Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture #19 Analysis of Designs with Random Factor Levels

Fermentation Process Experiment MGH Ex 10.17

Fermentation Process Experiment Proc GLM data=Ferment; class batch process; model response = batch process; random batch / test; lsmeans process / stderr pdiff; run;

Fermentation Process Experiment The GLM Procedure Dependent Variable: Response Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square Coeff Var Root MSE Response Mean Source DF Type I SS Mean Square F Value Pr > F Batch Process Source DF Type III SS Mean Square F Value Pr > F Batch Process

Fermentation Process Experiment Mason, Gunst, & Hess: Exercise The GLM Procedure Source Type III Expected Mean Square Batch Var(Error) + 4 Var(Batch) Process Var(Error) + Q(Process) The Random Statement Produces This Output

Fermentation Process Experiment Mason, Gunst, & Hess: Exercise The GLM Procedure Tests of Hypotheses for Mixed Model Analysis of Variance Dependent Variable: Response Source DF Type III SS Mean Square F Value Pr > F Batch Process Error: MS(Error) The Random Statement Produces This Output

Fermentation Process Experiment Mason, Gunst, & Hess: Exercise Least Squares Means Response Standard LSMEAN Process LSMEAN Error Pr > |t| Number F < F < F < F < Least Squares Means for effect Process Pr > |t| for H0: LSMean(i)=LSMean(j) Dependent Variable: Response i/j LSMEANS Standard Errors Only use the Fixed Effects Computing Formulas These are Incorrect (See Proc Mixed Results)

Estimation of Variance Components: Method of Moments Equate mean squares to their expected mean squares and solve Equate mean squares to their expected mean squares and solve Random Main Effects Model Method of Moments F Test: MS A / MS E

Fermentation Process Experiment Mason, Gunst, & Hess: Exercise The GLM Procedure Source Type III Expected Mean Square Batch Var(Error) + 4 Var(Batch) Process Var(Error) + Q(Process)

Estimation of Variance Components: Method of Moments Equate mean squares to their expected mean squares and solve Equate mean squares to their expected mean squares and solve Three-Factor Random Effects Model Method of Moments F Test: MS ABC / MS E

Estimation of Variance Components Three-Factor Random Effects Model F Test: MS AB / MS ABC

Estimation of Variance Components Three-Factor Random Effects Model F Test: No Exact Test

Estimation of Variance Components Confidence Intervals 

Estimation of Variance Components Confidence Intervals 

Estimation of Variance Components Confidence Intervals

Testing Variance Components Three-Factor Random Effects Model F Test: No Exact Test

Satterthwaite’s Approximate F Statistic Assumptions MS 1, MS 2,..., MS k are Pairwise Independent ANOVA Mean Squares

Satterthwaite’s Approximate F Statistic Approximation L ~  L    L 

Satterthwaite’s Approximate F Statistic Solution

Satterthwaite’s Approximate F Statistic Application Select to be Independent of L Regardless of Ho Under Ho

Satterthwaite’s Approximate F Statistic

Approximation #1 M 1 & F Can Be Negative M 1 & F Can Be Negative

Satterthwaite’s Approximate F Statistic Approximation #2 M 2 & F Are Positive M 2 & F Are Positive

Random Effects Testing Three-Factor Random Effects Model Source Mean SquareExpected Mean Square AMS A  e 2 + r  abc  + cr  ab  + br  ag  + bcr  a  ABMS AB  e 2 + r  abc  + cr  ab  ABCMS ABC  e + r  abc  ErrorMS E  e 2 Effects Not Necessarily Tested Against Error Test Main Effects Even if Interactions are Significant May Not be an Exact Test (Mixed Effects Models) Effects Not Necessarily Tested Against Error Test Main Effects Even if Interactions are Significant May Not be an Exact Test (Mixed Effects Models)

Random Effects Testing Three-Factor Random Effects Model Source Mean SquareExpected Mean Square AMS A  e 2 + r  abc  + cr  ab  + br  ag  + bcr  a  ABMS AB  e 2 + r  abc  + cr  ab  ABCMS ABC  e + r  abc  ErrorMS E  e 2 Proc GLM: Random... / Test Produces Satterthwaite Approximate Test Statistics Fixed Effects Standard Errors May be Incorrect Proc GLM: Random... / Test Produces Satterthwaite Approximate Test Statistics Fixed Effects Standard Errors May be Incorrect

Restricted Maximum Likelihood

Proc Mixed Proc Mixed data=Ferment Cl; class batch process; model response = process; random batch ; lsmeans process / adjust=tukey pdiff; run;

Fermentation Process Experiment Mason, Gunst, & Hess: Exercise The Mixed Procedure Covariance Parameter Estimates Cov Parm Estimate Alpha Lower Upper Batch Residual Balanced Design: Same as Method of Moments

Fermentation Process Experiment Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Process Least Squares Means Standard Effect Process Estimate Error DF t Value Pr > |t| Process F <.0001 Process F <.0001 Process F <.0001 Process F <.0001 Correct Standard Errors

Fermentation Process Experiment Differences of Least Squares Means Standard Effect Process _Process Estimate Error DF t Value Pr > |t| Process F1 F Process F1 F Differences of Least Squares Means Effect Process _Process Adjustment Adj P Process F1 F2 Tukey-Kramer Process F1 F3 Tukey-Kramer Other Pairwise Comparisons on the Next Output Page Note: Standard Error of a Difference is Smaller than Random Effects Cancel in y i1 – y i2 (Pairwise Balance Needed)

Randomized Complete Block Designs Factorial Structure with Main Effect for Blocks Factorial Structure with Main Effect for Blocks Nothing New

Latin Square Designs Control Two Sources of Variability Restrictions Factor of Interest and Two blocking Factors Each at k Levels No Interactions Among the Experimental and Blocking Factors Experiment Size Latin Square : n = k 2 Complete Factorial : n = k 3 + r

Analysis of Latin Square Designs ith Row Block Effect Error Variation From All Sources Except Blocks & Factor Main Effects jth Column Block Effect kth Factor Level Effect Main Effects Analysis of Variance Model Main Effects Analysis of Variance Model

Balanced Incomplete Block Designs Used when blocks contain fewer experimental units than the number of unique factor-level combinations b blocks f factor-level combinations k < f experimental units per block No interactions with the design factor(s)

Asphalt-Pavement Rating Study Purpose Assess the Deterioration of Highway Pavement Response Rating: 0 = No pavement remaining 100 = excellent condition Design Factor 16 District Engineers (Random) Blocking Factor 16 Road Segments (Random)

Asphalt-Pavement Rating Study b = 16 Road Segments (Blocks) f = 16 Engineers (Factor-Level Combinations) k = 6 Engineers/Road Segment

Asphalt-Pavement Rating Study Design Engineer RoadSegmentRoadSegment

Analysis of Variance with Unbalanced Data Reduction in Error Sums of Squares R(M 1 | M 2 ) = SS E2 - SS E1 Error Sums of Squares Models 1 & 2 are Hierarchical Model 2 has a Subset of Model 1 Terms SS E2 SS E1 Testing Effects df = 2 - 1

Balanced Incomplete Block Design Model 1 Model 2 Model 3 Block Effect: R(M 1 | M 2 ) = SS E2 - SS E1 Factor Effect: R(M 1 | M 3 ) = SS E3 - SS E1 SAS PROC GLM Type I Sums of Squares Two Model Fits SAS PROC GLM Type I Sums of Squares Two Model Fits

Asphalt-Pavement Rating Study Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE Rating Mean Source DF Type I SS Mean Square F Value Pr > F Road <.0001 Engineer Source DF Type III SS Mean Square F Value Pr > F Road <.0001 Engineer

Asphalt-Pavement Rating Study Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE Rating Mean Source DF Type I SS Mean Square F Value Pr > F Engineer Road <.0001 Source DF Type III SS Mean Square F Value Pr > F Engineer Road <.000

Asphalt-Pavement Rating Study Asphalt-Paving Rating Study Mason, Gunst, & Hess: Table 10.4 The GLM Procedure Source Type III Expected Mean Square Engineer Var(Error) Var(Engineer) Road Var(Error) Var(Road)

Asphalt-Pavement Rating Study The Mixed Procedure Convergence criteria met Covariance Parameter Estimates Cov Parm Estimate Alpha Lower Upper Road Engineer Residual

Allergic Reaction Study: Randomized Complete Block Design MGH Table 10.6 Not Additive Not Additive

Balanced Incomplete Block Designs Multiple Comparisons Use Adjusted Factor-Level Averages Multiple Comparisons Use Adjusted Factor-Level Averages Average of r Block Averages Containing Factor-Level i MGH Exhibit 10.5