4. Modeling Change with Covariates
• Analysts can use time –varying covariates at
level-1 to account for variation in observations
within individuals, and time-invariant
covariates(gender,race) at level-2 to account
for variation in growth parameters across
individuals.
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7. Modeling Change in growth rates
• Instantaneous rate of change at initial status.
• When positive,acceleration, the growth curve
is convex to the time.
• At least more than one time waves for the
growth parameters.
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8. • In multilevel polynomial growth models,
analysts interpret the highest-order term (e.g.,
cubic parameter)across the full range of the
time variable, whereas they interpret the
lower-order terms(intercept, linear, and
quadratic parameters) at the centering point.
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9. Modeling Change in Growth Rates
• Polynomial multilevel growth models
A quadratic growth curve includes the square
of the time variable, and the coefficient
represents the degree of acceleration or
deceleration in growth that occurs over time.
Typically, analysts test model fit by a likelihood
ratio test.
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23. Working with HLM3
• Level-1 file:EG1.SAV, 7242observations
collected on 1721children from grade1 to 6.
four level-1 variables:
year: year of the study minus 3.5
grade: the grade level minus 1.0
math:
retained: (1=retained, 0=not retained)
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24. Level-2 file: EG2.SAV
Three variables:
Female: (1=female, 0=male)
Black:(1=black,0=other)
Hispanic:(1=hispanic,0=other)
Level-3 file:
Three variables:
Size
Lowing
Mobile
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39. Conceptual and statistical background
for HMLM
學生 i (level-2 units) , 班級 j (leve-3 units),有m
個變數 Y1,……Ym被測量,這些測量值則是屬於
(level-1 units)(例如不同學科之測驗分數或不
同量尺的態度量表)
The dependent variable is denoted
Yhij is the measurement on the h’th variable for individual i in group j
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40. It is not necessary that, for each individual i in
each group j , an observation of each of the m
variables is available.
The complete data vector is denoted by
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41. The level-1 model relates the observed data Y, to
the complete data, Y*
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42. To make it clear ,consider T=5 and a person who
has data at occasions 1,2, and 4, but not at
occasions 3 and 5,
Or, in matrix notation
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