Linear Mixed Models for Longitudinal Data - Notes | STAT 587, Study notes of Statistics

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Linear Mixed Models for

Longitudinal Data

EdPsych/Psych/Stat 587C.J. Anderson

Linear Mixed Models for Longitudinal Data – p. 1/

Outline^ •^

Introduction • Approaches to Longitudinal Data Analysis • Longitudinal HLM by Example^ •^ The Riesby Data^ •^ Exploratory Analysis^ •^ Model Selection • Models for Serial Correlation

Linear Mixed Models for Longitudinal Data – p. 2/

Introduction^ •^

Purpose:

Study change

and the factors that

effect change. • Data:

Longitudinal data consist of repeated measurements on the same unit over time

-^ Models:

Hierarchical Linear Models

(linear

mixed models) with extensions for possibleserial correlation and non-linear pattern ofchange.

Linear Mixed Models for Longitudinal Data – p. 4/

Purpose: Change^ Goal:

Study change and the factors that effect intra- and inter-individual change.^ •^

Differences found in

cross-sectional data

often explained as reflecting change inindividuals. • A model for cross-sectional data

Yi^1

=^ β

βcs

xi^1

+^ ǫ

i^1

where

i^ = 1

,... , N

(individuals) and

xi^1

is

some time measure (e.g., age). • Interpretation:

βcs

=^

difference in

Y^ between

2 individuals that differ by 1 unit of time (

x). Linear Mixed Models for Longitudinal Data – p. 5/

Cross-Sectional Data

(continued)

Occasion 1:

̂reading

)= 111i^1

.^86

−^10

age

)i^1

Occasion 2:

̂reading

)= 140i^2

.^01

−^10

age

)i^2 Linear Mixed Models for Longitudinal Data – p. 7/

A Model for Longitudinal Data^ or repeated observations.

Yit^

=^ β

βcs

xi^1

+^ β

(xlit

−^ x

) +i 1

ǫit

-^ When

t^ = 1

, the model is the same as the

cross-sectional model. • β=l^

the expected change in

Y^ over time

per

unit change in the time measure

x^ (within

individual differences). • βcs^

still reflects differences between individuals. • βcs^

and

βreflect different processes.l^

Linear Mixed Models for Longitudinal Data – p. 8/

Advantages: Longitudinal Data^ More Powerful.^ •^

Inference regarding

βcs

is a comparison of

individuals with the same value of

x.

-^ Inference regarding

βis a comparison of anl^

individual’s response at two times —assuming

y^ changes systematically

with time

and retains it’s meaning

-^ Each individual is their own control group. •^ Often there is much more of variabilitybetween individuals than within individualsand the between variability is consistent overtime.

Linear Mixed Models for Longitudinal Data – p. 10/

Advantages: Longitudinal Data

(continued)

Distinguish Among Sources of Variation. Variation in

Y^ may be due

-^ Between individuals differences. •^ Within individuals:^ •

Measurement error & unobservedcovariates. • Serial correlation.

Linear Mixed Models for Longitudinal Data – p. 11/

Time^ •^

Time is a level 1 (micro level) predictor.The number of time points/occasions needed. • Measure of time should be^ •^ Reliable^ •^ Valid^ •^ Makes sense for outcome and researchquestions.

Linear Mixed Models for Longitudinal Data – p. 13/

Metric for Time^ Example from Singer & Willett:If you want to study the “longevity” ofautomobiles.^ •^

Change in appearance of cars

Age.

-^ Tire wear

Miles.

-^ Wear of ignition system

Trips (

#^ of

starts). • Engine wear

Oil changes.

Linear Mixed Models for Longitudinal Data – p. 14/

Three Major Approaches^ To analyzing longitudinal data.Classic reference: Diggle, Liang & Zeger^ •^

Marginal Analysis:

Only interested in average

response. • Transition Models:

Focus on how

Yit^

depends

on past values of

Y^ and other variables (i.e.,

a conditional model). • Random Effects Models:

Focus on how

regression coefficients vary over individuals.

Linear Mixed Models for Longitudinal Data – p. 16/

Marginal Analysis^ Focus on average of the response variable:

¯Y+t^

1 = N

N∑^ Y i=

it

and how the mean changes over time.^ •^ In HLM terms, only interested in the fixedeffects,

E(

Y) =it

Xi

-^ Observations are correlated, so need to makeadjustments to variance estimates, i.e.,var

(Y^ i

V^ i

(α)

where

α^ are parameters.

-^ “Sandwich estimator” or Robust estimation.

Linear Mixed Models for Longitudinal Data – p. 17/

Transitional Models

(continued)

We have focused on continuous/numerical

Y^ ’s,

but when

Y^ is categorical,

-^ “Stage sequential models” (e.g., must masteraddition and subtraction before can mastermultiplication). •^ The “gateway hypothesis” of drug use. Digression:

When

an event occurs is another

type discrete outcome variable, but we’re notconsidering such discrete variables in this class.

Linear Mixed Models for Longitudinal Data – p. 19/

Random Effects Models^ •^

Observations are correlated becauserepeated measurements are made on thesame individual. • Regression coefficients vary over individuals,i.e., E

(Yit

|βi^1

,... , β

) =ip

∑p k

β= xikikt

-^ One individual’s data does not containenough information to estimate

βik

’s;

therefore, we assume a distribution for

βik

’s,

β=i^

Zi

Γ^ +

U^ i

where

U^ i

∼ N

(^0 ,

T^ ).

Linear Mixed Models for Longitudinal Data – p. 20/