REPEATED MEASURES. Copyright c 2012 (Iowa State University) Statistics / 29
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1 REPEATED MEASURES Copyright c 2012 (Iowa State University) Statistics / 29
2 Repeated Measures Example In an exercise therapy study, subjects were assigned to one of three weightlifting programs i=1: The number of repetitions of weightlifting was increased as subjects became stronger (RI) i=2: The amount of weight was increased as subjects became stronger (WI) i=3: Subjects did not participate in weightlifting (XCont) Copyright c 2012 (Iowa State University) Statistics / 29
3 Measurements of strength (y) were taken on days 2, 4, 6, 8, 10, 12 and 14 for each subject Source: Littel, Freund, and Spector (1991), SAS System for Linear Models R code: RepeatedMeasuresR Copyright c 2012 (Iowa State University) Statistics / 29
4 A Linear Mixed-Effect Model y ijk = µ + α i + s ij + τ k + γ ik + e ijk Y ijk strength measurement for program i, subject j, time point k α i fixed program effect s ij random subject effect τ k fixed time effect γ ik fixed program time interaction e ijk random error Copyright c 2012 (Iowa State University) Statistics / 29
5 Initially, we will assume s ij iid N(0, σ 2 s ) independent of e ijk iid N(0, σ 2 e) Copyright c 2012 (Iowa State University) Statistics / 29
6 Average strength after 2k days on the ith program is µ ik = E(y ijk ) = E(µ + α i + s ij + τ k + γ ik + e ijk ) = µ + α i + E(s ij ) + τ k + γ ik + E(e ijk ) = µ + α i + τ k + γ ik for i = 1, 2, 3 and k = 1, 2,, 7 Copyright c 2012 (Iowa State University) Statistics / 29
7 The variance of any single observation is Var(y ijk ) = Var(µ + α i + s ij + τ k + γ ik + e ijk ) = Var(s ij + e ijk ) = Var(s ij ) + Var(e ijk ) = σs 2 + σe 2 Copyright c 2012 (Iowa State University) Statistics / 29
8 The covariance between an two different observations from the same subject is Cov(y ijk, y ijl ) = Cov(µ + α i + s ij + τ k + γ ik + e ijk, µ + α i + s ij + τ l + γ il + e ijl ) = Cov(s ij + e ijk, s ij + e ijl ) = Cov(s ij, s ij ) + Cov(s ij, e ijl ) +Cov(e ijk, s ij ) + Cov(e ijk, e ijl ) = Var(s ij ) = σ 2 s Copyright c 2012 (Iowa State University) Statistics / 29
9 The correlation between y ijk and y ijl is σ 2 s σ 2 s + σ 2 e ρ Observations taken on different subjects are uncorrelated Copyright c 2012 (Iowa State University) Statistics / 29
10 For the set of observations taken on a single subject, we have y ij1 σe 2 + σs 2 σs 2 σs 2 y ij3 Var = σs 2 σe 2 + σs 2 σ 2 s σs 2 y ij7 σs 2 σs 2 σs 2 σe 2 + σs 2 = σ 2 ei σ 2 s This is known as a compound symmetric covariance structure Copyright c 2012 (Iowa State University) Statistics / 29
11 Using n i to denote the number of subjects in the ith program, we can write this model in the form as follows y = Xβ + Zu + e Copyright c 2012 (Iowa State University) Statistics / 29
12 y 111 y 112 y 117 y 121 y 122 y 127 y 211 y 212 y 217 y 3n31 y 3n32 y 3n37 = columns for interaction µ α 1 α 2 α 3 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ 7 γ 11 γ 12 γ 37 + Copyright c 2012 (Iowa State University) Statistics / 29
13 s 11 s 12 s 3n3 + e 111 e 112 e 117 e 121 e 122 e 127 e 3n31 e 3n32 e 3n37 Copyright c 2012 (Iowa State University) Statistics / 29
14 In this case, G = Var(u) = σ 2 s I m m and R = Var(e) = σ 2 ei (7m) (7m), where m = n 1 + n 2 + n 3 is the total number of subjects Σ = Var(y) = ZGZ + R is a block diagonal matrix with one block of the form σs σei for each subject Copyright c 2012 (Iowa State University) Statistics / 29
15 LSMEAN for Program i and Time k ȳ i k = 1 n i n i j=1 y ijk = µ + α i + s i + τ k + γ ik + ē i k Var(ȳ i k ) = σ2 s + σ 2 e (6 se = 7 MS error + 1 ) 1 7 MS subj(prog) Cochran-Satterthwaite degrees of freedom = 658 n i n i Copyright c 2012 (Iowa State University) Statistics / 29
16 LSMEAN for Program i ȳ i k = ȳ i = µ + α i + s i + τ + γ i + ē i k=1 Var(ȳ i ) = 7σ2 s + σe 2 7n i MS subj(prog) se = 7n i Degrees of freedom = 54 because there are n 1 = 16, n 2 = 21, n 3 = 20 subjects in the three programs Copyright c 2012 (Iowa State University) Statistics / 29
17 LSMEAN for time k ȳ i k = µ + ᾱ i=1 se = 1 3 Var ( s i + τ k + γ k i=1 ) 3 ȳ i k = 1 9 i=1 3 i=1 ( 6 7 MS error MS subj(prog) σ 2 e + σ 2 s n i ) ( 3 i=1 3 i=1 ) 1 Cochran-Satterthwaite degrees of freedom = 658 n i ē i k Copyright c 2012 (Iowa State University) Statistics / 29
18 Difference between Strength Means at Times k and l (Averaging across Programs) (ȳ i k ȳ i l ) = 1 3 i=1 3 1 n i n i i=1 j=1 (y ijk y ijl ) = τ k τ l + γ k γ l n i (e ijk e ijl ) 3 i=1 n i j=1 Note that subject effects cancel out Copyright c 2012 (Iowa State University) Statistics / 29
19 The variance of the estimator is 2σe n i i=1 Thus, the standard error of the estimator is 2MS 3 error 1 9 n i i=1 Degrees of freedom = 7( ) ( ) = 324 Copyright c 2012 (Iowa State University) Statistics / 29
20 Difference between Strength Means for Program i and l at Time k ȳ i k ȳ l k = α i α l + γ ik γ lk + s i s l + ē i k ē l k ( 1 Var(ȳ i k ȳ l k ) = (σe 2 + σs 2 ) + 1 ) n i n l (6 se = 7 MS error + 1 ) ( 1 7 MS subj(prog) + 1 ) n i n l Cochran-Satterthwaite degrees of freedom = 658 Copyright c 2012 (Iowa State University) Statistics / 29
21 Difference between Strength Means at Times k and l within a Program i ȳ i k ȳ i l = τ k τ l + γ ik γ il + ē i k ē i l Var(ȳ i k ȳ i l ) = Var 1 n i (y ijk y ijl ) = 2σ2 e n i n i j=1 ( ) se = MS error 2ni Degrees of freedom = 324 Copyright c 2012 (Iowa State University) Statistics / 29
22 Difference in Strength Means between Programs i and l (Averaging across Time) ȳ i ȳ l = α i α l + γ i γ l + s i s l + ē i ē l ( 1 Var(ȳ i ȳ l ) = (7σs 2 + σe) ) 7n i 7n l ( 1 se = MS subj(prog) + 1 ) 7n i 7n l Degrees of freedom = 54 Copyright c 2012 (Iowa State University) Statistics / 29
23 Other Variance Matrices We began with the model where y ijk = µ + α i + s ij + τ k + γ ik + e ijk s ij iid N(0, σ 2 s ) independent of e ijk iid N(0, σ 2 e) Copyright c 2012 (Iowa State University) Statistics / 29
24 This model was expressed in the form where y = Xβ + Zu + e, u = s 11 s 3n3 contained the random subject effects Copyright c 2012 (Iowa State University) Statistics / 29
25 Here G = Var(u) = σ 2 s I, R = Var(e) = σ 2 ei, and Σ = Var(y) = ZGZ + R 11 = σs σ2 ei 11 σei 2 + σs 2 11 = σei 2 + σs 2 11 Copyright c 2012 (Iowa State University) Statistics / 29
26 If you are not interested in predicting subject effects (random subject effects are included only to introduce correlation among repeated measures on the same subject), you can work with an alternative expression of the same model by using the general linear model y = Xβ + ɛ, where Var(y) = Var(ɛ) = σ 2 ei + σ 2 s 11 σ 2 ei + σ 2 s 11 Copyright c 2012 (Iowa State University) Statistics / 29
27 More generally, we can replace the mixed model y = Xβ + Zu + e with the model y = Xβ + ɛ where W W 0 Var(y) = Var(ɛ) = 0 0 W Copyright c 2012 (Iowa State University) Statistics / 29
28 We can choose a structure for W that seems appropriate based on the design and the data One choice for W is a compound symmetric matrix like we have considered previously Another choice for W is an unstructured positive definite matrix A common choice for W when repeated measures are equally spaced in time is the first order autoregressive structure known as AR(1) Copyright c 2012 (Iowa State University) Statistics / 29
29 AR(1):First Order Autoregressive Covariance Structure W = σ 2 1 ρ ρ 2 ρ 3 ρ 4 ρ 5 ρ 6 ρ 1 ρ ρ 2 ρ 3 ρ 4 ρ 5 ρ 2 ρ 1 ρ ρ 2 ρ 3 ρ 4 ρ 3 ρ 2 ρ 1 ρ ρ 2 ρ 3 ρ 4 ρ 3 ρ 2 ρ 1 ρ ρ 2 ρ 5 ρ 4 ρ 3 ρ 2 ρ 1 ρ ρ 6 ρ 5 ρ 4 ρ 3 ρ 2 ρ 1 where σ 2 (0, ) and ρ ( 1, 1) are unknown parameters Copyright c 2012 (Iowa State University) Statistics / 29
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