Topic 25 - One-Way Random Effects Models. Outline. Random Effects vs Fixed Effects. Data for One-way Random Effects Model. One-way Random effects

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1 Topic 5 - One-Way Random Effects Models One-way Random effects Outline Model Variance component estimation - Fall 013 Confidence intervals Topic 5 Random Effects vs Fixed Effects Consider factor with numerous levels Want to draw inference on population of levels, not specifically concerned with comparing observed levels Clear example of difference (1=fixed, =random) 1. Compare reading ability of 10 nd grade classes in Indiana Go to each of the r = 10 specific classes of interest and randomly choose n students from each classroom.. Compare variability among all nd grade classes in Indiana Randomly choose r = 10 classes from population of nd grade classes. Then randomly choose n students from each classroom. Inference broader in random effects case Levels chosen randomly inference on pop of levels Data for One-way Random Effects Model Exact same data framework as fixed effects case... Y is the response variable Factor with levels i = 1,,..., r Y ij is the j th observation from cell i Consider j = 1,,..., n i but different statistical model. Topic 5 3 Topic 5 4

2 Example Page 1036 Interested in studying the variability in the rating of job applicants Two sources of variability Variability among applicants Variability among personnel officers Y is the job applicant rating Factor: officer/interviewer (r = 5) SAS Commands options nocenter; data a1; infile u:\.www\datasets55\ch5ta01.txt ; input rating officer; proc print data=a1; run; title1 Plot of the data ; symbol1 v=circle i=none c=black; proc gplot data=a1; plot rating*officer/frame; ***Scatterplot; proc means data=a1; output out=a mean=avrate; var rating; by officer; Interviewers selected at random from population of personnel officers (assume population large) Twenty applicants randomly and equally assigned (n = 4) to each personnel officer title1 Plot of the means ; symbol1 v=circle i=join c=black; proc gplot data=a; plot avrate*officer/frame; run; ***Means plot; Topic 5 5 Topic 5 6 Output Obs rating officer Obs rating officer Topic 5 7 Topic 5 8

3 Random Effects Model Expressed numerically Y ij = µ i + ε ij µ i N(µ,σµ) ε ij N(0,σ ) µ i and ε ij independent Implies Y ij N(µ,σµ + σ ) Cov(Y ij,y ik )=σµ Correlation between some obs Also called Model II Topic 5 9 Topic 5 10 Random Factor Effects Model Quantities of Interest Statistical model is Often interested in the percent of total variability due to factor Y ij = µ + τ i + ε ij σ µ σ µ + σ = σ µ σ Y µ - population mean τ i N(0,σµ) ε ij N(0,σ ) There are TWO parameters/variances in each model Cell means are random variables, not parameters Is also called the intraclass correlation coefficient because it describes the correlation between two observations from the same factor level ρ IC = Cov(Y ij, Y ik ) = σ µ Var(Yij )Var(Y ik ) σy Depending of example, may want this value small or large Topic 5 11 Topic 5 1

4 Least Squares Approach: ANOVA Table and EMS Terms and layout of ANOVA table the same as that in the fixed effects case The expected means squares (EMS) are different because of the different model assumptions This also means the hypotheses being tested are different Topic 5 13 Likelihood Approach: General Mixed Effect Model Consider expressing the model Y = Xβ + Zδ + ε β is a vector of fixed-effect parameters δ is a vector of random-effect parameters ε is the error vector δ and ε assumed uncorrelated means 0 covariance matrices G and R Topic 5 14 Likelihood Approach: General Mixed Effect Model Cov(Y ) = ZGZ + R If R = σ I and Z = 0, back to standard linear model SAS Proc Mixed allows one to specify G and R G through RANDOM, R through REPEATED Likelihood Approach: General For known G and R, Mixed Effect Model ˆβ = (X Σ 1 X) 1 X Σ 1 Y ˆδ = GZ Σ 1 (Y X ˆβ) For unknown G and R, their REML estimates can be substituted into these expressions REML uses likelihood to take into account loss of DF log L = (n p)log(π) + log( Σ ) + r Σ 1 r + log( X Σ 1 X ) where r = Y X ˆβ Topic 5 15 Topic 5 16

5 Random Effects Model Model Estimates The hypotheses are: H 0 : σµ = 0 H a : σµ > 0 Same breakdown of Total SS but E(MSE)=σ E(MSTR)=σ + nσµ Under H 0, F 0 F α,r 1,nT r Same F test as before Conclusion pertains to entire population of levels Typically interested in estimating variances and functions of these variances Under ANOVA/least squares approach, use mean squares ˆσ = MSE ˆσ µ = (MSTR MSE)/n If unbalanced, replace n with n 0 = (( n i ) n i )/((r 1) n i ) Under this approach, estimate of σ µ can be negative - Supports H 0 so use zero as estimate? - If σ µ small, chance variation can result in negative estimate - Bayesian approach (nonnegative prior) - Residual maximum likelihood (nonnegative restriction) Topic 5 17 Topic 5 18 Output proc glm data=a1; class officer; model rating=officer; random officer; run; proc mixed data=a1 cl; class officer; model rating=; random officer/vcorr; run; SAS Commands Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square Coeff Var Root MSE rating Mean Source DF Type I SS Mean Square F Value Pr > F officer Source DF Type III SS Mean Square F Value Pr > F officer Source officer Type III Expected Mean Square Var(Error) + 4 Var(officer) Topic 5 19 Topic 5 0

6 Output Confidence intervals Estimated V Correlation Matrix for Subject 1 Row Col1 Col Col3 Col Covariance Parameter Estimates Cov Parm Estimate Alpha Lower Upper officer Residual Fit Statistics - Res Log Likelihood 145. AIC (smaller is better) 149. AICC (smaller is better) BIC (smaller is better) σ : Page 1041 σ µ: Page 1043 so r(n 1)MSE σ r(n 1)MSE χ α/,r(n 1) σ (r 1)MSTR σ + nσ µ χ r(n 1) r(n 1)MSE χ 1 α/,r(n 1) χ r 1 f(σ µ) = σ + nσ τ n(r 1) χ r 1 σ nr(n 1) χ r(n 1) No closed form expression for this distribution Satterthwaite Procedure page 1043 (Proc Mixed) MLS Procedure page 1045 Topic 5 1 Topic 5 Confidence intervals Intraclass Correlation Coefficient : Page 1040 Uses ratio of previous two χ distributions (i.e., F dist) L L + 1 σµ σ + σµ U U + 1 L = 1 n U = 1 n ( ( MS Trt MS E F α/,a 1,N a 1 MS Trt MS E F 1 α/,a 1,N a 1 Population mean -µ : Page ) ) Background Reading KNNL Section 5.1 knnl1036.sas KNNL Sections Y.. = 1 r (Y 1. + Y Y r. ) Y i. N ( µ, σ µ + σ n ) Topic 5 3 Topic 5 4