Longitudinal data analysis in hlm
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Longitudinal data analysis in HLM. Longitudinal vs cross-sectional HLM. Similar things: Fixed effects Random effects Difference: Cross-sectional HLM: individual, school,… Longitudinal HLM: observations over time, individual,…. Characteristics in longitudinal data.

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Longitudinal vs cross sectional hlm
Longitudinal vs cross-sectional HLM

  • Similar things:

  • Fixed effects

  • Random effects

  • Difference:

  • Cross-sectional HLM: individual, school,…

  • Longitudinal HLM: observations over time, individual,…


Characteristics in longitudinal data
Characteristics in longitudinal data

  • Source of variations

  • Within-subject variation (intra-individual variation)

  • Between-subject variation (inter-individual variation)

  • Often incomplete data or unbalanced data

  • OLS regression is not suitable to analyze longitudinal data because its assumptions are violated by the data.


Limitations of traditional approach for modeling longitudinal data
Limitations of traditional approach for modeling longitudinal data

  • Univariate repeated measure ANOVA

  • Person effects are random, time effects and other factor effects are fixed – it reduces residual variance by considering the person effects.

  • Fixed time point (evenly or unevenly spaced)

  • It assumes a unique residual variance-covariance structure (compound symmetry), which assume equal variance over time among observations from the same person and a constant covariance.


Limitations of traditional approach for modeling longitudinal data1
Limitations of traditional approach for modeling longitudinal data

  • Univariate repeated measure ANOVA

  • An alternative assumption, sphericity: it assumes equal variance difference between any two time points.


Limitations of traditional approach for modeling longitudinal data2
Limitations of traditional approach for modeling longitudinal data

  • Univariate repeated measure ANOVA

  • The assumptions could not be held for longitudinal data

  • People change at varied rates, so that variances often change over time

  • Covariances close in time usually greater than covariances distil in time

  • Test of variance-covariance structure is necessary to validate significance tests


Limitations of traditional approach for modeling longitudinal data3
Limitations of traditional approach for modeling longitudinal data

  • Multivariate repeated measure ANOVA

  • Use generalized method – no specific assumptions about variances and covariances (unstructured).

  • It does not allow any other structure, so when the repeated measures increase, it causes over-parameterization.

  • Subjects with missing data on any time point will be deleted from analysis.


Limitations of traditional approach for modeling longitudinal data4
Limitations of traditional approach for modeling longitudinal data

  • In addition, none of them allow time-varying predictors


Advantage of longitudinal data analysis in hlm
Advantage of longitudinal data analysis in HLM longitudinal data

  • Ability to deal with missing data (missing at random, MAR)

  • No assumptions about compound symmetry

  • More flexible:

  • Unequal numbers of measurement or unequal measurement intervals

  • Includes time-varying covariate


Research questions
Research questions longitudinal data

  • Is there any effect of time on average (fixed effect of time significant)?

  • Does the average effect of time vary across persons (random effect of time significant)?


A linear growth model
A Linear Growth Model longitudinal data

  • Level 1 (within subject model)

    Yti is the measurement of ith subject at tthtime point

  • Level 2 (between subject model)


An example
An example longitudinal data

covariance

Residual

Sample

Intercept

(Grand Mean)

Sample

slope

(Grand Mean)

Individual

Intercept

Deviation

Individual

slope

Deviation


In the model
In the model longitudinal data

  • Six Parameters:

  • Fixed Effects: β00and β10, level 2

  • Random Effects:

  • Variances of r0iand r1i(τ002, τ112), level 2

  • Covariance of r0iand r1i(τ01), level 2

  • Residual Variance of eti (σe2), level 1


Average growth trend
Average growth trend longitudinal data

Growth rate, average English increase at one unit of time increment is 1.50

Initial status, average English score at time 0 is 235.62

Final estimation of fixed effects:

----------------------------------------------------------------------------

Standard Approx.

Fixed Effect Coefficient Error T-ratio d.f. P-value

----------------------------------------------------------------------------

For INTRCPT1, P0

INTRCPT2, B00 235.619409 0.344737 683.476 6822 0.000

For TIME slope, P1

INTRCPT2, B10 1.500423 0.033980 44.156 6822 0.000


Random intercept slope
Random intercept-slope longitudinal data

Growth rates are different among different students, various slopes. A student whose growth is 1 SD above average is expected to grow at the rate of 1.50+0.93=2.43 per time unit

Initial status, students vary significantly in English score at time 0.

Final estimation of variance components:

-----------------------------------------------------------------------------

Random Effect Standard Variance df Chi-square P-value

Deviation Component

-----------------------------------------------------------------------------

INTRCPT1, R0 19.99410 399.76396 6703 11430.55547 0.000

TIME slope, R1 0.930350.86556 6703 9568.06880 0.000

level-1, E 24.81882 615.97397


Reliability
Reliability longitudinal data

  • Ratio of the “true” parameter variance to the “total” observed variance. Close to zero means observed score variance must be due to error.

  • Without knowledge of the reliability of the estimated growth parameter, we might falsely draw a conclusion due to incapability of detecting relations.

----------------------------------------------------

Random level-1 coefficient Reliability estimate

----------------------------------------------------

INTRCPT1, B0 0.423

TIME, B1 0.108

----------------------------------------------------


Correlation of change with initial status
Correlation of change with initial longitudinal datastatus

  • Choose “print variance-covariance matrices” under output settings.

  • Students who have higher English score at initial point tend to have a faster growth rate.

Tau (as correlations)

INTRCPT1,B0 1.000 0.413

TIME,B1 0.413 1.000


We could make it more complicated
We could make it more complicated longitudinal data

  • An intercepts- and Slopes-as-outcomes model

  • Level 1 (within subject model)

    Yti is the measurement of ith subject at tthtime point

  • Level 2 (between subject model)


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