransn prbables dffer acrss laen sauses m (see Nylund, 2007 fr examples). Tesng fr a saus by x neracn culd be accmplshed by H : 0 γm = γ. (3) Hlsm prncple. Lmed esng f he hlsm prncple n he laen Marv mdel wuld be pssble f he lngudnal sequence f anher, enrely dfferen, behavr were mdeled smulaneusly (n mulple ndcars f he same repeaed cnsruc as n laen ransn analyss). Ths s called an asscave laen Marv mdel (Flahery, 2008). Suppse he secnd behavr had V laen sauses a me 1, W a me 2, X a me 3, and Z laen sauses a me 4. Then, n he asscave laen Marv mdel, δ wuld be esmaed as n Equan (8), bu nw ransn prbables wuld be cndnal n curren saus n he secnd behavr as well (.e. τ, τ,and τ ), and respnse prbables fr each em wuld be cndnal n curren mv, mnw, nx, laen sauses fr bh behavrs (.e. ρv,, ρmw,, ρnx,,and ρ z, ). As well, nal saus fr he secnd behavr wuld be cndnal n nal saus f he frs behavr (.e. δ v ), and ransn prbables fr he secnd behavr wuld be cndnal n prr and curren laen saus fr he frs behavr (.e., τ wv,, m, τxw, m, n, τ zx, n, ; see Flahery s 2008 Appendx fr smlar mdel). Then, esng he recprcy aspec f he hlsm prncple frm mepn 1 2, fr example, culd nvlve evaluang: H0 : τ wv,, m = τwv and τm, v= τ m (.e. ha ransn prbables n ne behavr d n depend n curren and/r prr laen saus membershp n he her behavr). Tesng he nerdependency aspec f he hlsm prncple a mepn 1, fr example, culd nvlve evaluang H0 : δv = δv(.e. ha nal laen saus prbables n he secnd behavr d n depend n nal laen saus prbables n he frs behavr). (Ne: (9) A6

7 Appendx: Persn-rened mehds and hery alhugh such mdels can n prncple be f n srucural equan mdelng prgrams, esman prblems can arse wh ncreasng numbers f saes/mepn and mepns.) Hybrd classfcan mehds Grwh mxure mdel (GMM). The measuremen mdel fr p=4 repeaed measures n persn n laen class frm Fgure 1 Panel D s: y = υ + Λη + ε where fen ε ~ N(0, Θ ). (10) The srucural mdels are η = α + ς where ς ~ N(0, Φ ) (11) π = exp( ν ) K = 1 exp( ν ) All nan s as defned n he LCGM excep fr ς, whch s a q x 1 vecr f class-specfc ndvdual devans frm grwh facr means and q x q varance-cvarance marx f ς fr class. Φ, whch s a ypcally-unsrucured Manuscrp Table 2 lss whch persn-rened prncples are esable wh he GMM. Nex we gve examples f hw hese prncples culd be esed. (1) Paern summary. Assumng ha rajecry classes represen ppulan subgrups, esng he paern summary prncple n GMM culd enal H : 0 0 K =, bu see qualfcans/cauns n he ex. (2) Paern parsmny. Under he same assumpn, esng he paern parsmny prncple culd enal H : 0 K < predefned small number, bu see qualfcans n he ex. (3) Inerndvdual dfferences/nrandvdual change. Under he same assumpn, an example f esng wheher here s remanng nerndvdual varably n change, ver and abve ha whch was accuned fr by α dfferences wuld be H0 :( φ 11) = 0, n Fgure 1 Panel D. (4) Indvdual specfcy prncple. Under he same assumpn, an example f esng wheher here s remanng ndvdual specfcy, afer accunng fr α dfferences, s H : 0. 0 Φ = (5) Cmplex neracns prncple. Under he same assumpn, esng he cmplex-neracns prncple n he GMM n Fgure 1 Panel D culd nvlve bh adpng sraeges emplyed fr A7

8 Appendx: Persn-rened mehds and hery deecng explc neracns n he predcn f grwh facrs frm LGM (.e. ncludng a vecr f persn-level predcr(s) x = [ x1, x2, x1 x 2 ] f grwh facrs whn-class): η = α + Γx + ς (12) and sraeges emplyed fr deecng mplc neracns n he predcn f grwh parameer values frm LCGM (.e. ncludng a vecr f persn-level predcr(s) x f class membershp): ( π ) = K exp( ν + δ x ) = 1 exp( ν + δ x ) (13) Then we culd, fr example, es H0 :( γ 23) = 0 (.e. ha here s n me by x1 by x2 neracn) frm Equan (12) and es H : 0 δ = δfrm Equan (13). (6) Hlsm prncple. Fnally, n hery, lmed esng f he hlsm prncple n GMM culd be pssble usng he same prcedures dscussed fr he LCGM mdel. Mxed laen Marv mdel. The mxed Laen Marv mdel fr a respnse paern n ne bnary varable measured a 4 mepns (e.g. 1,0,1,1 r 0,0,0,1 r 1,1,0,0), as shwn n Fgure 2 Panel E, s C K M N O Py ( ) π δ ρ τ ρ τ ρ τ ρ = (14) c= 1 = 1 m= 1 n= 1 = 1 c c c m, c mc mn, c nc n, c c Here, here are C laen chans whch allw fr acrss-chan heergeney n lngudnal saus-saus behavral sequences. The prprn f membershp n chan c s dened π c and all her mdel parameers are as defned n he laen Marv mdel excep ha hey are nw cndned n chan membershp als. Ne ha, n hs mdel, parameers are fen cnsraned equal acrss chan r fxed n ne chan / free he her. Manuscrp Table 2 lss whch persn-rened prncples are esable wh he mxed laen Marv mdel. Nex we gve examples f hw hese prncples culd be esed. (1) Paern parsmny prncple. Assumng ha he paern summary prncple s vald, esng he paern parsmny prncple n he mxed laen Marv mdel culd nvlve cmparng he f f K=2, 3 sauses/chan, M=2, 3 sauses/chan, N=2, 3 sauses/chan, O=2, 3 sauses/chan asceranng wheher he pmally fng number f sauses/mepn n each chan s < a predefned small number. A8

9 Appendx: Persn-rened mehds and hery (2) Indvdual specfcy prncple. Under he same assumpn, esng he ndvdual specfcy prncple culd nvlve H : C =1. 0 (3) Inerndvdual dfferences/nrandvdual change prncple. Under he same assumpn, esng he nerndvdual dfferences/nrandvdual change prncple culd nvlve he mre specfc hyphess ha he ransn prbables are he same acrss chan: H : τ = τ ; τ = τ ; τ = τ. 0 m, c m mn, c mn n, c n (4) Cmplex neracns prncple. Under he same assumpn, esng he cmplex neracns prncple n he mxed laen Marv mdel culd nvlve addng persn-level predcrs f whn-chan laen ransn prbables (much le n Equan (9)). (5) Hlsm prncple. Fnally, alhugh lmed esng f he hlsm prncple usng smlar prcedures hse descrbed n he laen Marv secn s n prncple pssble, n pracce s unlely ha mulple chans and mulple Marv prcesses whn chan wuld be esmable. Sngle subjec mehds P-echnque facr mdel. The measuremen mdel fr p varables n ccasns fr ne persn s y = Λη + ε where fen ε ~ N(0, Θ ). (15) The srucural mdel s η = ς where ς ~ N( 0, Φ ). (16) Ne ha he cnvennal p-echnque mdel has n mean srucure. Here y s a p x 1 vecr f bserved varables, where n Fgure 1 Panel F p=20. q=number f prcess-facrs, whch n Fgure 1 Panel F s 2. Λ s a p x q marx f prcess-facr ladngs, where q s he number f prcess-facrs. η s a q x 1 vecr f prcess-facr scres ha vary acrss mepns. ε s a p x 1 vecr f resduals. Θ s a ypcally-dagnal p x p cvarance marx f ε. ς s a q x 1 vecr f me-specfc devans frm prcess facr means; (hese means are assumed be 0). Φ s a ypcally-unsrucured q x q cvarance marx fς. A9

10 Appendx: Persn-rened mehds and hery We nly descrbe esng persn-rened prncples wh respec he dynamc facr mdel belw, as he p-echnque mdel was nly presened as an nermedae sep buld up he dynamc facr mdel. Dynamc facr mdel. The measuremen mdel fr p varables n ccasns fr ne persn and fr nly 1 lag (as n Fgure 2 Panel G) s y = Λη + ε where ε~ N(0, Θ ). (17) The srucural mdel allwng fr mean rend (Mlenaar, de Gjer, & Schmz, 1992) (n shwn n Fgure 1 Panel G) s: η= γτ + ς where ς ~ N(0, Φ ). (18) Ths parcular dynamc facr mdel s fen called a whe nse facr mdel wh nnsanary f means r a shc facr mdel wh nnsanary f means (Brwne and Nesselrade, 2005). Hwever, hs mdel sll requres ha here be n sysemac rend n varances/cvarances f he repeaed measures, r ha such a rend has been remved. Here y y = s a 2p x 1 vecr whch cnans y, a p x 1 vecr f lag-0 measured varables, y 1 saced n p f y 1, a p x 1 vecr f lag-1 measured varables. In Fgure 2 Panel G y wuld be f dmensn 40 x 1, as here are 20 lag-0 measures cnsung he vecr y and 20 lag-1 cunerpars cnsung he vecr y 1. q s he number f prcess facrs, where n Fgure 2 (0) Panel G q=2. Λ s dmensn 2p x 3q and cnans lag-0 p x q facr ladng marx Λ and lag- (1) 1 p x q facr ladng marx Λ n he fllwng paern: η (0) (1) Λ Λ 0 Λ = (0) (1) 0 Λ Λ. η= η 1 s a 3q x 1 vecr whch cnans facr scres fr prcess η 2 facrs a lag-0,.e. η and lag-1,.e. η 1 and lag-2 η 2. Ne ha η 2 are ncluded even hugh hs s nly a 1-lag mdel because hey are needed specfy he nal cndn/hsry f he ε w prcesses prr he frs measuremen ccasn. ε = s a 2p x 1 vecr f resduals. ε 1 (0) (1) Θ Θ Θ = (1) (0) s a 2p x 2p cvarance marx f he ε s, and has a specalzed (blc- Θ Θ A10

11 Appendx: Persn-rened mehds and hery Teplz) frm such ha (0) Θ =COV ( ε, ε ) =COV( ε 1, ε 1 ), whch s p x p and dagnal, and (1) Θ =COV( ε, ε 1 ), whch s p x p and dagnal. Ths allws resduals be crrelaed acrss bu n whn lag. γ s a 3q x 1 vecr f slpes relang prcess facrs me. τ s a scalar me ς varable denng he ccasn. ς = ς 1 s a 3q x 1 vecr f schasc erms. Φ s a 3q x 3q ς 2 blc-dagnal cvarance marx f he ς, wh equal blcs, where ς s have varances f 1 and are allwed be crrelaed nly whn lag. Fng hs mdel usng srucural equan mdelng sfware requres frs addng a me varable τ (e.g. wh values 1 71 f here were 71 ccasns) he ccasn by varables daa marx and hen cnverng hs marx n a Blc Teplz frm; (SAS Macr fr dng s s avalable frm Wd & Brwn, 1994). See Hershberger (1998) fr cde fr fng hs mdel. Manuscrp Table 2 lss whch persn-rened prncples are esable wh he dynamc facr mdel. Nex we gve examples f hw hese prncples culd be esed. (1) Paern summary prncple. If we had p varables n ccasns fr mre han ne persn, we culd dene hs as y = Λη + ε where ε ~ N( 0, Θ ). (19) η = γτ + ς where ς ~ N( 0, Φ ). (20) Then we culd es wheher here s evdence f measuremen nvarance f nra-ndvdual prcesses acrss persns. Tha s, we culd es (a) H : he same number f prcess facrs q s bes-fng acrss persns. If s, we culd es (b) H : Λ = Λ, (.e. ha he magnude f lag-0 and lag-1 ladngs n Λ s equal acrss persns). If s, we culd es (c) H : Θ = Θ, (.e. resdual varances n Θ are equal acrss persns). If he abve hree hypheses (a)-(c) were suppred whn grups f persns, bu n acrss grups f persns, hs yelds evdence fr he paern summary prncple. (2) Paern parsmny prncple. The paern-parsmny prncple s suppred he exen ha he number f grups (.e. number dfferen bes-fng mdels) s much less han he number f persns whse daa were mdeled. A11

12 Appendx: Persn-rened mehds and hery (3) Indvdual-specfcy prncple. We may furher es wheher srucural parameers, fr example, sll vary acrss persns whn each grup,.e. H : Φ = Φand H : γ = γ, whch wuld be ndcave f sme remanng ndvdual-specfcy. Als, f ceran ndvduals have her wn unque bes-fng dynamc facr mdel, hs supprs he ndvdual specfcy prncple. (4) Inerndvdual dfferences/nrandvdual change. Alhugh n varance rends are allwed here, f mean rends were fund and ncluded n he mdel, we culd es wheher hese nrandvdual mean changes had nerndvdual varably wh H : γ = γ. A12