What is a Launch relationship

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What is a Launch relationship?
Definition Individual
differences in the rate of change of a target
variable is predicted from individual differences
in the initial level of an antecedent variable.
Target Rate of change of the outcome Predictor
Initial level of antecedent
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Example Individual differences in attachment at
18 months of age LAUNCHES the trajectory of the
quality of relationships with peers throughout
life.
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What is an Ambient Level relationship?
Definition Individual differences in the average
level of an antecedent variable present across
time is important in shaping individual
differences in the rate of change in target
variable.
Target Rate of change of the outcome
Predictor Average level of antecedent
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Example A certain ambient level of parental
warmth is necessary for optimal development of
the psychosocial functioning of a child.
Increases in parental warmth are not necessary,
but decreasing parental warmth may be detrimental.
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What is a Change-to-Change relationship?
Definition Individual
differences in the pattern of change in an
antecedent variable predicts individual
differences in the trajectory of a target
variable.
Target Rate of change of the outcome
Predictor Rate of change of the antecedent
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Example As self-esteem improves, positive affect
also improves. As self-esteem decays, positive
affect also decays. It is the pattern of change
in self-esteem that predicts positive affect.
Self-esteem
Self-esteem
Positive Affect
Positive Affect
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What do you need to test Launch,
Change-to-Change, and Ambient Level Hypotheses?
1. Longitudinal data 2. At least 3 times of
measurement (although you can have missing
data) 3. Estimates of the Intercept, Slope, and
Ambient Level of the trajectory of
antecedent and/or target variables FOR EACH
STUDY PARTICIPANT, where a) The
Intercept of the Developmental Trajectory of an
Antecedent Variable is an indicator of
LAUNCH.
Level of an antecedent variable
Intercept of the Trajectory
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b) The Slope of the Developmental Trajectory of
an Antecedent or Target Variable is the
indicator of CHANGE
Slope
Slope is the rate of change
Level of the antecedent or target variable
(Slope Rise/Run)
Rise
Run
and c) the Mean Level of the Developmental
Trajectory of an Antecedent Variable is the
indicator of AMBIENT LEVEL.
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How do you estimate the Intercept and Slope of a
Developmental Trajectory of a Variable for each
individual study participant?

• Use hierarchical linear modeling.
• Hierarchical linear modeling (HLM), which is
  also called a random effects model or the general
  linear mixed model, can be used to estimate both
  fixed and random effects simultaneously. Fixed
  effects are variables that are assumed to
  represent all possibilities in the entire
  population of interest. Random effects are those
  that are assumed to have been randomly selected
  from all possibilities. In this case, times of
  measurement and study participants are assumed to
  have been randomly selected from the entire
  possible collection of times and participants.
• When intercept and time of measurement are
  specified as random effects in HLM, and there are
• repeated measures of variables of interest for
  each subject, estimates of the intercept and
  slope of the developmental trajectory of each
  participant can be made based upon information
  available about the average study population
  trajectory and the individuals own repeated
  measures of the variable of interest.

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In the following example the MIXED procedure
available in SAS/STAT software was used to 1)
assess the population trajectory of Perceived
Control in children in 3rd to 7th grade and to
determine if there were group differences in
trajectories. 2) output estimates of the
intercept and slope of each individuals
developmental trajectory of perceived control,
and 3) output values of perceived control
predicted by the HLM model. This example is a
portion of a cross-sectional sequential study
that had LAUNCH, AMBIENT LEVEL, and
CHANGE-TO-CHANGE hypotheses.
In this study, two waves of children, initially
in grades 3 to 6, participated in a study
of perceived control and motivation in the
classroom. Children completed a measure
of perceived control (as well as other measures)
in the fall and spring of the school year for up
to three consecutive years. The numbers of
children participating at each time of
measurement are shown in Figure 1.
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1) Assessing the Population Trajectory of
Perceived Control and Between Group Differences
in Developmental Trajectories
In order to use HLM to estimate the population
trajectory of perceived control (or subsample
trajectories) the data are structured as in the
following example. As you can see, there is a
record for each subject at each time of
measurement that an assessment was completed .
Also, not all participants had data at all times
of measurement.
SUBNUM SEX WAVE GRADE TIME
CON 4410 2 1
6 7 30.11 4410
2 1 6
8 31.39 4410 2
1 7 10 34.42
6340 2 1
6 7 32.14 6340
2 1 6
8 23.58 6340 2 1
7 9 19.10
6340 2 1
7 10 22.58 7598 2
1 6 8
29.33 7598 2 1
7 10 9.75 8360
1 2 3
1 28.25 8360 1
2 3 2
22.46 8360 1 2
4 3 39.00 8360
1 2 4
4 21.67 ETC...
TIME indicates the time of measurement and is
coded as follows 1 Measurement of a 3rd grade
student in the fall of the school
year 2 3rd grade spring measure 3 4th grade
fall measure ... 9 7th grade fall measure 10
7th grade spring measurement. CON is the measure
of Perceived Control.
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The following is an example SAS program using
PROC MIXED. Lines were numbered on the right
to provide points of reference for the following
section. PROC MIXED DATASASUSER.GW36 CLASS
SUBNUM SEX WAVE 1 MODEL CONTIME SEX WAVE
TIMESEX TIMEWAVE / SOLUTION CHISQ 2 RANDOM
INTERCEPT TIME / TYPEUN SUBJECTSUBNUM 3 RUN
In line 1, SUBNUM, SEX, and WAVE were
specified as CLASS variables. This indicates
that these are nominal variables. Line 2
contains the MODEL statement. CON (perceived
control) is the dependent variable. CON is
measured at multiple points in TIME. Therefore,
the first independent variable specified is
TIME which specifies the time of measurement.
SEX and WAVE are additional independent
variables. These effects will test whether the
level of the trajectories of perceived control
differed between participants grouped by sex or
wave. TIMESEX and TIMEWAVE are interaction
effects which test whether changes in perceived
control over time differ by sex or by wave.
SOLUTION requests a solution for the fixed effect
parameters be printed and a CHISQ requests of
Chi-square test of these effects in addition to
the F test. Higher level effects may be added
to test whether a development trajectory has
a quadratic, cubic, etc. shape. For example, to
test a quadratic shape, TIMETIME would be
included as an additional model effect. Line 3
contains the RANDOM statement which indicates
that the intercept and time are random
effects. TYPE allows for selection of the
covariance structure of random effects. TYPEUN
requests an unstructured covariance matrix. This
is recommended for a correlated random
coefficient model. SUBJECTSUBNUM identifies the
subjects in you database.
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The SAS output can be interpreted as
follows Since this is an iterative method, the
iterations are first printed. The Covariance
Parameter Estimates are computed using the
default residual maximum likelihood method
(REML). UN(1,1) is the test of whether the
variance in the intercepts of perceived
control trajectories is significantly different
from 0. The test performed is the Wald Z.
Since, Pr gt Z is less than .05, the variance of
the intercepts of the developmental trajectories
of perceived control it is significantly
different from 0. UN(2,2) tests the variation
among the slopes of the developmental
trajectories of perceived control. Since, Pr gt
Z is less than .05, there is also significant
variation in the slope of the trajectories of
perceived control. Model Fitting Information for
CON gives various pieces of information on the
fit of the model including the number of
observations, the residual variance estimate and
the square root of this estimate. The Solution
for Fixed Effects table is next printed. This
information indicates that the intercept of the
average perceived control trajectory of SEX 2 is
37.83 (the Intercept Estimate) while the
intercept for SEX 1 is 37.83 .81 38.64. In
addition, the estimate of the slope of the
perceived control trajectory of SEX 2 is -2.02
(the Time Estimate), while the slope for SEX 1 is
1.06 less (-2.02-1.06-3.08). Thus, group SEX 1
starts with a higher initial level of perceived
control than group SEX 2, but declines more
rapidly. (However, in the next section it is
shown that these differences are not significant.)
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The Tests of Fixed Effects table report s tests
of the unique effect of each fixed effect
specified. Neither the TIMESEX interaction or
the TIMEWAVE interaction are significant at the
plt.05 level (see PrgtF). These results indicate
that there was no significant differences in the
slopes of perceived control between participants
grouped by sex or grouped by wave. The overall
effects of SEX and WAVE are also not significant
(although, the interactions should be removed and
the model re-fit to draw this conclusion).
Therefore, in this model only the effect of TIME
is significant indicating that perceived control
significantly changed from 3rd grade to 7th
grade. In addition, since the parameter estimate
of TIME equals -2.02, there is a significant
downward slope of perceived control from 3rd to
7th grade for this population.
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2) Output Estimates of the Intercept and Slope
of Each Individuals Developmental Trajectory of
Perceived Control
An additional option is added to the PROC MIXED
program to request a solution to the random
effects designated. This solution provides
estimates of the intercepts and slopes of the
trajectory of the dependent variable for each
study participant. The option is underlined in
the following example SAS program PROC MIXED
DATASASUSER.GW36 CLASS SUBNUM SEX WAVE
MODEL CONTIME SEX WAVE TIMESEX TIMEWAVE /
SOLUTION CHISQ RANDOM INTERCEPT TIME /
TYPEUN SUBJECTSUBNUM SOLUTIONRUN
The following is an example of the output
received by including the SOLUTION option (as
well as some additional programming to actually
output the information to a file)
SUBNUM INTERCEPT INTSE TIME
TIMESE 4410 4.93949332
6.80477868 2.00358954
2.38626018 6340 4.44533301
6.76336746 -0.60975877 2.33977441 7598
6.79776566 9.50030688
-2.52123912 2.83436687 8360
10.50907960 6.76349884 1.14906584
2.33977923
INTERCEPT is the estimated intercept of each
subjects perceived control trajectory. TIME is
the estimated slope of the trajectory of
perceived control for each subject. For example,
the developmental trajectory of perceived control
for SUBNUM 4410 had an estimated intercept of
4.94 and an estimated slope of 2.00. INTSE and
TIMESE are the standard errors of the intercept
and slope estimates, respectively, for each
subject. Figure 2 illustrates the trajectories of
perceived control of a few study participants
and the estimates of the intercepts and slopes of
their trajectories.
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The intercepts and slopes can then be used as
usual individual difference variables in
more traditional correlation or regression
analyses. for example, in Table 1 two multiple
regressions examining relationships between the
intercept and/or slope of the trajectories of
perceived control and the slope parameter of the
developmental trajectory of engagement in the
classroom are shown. These regression results
indicate that there was no LAUNCH relationship
between student perceptions of perceived control
in the academic domain (CON) and engagement in
the classroom (?.01). However, there was a
CHANGE-TO-CHANGE relationship (? .22, plt.001).
Note, that in the 2nd regression model, the
intercept of the trajectory of engagement in the
classroom was included as an independent
variable. This is to account for possible
ceiling effects and to determine the unique
LAUNCH and CHANGE-TO-CHANGE relationships between
perceived control and engagement
after partialling out the intercept of engagement.
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3) Output Values of Perceived Control Predicted
by the HLM Model.
Finally, HLM can be used to output values of a
variable of interest at each time of measurement
for all participants. PROC MIXED
DATASASUSER.GW36 CLASS SUBNUM SEX WAVE
MODEL CONTIME SEX WAVE TIMESEX TIMEWAVE /
SOLUTION CHISQ PREDICTED RANDOM INTERCEPT
TIME / TYPEUN SUBJECTSUBNUM ID SUBNUM
TIME MAKE PREDICTED OUTSASUSER.CONPREDRUN
The PREDICTED option requests predicted values
for the dependent variable for each subject at
each time of measurement. The input dataset must
include a record for each subject and time for
which you want to output predicted values. For
example
SUBNUM SEX WAVE GRADE TIME
CON 4410 2 1
3 1 . 4410
2 1 3
2 . 4410 2 1
4 3 34.42
4410 2 1
4 4 32.14 4410 2
1 5 5
23.58 4410 2 1
5 6 19.10
4410 2 1
6 7 . 4410 2
1 6 8
18.00 4410 2 1
7 9 . 4410
2 1 7
10 18.50 ETC...
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The ID statement asks that each record in the
output dataset contain these variables
for identification. The MAKE statement asks SAS
to output the predicted values of the
dependent variable to a dataset name
CONPRED. The following is an example of the
output received by including the P option and
MAKE statement
SUBNUM TIME CONOBS CONPRED CONV_PR
CONSE_P CONL95M CONU95M CONRES 4410
1 . 35.01 2.05
4410 2 . 35.06 1.54
4410 3 34.42 34.32 2.34
ETC... 4410 4 32.14 32.25
2.21 4410 5
23.58 26.45 2.10 4410 6
19.10 19.13 1.78 4410 7
. 18.67 1.79 4410 8
18.00 18.10 1.87 4410 9
. 18.25 1.76 4410
10 18.50 18.43 1.87 ETC...
CONOBS is the actual score for perceived control,
CONPRED is the value predicted by HLM.
In addition, the output table includes the
variance of the predicted value, the standard
error of the predicted value, the lower and upper
95 confidence interval of the predicted value,
and the residual. These predicted values can be
used to plot complete trajectories of perceived
control from fall of the 3rd grade to spring of
the 7th grade whether or not a participant had
missing data (see Figure 2). These could also be
used to compute ambient levels. The ambient
level for each participant would be the average
of his/her 10 predicted values of CON.
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For further information seeBurchinal, M.,
Appelbaum, M.I. (1991). Estimating individual
developmental function Methods and their
assumptions. Child Development, 62,
23-43.Bryk, A.S., Raudenbush, S.W. (1992).
Hierarchical linear models Application and data
analysis methods. Newbury Park, CA
Sage.Francis, D.J., Fletcher, J.M., Steubing,
K.K., Davidson, K.C., Thompson, N.M. (1991).
Analysis of change Modeling individual growth.
Journal of consulting and Clinical
Psychology, 59, 27-37.Bailey, Jr., D.B.,
Burchinal, M.R., McWilliam, R.A. (1993). Age
of peers and early childhood development.
Child Development, 64, 848-862.Skinner, E.A.
(1995). Perceived control, motivation, and
coping. Newbury Park, CA Sage
Publications.Wellborn, J.G., Connell, J.P.,
Skinner, E.A. (1989). The Students Perceptions
of Control Questionnaire (SPOCQ) Academic
domain. Technical report, University of
Rochester, New York.Willet, J.B., Ayoub,
C.C. Robinson, D. (1991). Using growth
modeling to examine systematic differences in
growth An example of change in the functioning
of families at risk of maladaptive parenting,
child abuse, and neglect. Journal of Consulting
and Clinical Psychology, 59, 38-47.