Lecture 10: Experiments with Random Effects

Transcription

1 Lecture 10: Experiments with Random Effects Montgomery, Chapter 13 1 Lecture 10 Page 1

2 Example 1 A textile company weaves a fabric on a large number of looms. It would like the looms to be homogeneous so that it obtains a fabric of uniform strength. A process engineer suspects that, in addition to the usual variation in strength within samples of fabric from the same loom, there may also be significant variations in strength between looms. To investigate this, she selects four looms at random and makes four strength determinations on the fabric manufactured on each loom. The layout and data are given in the following. Observations looms Lecture 10 Page 2

3 Random Effects vs Fixed Effects Consider factor with numerous possible levels Want to draw inference on population of levels Not concerned with any specific levels Example of difference (1=fixed, 2=random) 1. Compare reading ability of 10 2nd grade classes in NY Select a = 10 specific classes of interest. Randomly choose n students from each classroom. Want to compare τ i (class-specific effects). 2. Study the variability among all 2nd grade classes in NY Randomly choose a = 10 classes from large number of classes. Randomly choose n students from each classroom. Want to assess στ 2 (class to class variability). Inference broader in random effects case Levels chosen randomly inference on population 3 Lecture 10 Page 3

4 Random Effects Model (CRD) Similar model (as in the fixed case) with different assumptions µ - grand mean y ij = µ + τ i + ϵ ij τ i - ith treatment effect (random) ϵ ij N(0, σ 2 ) Instead of τ i = 0, assume τ i N(0, σ 2 τ ) {τ i } and {ϵ ij } independent Var(y ij ) = σ 2 τ + σ 2 i = 1, 2... a j = 1, 2,... n i σ 2 τ and σ 2 are called variance components 4 Lecture 10 Page 4

5 The basic hypotheses are: Statistical Analysis H 0 : σ 2 τ = 0 vs. H 1 : σ 2 τ > 0 Same ANOVA table (as before) Source SS DF MS F 0 Between SS tr a 1 MS tr F 0 = MS tr MS E Within SS E N a MS E Total SS T = SS tr + SS E N 1 E(MS E )=σ 2 E(MS tr )=σ 2 + nσ 2 τ Under H 0, F 0 F a 1,N a Same test as before Conclusions, however, pertain to entire population 5 Lecture 10 Page 5

6 Estimation Usually interested in estimating variances Use mean squares (known as ANOVA method) ˆσ 2 = MS E ˆσ 2 τ = (MS tr MS E )/n If unbalanced, replace n with n 0 = 1 a 1 ( a i=1 n i Estimate of σ 2 τ can be negative ai=1 ) n 2 i ai=1 n i - Supports H 0? Use zero as estimate? - Validity of model? Nonlinear? - Other approaches (MLE, Bayesian with nonnegative prior) 6 Lecture 10 Page 6

7 σ 2 : Same as fixed case Confidence intervals (N a)ms E σ 2 χ 2 N a (N a)ms E χ 2 α/2,n a σ 2 (N a)ms E χ 2 1 α/2,n a σ 2 τ : Linear combination of χ 2 (a 1)MS tr σ 2 + nσ 2 τ χ 2 a 1 ˆσ 2 τ = (MS tr MS E )/n, so ˆσ 2 τ σ2 + nσ 2 τ n(a 1) χ2 a 1 σ 2 n(n a) χ2 N a No closed form expression for this distribution. Can use approximation 7 Lecture 10 Page 7

8 Recall For σ 2 τ /σ 2 : MS tr /(σ 2 + nσ 2 τ ) MS E /σ 2 F a 1,N a L = L σ2 τ σ U 2 ( ) ( ) MS tr MS MS E 1 /n and U = tr F α/2,a 1,N a MS E 1 /n F 1 α/2,a 1,N a For σ 2 τ /(σ 2 + σ 2 τ ) : L L + 1 σ2 τ σ 2 + σ 2 τ U U + 1, or F 0 F α/2,a 1,N a F 0 + (n 1)F α/2,a 1,N a σ2 τ σ 2 + σ 2 τ F 0 F 1 α/2,a 1,N a F 0 + (n 1)F 1 α/2,a 1,N a 8 Lecture 10 Page 8

9 Loom Experiment (continued) Source of Sum of Degrees of Mean F 0 Variation Squares Freedom Square Between Within Total Highly significant result (F.05,3,12 = 3.49) ˆσ τ 2 = ( )/4 = % (=6.98/( )) is attributable to loom differences Time to improve consistency of the looms 9 Lecture 10 Page 9

10 95% CI for σ 2 Loom Experiment Confidence Intervals SS E χ 2.025,12 σ 2 SS E χ 2.975,12 = c(22.75/23.34, 22.75/4.40) = (0.97, 5.17) 95% CI for στ 2 /(στ 2 + σ 2 ) ( ) (4 1)4.47, (1/14.34) (4 1)(1/14.34) F 0.025,3,12 = 4.47, F.975,3,12 = 1/14.34 using property that = (0.385, 0.982) F 1 α/2,v1,v 2 = 1/F α/2,v2,v 1 the variability between looms is not negligible. 10 Lecture 10 Page 10

11 options nocenter ps=35 ls=72; Using SAS data example; input loom strength; datalines; ; proc glm; class loom; model strength=loom; random loom; output out=diag r=res p=pred; proc plot; 11 Lecture 10 Page 11

12 plot res*pred; proc varcomp method = type1; class loom; model strength = loom; proc mixed CL; class loom; model strength = ; random loom; run; 12 Lecture 10 Page 12

13 SAS Output Dependent Variable: strength Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total Source DF Type I SS Mean Square F Value loom Source Type III Expected Mean Square loom Var(Error) + 4 Var(loom) Variance Components Estimation Procedure Dependent Variable: strength Sum of Source DF Squares Mean Square loom Lecture 10 Page 13

14 Error Corrected Total Source loom Error Expected Mean Square Var(Error) + 4 Var(loom) Var(Error) Variance Component Estimate Var(loom) Var(Error) Lecture 10 Page 14

15 SAS Output (Continued) The Mixed Procedure Iteration History Iteration Evaluations -2 Res Log Like Criterion Convergence criteria met. Covariance Parameter Estimates Cov Parm Estimate Alpha Lower Upper loom Residual Fit Statistics -2 Res Log Likelihood 63.2 AIC (smaller is better) 67.2 AICC (smaller is better) 68.2 BIC (smaller is better) Lecture 10 Page 15

16 A Measurement Systems Capability Study A typical gauge R&R experiment is shown below. An instrument or gauge is used to measure a critical dimension of certain part. Twenty parts have been selected from the production process, and three randomly selected operators measure each part twice with this gauge. The order in which the measurements are made is completely randomized, so this is a two-factor factorial experiment with design factors parts and operators, with two replications. Both parts and operators are random factors. Parts Operator 1 Operator 2 Operator Variance components equation: σ 2 y = σ 2 τ + σ 2 β + σ 2 τβ + σ 2 Total variability=parts + Operators + Interaction + Experimental Error =Parts + Reproducibility + Repeatability 16 Lecture 10 Page 16

17 Statistical Model with Two Random Factors i = 1, 2,..., a y ijk = µ + τ i + β j + (τβ) ij + ϵ ijk j = 1, 2,..., b k = 1, 2,..., n τ i N(0, σ 2 τ ) β j N(0, σ 2 β) (τβ) ij N(0, σ 2 τβ) Var(y ijk ) = σ 2 + σ 2 τ + σ 2 β + σ 2 τβ Expected MS s similar to one-factor random model E(MS E )=σ 2 ; E(MS A ) = σ 2 + bnσ 2 τ + nσ 2 τβ E(MS B ) = σ 2 + anσ 2 β + nσ 2 τβ ; E(MS AB) = σ 2 + nσ 2 τβ EMS determine what MS to use in denominator H 0 : στ 2 = 0 MS A /MS AB H 0 : σβ 2 = 0 MS B /MS AB H 0 : στβ 2 = 0 MS AB /MS E 17 Lecture 10 Page 17

18 Estimating Variance Components Using ANOVA method ˆσ 2 = MS E ˆσ τ 2 = (MS A MS AB )/bn ˆσ β 2 = (MS B MS AB )/an ˆσ τβ 2 = (MS AB MS E )/n Sometimes results in negative estimates Proc Varcomp and Proc Mixed compute estimates Can use different estimation procedures ANOVA method - Method = type1 RMLE method - Method = reml (default) Proc Mixed Variance component estimates Hypothesis tests and confidence intervals 18 Lecture 10 Page 18

19 Gauge Capability Example in Text 13-2 options nocenter ls=75; data randr; input part operator resp cards; ; proc glm; class operator part; model resp=operator part; random operator part operator*part / test; test H=operator E=operator*part; test H=part E=operator*part; 19 Lecture 10 Page 19

20 proc mixed cl maxiter=20 covtest method=type1; class operator part; model resp = ; random operator part operator*part; proc mixed cl maxiter=20 covtest; class operator part; model resp = ; random operator part operator*part; run; quit; 20 Lecture 10 Page 20

21 Dependent Variable: resp Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error CorreTotal Source DF Type III SS Mean Square F Value Pr > F operator part <.0001 operator*part Source operator part operator*part Type III Expected Mean Square Var(Error) + 2 Var(operator*part) + 40 Var(operator) Var(Error) + 2 Var(operator*part) + 6 Var(part) Var(Error) + 2 Var(operator*part) Tests of Hypotheses Using the Type III MS for operator*part as an Error Term Source DF Type III SS Mean Square F Value Pr > F 21 Lecture 10 Page 21

22 operator part <.0001 Tests of Hypotheses for Random Model Analysis of Variance Dependent Variable: resp Source DF Type III SS Mean Square F Value Pr > F operator part <.0001 Error Error: MS(operator*part) Source DF Type III SS Mean Square F Value Pr > F operator*part Error: MS(Error) Lecture 10 Page 22

23 The Mixed Procedure Type 1 Analysis of Variance Sum of Source DF Squares Mean Square operator part operator*part Residual Type 1 Analysis of Variance Error Source Expected Mean Square Error Term DF operator Var(Residual) + 2 Var(operator*part) MS(operator*part) Var(operator) part Var(Residual) + 2 Var(operator*part) MS(operator*part) Var(part) operator*part Var(Residual) + 2 Var(operator*part) MS(Residual) 60 Residual Var(Residual). 23 Lecture 10 Page 23

24 Source F Value Pr > F operator part <.0001 operator*part Covariance Parameter Estimates Standard Z Cov Parm Estimate Error Value Pr Z Alpha Lower Upper operator part operator*part Residual < Lecture 10 Page 24

25 The Mixed Procedure Estimation Method REML Iteration History Iteration Evaluations -2 Res Log Like Criterion Convergence criteria met. Covariance Parameter Estimates Standard Z Cov Parm Estimate Error Value Pr Z Alpha Lower Upper operator E12 part operator*part Residual < Lecture 10 Page 25

26 Confidence Intervals for Variance Components Can use asymptotic variance estimates to form CI Known as Wald s approximate CI Mixed: option CL=WALD or METHOD=TYPE1 Use standard normal 95% CI uses 1.96 ˆσ β 2 ± 1.96(.0330) = ( 0.05, 0.08) ˆσ τ 2 ± 1.96(3.3738) = (3.67, 16.89) In general Proc Mixed uses Satterthwaite CI Default method - REML Versions < 6.12 computed Wald CI Current uses Satterthwaite s Approximation Will discuss this CI construction later on. 26 Lecture 10 Page 26

27 Rules For Expected Mean Squares In models so far, EMS fairly straightforward Could calculate EMS using brute force method For mixed models, good to have formal procedure Montgomery describes procedure for restricted model 0 Write the error term in the model as ϵ (ij..)m, where m represents the replication subscript 1 Write each variable term in model as a row heading in a two-way table 2 Write the subscripts in the model as column headings. Over each subscript write F if factor fixed and R if random. Over this, write down the levels of each subscript 3 For each row, copy the number of observations under each subscript, providing the subscript does not appear in the row variable term 4 For any bracketed subscripts in the model, place a 1 under those subscripts that are inside the brackets 5 Fill in remaining cells with a 0 (if subscript represents a fixed factor) or a 1 (if random factor). 27 Lecture 10 Page 27

28 6 To find the expected mean square of any term (row), cover the entries in the columns that contain non-bracketed subscript letters in this term in the model. For those rows with at least the same subscripts, multiply the remaining numbers to get coefficient for corresponding term in the model. 28 Lecture 10 Page 28

29 Two-Factor Fixed Model: y ijk = µ + τ i + β j + (τβ) ij + ϵ ijk F F R a b n term i j k EMS τ i 0 b n σ 2 + bnστ 2 i a 1 β j a 0 n σ 2 + anσβ2 j b 1 (τβ) ij 0 0 n σ 2 + nσσ(τβ)2 ij (a 1)(b 1) ϵ (ij)k σ 2 29 Lecture 10 Page 29

30 Two-Factor Random Model: y ijk = µ + τ i + β j + (τβ) ij + ϵ ijk R R R a b n term i j k EMS τ i 1 b n σ 2 + nστβ 2 + bnστ 2 β j a 1 n σ 2 + nστβ 2 + anσβ 2 (τβ) ij 1 1 n σ 2 + nστβ 2 ϵ (ij)k σ 2 30 Lecture 10 Page 30

31 Two-Factor Mixed Model (A Fixed): y ijk = µ + τ i + β j + (τβ) ij + ϵ ijk F R R a b n term i j k EMS τ i 0 b n σ 2 + nσ 2 τβ + bnστ 2 i a 1 β j a 1 n σ 2 + anσ 2 β (τβ) ij 0 1 n σ 2 + nσ 2 τβ ϵ (ij)k σ 2 31 Lecture 10 Page 31

32 Three-Factor Mixed Model (A Fixed): y ijkl = µ + τ i + β j + γ k + (τβ) ij + (τγ) ik + (βγ) jk + (τβγ) ijk + ϵ ijkl F R R R a b c n term i j k l EMS τ i 0 b c n σ 2 + cnσ 2 τβ + bnσ 2 τγ + nσ 2 τβγ + bcnστ 2 i a 1 β j a 1 c n σ 2 + anσ 2 βγ + acnσ 2 β γ k a b 1 n σ 2 + anσ 2 βγ + abnσ 2 γ (τβ) ij 0 1 c n σ 2 + nσ 2 τβγ + cnσ 2 τβ (τγ) ik 0 b 1 n σ 2 + nσ 2 τβγ + bnσ 2 τγ (βγ) jk a 1 1 n σ 2 + anσ 2 βγ (τβγ) ijk n σ 2 + nσ 2 τβγ ϵ ijkl σ 2 32 Lecture 10 Page 32

33 Two-Factor Mixed Effects Model y ijk = µ + τ i + β j + (τβ) ij + ϵ ijk Assume A fixed and B random 1 τ i = 0 and β N(0, σβ) 2 usual assumptions 2 (τβ) ij N(0, (a 1)στβ/a) 2 (a 1)/a simplifies EMS 3 (τβ) ij = 0 for β level j added restriction Due to added restriction Not all (τβ) ij indep, Cov((τβ) ij, (τβ) i j) = 1 a σ2 τβ Cov(y ijk, y i jk ) = σ2 β 1 a σ2 τβ, i i. Known as restricted mixed effects model This model coincides with EMS algorithm E(MS E )=σ 2 E(MS A ) = σ 2 + bn τi 2 /(a 1) + nστβ 2 E(MS B ) = σ 2 + anσβ 2 E(MS AB ) = σ 2 + nστβ 2 33 Lecture 10 Page 33

34 Hypotheses Testing and Diagnostics Testing hypotheses: H 0 : τ 1 = τ 2 =... = 0 MS A /MS AB H 0 : σβ 2 = 0 MS B /MS E H 0 : στβ 2 = 0 MS AB /MS E Variance Estimates ( Using ANOVA method) ˆσ 2 = MS E ˆσ β 2 = (MS B MS E )/an ˆσ τβ 2 = (MS AB MS E )/n Diagnostics Histogram or QQplot Normality or Unusual Observations Residual Plots Constant variance or Unusual Observations 34 Lecture 10 Page 34

35 Multiple Comparisons for Fixed Effects y ijk = µ + τ i + β j + (τβ) ij + ϵ ijk y i.. = µ + τ i + β. + (τβ) i. + ϵ i.. Var(y i.. ) = σ 2 β/b + (a 1)σ 2 τβ/ab + σ 2 /bn y i.. y i.. = τ i τ i + (τβ) i. (τβ) i. + ϵ i.. ϵ i.. Var(y i.. y i.. ) = 2σ2 τβ/b + 2σ 2 /bn = 2(nστβ 2 + σ 2 )/bn Need to plug in variance estimates to compute Var(y i.. ) What are the DF? For pairwise comparisons, use estimate 2MS AB /bn Use df AB for t-statistic or Tukey s method. 35 Lecture 10 Page 35

36 Gauge Capability Example in Text 12-3 options nocenter ls=75; data randr; input part operator resp cards; ; proc glm; class operator part; model resp=operator part; run; ================================================== Dependent Variable: resp Sum of 36 Lecture 10 Page 36

37 Source DF Squares Mean Square F Value Pr > F Model <.0001 Error CorrTotal Source DF Type I SS Mean Square F Value Pr > F operator part <.0001 operator*part Lecture 10 Page 37

38 H 0 : τ 1 = τ 2 = τ 3 = 0: P-value based on F 2,38 : H 0 : σ 2 β = 0: P-value based on F 19,60 : H 0 : σ 2 τβ = 0: P-value based on F 38,60 : 0.86 Variance components estimates: Gauge Capability Example F 0 = MS A MS AB = = 1.84 F 0 = MS B MS E = = F 0 = MS AB MS E = = 0.72 ˆσ 2 β = (3)(2) = Lecture 10 Page 38

39 ˆσ 2 τβ = ˆσ 2 = 0.99 =.14( 0) Pairwise comparison for τ 1, τ 2 and τ Lecture 10 Page 39

40 Unrestricted Mixed Model SAS uses unrestricted mixed model in analysis y ijk = µ + τ i + β j + (τβ) ij + ϵ ijk τi = 0 and β j N(0, σ 2 β) (τβ) ij N(0, σ 2 τβ) Expected mean squares: E(MS E )=σ 2 E(MS A ) = σ 2 + bn τ 2 i /(a 1) + nσ 2 τβ E(MS B ) = σ 2 + anσ 2 β + nσ 2 τβ E(MS AB ) = σ 2 + nσ 2 τβ random statement in SAS also gives these results Differences E(MS B ) Test H 0 : σ 2 β = 0 using MS AB in denominator Cov(y ijk, y i jk )=σ2 β, i i. 40 Lecture 10 Page 40

41 Connection (τβ).j = ( i (τβ) ij )/a y ijk = µ + τ + (β j + (τβ).j ) + ((τβ) ij (τβ).j ) + ϵ ijk Check the model above satisfies the conditions of restricted mixed model Restricted model is slightly more general. 41 Lecture 10 Page 41

42 Gauge Capability Example (Unrestricted Model) options nocenter ls=75; data randr; input part operator resp cards; ; proc glm; class operator part; model resp=operator part; random part operator*part / test; means operator / tukey lines E=operator*part; lsmeans operator / adjust=tukey E=operator*part tdiff stderr; 42 Lecture 10 Page 42

43 proc mixed alpha=.05 cl covtest; class operator part; model resp=operator / ddfm=kr; random part operator*part; lsmeans operator / alpha=.05 cl diff adjust=tukey; run; quit; 43 Lecture 10 Page 43

44 Approximate F Tests For some models, no exact F-test exists Recall 3 Factor Mixed Model (A - fixed) No exact test for A based on EMS Assume a = 3, b = 2, c = 3, n = 2 and following MS were obtained 44 Lecture 10 Page 44

45 Source DF MS EMS F P A ϕ A + 6σAB 2 + 4σAC 2?? +2σABC 2 + σ 2 B σB 2 + 6σBC 2 + σ AB σAB 2 + 2σABC 2 + σ C σC 2 + 6σBC 2 + σ AC σAC 2 + 2σABC 2 + σ BC σBC 2 + σ ABC σABC 2 + σ Error σ 2 Possible approaches: Could assume some variances are negligible, not recommended without conclusive evidence Pool (insignificant) means squares with error, also risky, not recommended when df for error is already big. 45 Lecture 10 Page 45

46 Satterthwaite s Approximate F-test H 0 : effect = 0, e.g., H 0 : τ 1 = = τ a = 0 or equivalently H 0 : τi 2 = 0. No exact test exists. Get two linear combinations of mean squares MS = MS r ±... ± MS s MS = MS u ±... ± MS v such that 1) MS and MS do not share common mean squares; 2) E(MS ) E(MS ) is a multiple of the effect. approximate test statistic F : F = MS MS = MS r±...±ms s MS u ±...±MS v F p,q where p = (MS r±...±ms s ) 2 MS 2 r /f r+...+ms 2 s /f s and q = (MS u±...±ms v ) 2 MS 2 u /f u+...+ms 2 v /f v f i is the degrees of freedom associated with MS i p and q may not be integers, interpolation is needed. SAS can handle noninteger dfs. Caution when subtraction is used 46 Lecture 10 Page 46

47 H 0 : τ 1 = τ 2 = τ 3 = 0 MS = MS A Example: 3-Factor Mixed Model (A Fixed) MS = MS AB + MS AC MS ABC E(MS MS ) = 12ϕ A = 12 Στ 2 i 3 1 F = MS A MS AB + MS AC MS ABC = = 57.0 p = 2 q = / / /4 = 4.15 Interpolation needed P(F 2,4 > 57) =.0011 P(F 2,5 > 57) =.0004 P =.85(.0011) +.15(.0004) = Lecture 10 Page 47

48 SAS can be used to compute P-values and quantile values for F and χ 2 values with noninteger degrees of freedom. Upper Tail Probability: probf(x,df1,df2) and probchi(x,df) Quantiles : finv(p,df1,df2) and cinv(p,df) data one; p=1-probf(57,2.0,4.15); f=finv(.95,2.0,4.15); c1=cinv(.025,18.57); c2=cinv(.975,18.57); proc print data=one; OBS P F C1 C Lecture 10 Page 48

49 Another Approach to Testing H 0 : τ 1 = τ 2 = τ 3 = 0 MS = MS A + MS ABC MS = MS AB + MS AC E(MS MS ) =? F = MS A + MS ABC MS AB + MS AC = = p = / /4 = 2.01 q = / /2 = 6.00 P-value= P (F > 48.41) = This is again found significant Avoid subtraction,summation should be preferred. 49 Lecture 10 Page 49

50 Suppose we are interested in σ 2 x. Approximate Confidence Intervals Case 1: there exists a mean square MS x with df x such that E(MS x ) = σ 2 x. Then ˆσ 2 x = MS x, and Exact 100(1-α)% CI: Case 2: there exist df x MS x χ 2 α/2,dfx df x MS x σ 2 x σ 2 x χ 2 (df x ) df xms x χ 2 1 α/2,dfx MS = MS r MS s and, MS = MS u MS v such that E(MS MS ) = kσ 2 x. Then ˆσ 2 x = MS MS k, and df xˆσ 2 x σ 2 x χ 2 (df x ) where df x = (ˆσ2 x) 2 MSi k 2 f i = (MS r + + MS s MS u MS v ) 2 MS 2 r/f r + + MS 2 s/f s + MS 2 u/f u + + MS 2 v/f v 50 Lecture 10 Page 50

51 Approximate 100(1-α)% CI: df xˆσ 2 x χ 2 α/2,df x σ 2 x df xˆσ 2 x χ 2 1 α/2,df x 51 Lecture 10 Page 51

52 Gauge Capability Example (Both Factors are Random) Dependent Variable: RESP Source DF Sum of Squares Mean Square F Value Pr > F Model Error CorrecTotal Source DF Type III SS Mean Square F Value Pr > F OPERATOR PART OPERATOR*PART ˆσ 2 τ = MS A MS AB bn = ( )/6 = df = ( ) / /38 = CI: (18.57(10.28)/32.28, 18.57(10.28)/8.61) = (5.91, 22.17) 52 Lecture 10 Page 52

53 ˆσ 2 β = MS B MS AB an = ( )/40 = df = ( ) / /38 =.413 CI: (.413(.015)/3.079,.413(.015)/ ) = (.002, ) 53 Lecture 10 Page 53

54 General Mixed Effect Model In terms of linear 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 (allows correlation) Cov(Y ) = ZGZ + R If R = σ 2 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 Unrestricted linear mixed model is default 54 Lecture 10 Page 54