4/1/2010 36-402/608 ADA-II H. SeltmanBreakout #19 ResultsThese data come from Snijders and Bosker’s book “Multilevel Analysis”, chapter 5 (partlyfrom: http://www.ats.ucla.edu/stat/sas/examples/mlm ma snijders/ch5.htm).There are multiple students within each school, and school is identified with the “schoolnr”variable. The outcome is a score on a language test labeled “langpost”.The per-school explanatory variables are “GndC size” (a centered variable indicatingschool size); “GrpMC verb” (a centered average verbal IQ for each school); and “GrpMC ses”(a centered variable indicating the mean socio-economic status for each school).Different from the above is “GndC verb” which is a (centered) student level verbal IQvariable indicating each student’s IQ relative to the mean for his or her school.Question 1: List some of the unmeasured school-level variables.Question 2: Now ignoring those variables, very roughly sketch a plot of themeans model whose fixed effects are defined byMODEL langpost = GndC size GrpMC verb GrpMC ses GndC verb;You should invent signs for the βs and not worry about the magnitudes. Oneeffective technique is to plot the means model for the 1/3 and 2/3 quantilevalues of GndC size, GrpMC verb, GrpMC ses.If you used the “tercile” method suggested above, you will have 8 lines on a plot ofGndC verb vs. langpost, with each line representing a different kind of school (andother kinds of schools un-plotted but imagined between or beyond the plotted lines). Toconsider random effects, think about several schools all with the same values of the threeschool-level variables. You can do this by picking any one of the 8 lines on your plot fromquestion 2. If there are no important student-level variables, note that for an “average”school all of the individual student points would fall on the line you picked. If there areimportant student-level variables, then the points randomly fall above or below that linewith a spread based on the residual variance, σ2.Now consider what will happen if these several schools differ on important school levelvariables. Initially, to simplify things, again imagine that there are no important student-level variables. Question 3: Add “prediction” lines for the several schools allwith the same levels of the measured school-level variables.You should see that you drew lines with different intercepts and slopes, but with a meanintercept and slope that gives the original line.This exercise shows how to think about which random effects are possible, and how theyrelate to the fixed effects models (and why random effects are defined to have mean zero).Now relate what you learned here to the idea that a random effect can be representedas adding a mean-zero per-group (per-school, here), random variable to any beta valuein the (fixed) means model, assuming that it makes sense for that quantity to vary fromgroup to group.Here is a simple model:OPTIONS LINESIZE=66;LIBNAME here ".";TITLE ’Snijders and Bosker School Data, Chapter 5’;TITLE2 "Random slope by IQ model";proc mixed data=schools2 covtest noitprint noclprint method=ml;class schoolnr;model langpost = GndC_verb GrpMC_verb / solution outpred=RSres;random intercept GndC_verb / subject=schoolnr type=un;run;2DimensionsCovariance Parameters 4Columns in X 3Columns in Z Per Subject 2Subjects 131Max Obs Per Subject 35Number of Observations Used 2287Covariance Parameter EstimatesStandard ZCov Parm Subject Estimate Error Value Pr ZUN(1,1) schoolNR 7.9177 1.3287 5.96 <.0001UN(2,1) schoolNR -0.8198 0.2914 -2.81 0.0049UN(2,2) schoolNR 0.1994 0.1003 1.99 0.0234Residual 41.3525 1.2902 32.05 <.0001BIC (smaller is better) 15247.7Solution for Fixed EffectsStandardEffect Estimate Error DF t Value Pr > |t|Intercept 40.7498 0.2859 129 142.54 <.0001GndC_verb 2.4588 0.08315 130 29.57 <.0001GrpMC_verb 1.4052 0.3214 2025 4.37 <.0001Question 4: What are there 131 of?Question 5: Make a rough fixed effect plot based on the model fit here, plottingGndC verb at -2 and +2 (roughly IQs 80 and 120). Ignoring the difficulty ofmaking a plot that incorprates the negative correlation of the intercept andslope, pick the mean line representing one particular level of school mean IQand draw several lines for separate schools consistent with the CovarianceParameter Estimates.3In practice we would do a lot of model selection and checking of residual plots at thisstage. For the Breakout, we will just look at the best model, which also include theindicator variable for whether the school uses mixed-grade classrooms.proc mixed data=schools2 covtest noclprint noitprint method=ml dfbw;class schoolnr;model langpost = GndC_verb GndC_ses GrpMC_verb mixedgra GndC_verb*mixedgra /solution outpred=RSparsRes;random intercept GndC_verb / subject=schoolnr type=un;run;DimensionsCovariance Parameters 4Columns in Z Per Subject 2Covariance Parameter EstimatesStandard ZCov Parm Subject Estimate Error Value Pr ZUN(1,1) schoolNR 7.5574 1.2562 6.02 <.0001UN(2,1) schoolNR -0.5890 0.2587 -2.28 0.0228UN(2,2) schoolNR 0.1277 0.08389 1.52 0.0640Residual 39.3402 1.2253 32.11 <.0001BIC (smaller is better) 15142.1Solution for Fixed EffectsStandardEffect Estimate Error DF t Value Pr > |t|Intercept 41.3213 0.3490 129 118.40 <.0001GndC_verb 2.1134 0.09245 2152 22.86 <.0001GndC_ses 0.1555 0.01464 2152 10.63 <.0001GrpMC_verb 0.8754 0.3237 129 2.70 0.0078mixedgra -1.3961 0.5743 2152 -2.43 0.0151GndC_verb*mixedgra 0.4472 0.1701 2152 2.63 0.0086Question 6: Summarize what this model