Chapter 5 Multlevel Models 5.1 Cross-sectonal multlevel models 5.1.1 Two-level models 5.1.2 Multple level models 5.1.3 Multple level modelng n other felds 5.2 Longtudnal multlevel models 5.2.1 Two-level models 5.2.2 Multple level models 5.3 Predcton 5.4 Testng varance components Appendx 5A Hgh order multlevel models
Multlevel Models Multlevel models - a condtonal modelng framework that takes nto account herarchcal and clustered data structures. Used extensvely n educatonal scence and other dscplnes n the socal and behavoral scences. A multlevel model can be vewed as a lnear mxed effects model and hence the statstcal nference technques ntroduced n Chapter 3 are readly applcable. By consderng multlevel data and models as a separate unt we expand the breadth of applcatons that lnear mxed effects models enoy. Also known as herarchcal models
5.1 Cross-sectonal multlevel models Two-level model example Level 2 (Schools) Level 1(Students wthn a school) Level 1 Model (student replcatons ) y = β 0 + β 1 z + ε y - student s performance on an achevement test z - total famly ncome Level 2 Model Thnkng of the schools as a random sample we model {β 0 β 1 } as random quanttes. β 0 = β 0 + α 0 and β 1 = β 1 + α 1 where α 0 α 1 are mean zero random varables.
Two-level model example The combned level 1 and level 2 models form: y = (β 0 + α 0 ) + (β 1 + α 1 ) z + ε = α 0 + α 1 z + β 0 + β 1 z + ε. The two-level model may be wrtten as a sngle lnear mxed effects model. Specfcally we defne α = (α 0 α 1 ) z = (1 z ) β = ( β 0 β 1 ) and x = z to wrte y = z α + x β + ε. We model and nterpret behavor through the successon of level models. We estmate the combned levels through a sngle (lnear mxed effects) model.
Two-level model example - varaton Modfy the level-2 model so that β 0 = β 0 + β 01 x + α 0 and β 1 = β 1 + β 11 x + α 1 where x ndcates whether the school was a Catholc based or a publc school. The combned level 1 and (new) level 2 models form: y = α 0 + α 1 z + β 0 + β 01 x + β 1 z + β 11 x z + ε. That we can agan wrte as y = z α + x β + ε. by defnng α = (α 0 α 1 ) z = (1 z ) β = ( β 0 β 01 β 1 β 11 ) and x =(1 x z x z ). The term β 11 x z nteractng between the level-1 varable z and the level-2 varable x s known as a cross-level nteracton. Many researchers argue that understandng cross-level nteractons s a maor motvaton for analyzng multlevel data.
Three Level Models Level 1 model (students) y k = z 1k β + x 1k β 1 + ε 1k The predctors z 1k and x 1k may depend on the student (gender famly ncome and so on) classroom (teacher characterstcs classroom facltes and so on) or school (organzaton structure locaton and so on). Level 2 model (classroom) β = Z 2 γ + X 2 β 2 + ε 2. The predctors Z 2 and X 2 may depend on the classroom or school but not students. Level 3 model (School) γ = Z 3 β 3 + ε 3. The predctors Z 3 may depend on the school but not students or classroom.
Combned Model The combned level 1 2 and 3 models form: y k = z 1k ( Z 2 (Z 3 β 3 + ε 3 ) + X 2 β 2 + ε 2 ) + x 1k β 1 + ε 1k = x k β + z k α + ε 1k where = 3 2 1 β β β β = k k k k 1 2 3 1 2 1 z Z Z z X x x = k k k 1 2 1 z Z z z = 3 2 ε ε α
Motvaton for multlevel models Multlevel modelng provdes a structure for hypotheszng relatonshps n a complex system. The ablty to estmate cross-level effects s one advantage of multlevel modelng when compared to an alternate research strategy callng for the analyss of each level n solaton of the others. Second and hgher levels of multlevel models also provde use wth an opportunty to estmate the varance structure usng a parsmonous parametrc structure. One typcally assumes that dsturbance terms from dfferent levels are uncorrelated.
5.2 Longtudnal multlevel models Smlar to cross-sectonal multlevel models except: Use a t subscrpt to denote the Level 1 replcaton for tme Allow for correlaton among Level 1 observatons to represent seral patterns. Possbly nclude functons of tme as Level 1 predctors. Typcal example use students as Level 2 unt of analyss and tme as Level 1 unt of analyss. Growth curve models are a classc example: we seek to montor the natural development or agng of an ndvdual. Ths development s typcally montored wthout nterventon and the goal s to assess dfferences among groups. In growth curve modelng one uses a polynomal functon of age or tme to track growth.
Example Dental Data Orgnally due to Potthoff and Roy (1964); see also Rao (1987). y s the dstance measured n mllmeters from the center of the ptutary to the pterygomaxllary fssure. Measurements were taken on 11 grls and 16 boys at ages 8 10 12 and 14. The nterest s n how the dstance grows wth age and whether there s a dfference between males and females.
Measure 32 30 28 26 24 22 20 18 16 8 10 12 14 Age Fgure 5.1 Multple Tme Seres Plot of Dental Measurements. Open crcles are grls sold crcles are boys
Dental Model Level 1 model y t = β 0 + β 1 z 1t + ε t z 1t s age. Level 2 model β 0 = β 00 + β 01 GENDER + α 0 and β 1 = β 10 + β 11 GENDER + α 1. GENDER - 1 for females and 0 for males.
Three Level Example Chldren Mental Health Assessment by Guo and Hussey (1999) The Level 1 model (tme replcatons) s y t = z 1t β + x 1t β 1 + ε 1t Assessment y s the Deveroux Scale of Mental Dsorders a score made up of 111 tems. x 1t = PROGRAM t -1 f the chld was n program resdence at the tme of the assessment and 0 f the chld was n day treatment or day treatment combned wth treatment foster care. z 1t = (1 TIME t ). TIME t s measured n days snce the ncepton of the study. Thus the level-1 model can be wrtten as y t = β 0 + β 1 TIME t + β 1 PROGRAM t + ε 1t.
Chldren Mental Health Assessment The level 2 model (chld replcatons) s β = Z 2 γ + X 2 β 2 + ε 2 where there are =1 n chldren and = 1 J raters. The level 2 model of Guo and Hussey can be wrtten as β 0 = β 00 + β 001 RATER + ε 2 and β 1 = β 20 + β 21 RATER. The varable RATER = 1 f rater was a teacher and = 0 f the rater was a caretaker. The level 3 model (rater replcatons) s γ = Z 3 β 3 + ε 3 β 00 = β 000 + β 010 GENDER + ε 3.
5.3 Predcton Recall that we estmate model parameters and predct random varables. Consder a two-level longtudnal model Level 1 model (replcaton on tme) y t = z 1t β + x 1t β 1 + ε 1t Level 2 model - β = Z 2 β 2 + α. Lnear mxed model s y t = z 1t (Z 2 β 2 + α )+ x 1t β 1 + ε 1t The best lnear unbased predctor (BLUP) of β s b BLUP = a BLUP + Z 2 b 2GLS wherea BLUP = D Z V -1 (y - X b GLS ).
Three-Level Model Predcton Estmate model parameters Next compute BLUP resduals Use the formula a BLUP = D Z V -1 (y - X b GLS ). Ths yelds the BLUPsfor α = (ε 2 ε 3 ) say a BLUP = (e 2BLUP e 3BLUP ). Then compute BLUP predctors of γ and β g BLUP = Z 3 b 3GLS + e 3BLUP b BLUP = Z 2 g BLUP + X 2 b 2 GLS + e 2BLUP. Forecasts are also straghtforward: for AR(1) level-1 dsturbances ths smplfes to yˆ T + L = z1 T + L b BLUP + x1 T + L b1 GLS + ρ L e T BLUP
5.4 Testng varance components For the error components model do we wsh to pool? We can express ths as an hypothess of the form H 0 : σ α2 = 0. For a two-level model do the data provde evdence that our 2 nd level model s vable? We may wsh to test H 0 : Var α = 0. Unfortunately the usual lkelhood rato testng procedure s not vald for testng many varance components of nterest. In partcular the concern s for testng parameters where the null hypothess s on the boundary of possble values. That s 0 σ 2 α < σ α2 = 0 s on the boundary. Usual approxmatons are not vald when we use the boundary restrcton n our defnton of the estmator. As a general rule the standard hypothess testng procedures favors the smpler null hypothess more often than t should.
Alternatve Testng Procedures Suppose that we wsh to test the null hypothess H 0 : σ 2 = σ 02 where σ 02 s a known postve constant. Standard lkelhood rato test s okay. Ths procedure s not avalable when σ 2 0 = 0 because the log-lkelhood under H 0 s not well defned. However H 0 : σ 2 = 0 s stll a testable hypothess A smple test s to reect H 0 f the maxmum lkelhood estmator 1 n 2 n y = 1 exceeds zero. Ths test procedure has power 1 versus all alternatves and a sgnfcance level of zero a good test!!!
Error Components Model Consder the lkelhood rato test statstc for assessng H 0 : σ 2 α = 0. 1 The asymptotc dstrbuton turns out to be χ 2 (1) 2 Typcally the asymptotc dstrbuton of the lkelhood rato 2 test statstc for one parameter s χ (1) Ths means that usng nomnal values we wll accept the null hypothess more often than we should; thus we wll sometmes use a smpler model than suggested by the data. Suppose that one allows for negatve estmates. Then the 2 asymptotc dstrbutons turns out to be the usual χ (1)
Recommendatons No general theory s avalable. Some addtonal theoretcal results are avalable. Ths can be mportant for some applcatons. Smulaton methods are always possble. You have to know what your software package s dong It may be gvng you the approprate test statstcs or t may gnore the boundary ssue. If t ncorrectly gnores the boundary ssue the test procedures are based towards smpler models.