Estimation and Centering

Transcription

1 Estimation and Centering PSYED 3486 Feifei Ye University of Pittsburgh

2 Main Topics Estimating the level-1 coefficients for a particular unit Reading: R&B, Chapter 3 (p85-94) Centering-Location of X Reading Chapters 2 (p31-35) & 5 (p ) Kreft1995 Hofmann DA & Gavin MB (1998) Centering decisions in hierarchical linear models: Implications for research in organizations Journal of Management, 23, Estimation and Centering p1/32

3 General Two-Level HLM Model Level 1: p Y ij = β 0j + β kj X kij + r ij (1) k=1 r ij N(0,σ 2 ) (2) Level 2: β kj = γ k0 + q γ kl W lj + u kj, for k = 0,1,2,,p (3) l=1 Estimation and Centering p2/32

4 General Two-Level HLM Model u 0j u 1j u 2j u pj N , τ 00 τ 01 τ 02 τ 0p τ 10 τ 11 τ 12 τ 1p τ 20 τ 21 τ 22 τ 2p τ p0 τ p1 τ p2 τ pp (4) and independent between macro units and independent of r ij (5) Estimation and Centering p3/32

5 Types of Parameters Fixed effects (γs) Variance-covariance components σ 2 τ 00, τ 11, τ 01 etc Random level one coefficients (βs) Estimation and Centering p4/32

6 Methods of Estimation Generalized Least Squares for fixed effects (γs) Maximum Likelihood for variance-covariance components FIML: Full Maximum Likelihood One shortcoming of the FEML is that the estimates are conditional on point estimates of the fixed effects REML: Restricted Maximum Likelihood REML estimates adjust for uncertainty about the fixed effects; FIML estimates do not If J is large, the two procedures are often similar If J is small, the FIML estimates for variance components tend to be smaller than the REML estimates therefore, the FIML components are biased in small samples Empirical Bayes for the randomly varying level 1 coefficients (βs) Estimation and Centering p5/32

7 Reliability Definition Reliability = λ kj = Reliability( ˆ β kj ) = Variance of True Scores Variance of observed scores (6) τ kk τ kk + σ 2 /n j, for k = 0,1,,p (7) Estimation and Centering p6/32

8 Factors impacting reliability Larger amounts of between school variance (relative to within school variance) increase reliability Larger n j s give you higher reliability Larger ICCs increases reliability: Larger ICC means that within school group variance is small relative to between group variability The reliability will be higher when the group means vary substantially across level-2 units (holding constant the sample size per group) So increasing group size, increasing homogeneity within clusters, and increasing heterogeneity between clusters will increase reliability Very low reliabilities ( 10) may indicate that a random coefficient could be fixed in subsequent analyses (HLM 6 Users Manual, p 82) Or, 05 in R&B, page 125 Estimation and Centering p7/32

9 Consequence of Reliability for Empirical Bayes Estimation Different groups have different precision (reliability) in the estimation of the group mean and level-1 slopes EB estimates borrows strength from information contained in other units or groups EB estimates optimally combine two alternative but feasible estimates for the level-1 coefficients by weighting the estimates based on the reliability (precision) of the coefficients substantially across level-2 units (holding constant the sample size per group) Estimation and Centering p8/32

10 EB estimates EB estimate of β 0j for random effects ANOVA β0j = λ j Ȳ j + (1 λ j ) γˆ 00 (8) shrinkage estimators : the Y-bar estimate is shrunk towards the model-based estimate The higher the reliability, the less shrinkage; the lower the reliability, the more shrinkage Reliability of 1= No shrinkage; Reliability of 0 = Complete Shrinkage Estimation and Centering p9/32

11 Empirical Bayes Estimates EB estimate is based not only on its own data but also takes into account estimates for other groups and the characteristics groups share Shrinkage estimates of level 1 coefficients- EB shrinks the group specific estimate towards the overall mean (although when the overall mean is greater than the group specific estimate, the shrunken or empirical Bayes estimate may actually be greater than the group specific estimate) Lower Reliability = Greater Shrinkage Estimation and Centering p10/32

12 Empirical Bayes Estimates Useful in estimating parameters for a group with few within group observations Weighted average shifts the group specific estimate toward the mean for similar groups The less precise the group specific estimate and the less the variability observed across groups, the greater the shift towards the overall group mean a a From Rouxs Glossary of Multilevel Termshttp://wwwpahoorg/English/DD/AIS/be v24n4-multilevelhtm Estimation and Centering p11/32

13 Centering Variables Centering refers to the act of shifting the location of a predictor to some other value such as zero by adding or subtracting a constant In standard linear regression, centering variables leads to the same model (ie, same fit), but centering affects the interpretation of the lower-order terms In a model with only main effects, centering of predictors influences only the interpretation of the intercept In a model that includes interactions, centering of predictors also influences the interpretation of the main effects Estimation and Centering p12/32

14 Effects of Centering Impacts the estimation of parameters Affects their variances and covariances Alters their interpretation No simple answer or one way to always approach centering decisions Make decisions early, before the analysis, to avoid misinterpretation, misspecification of the model, or drawing conclusions that dont mirror the overall goals for the analysis Estimation and Centering p13/32

15 What is Centering Centering issues usually focus on how the level one explanatory variables are treated in the model Usually done for continuous IVs although it is possible to center dummy coded variables For level 1 predictor, there are typically three choices Raw-score or uncentered data (no transformation RAW): X ij Group-mean centering (centering within contexts CWC): X ij X j Grand-mean centering (centering around the grand mean CGM): X ij X Estimation and Centering p14/32

16 Raw Scores Maintaining the raw scale for level 1 predictors can be useful if the predictors have a natural and meaningful scale The intercept of a model (here β 0j ) is always interpreted as the expected value when all predictors are equal to zero If zero is outside of the logical score range of the predictor, the intercept will be difficult to interpret eg, SAT can serve as a predictor The intercept will be the expected outcome for a student in school j who has an SAT of zero Intercept parameter is meaningless because the lowest SAT score is 200 Correlation between the intercept and slope will tend toward -10 The intercept is essentially determined by the slope Schools with strong positive SAT-outcome slopes will tend to have very low intercepts Schools with negligible slopes will have higher intercepts Meaningful X value of zero: egl, drug dosage Estimation and Centering p15/32

17 Grand-mean Centering Under grand mean centering, we subtract the overall mean from each persons value of the IV: X ij X The intercept, β 0j, is the expected outcome for a subject whose value on X ij is equal to the grand mean This is the standard choice of location for X ij in the classical ANCOVA model Grand-mean centering yields an intercept that can be interpreted as an adjusted mean for group j β 0j = µ Yj β 1j ( X j X ) (9) Variance of intercept, τ 00, is the variance among the level 2 units in the adjusted means (adjusted for level one predictors) Estimation and Centering p16/32

18 Grand-mean Centering Y X = 0 GM X Estimation and Centering p17/32

19 Group-mean Centering Center the original predictors around their corresponding level 2 unit means: X ij X j The zero point for the predictor differs for each group Two individuals may have the same X ij, but X ij X j will be dependent on group mean The group intercept, β 0j, is the unadjusted mean for the group β 0j = µ Yj (10) Variance of intercept, τ 00, is the variance among the level 2 unit means (µ Yj ) Estimation and Centering p18/32

20 Group-mean Centering Y Group Means X = 0 X Predictions for each GROUP when X is at the GROUP MEAN Estimation and Centering p19/32

21 Group-mean Centering In organizational research (using HSB as an example), coefficient of interest is the pooled within-group relationship between math achievement and student SES We want to estimate the level-one relationship net of any group membership effects That means we want an unbiased estimate of the slope for SES on math achievement BUT: frog pond effect Estimation and Centering p20/32

22 Frog Pond Effects The overall SES for a school may also be related to the outcome of math achievement For the same frog, the effect of being in a pond with big frogs is different from the being in a pond with small frogs The effect of SES on math achievement is dependent on relative standing of the student within his or her own school Group mean centering addresses the frog pond effect, but it also changes the model in a complex way Estimation and Centering p21/32

23 Group-mean Centering We need to consider how the aggregate of the person-level SES is related to the outcome, even after controlling for the effect of individual SES Compositional effect: difference between the group or organizational level effect and the person-level effect When aggregate is added back in at level two, we can get estimates of both effects through group mean centering Group mean centering removes between group variability from the model (since we are using as predictors just the deviations from the group mean) Thus, to examine contextual effects we need to reintroduce the means at level two, otherwise the between and within group relationship is confounded (only looking at relative standing) Estimation and Centering p22/32

24 Within- and between-group effects Y between-group regression line regression line within group 2 regression line within group 3 regression line within group 1 X Estimation and Centering p23/32

25 Confounding of Within- and between-group effects Raw and Grand mean centering: within-group and between-group effects are confounded when Level 1 predictors are raw scale or grand-mean centered Combined model for grand-mean centering Y ij = γ 00 + γ 10 (X ij X ) + u 0j + r ij (11) = γ 00 + γ 10 (X ij X j + X j X ) + u 0j + r ij (12) = γ 00 + γ }{{} 10 (X ij X j ) + γ 10 ( }{{} X j X ) + u 0j + r ij within between (13) Estimation and Centering p24/32

26 Within- and between-group effects For raw and Grand mean centering, including the group mean as a predictor separates the between-group effect from the within group effect Level 1: Y ij = β 0j + β 1j (X ij X ) + r ij (14) Level 2: β 0j = γ 00 + γ 01 X j + u 0j (15) β 1j = γ 10 (16) Combined: Y ij = γ 00 + γ }{{} 10 (X ij X ) + γ 01 X j + u }{{} 0j + r ij β w β b β w β c = β b β w is called the context or compositional effect (17) Estimation and Centering p25/32

27 Within- and between-group effects For group mean centering, within and between group effects are orthogonal Including a group mean centered predictor will only decrease level 1 variance; including the group means will only decrease level 2 variance Level 1: Y ij = β 0j + β 1j (X ij X j ) + r ij (18) Level 2: β 0j = γ 00 + γ 01 X j + u 0j (19) β 1j = γ 10 (20) Combined: Y ij = γ 00 + γ }{{} 10 (X ij X j ) + γ }{{} 01 β w β b X j + u 0j + r ij (21) Estimation and Centering p26/32

28 Compositional Effects Compositional effect: when the aggregate of a person level predictor (here, MeanSES determined from the sample of children within each school) is related to the outcome Y, even after controlling for the effect of individual characteristics CWC CGM Y ( SES SES ) r ij 0 j 1 j ij j ij ( MeanSES) u 0 j j 0 j u 1 j 10 1 j Y ( SES SES ) r ij 0 j 1 j ij ij ( MeanSES) u 0 j j 0 j u 1 j 10 1 j how much does the organizational level effect (ie, between-groups) differ from the individual level effect (ie, within-groups)? γ cwc 01 = β between group γ cwc 10 = β within group (22) γ cgm 01 = γ cwc 01 γ cwc 10 = β compositional effect (23) Estimation and Centering p27/32

29 Summary of centering predictors Scaling of predictors has direct implications on the ability to disentangle between and within effects Raw-scale predictor Intercept represents the group mean when the predictor equals zero Intercept is only directly interpretable if the scale has a natural zero-point Grand-mean centered predictor Intercept represents the adjusted group mean, holding the predictor constant at the grand mean for all groups Group-mean centered predictor Intercept represents the unadjusted group mean Inclusion of group mean as level 2 predictors disentangles between and within group effects Estimation and Centering p28/32

30 Summary Centering decisions impact both the intercept and the slope estimates in level two (our examples will make this clear) Group mean centering with reintroduction of the aggregates at level two will always provide an unbiased estimate of the within group slope Need to carefully consider the needs of your analysis and your assumptions (theory) about how contextual variables are measured, how they relate to level one variables and your outcome Be aware of how your choice affects the model and the question you are trying to answer Estimation and Centering p29/32

31 Considerations Researchers tend to differ in how they approach centering primarily because the decision hinges on goals for the analysis Theres not one way thats correct All approaches are correct if they accommodate your theory regarding contextual effects of variables and address the research question you really want to be asking! Estimation and Centering p30/32

32 Some guidelines For raw score (uncentered models) generally the interest is more in effects of variables on individuals rather than group effects For centering without reintroducing the means (aggregates of level one variables) at level two, interest is generally for situations where that aggregate is not of interest, but other level two variables might be For centering with introduction of the aggregates at level two, interest is on separating out individual effects of a variable (like SES) from group-level effects of that variable (like MeanSES for a particular school) Estimation and Centering p31/32

33 Recommendation When to use raw scores or grand mean centering If you are more interested in the effects in individuals performance than in group effects eg, clustering is a nuisance factor When raw score does not allow a meaningful interpretation of the intercept, use grand mean centering When to use group mean centering Theory says that individual and group effects are separate For technical reasons Smaller correlation between random intercept and random slope Smaller correlation between level 1 and level 2 variables and cross-level interactions This will stabilize model (coefficients are more or less independent estimates) Estimation and Centering p32/32