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 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

  • 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

  • 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 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 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 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 data

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


Advantage of longitudinal data analysis in HLM

  • 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

  • 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

  • Level 1 (within subject model)

    Yti is the measurement of ith subject at tthtime point

  • Level 2 (between subject model)


An example

covariance

Residual

Sample

Intercept

(Grand Mean)

Sample

slope

(Grand Mean)

Individual

Intercept

Deviation

Individual

slope

Deviation


In the model

  • 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

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

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

  • 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

  • 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

  • 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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