Experimental Statistics week 9 Chapter 17 Models with

Experimental Statistics - week 9 Chapter 17: Models with Random Effects Chapter 18: Repeated Measures 1

Discussion of Comments • upset about HW grade – I will drop one HW • availability of slides • HW - do by hand • in-class examples 2

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2 -Factor Mixed Effects Model Assumptions: fixed random Sum-of-Squares obtained as before 4

Expected Mean Squares for 2 -Factor ANOVA with Mixed Effects: SAS Expected MS (fixed) A (random) B Book’s Expected MS AB Error 5

Mixed-Effects Model To Test: use F = SAS uses F = use F = Again: Test each of these 3 hypotheses as in random-effects case. 6

2 -Factor Mixed-Effects ANOVA Table (using SAS Expected MS) Source SS df MS F Main Effects A SSA a - 1 B SSB b- 1 Interaction AB SSAB (a - 1)(b- 1) Error SSE ab(n - 1) Total TSS abn - 1 7

Estimating Variance Components 2 -Factor Mixed-Effects Model (based on SAS Expected MS) Note: A is a fixed effect 8

(F)ull Military Inspect. Product Inspection Response – fatigue of mechanical part A – type of inspection (a = ) B – inspector (randomly selected) (b = ) n= Inspector (R)educed Military Inspect. (C)ommercial 7. 50 7. 08 6. 15 7. 42 6. 17 5. 52 1 5. 85 5. 65 5. 48 5. 89 5. 30 5. 48 5. 35 5. 02 5. 98 7. 58 7. 68 6. 17 6. 52 5. 86 6. 20 2 6. 54 5. 28 5. 44 5. 64 5. 38 5. 75 5. 12 4. 87 5. 68 7. 70 7. 19 6. 21 6. 82 6. 19 5. 66 3 6. 42 5. 85 5. 36 5. 39 5. 35 5. 90 5. 35 5. 01 6. 12 9

Mixed-Effects Data DATA one; INPUT insp$ level$ fatigue; DATALINES; 1 F 7. 50 1 F 7. 42 1 F 5. 85 1 F 5. 89 . . . 2 C 5. 68 3 C 6. 21 3 C 5. 66 3 C 5. 36 3 C 5. 90 3 C 6. 12 ; PROC GLM; CLASS insp level; MODEL fatigue= level insp level*insp; TITLE 'Mixed-Effects Model'; RANDOM insp level*insp/test; RUN; PROC MEANS mean var; CLASS level; VAR fatigue; RUN; 10

SAS Mixed-Effects Output Mixed-Effects Model The GLM Procedure Dependent Variable: fatigue Sum of Source DF Squares Mean Square F Value Pr > F Model 8 2. 70711111 0. 33838889 0. 53 0. 8282 Error 36 23. 11448000 0. 64206889 Corrected Total 44 25. 82159111 R-Square Coeff Var Root MSE fatigue Mean 0. 104839 13. 35141 0. 801292 6. 001556 Source DF Type III SS Mean Square F Value Pr > F level 2 2. 58739111 1. 29369556 2. 01 0. 1481 insp 2 0. 02523111 0. 01261556 0. 02 0. 9806 insp*level 4 0. 09448889 0. 02362222 0. 04 0. 9973 11

SAS Mixed-Effects Output - Continued Mixed-Effects Model The GLM Procedure Source Type III Expected Mean Square level Var(Error) + 5 Var(insp*level) + Q(level) insp Var(Error) + 5 Var(insp*level) + 15 Var(insp) insp*level Var(Error) + 5 Var(insp*level) Mixed-Effects Model The GLM Procedure Tests of Hypotheses for Mixed Model Analysis of Variance Dependent Variable: fatigue Source DF Type III SS Mean Square F Value Pr > F level 2 2. 587391 1. 293696 54. 77 0. 0012 insp 2 0. 025231 0. 012616 0. 53 0. 6229 Error 4 0. 094489 0. 023622 Error: MS(insp*level) Source DF Type III SS Mean Square F Value Pr > F insp*level 4 0. 094489 0. 023622 0. 04 0. 9973 Error: MS(Error) 36 23. 114480 0. 642069 12

Multiple Comparisons for Fixed Effect (Inspection Level) -- Use MSAB in place of MSE where ▪ N denotes the # of observations involved in the computation of a marginal mean ▪ v denotes the df associated with AB interaction 13

SAS Mixed-Effects Output – Output from PROC Means The MEANS Procedure Analysis Variable : fatigue N level Obs Mean Variance ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ C 15 5. 8066667 0. 0981810 F 15 6. 3393333 0. 8208638 R 15 5. 8586667 0. 7405410 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 14

Mixed-Effects Example Results and Conclusions: 15

Repeated Measures Designs Setting: 1. Random sample of “subjects” 2. Each subject is measured at t different time points 3. Interested in the effect of treatment over time 16

Repeated Measures with a Single Factor Time Subject Reading for ith time period jth subject 17

Single Factor Repeated Measures Designs • single factor repeated measures model is similar to the randomized complete block model - i. e. 2 factors (subject and time) with one observation cell - since there is only one observation per cell, we cannot estimate an interaction term • typically: - subject is a random effect - time is a fixed effect time subject 18

ANOVA Table for Repeated Measure Design with Single Factor Source SS df MS EMS F Between subjects SSP n - 1 MSP MSP/MSE Time SSA a - 1 MSA/MSE Error SSE (n - 1)(a- 1) MSE Total TSS an - 1 19

Data – 5 subjects take tablet -- blood samples taken. 5, 1, 2, 3, and 4 hours after ingestion Goal: understand rate at which medicine enters blood Time Subject. 5 1 2 3 4 1 50 75 120 60 30 2 40 80 135 70 40 3 55 75 125 85 50 4 70 85 140 90 40 5 60 90 150 95 50 20

Dependent Variable: conc Sum of Source DF Squares Mean Square F Value Pr > F Model 8 26442. 00000 3305. 25000 66. 60 <. 0001 Error 16 794. 00000 49. 62500 Corrected Total 24 27236. 00000 R-Square Coeff Var Root MSE conc Mean 0. 970847 8. 985333 7. 044501 78. 40000 Source DF Type III SS Mean Square F Value Pr > F subject 4 1576. 00000 394. 00000 7. 94 0. 0010 time 4 24866. 00000 6216. 50000 125. 27 <. 000 21

The GLM Procedure t Tests (LSD) for conc NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha 0. 05 Error Degrees of Freedom 16 Error Mean Square 49. 625 Critical Value of t 2. 11991 Least Significant Difference 9. 4449 Means with the same letter are not significantly different. t Grouping Mean N time A 134. 000 5 2 B 81. 000 5 1 B 80. 000 5 3 C 55. 000 5 0. 5 D 42. 000 5 4 22

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Results: 24

Residual Diagnostics – 1 -factor Repeated Measures Data 25

Two-Factor Repeated Measure Data (p. 1033) Data – 10 subjects (5 take tablet, 5 take capsule) -- blood samples. 5, 1, 2, 3, and 4 hours after ingestion Goal: compare blood concentration patterns of the two methods of administration Tablet Time Subject. 5 1 2 3 4 1 50 75 120 60 30 2 40 80 135 70 40 3 55 75 125 85 50 4 70 85 140 90 40 5 60 90 150 95 50 Capsule Time Subject. 5 1 2 3 4 1 30 55 80 130 65 2 25 50 75 125 60 3 35 65 85 140 85 4 45 70 90 145 80 5 50 75 95 160 90 26

2 -Factor with Repeated Measure -- Model type subject within type time type by time interaction NOTE: type and time are both fixed effects in the current example 27

PROC GLM; CLASS type subject time; MODEL conc=type subject(type) time type*time; TITLE 'Repeated Measures – 2 factors'; OUTPUT out=new r=resid; MEANS type time/LSD; RANDOM subject(type)/test; 28

2 -Factor Repeated Measures – ANOVA Output The GLM Procedure Dependent Variable: conc Sum of Source DF Squares Mean Square F Value Pr > F Model 17 57720. 50000 3395. 32353 110. 87 <. 0001 Error 32 980. 00000 30. 62500 Corrected Total 49 58700. 50000 R-Square Coeff Var Root MSE conc Mean 0. 983305 6. 978545 5. 533986 79. 30000 Source DF Type III SS Mean Square F Value Pr > F type 1 40. 50000 1. 32 0. 2587 subject(type) 8 3920. 00000 490. 00000 16. 00 <. 0001 time 4 34288. 00000 8572. 00000 279. 90 <. 0001 type*time 4 19472. 00000 4868. 00000 158. 96 <. 0001 29

2 -factor Repeated Measures Source Type III Expected Mean Square type Var(Error) + 5 Var(subject(type)) + Q(type, type*time) subject(type) Var(Error) + 5 Var(subject(type)) time Var(Error) + Q(time, type*time) type*time Var(Error) + Q(type*time) The GLM Procedure Tests of Hypotheses for Mixed Model Analysis of Variance Dependent Variable: conc Source DF Type III SS Mean Square F Value Pr > F * type 1 40. 500000 0. 08 0. 7810 Error 8 3920. 000000 490. 000000 Error: MS(subject(type)) * This test assumes one or more other fixed effects are zero. Source DF Type III SS Mean Square F Value Pr > F subject(type) 8 3920. 000000 490. 000000 16. 00 <. 0001 * time 4 34288 8572. 000000 279. 90 <. 0001 type*time 4 19472 4868. 000000 158. 96 <. 0001 Error: MS(Error) 32 980. 000000 30. 625000 30

NOTE: Since time x type interaction is significant, and since these are fixed effects we DO NOT test main effects – we compare cell means (using MSE) Cell Means C T . 5 37 55 1 63 81 2 85 134 3 140 80 4 76 42 31

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Diagnostic Plots for 2 -Factor Repeated Measures Data 33