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Longitudinal data analysis in HLM

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Longitudinal data analysis in HLM

- Similar things:
- Fixed effects
- Random effects
- Difference:
- Cross-sectional HLM: individual, school,…
- Longitudinal HLM: observations over time, individual,…

- 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.

- 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.

- Univariate repeated measure ANOVA
- An alternative assumption, sphericity: it assumes equal variance difference between any two time points.

- 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

- 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.

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

- 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

- 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)?

- Level 1 (within subject model)
Yti is the measurement of ith subject at tthtime point

- Level 2 (between subject model)

covariance

Residual

Sample

Intercept

(Grand Mean)

Sample

slope

(Grand Mean)

Individual

Intercept

Deviation

Individual

slope

Deviation

- 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

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:

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Standard Approx.

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

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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

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:

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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

- 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.

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Random level-1 coefficient Reliability estimate

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INTRCPT1, B0 0.423

TIME, B1 0.108

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- 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

- 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)