Multilevel Analysis with Informative Weights
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1 Secton on Survey Research Methods JSM 2008 Multlevel Analyss wth Informatve Weghts Frank Jenkns Westat, 650 Research Blvd., Rockvlle, MD Abstract Multlevel modelng has become common n large scale assessments wth multstage samplng and unequal probabltes of selecton (Raudenbush, 2000, Koretz & McCaffrey, 200). Pseudo-lkelhood estmaton wth parttoned weghts produce asymptotcally unbased estmates of parameters n many applcatons (Pfeffermann, et al, 998). However, a case often found n evaluatons, longtudnal data of ndvduals nested wthn groups, has gotten lttle scrutny. In ths paper smulaton studes wll be used to llustrate the effect of nformatve weghts wth nonresponse correctons on parameter estmaton n the nested longtudnal case wth correlated ndvdual-level effects. Contnuous and categorcal outcomes wll be examned. Key Words: Samplng weghts, multlevel, longtudnal. Introducton Multlevel models are a natural choce for analyzng large-scale clustered multstage samples. These models account for clusterng of responses n estmatng standard errors of regresson coeffcents. They also allow for modelng dfferent effects for dfferent groups of students (.e. random group effects) and the predcton of group effects from group-level covarates (.e. specfyng fxed cross-level nteractons). A problem wth usng multlevel models n large-scale studes s the proper ncorporaton of samplng weghts nto the analyss. In a sngle-level regresson, samplng weghts are appled to sums of squares and cross products. Wth a multlevel analyss the standard use of weghts leads to nconsstent estmates of fxed and random effects (Koretz & McCaffrey, 200). Recently, technques have been developed to apply samplng weghts to the lkelhood. The resultng multlevel pseudolkelhood s maxmzed to yeld maxmum lkelhood estmates of model parameters (see Pfeffermann, et al, 998). The resultng estmates are asymptotcally unbased over a broad range of lkely analyss scenaros. Smulaton studes to gauge parameter recovery of multlevel software that utlzes the pseudo maxmum lkelhood approach, have usually focused on a lmted range of models: Models wth only two levels of nestng Wth categorcal outcomes, usually only the ntercept term s modeled at hgher levels of nestng. In response to ths lmtaton, the current study wll examne how a popular multlevel analyss program, HLM (Raudenbush, et al, 2006) recovers parameters n a typcal model of ndvdual growth. Examples of contnuous and categorcal outcomes wll be examned n subsequent sectons. HLM results wll be compared to the adaptve quadrature approaches of SAS NLMxed and Stata s GLLAMM programs. 2226
2 Secton on Survey Research Methods JSM The Generatng Model The generatng model for the contnuous outcome case s depcted n fgure, Level : Tme w/n Person y = π + π * Tme + e t 0 t t Level 2 : Person w/n School π0 = β0 + ζ 0 π = β0 + β * Treat + ζ Level3 : School β0 + γ000 + γ00 * SES + ζ Fgure : Generatng model for contnuous outcome case Where y s the outcome for person n group at tme t, ζ and ζ are the random effects for the ntercept and slope at level 2, 0 ζ s the random effect of the ntercept at level 3 and the level 2 random effects are correlated. 0 2 The Varances of the random effects are: var( e ) = σ, var( ζ ) = ψ, var( ζ ) = ψ, and var( ζ ) = ψ. The model s t 0 0 standardzed so the varance parttonng s n terms of percent of total varance and the sze of fxed effects s n terms of standard devaton unts. The factors assumed for the model are based on a couple of large scale, early chldhood longtudnal evaluatons..2 Multlevel Lkelhood ML estmaton requres the evaluaton of a multlevel level Lkelhood nvolvng complex ntegraton over the random effects. In the normal case an explct soluton s avalable for the ntegraton (Pnhero & Bates, 2000) so that parameters can be maxmzed wth respect to ths lkelhood. The multlevel lkelhood s calculated for each level of nestng. Frst there s the lkelhood of the repeated t measures of person, = T L(y ) Py ( t, tt, ) t = % ζ θ ζ, where θ are the ndvdual lnear growth parameters for the person and t s an ndcator of tme. Next, to get the ndependent contrbuton of a person to the lkelhood, person-level random effects are ntegrated out. L(y ) (L(y ζ ) φ( ζ ) dζ, % % where φ (.) s the normal densty functon. The condtonal lkelhood for a school s the product of person lkelhoods, = n L( school ζ m ) L(y ) = = % And the ndependent contrbuton of each school to the lkelhood s assessed by ntegratng out school-level random effects, L( school ) = (L( school ζ ) φ( ζ ) dζ, m wth the fnal lkelhood beng the product of school lkelhoods: 2227
3 Secton on Survey Research Methods JSM 2008 J = = L L( school ) Combnng the expresson for the person and school lkelhoods yelds, J n L = { [ ( L(y ζ ) φ( ζ ) dζ ) ] φ( ζ ) dζ } = = %.3 Weghted Analyss Pfeffermann, et al (998) devsed a scheme for applyng samplng weghts to multlevel samples by defnng a pseudolkelhood. The procedure s the followng:. Partton the weghts by the varous levels of the model (e.g. defne separate person wthn group & group weghts) 2. Normalzed the weghts at each level. In one common approach the weghts are normalzed so that they sum to the number of unts. Ths normalzaton scheme s the one adopted by several packages (e.g. HLM, LISREL) because t produces relatvely unbased results for fxed and random effects n smulaton studes (Pfeffermann, et al, 998). 3. Apply normalzed weghts to each level of the multlevel lkelhood to create a pseudo lkelhood. The combned the pseudo lkelhood s defned by applyng the weghts as exponents of the person and school lkelhoods. These weghts are the exponents w and w n equaton, w w ζ φ ζ ζ φ ζ ζ, () J n L = { [ ( L(y ) ( ) d ) ] ( ) d } = = % where w s the student weght condtonal on membershp n school, and w s the school weght. Pseudo maxmum lkelhood estmates of parameters are acheved by maxmzng the pseudolkelhood, whch s accomplshed n HLM wth an applcaton of the EM algorthm. 2. Defnng Informatve Weghts and Nonresponse Adustments 2. Informatve Weghts In the study we smulated nformatve selecton probabltes wth a logstc functon of the outcome followng Asparouhov (2006) and others. If the outcome for an ndvdual or group s x, the selecton probablty s gven by the followng, π = + e x, where x s the outcome and π s the selecton probablty. Usng a logstc functon returns a value between 0 and, nterpretable as a probablty, that s correlated wth the outcome, x. The samplng weght s defned as the nverse of the selecton probablty: w = / π. In a smple random sample the correlaton of the outcome wth the weght wll be about.80. The degree of correlaton between the outcome and the weght can be controlled by substtutng x * for x n the logstc functon, where x * s a varate havng a pre-defned correlaton wth x. For example f the correlaton of x * and x s set to.80, the correlaton of the orgnal outcome, x wth the weght wll be about.62. An ndex of nformatveness of the weghts can be estmated from the data. In the present study we use an ndex somewhat dfferent from that suggested n Asparouhov & Muthen (2006). The ndex we use s 2228
4 Secton on Survey Research Methods JSM 2008 I = θ ˆ θ / θ, θ where θ s the true parameter (e.g. a regresson weght) and ˆ θ s the weghted estmate of the parameter. Ths ndex s only useful n smulatons snce the true parameter must known. The present study s conducted by smulatng nformatve student wthn school and school selecton probabltes. Indvduals and groups are selected accordng to these probabltes and samplng weghts are defned as the nverse of the selecton probabltes. These weghts are normalzed (wthn the HLM program) and appled to lkelhood as n equaton Nonresponse Weghts Nonresponse weghts for ndvduals are used n most large scale studes to mtgate the bas of nonrandom nonresponse. In the herarchcal modelng context, the use of nonresponse weghts volates the assumpton that weghts for ndvduals wthn a cluster (e.g. school) are condtonal only on cluster membershp. Nonresponse adustments are calculated as a functon of the response rates of demographc groups wthn the populaton. These weghts are then appled to ndvdual-wthn-cluster weghts and the resultng weghts are no longer strctly condtonal on cluster membershp. The smulaton study wll address the queston of whether the use of nonresponse weghtng factors bases parameter estmates n multlevel models. To smulate nonresponse weghts, the followng steps are taken. Frst ndvduals are sorted nto sample octles of the outcome, smulatng the classfcaton of ndvduals nto demographc nonresponse cells. Indvduals are selected n each octle accordng to an a pror cell nonresponse probablty. The nonresponse weght factor s the nverse of ths probablty, whch s multpled wth the ndvdual-wthn-cluster weght. Note that there s no classfcaton error n the way of smulatng nonresponse adustments, snce the focus was on whether multlevel modelng estmates were robust to the use of nonresponse weghts 3. Smulatons wth Informatve Samplng Weghts and a Contnuous Outcome In ths secton we present results for a smulaton of a normal categorcal outcome accordng to the generatng model n fgure. The parameters for all smulatons are lsted n table. The factors are n table 2, consstng of effect sze (small or large), number of tme ponts (3 or 6), ndvduals per school (8 or 24), level of nformatveness of the weghts (hgh or low), and whether nonresponse weghts were used, and. Ths results n 32 cells, smulated to a depth of 500 repettons. Table : Parameters of smulatons Level Varance random Varance random Correlaton level 2 Varance of level varance effects ntercept effects slope random effects 3 random effects N of schools Parameter 2 σ ( 2) ( 2) ψ o ψ ζ 0, ζ ( 3 ψ ) J Values Table 2: Factors vared n the smulaton Number of Fxed effect level Fxed effect level 2 Fxed effect level 3 tme ponts N per school Informatve wts: h/low Factor Tme Treat SES n t n I Value Value
5 Secton on Survey Research Methods JSM Bas n Parameters Table 3 shows relatve bas for four fxed and four varance parameters for the varous factors n the smulaton. The relatve bas s the bas as a proporton of the parameter value (except for the ntercept, where the denomnator s fxed to ). Table 3: Relatve bas n parameters for man effects of smulaton factors-contnuous outcome Bas fxed effect parameters Bas varance parameters Intercept Tme Treat SES Level Level 2 Level 2 Level 3 Nonresponse Wt Sgma Ps0 Ps Ps Yes No Effect sze Small Large Group sze T3/N T3/N # Tme ponts* T3/N T6/N Informatve weghts Hgh Low * The combnaton T6/n24 was not ncluded, snce T6 and N24 separately yelded relatvely unbased results. Although the results are not 00 percent consstent some general trends are apparent. All smulaton factors reduce bas n the expected drectons,.e. larger sample sze, usng less nformatve weghts and not usng nonresponse weghts decrease bas. The effect of specfc factors can be summarzed below:. Informatveness: a) When weghts are hghly nformatve, usng weghts dramatcally reduces the bas n fxed effects. b) When weghts are of low nformatveness, unweghted estmates have lttle bas but level 3 varances are based upward to some degree. 2. N of Cases: Increasng the number of tme ponts s more benefcal for decreasng bas than a smlar ncrease n the number of groups. Ths s probably because; regardng the effect of most nterest, treatment, and the unt of assgnment s the ndvdual rather than the group. 3. Nonresponse Weghts: Fxed effect estmates are robust to the use of nonresponse weghts. Varance estmates show a small ncrease n bas when these weghts are used. 3.2 Power Table 4 shows the power of the fxed effects assocated wth the varous desgn factors. Power s smply the percent of the 500 smulatons n whch the effect was sgnfcant. As wth bas, the desgn factors ncrease power n expected drectons,.e. larger sample sze, usng less nformatve weghts and not usng nonresponse weghts ncrease power. Note that snce the ntercept s zero, smaller percent sgnfcant s better. Also, the tme effect was large enough that t was always sgnfcant, not an unexpected result for early chldhood longtudnal studes. When weghts are relatvely unnformatve, there s a large loss of power when usng weghts for a small reducton n bas. 2230
6 Secton on Survey Research Methods JSM 2008 Table 4: Power for fxed effects by smulaton factors-contnuous outcome Intercept % Tme % Treat % SES % Nonresponse Wt Yes No Effect Sze Small Large Group Sze T3/N T3/N #Tme Ponts T3/N T6/N Informatveness Hgh Low Smulatons wth Informatve Samplng Weghts and a Dchotomous Outcome Pseudolkelhood estmaton n multlevel models when the outcome s dchotomous s problematcal because the there s no explct soluton to the multvarate ntegraton of the lkelhood. There are three approxmatons that can be appled n ths case:. Penalzed Quas-Lkelhood (PQL): Ths nvolves Lnearzng the dscrete outcome wth a Taylor seres expanson (2 nd order). Then the outcome s treated as f t was contnuous and the standard EM estmaton algorthm s appled. 2. Multvarate Laplace Approxmaton: In ths approach, the ntegrand of the lkelhood s approxmated wth a Taylor expanson. For the HLM program Raudenbush and colleagues derved a 6 th order Taylor seres expanson of ntegrand (Raudenbush, et al, 2000). Ths s relatvely fast n computng tme and accurate for moderately large sample szes. PQL s used for startng values. 3. Adaptve Gauss Quadrature (AGQ): The ntegraton s approxmated by performng multdmensonal dscrete quadrature. The great gan n accuracy s pad for n computng tme, whch can become ntractable when there are several nested levels and multvarate random effects. 4. Generatng Model for Dscrete Outcome Smulaton Fgure two gves the generatng model for the multvarate logstc regresson. The outcome s dchotomous and there s a logt lnk functon. The rest of the model s the same as for the contnuous case. Level: y ~ Bernoull( p ) t Logt( p ) = π + π * Tme Level 2: π = β + ζ 0 00 t 0 0 π = β + β * Treat + ζ 0 Level 3: β = γ + γ * SES + ζ t t Fgure 2: Generatng model for the dchotomous outcome smulaton 223
7 Secton on Survey Research Methods JSM 2008 Problems were encountered n usng the Laplace approxmaton n HLM for a 3-level model. The only 3-level model estmable was one wthout a random slope effect, ζ, at level 2. As a result we explored the behavor of HLM estmates wth a 2-level model that had correlated random effects at level 2. The 3-level model was explored the Stata GLLAMM software, whch uses an adaptve quadrature. Table 5 gves the result of the 2-level smulaton run wthout weghts compared wth the adusted Gauss approxmaton approach usng the SAS Proc NLMxed program. Table 5: Bas n unweghted estmates for 2-level smulaton- HLM compared to NLMxed (2-Level results: No weghts: Proportonal bas (bas/theta)) (Intercept=0, Tme=.5, Treat=.08, V(Intercept)=.50, V(Slope)=.5, R=.70. NSchools= 200) Model HLM HLM HLM NLMxed Laplace Laplace Laplace AGQ # Tm Pts Bas Fxed Intercept Tme Treatment Bas Var. ψ ψ HLM Laplace approxmaton only performs satsfactorly wth 2-tme ponts. Whle the adaptve Gauss-Hermte quadrature performs well wth only 3-tme ponts. The 3-level weghted model was explored wth GLLAMM (Rabe-Hesketh & Skrondal, 2008). Table 6 gves the bas of estmates wth 3-tme ponts. However, the computng tme for ths model was at least 00 tmes that of HLM. A modest smulaton as a proof of concept was conducted wth ust 00 repettons and 25 groups (vs. 200 n the prevous smulaton). Results are mxed. Frst consder a smulaton wth three tme ponts, labeled as (T3) n the table. Usng weghts yelds consderable bas n the ntercept and the level 3 varance. But the parameters of most nterest, the fxed effects and the level 2 varances, are estmated well consderng relatvely small sample sze (000 cases n 25 groups). The last two columns lst the results when 6 tme ponts per person were smulated. In ths case, there was about /3 less bas n the ntercept and level 3 varance estmates. Note that unequal group sze was mposed by the nonresponse selecton. However GLLAMM doesn t normalze the weghts as s recommended by Pfeffermann, et al (998). An analyss wth pre-normalzed weghts would probably yeld less based results. Table 6: Gauss quadrature estmates of 3-level logstc regresson va GLLAMM Effect Name True Value Estmate (T3) Bas (T3) Estmate (T6) Bas (T6) Fxed Intercept Tme Treat SES Random ψ ψ ψ
8 Secton on Survey Research Methods JSM Conclusons For contnuous outcomes wth three tme ponts, HLM yelds relatvely unbased estmates of fxed effects but demonstrates negatve bas for level 2 varances and postve bas for Level 3 varance. Power for fxed effects at levels 2 and 3 are poor except when group szes are 24 or there are 6 or more tme ponts. Fnally, when weghts are not hghly nformatve, unweghted analyss results n lttle bas and more power than weghted analyss. Surprsngly, estmates were farly robust wth respect to the use of nonresponse weghts. For dchotomous outcomes, analyss usng adaptve quadrature outperforms Laplace approxmaton n unweghted two-level analyses. For weghted 3-level analyses wth the standard longtudnal model explored n ths paper, adaptve quadrature va GLLAMM performs reasonably well even wth relatvely small sample szes. We conclude that unless the number of tme ponts or ndvduals s large, adaptve quadrature s the recommended analyss approach for dchotomous outcomes, even though there s a large computng overhead. It s dffcult to assess how generalzable these results are. For longtudnal data and two levels of nestng, the number of possble models s unlmted, wth dfferent numbers and magntudes of fxed and random effects. It s recommended that for large scale evaluatons, ssues of power and bas should be explored through smulaton studes smlar to those presented n ths paper. Plausble ranges of effect szes and varances of random effects can be posted. Also estmates of the nformatveness of weghts as well as the magntude of nonresponse can be bult nto smulatons to gve plausble ranges of power and bas for a partcular study. Effcent easy to use utlty programs need to be devsed that smulate data for a wde range of longtudnal desgns. For dchotomous outcomes the assessment of power s made problematcal by the large computng tme assocated wth adaptve quadrature. However, the relatve effcency of dfferent desgns can be explored usng HLM and Laplace approxmatons, and fnal model can be assessed wth adaptve quadrature analyss. References Asparouhov, T. (2006). General multlevel modelng wth samplng weghts, Communcaton n Statstcs- Theory and Method, 35, Asparouhov, T. & Muthen, B. (2006). Multlevel modelng of complex survey data, Proceedngs of the Amercan Statstcal Assocaton, Seattle, WA: Amercan Statstcal Assocaton. Koretz, D. & McCaffrey, D. (200). Usng TIMSS to analyze correlates of performance varaton n mathematcs, US department of educaton, OERI, Workng Paper seres, No Pnhero, J.C. & Bates, D. M. (2000). Mxed-Effects Models n S and S-Plus, Sprnger. Pfeffermann, D.; Sknner, C., Goldsten, H. & Rasbash, J. (998). Weghtng for unequal selecton probabltes n multlevel models, Journal of the Royal Statstcal Socety, Seres B, 60, part, pp Rabe-Hesketh, S. and Skrondal, A. (2008). Multlevel and Longtudnal Modelng Usng Stata. (Second Edton). College Staton, TX: Stata Press. Raudenbush, S.W. (2000). Syntheszng Results for NAEP Tral State Assessment. In Grssmer, D.W. and Ross, Mchael (Ed.), Analytc Issues n the Assessment of Student Achevement, Washngton, DC: Natonal center for Educatonal Statstcs. Raudenbush, S.; Bryk, A.; Cheong, Y.; & Congdon, R.T. (2006). HLM 6: Herarchcal lnear and nonlnear modelng. Chcago: Scentfc Software Internatonal. Raudenbush, S.; Yang, M. & Yosef, M.(2000). Maxmum lkelhood for generalzed lnear models wth nested random effects va hgh-order, multvarate Laplace approxmaton, Journal of computatonal and Graphcal Statstcs, 9(),
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