Linear Mixed Models for Longitudinal Data: Modeling Serial Correlation - Prof. Carolyn J. , Study notes of Statistics

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Linear Mixed Models forLongitudinal DataModeling Serial Correlation^ EdPsych/Psych/Stat 587C.J. Anderson

Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 1/

Model for Level 1 Residuals^ There are three sources of possible variance^ •^ Between individuals, modeled by

ZU^ whereii^ U^ ∼ N^ (^0 ,^ T^ ).i^ • Within individuals,

R=^ e+^ eit^ (1)it^ (2)it

-^ Random (measurement error, variables notincluded, etc.),^

(^2) e∼ N (0, σ).(1)it

-^ Autocorrelated errors,

e... need model(2)it for this.

Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 2/

Possible models for

Rit

In an HLM/linear mixed model,^ Y^ Xii=^ (r×^ 1)^ (r×^ i^ i^

Z^ U^ Γii p) (p × 1) + (r×^ q)^ (q^ ×^ 1)^ i^ Ri + (r×^ 1)i^ where^ i^ = 1,... , N

and^ r=^ number of timei^ points for individual

i.

-^ U^ ∼ N^ (^0 ,^ T^ ).i^2 •^ R∼ N^ (^0 , σΩi^ i

-^ var(Y^ ) =^ V^ =ii^

′^2 ZT Z+^ σΩ.ii i^ Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 4/

Possible Models for Serial Correlation^ •^ Autoregressive (AR).^ •^ Moving average (MA).^ •^ Autoregressive with a moving average(ARMA).^ •^ Toeplitz.^ •^ Others.

Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 5/

Autoregressive Errors^ First order autoregressive process, AR1:^ R

=^ ρR+^ ǫit iti(t−1)^ where^2 •^ ǫ∼ N^ (0, σ)^ it^ ǫ^

i.i.d. • ρ is autocorrelation coefficient,^

0 ≤ |ρ|^ <^1.

-^ Stationarity:^ variance of

Rand covarianceit^ between^ Rand^ it^

′^ Rare independent ofit

t.

-^ Resulting error variance structure...

Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 7/

Autoregressive Errors^2 σ^ ǫ^2 σΩ^ =^2 (1^ −^ ρ)

02 1 ρ^ ρ.. .BBBBBBBBB^

1 r− 1 i ρCCr− 2 i ρ 1 ρ... ρCC 2 r− 3 Ci ρρ 1... ρ.CC .... C.... C. ..... .A^ r− 1 r− 2 r− 3 i i iρρρ... 1

In SAS, AR1 is parameterized as^01 ρ^ BBBBB^2 ∗^2 ∗ σΩ^ =^ σB^ ǫBBB^

12 r− 1 i ρ... ρCCr− 2 i ρ 1 ρ... ρCC 2 r− 3 Ci ρρ 1... ρwhereCC .... C.... C. ..... .A^ r− 1 r− 2 r− 3 iii ρρρ... 1

2 ∗^22 σ=^ σ/(1^ −^ ρ). ǫ^ ǫ^ Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 8/

Simulated Data: No Serial Correlation^ Y= 10 +^ (time)it^

+^ U+^ U(time)^0 j^1 j^ it

+^ Ritit where^ •^ (time)=^ t^ = 1^... ,

20 , and^ N^ = 50^ individuals. • U ∼ N ( 0 , T ) withi ^ ^4 −^2 ^ ^ T^ =^ −^2 4 • R∼ N ( 0 , 4).it Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 10/

No Serial Correlation: Data^2 R∼ N^ (0, σ)^ iid.it^

Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 11/

No Serial Correlation:

Rit

(^2) R∼ N (0, σ)^ iid.it

Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 13/

Simulated AR1 Data^ Y= 10+(time)it^ it

+U+U(time)^0 j^1 j^ it

+.^75 R+ǫiti(t−1)^ where^ •^ (time)=^ t^ = 1^... ,

20 , and^ N^ = 50^ individuals. • U ∼ N ( 0 , T ) withi ^ ^4 −^2 ^ ^ T^ =^ −^2 4 • ǫ∼ N ( 0 , 4).it (^) • R=. 75 R+^ ǫ.it i(t−1) it^ Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 14/

Eg of AR1: The

Rit^ Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 16/

Eg of AR1: Some

Rit^ Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 17/

Eg of AR1: OLS

2 ˆ R it^ Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 19/

Eg of AR1: OLS mean

2 ˆ R it Linear Mixed Models for Longitudinal DataModeling Serial Correlation – p. 20/