growth curves and extensions using mplus - UCLA Integrated ...

GROWTH CURVES AND EXTENSIONS USING MPLUS
Alan C. Acock
alan.acock@oregonstate.edu
Department of HDFS
322 Milam Hall
Oregon State University
Corvallis, OR 97331
7/2008
This document and selected references, data, and programs can be downloaded from
http://oregonstate.edu/~acock/growth
Growth Curve and Related Models, Alan C. Acock 1

GROWTH CURVES AND EXTENSIONS USING MPLUS
Outline
Topic Page
1 Brief summary of topics 3
2 A growth curve 4
3 Quadratic terms in growth curves 17
4 An alternative—developmental time 23
5 Working with missing values 24
6 Multiple cohort growth model with missing waves 25
7 Multiple group models with growth curves 27
8 Alternative to multiple group analysis 35
9 Growth curves with time invariant covariates 41
10 Mediational models with time invariant covariates 49
11 Time varying covariates 50
12 References 51
Goal of the Workshop
The goal of this workshop is to explore a variety of applications of latent growth curve models
using the Mplus program. Because we will cover a wide variety of applications and extensions of
growth curve modeling, we will not cover each of them in great detail. At the end of this
workshop it is hoped that participants will be able to run Mplus programs to execute a variety of
growth curve modeling applications and to interpret the results correctly.
Assumed Background
Participants should be familiar with the content in Introduction to Mplus that is located at
www.oregonstate.edu/~acock/growth . It will be assumed that participants in the workshop have
some background in Structural Equation Modeling. Background in multilevel analysis will also be
useful, but is not assumed. It is possible to learn how to estimate the specific models we will cover
Growth Curve and Related Models, Alan C. Acock 2

without a comprehensive knowledge of Mplus, but some background using an SEM program is
useful.
Introduction to Growth Curve Modeling
1 Brief Summary of Topics
Growth Curves are a new way of thinking that is ideal for longitudinal studies. Instead of
predicting a person’s score on a variable (e.g., mean comparison among scores at different time
points or relationships among variables at different time points), we predict their growth
trajectory—what is their level on the variable AND how is this changing. We will present a
conceptual model, show how to apply the Mplus program, and interpret the results. Once we can
estimate growth trajectories, the more interesting issues of explaining individual differences in
trajectories (why some people go up, down, or stay the same). More advanced topics we will
introduce include:
1. Growth Curves with Limited Outcome Variables
Sometimes a researcher is interested in growth on variable that is not continuous.
• It may be a binary variable.
o Ever drinking alcohol for adolescents.
o Retention of program participants across 10 sessions.
• Some times a researcher is interested in a count variable that involves a relatively rare
event
o Number of days an adolescent has 5+ drinks of alcohol in the last 30 days.
o Number of times a couple has severe verbal conflict in last week.
• Sometimes we are interested in both types of variables at the same time.
o Onset of sexual intercourse between 12 – 18 cf. frequency of sexual intercourse.
o Onset of binge drinking cf. frequency of binge drinking.
o Different variables may predict the binary aspect (onset) of the variable than
predict the count aspect of the variable.
• These applications can be extremely CPU intensive, taking hours to converge. Mplus
utilizes multiple processors.
2. Growth Mixture Models
It is possible to use Mplus to do an exploratory growth curve analysis where our focus is on
the person and not the variable. We can locate clusters of people who share similar growth
trajectories. This is exploratory research and the standards for it are still evolving.
Growth Curve and Related Models, Alan C. Acock 3

• A study of alcohol consumption from age 15 to 30. It is possible to empirically identify
different clusters of people.
o One cluster may never drink or never drink very much.
o A second cluster may have increasing alcohol consumption up to about 22 or 23
and then a gradual decline.
o A third cluster may be very similar to the second cluster but not decline after 23.
• A study of marital conflict across the first 5 years of marriage
o One cluster may have persistently low conflict
o One cluster may have monotonic escalating conflict
o One cluster may have persistently high conflict
o One cluster may have initially low conflict, peak at the end of the first year, and
then have monotonically declining conflict.
After deriving these clusters of people who share growth trajectories, it is possible to compare
them to find what differentiates membership in the different clusters.
• What covariates predict cluster membership?
• What outcome variables does cluster membership predict?
2 A Simple Growth Curve
Estimating a basic growth curve using Mplus is quite easy. When developing a complex model it
is best to start easy and gradually build complexity.
• Starting easy should include data screening to evaluate the distributions of the variables,
patterns of missing values, and possible outliers.
• Even if you have a theoretically specified model that is complex, always start with the
simplest model and gradually add the complexity.
• Here we will show how structural equation modeling conceptualizes a latent growth curves.
Before showing a figure to represent a growth curve, we examine a small sample of our
observations:
• Data is from the National Longitudinal Survey of Youth that started in 1997.
• We use the cohort that was 12 years old in 1997 and examine their trajectory for the BMI.
• Some may not like using the BMI on this age group, but this is only to illustrate an
application of growth curve modeling.
• The following graph of 10 randomly selected kids was produced by Mplus
Growth Curve and Related Models, Alan C. Acock 4

• A BMI value of 25 is considered overweight and a BMI of 30 is considered obese (I’m
aware of problems with the BMI as a measure of obesity and with its limitations when used
for adolescents)
• With just 10 observations it is hard to see much of a trend, but it looks like people are
getting a bigger BMI score as they get older.
• The X-axis value of 0 is when the adolescent was 12 years old; the 1 is when the adolescent
was 13 years old, etc. We are using seven waves of data (labeled 0 to 6) from the panel
study.
A growth curve requires us to have a model and we should draw this before writing the Mplus
program. Figure 1 shows a model for our simple growth curve:
Growth Curve and Related Models, Alan C. Acock 5

This figure is much simpler than it first appears.
• The key variables are the two latent variables labeled the Intercept and the Slope.
• The Intercept
a. The intercept represents the initial level and is sometimes called the initial level for this
reason. It is the estimated initial level and its value may differ from the actual mean for
BMI97 because in this case we are imposing a linear growth model.
b. It may differ from the mean of BMI97 when covariates are added, especially when a zero
value on covariates is rare and covariates are not centered (household income)
c. Unless the covariates are centered, it usually makes sense to just call it an intercept rather
than the initial level.
Growth Curve and Related Models, Alan C. Acock 6

d. The intercept is identified by the constant loadings of 1.0 going to each BMI score. Some
programs call the intercept the constant, representing the constant effect to which other
effects are added.
e. It is possible to shift the intercept by how the waves are coded, e.g., -2 -1 0 1 2.
• The slope
a. Is identified by fixing the values of the paths to each BMI variable. In a publication you
normally would not show the path to BMI97, since this is fixed at 0.0.
b. We fix the other paths at 1.0, 2,0, 3.0, 4.0, 5.0, and 6.0. Where did we get these values? The
first year is the base year or year zero. The BMI was measured each subsequent year so
these are scored 1.0 through 6.0.
c. Other values are possible. Suppose the survey was not done in 2000 or 2001 so that we had
5 time points rather than 7. We would use paths of 0.0, 1.0, 2.0, 5.0, and 6.0 for years 1997,
1998, 1997, 2002, and 2003, respectively.
d. It is also possible to fix the first couple years and then allow the subsequent waves to be
free.
- This might make sense for a developmental process where the yearly intervals may not
reflect the developmental rate. Developmental time may be quite different than
chronological time.
- This has the effect of “stretching” or “shrinking” time to the pattern of the data (Curran
& Hussong, 2003).
- An advantage of this approach is that it uses fewer degrees of freedom than adding a
quadratic slope.
- Mplus has a feature that allows each participant to have a different interval which is
important when the time between waves varies.
• Residual Variance and Random Effects
a. The individual variation around the Intercept and Slope are represented in Figure 1 by the RI
and R2. These are the variance in the intercept and slope around their respective means.
b. We expect substantial variance in both of these as some individuals have a higher or lower
starting BMI and some individuals will increase (or decrease) their BMI at a different rate
than the average growth rate.
c. In addition to the mean intercept and slope, each individual will have their own intercept
and slope. We say the intercept and the slope are random effects since they may vary
across individuals.
- They are random in the sense that each individual may have a steeper or flatter slope
than the mean slope and
- Each individual may have a higher or lower initial level than the mean intercept.
d. In our sample of 10 individuals shown above, notice one adolescent starts with a BMI
around 12 and three adolescents start with a BMI around 30. Some children have a BMI that
increases and others do not.
Growth Curve and Related Models, Alan C. Acock 7

e. The variances, RI and R2 are critical if we are going to explore more complex models with
covariates (e.g., gender, psychological problems, race, household income, physical activity)
that might explain why some individuals have a steeper or less steep growth rate than the
average.
• The ei terms represent individual error terms for each year. Some years may move above or
below the growth trajectory described by our Intercept and Slope.
• Sometimes it might be important to allow error terms to be correlated, especially subsequent
pairs such as e97-e98, e98-e99, etc.
Here is the Mplus program for a simple growth model:
Title: bmi_growth.inp
Basic growth curve
Data:
File is "C:\Mplus examples\bmi_stata.dat" ;
Analysis: Processors = 2;
Variable:
Names are
id grlprb_y boyprb_y grlprb_p boyprb_p
male race_eth bmi97 bmi98 bmi99 bmi00
bmi01 bmi02 bmi03 white black hispanic
asian other;
Missing are all (-9999) ;
! Notice usevariables is limited to bmi variables
Usevariables are bmi97 bmi98 bmi99 bmi00
bmi01 bmi02 bmi03 ;
Model:
i s | bmi97@0 bmi98@1 bmi99@2 bmi00@3
bmi01@4 bmi02@5 bmi03@6;
Output:
Sampstat Mod(3.84);
Plot:
Type is Plot3;
Series = bmi97 bmi98 bmi99 bmi00 bmi01
bmi02 bmi03(*);
What is new compared to an SEM program?
• Usevariables are: subcommand to only include the BMI variables since we are doing a
growth curve for these variables.
Growth Curve and Related Models, Alan C. Acock 8

• We drop the Analysis: section if we have a single processor because we are doing basic
growth curve and can use the default options. With multiple processors, this is included to tell
Mplus how many processors to utilize. (This model ran 40% faster on a 2 processor IMac
running Windows under Parallels.)
• We have a Model: section because we need to describe the model. Mplus was designed after
growth curves were well understood. There is a single line to describe our model:
i s | bmi97@0 bmi98@1 bmi99@2 bmi00@3 bmi01@4 bmi02@5 bmi03@6;
a. In this line the “i” and “s” stand for the intercept and slope, respectively. We could have
called these anything such as intercept and slope or initial and trend. The
vertical line, | (sometimes called “oar bar”, tells Mplus that it is about to define an
intercept and slope.
b. Defaults
- The intercept is defined by a constant of 1.0 for each bmi variable. Intercept�bmij path
is 1.0. Therefore, we do not need to mention this.
- The slope is defined by fixing the path from the slope to bmi97 at 0, the path to bmi98
at 1, etc. The @ sign is used for “at.” Don’t forget the semi-colon to end the command.
- Mplus assumes that there is a residual variance for both the intercept and slope (RI and
R2) and that these covary. Therefore, we do not need to mention this
- Mplus assumes there is uncorrelated random error, ei for each observed variable
c. To allow e97 and e98 to be correlated, we would need to add a line saying bmi97 with
bmi98; .
- This may seem strange because we are not really correlating bmi97 with bmi98, but
e97 with e98. Mplus knows this and we do not need to generate a separate set of names for
the error terms.
The last additional section in our Mplus program is for selecting what output we want Mplus to
provide. There are many optional outputs of the program and we will only illustrate a few of these.
The Output: section has the following lines
Output:
Sampstat Mod(3.84);
Plot:
Type is Plot3;
Series = bmi97 bmi98 bmi99 bmi00
bmi01 bmi02 bmi03(*);
Growth Curve and Related Models, Alan C. Acock 9

• The first line, Sampstat Mod(3.84) asks for sample statistics and modification indices for
parameters we might free, as long as doing so would reduce chi-square by 3.84 (corresponding
to the .05 level). We do not bother with parameter estimates that would have less effect than
this. The default value is 10.0.
• Next comes the Plot: subcommand and we say that we want Type is Plot3; for our
output. This gives us the descriptive statistics and graphs for the growth curve.
• The last line of the program specifies the series to plot. By entering the variables with an (*)
at the end we are setting a path at 0.0 for bmi97, 1.0 for bmi98, etc.
Annotated Selected Growth Curve Output
The following is selected output with comments:
Mplus VERSION 5.1
MUTHEN & MUTHEN
07/01/2008 2:40 PM
*** WARNING
Data set contains cases with missing on all variables.
These cases were not included in the analysis.
Number of cases with missing on all variables: 3
1 WARNING(S) FOUND IN THE INPUT INSTRUCTIONS
Mplus uses all available data assuming MAR. There were three cases that were
dropped because they had no BMI report for any wave.
bmi_growth.inp
Basic growth curve
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 1768
With listwise deletion we would have an N = 1102
Number of dependent variables 7
Number of independent variables 0
Number of continuous latent variables 2
Observed dependent variables
Continuous
BMI97 BMI98 BMI99 BMI00 BMI01 BMI02
BMI03
Continuous latent variables
I S
Growth Curve and Related Models, Alan C. Acock 10

The following is a very nice analysis of patterns of missing values.
Estimator ML
Information matrix OBSERVED
SUMMARY OF DATA
Number of missing data patterns 81
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value 0.100
PROPORTION OF DATA PRESENT
Covariance Coverage
BMI97 BMI98 BMI99 BMI00 BMI01
________ ________ ________ ________ ________
BMI97 0.925
BMI98 0.847 0.902
BMI99 0.850 0.856 0.910
BMI00 0.842 0.846 0.864 0.906
BMI01 0.839 0.837 0.854 0.859 0.904
BMI02 0.796 0.794 0.805 0.811 0.817
BMI03 0.777 0.775 0.788 0.788 0.801
Covariance Coverage
BMI02 BMI03
________ ________
BMI02 0.861
BMI03 0.774 0.840
Check means to see if there is a clear overall trajectory
SAMPLE STATISTICS
ESTIMATED SAMPLE STATISTICS
Means
BMI97 BMI98 BMI99 BMI00 BMI01
________ ________ ________ ________ ________
1 20.572 21.839 22.651 23.305 23.846
Means
BMI02 BMI03
________ ________
1 24.390 24.935
Correlations
BMI97 BMI98 BMI99 BMI00 BMI01
________ ________ ________ ________ ________
BMI97 1.000
BMI98 0.764 1.000
BMI99 0.765 0.850 1.000
BMI00 0.721 0.812 0.853 1.000
BMI01 0.709 0.799 0.853 0.856 1.000
BMI02 0.652 0.720 0.745 0.752 0.813
BMI03 0.651 0.707 0.737 0.751 0.815
Correlations
BMI02 BMI03
________ ________
Growth Curve and Related Models, Alan C. Acock 11

BMI02 1.000
BMI03 0.766 1.000
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value 268.041
Degrees of Freedom 23
P-Value 0.0000
Chi-Square Test of Model Fit for the Baseline Model
Value 11502.912
Degrees of Freedom 21
P-Value 0.0000
CFI/TLI
CFI 0.979
TLI 0.981
Loglikelihood
H0 Value -27739.720
H1 Value -27605.699
Information Criteria
Number of Free Parameters 12
Akaike (AIC) 55503.439
Bayesian (BIC) 55569.171
Sample-Size Adjusted BIC 55531.048
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.078
90 Percent C.I. 0.069 0.086
Probability RMSEA

BMI98 1.000 0.000 999.000 999.000
BMI99 2.000 0.000 999.000 999.000
BMI00 3.000 0.000 999.000 999.000
BMI01 4.000 0.000 999.000 999.000
BMI02 5.000 0.000 999.000 999.000
BMI03 6.000 0.000 999.000 999.000
Intercept and slope have significant covariance
S WITH
I 0.408 0.073 5.559 0.000
Means
I 21.035 0.100 210.352 0.000
S 0.701 0.017 40.663 0.000
Growth curve is BMI’ = 21.035 + .701×Year
Intercepts
BMI97 0.000 0.000 999.000 999.000
BMI98 0.000 0.000 999.000 999.000
BMI99 0.000 0.000 999.000 999.000
BMI00 0.000 0.000 999.000 999.000
BMI01 0.000 0.000 999.000 999.000
BMI02 0.000 0.000 999.000 999.000
BMI03 0.000 0.000 999.000 999.000
Variances
I 15.051 0.597 25.209 0.000
S 0.255 0.018 14.228 0.000
There is a big random intercept and random slope effect. The standard
deviation is sqrt(.255) = .50. Putting plus or minus two standard deviations
around the slope of .70 shows how big the variance is. The standard
deviation for the intercept is sqrt(15.051) = 3.880. BMI is probably skewed
positively.
Residual Variances
BMI97 5.730 0.268 21.413 0.000
BMI98 3.276 0.164 19.942 0.000
BMI99 3.223 0.146 22.009 0.000
BMI00 4.361 0.185 23.538 0.000
BMI01 2.845 0.150 19.005 0.000
BMI02 9.380 0.397 23.622 0.000
BMI03 8.589 0.422 20.345 0.000
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.656E-02
(ratio of smallest to largest eigenvalue)
Growth Curve and Related Models, Alan C. Acock 13

MODEL MODIFICATION INDICES
Minimum M.I. value for printing the modification index 3.840
We don’t want to change the intercept loadings of 1.0. We might think about
a nonlinear growth. We might think about correlating adjacent error terms.
The suggested correlation between E1 and E7 indicates a straight line is
missing both ends, hence a curve of some kind? Muthen suggests to try not to
mess with the intercepts.
M.I. E.P.C. Std E.P.C. StdYX E.P.C.
BY Statements
I BY BMI97 112.472 -0.038 -0.147 -0.032
I BY BMI98 6.440 0.007 0.027 0.006
I BY BMI99 33.234 0.014 0.054 0.012
I BY BMI00 13.026 0.010 0.037 0.008
I BY BMI02 4.015 -0.008 -0.032 -0.005
I BY BMI03 28.212 -0.023 -0.091 -0.015
S BY BMI97 70.828 -0.825 -0.417 -0.091
S BY BMI99 18.208 0.276 0.139 0.030
S BY BMI00 8.062 0.204 0.103 0.021
S BY BMI03 38.314 -0.755 -0.382 -0.062
WITH Statements
BMI99 WITH BMI98 12.747 0.449 0.449 0.138
BMI00 WITH BMI97 9.699 -0.511 -0.511 -0.102
BMI00 WITH BMI99 26.084 0.641 0.641 0.171
BMI01 WITH BMI97 6.914 -0.388 -0.388 -0.096
BMI01 WITH BMI98 11.566 -0.403 -0.403 -0.132
BMI01 WITH BMI00 5.456 0.310 0.310 0.088
BMI02 WITH BMI97 8.645 0.715 0.715 0.098
BMI02 WITH BMI99 9.066 -0.544 -0.544 -0.099
BMI02 WITH BMI00 9.560 -0.633 -0.633 -0.099
BMI03 WITH BMI97 37.342 1.564 1.564 0.223
BMI03 WITH BMI99 22.526 -0.874 -0.874 -0.166
BMI03 WITH BMI00 11.717 -0.724 -0.724 -0.118
BMI03 WITH BMI02 11.053 1.083 1.083 0.121
Means/Intercepts/Thresholds
[ BMI97 ] 97.476 -0.754 -0.754 -0.165
[ BMI98 ] 7.230 0.155 0.155 0.035
[ BMI99 ] 25.098 0.257 0.257 0.056
[ BMI00 ] 10.542 0.185 0.185 0.038
[ BMI02 ] 4.646 -0.189 -0.189 -0.032
[ BMI03 ] 22.536 -0.448 -0.448 -0.073
There are a number of plots available. These are not bad, but Stata or some
other package, even Excel, could do nicer graphs.
PLOT INFORMATION
The following plots are available:
Growth Curve and Related Models, Alan C. Acock 14

Histograms (sample values, estimated factor scores, estimated values)
Scatterplots (sample values, estimated factor scores, estimated values)
Sample means
Estimated means
Sample and estimated means
Observed individual values
Estimated individual values
Here are Some of the Available Plots
It is often useful to show the actual means for a small random sample of participants. These are
Sample Means.
• Click on Graphs
• Observed Individual Values
This gives you a menu where you can make some selections. I used the clock to seed a random
generation of observations.
Here I selected Random Order and for 20 cases. This results in the following graph:
Growth Curve and Related Models, Alan C. Acock 15

This shows one person who started at an obese BMI = 30 and then dropped down. However, most
people increased gradually.
Next, let’s look at a plot of the actual means and the estimated means using our linear growth
model. Click on
• Graphs and then select View graphs
• Sample and estimated means.
• Demonstrate how to edit the graph.
Growth Curve and Related Models, Alan C. Acock 16

Notice that there is a clear growth trend in BMI. A BMI of 15-20 is considered healthy and a BMI
of 25 is considered overweight. Notice what happens to American youth between the age of 12
and the age of 18.
3 A Growth Curve with a Quadratic Term
This graph is useful to seeing if there is a nonlinear trend. Changing the scale of the Y-axis can
clarify this. It is simple to add a quadratic term, if the curve is departing from linearity.
• Looking at the graph it may seem that the linear trend works very well, but our RMSEA
was a bit big.
• The estimated initial BMI is higher than the observed mean.
• The estimated BMI at 2003 is also higher than the observed mean
• A quadratic might pick this up by having a curve that drops slightly to pick up the BMI97
mean and the BMI2003 mean.
• Estimation of three terms (Intercept, Lin1ear trend, Quadratic trend) requires at least four
waves of data, but more waves are highly desirable for a good test of the quadratic term.
The conceptual model in Figure 1 will be unchanged except a third latent variable is added.
• We will have the Intercept, Slope, now called linear trend, and the new latent variable
called the Quadratic trend.
• Like the first two, the Quadratic trend will have a residual variance (R3) that will freely
correlated with R1 and R2.
• The paths from the quadratic trend to the individual BMI variables will be the square of
the path from the Linear trend to the BMI variables. Hence
a. The values for the linear trend will remain 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, and 6.0.
b. For the quadratic these values will be 0.0, 1.0, 4.0, 9.0, 16.0, 25.0, and 36.0.
Growth Curve and Related Models, Alan C. Acock 17

You really appreciate the defaults in Mplus when you see what we need to change in the Mplus
program when we add a quadratic slope. Here is the only change we need to make:
Model:
i s q| bmi97@0 bmi98@1 bmi99@2 bmi00@3 bmi01@4 bmi02@5 bmi03@6;
Mplus will know that the quadratic, q (we could use any name) will have values that are the
square of the values for the slope, s.
Title: bmi_guadratic.inp
Quadratic growth curve
Data:
File is "C:\Mplus examples\bmi_stata.dat" ;
Variable:
Names are
id grlprb_y boyprb_y grlprb_p boyprb_p
male race_eth bmi97 bmi98 bmi99 bmi00
Growth Curve and Related Models, Alan C. Acock 18

mi01 bmi02 bmi03 white black hispanic
asian other;
Missing are all (-9999) ;
! usevariables is limited to bmi variables
Usevariables are bmi97 bmi98 bmi99 bmi00
bmi01 bmi02 bmi03 ;
Model:
i s q| bmi97@0 bmi98@1 bmi99@2 bmi00@3
bmi01@4 bmi02@5 bmi03@6;
Output:
Sampstat Mod(3.84);
Plot:
Type is Plot3;
Series = bmi97 bmi98 bmi99 bmi00 bmi01
bmi02 bmi03(*);
Here is selected output:
TESTS OF MODEL FIT
We had 23 degrees of freedom with the linear growth curve and a chi-square
of 268.041. Now we have 19 degrees of freedom and a chi-square of 73.121.
Where did we lose four degrees of freedom?
• Mean for the quadratic
• Variance of the quadratic
• Covariance of quadratic residual with intercept residual
• Covariance of quadratic residual with slope residual
Did we improve our fit?
• 268.041-73.121 = 194.92 with 4 degrees of freedom, p <
.001
Does our model fit?
• Chi-square (19) = 73.121, p < .001, but
• CFI = .995
• RMSEA = .040
Chi-Square Test of Model Fit
Value 73.121
Degrees of Freedom 19
P-Value 0.0000
Chi-Square Test of Model Fit for the Baseline Model
Growth Curve and Related Models, Alan C. Acock 19

CFI/TLI
Loglikelihood
Value 11502.912
Degrees of Freedom 21
P-Value 0.0000
CFI 0.995
TLI 0.995
H0 Value -27642.260
H1 Value -27605.699
Information Criteria
Number of Free Parameters 16
Akaike (AIC) 55316.520
Bayesian (BIC) 55404.161
Sample-Size Adjusted BIC 55353.330
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.040
90 Percent C.I. 0.031 0.050
Probability RMSEA

Means
I 20.713 0.101 204.728 0.000
S 1.060 0.044 23.834 0.000
Q -0.063 0.007 -8.585 0.000
Variances
I 14.273 0.638 22.382 0.000
S 1.141 0.139 8.184 0.000
Q 0.029 0.004 7.730 0.000
Residual Variances
BMI97 4.635 0.306 15.132 0.000
BMI98 3.340 0.162 20.643 0.000
BMI99 2.852 0.143 19.954 0.000
BMI00 3.994 0.182 21.926 0.000
BMI01 2.880 0.154 18.762 0.000
BMI02 9.343 0.394 23.690 0.000
BMI03 5.677 0.507 11.192 0.000
MODEL MODIFICATION INDICES
Minimum M.I. value for printing the modification index 3.840
M.I. E.P.C. Std E.P.C. StdYX E.P.C.
BY Statements
I BY BMI97 24.292 -0.024 -0.090 -0.021
I BY BMI98 9.860 0.008 0.032 0.007
I BY BMI99 5.419 0.006 0.022 0.005
I BY BMI01 12.777 -0.009 -0.035 -0.007
I BY BMI03 14.857 0.024 0.092 0.016
S BY BMI97 18.253 -0.363 -0.388 -0.089
S BY BMI98 7.381 0.126 0.135 0.031
S BY BMI01 9.168 -0.137 -0.147 -0.029
S BY BMI03 10.308 0.349 0.373 0.063
Q BY BMI97 12.444 3.442 0.589 0.136
Q BY BMI99 4.868 -1.114 -0.191 -0.041
Q BY BMI01 11.767 1.725 0.295 0.058
Q BY BMI03 13.934 -5.610 -0.961 -0.163
ON/BY Statements
Q ON I /
I BY Q 999.000 0.000 0.000 0.000
WITH Statements
BMI98 WITH BMI97 11.493 -1.044 -1.044 -0.265
BMI99 WITH BMI98 8.019 0.361 0.361 0.117
BMI01 WITH BMI98 8.978 -0.354 -0.354 -0.114
Growth Curve and Related Models, Alan C. Acock 21

BMI02 WITH BMI01 12.482 0.694 0.694 0.134
BMI03 WITH BMI02 5.261 -1.040 -1.040 -0.143
Means/Intercepts/Thresholds
[ BMI97 ] 23.635 -0.492 -0.492 -0.113
[ BMI98 ] 11.403 0.191 0.191 0.043
[ BMI01 ] 9.093 -0.166 -0.166 -0.032
[ BMI03 ] 13.777 0.495 0.495 0.084
PLOT INFORMATION
The following plots are available:
Histograms (sample values, estimated factor scores, estimated values)
Scatterplots (sample values, estimated factor scores, estimated values)
Sample means
Estimated means
Sample and estimated means
Observed individual values
Estimated individual values
• The fit is so good because the estimated means and observed means are so close.
• However, there is still significance variance (random effects for both the intercept and the
slope) among individual adolescents that still needs to be explained.
• Here are 20 estimated individual growth curves.
a. Notice that each of these is a curve, but they start at different initial levels and have
different trajectories.
b. Next, we want to use covariates to explain these differences in the initial levels and growth
trajectories.
Growth Curve and Related Models, Alan C. Acock 22

4 An Alternative to Use of a Quadratic Slope
An alternative to adding a quadratic slope is to allow some of the time loadings to be
free.
• We have used loadings of 0, 1, 2, 3, 4, 5, and 6 for the linear slope and 0, 1, 4,
9, 16, 25, and 36 for the quadratic slope. Alternatively
• We could allow all but two of the loadings to be free. We might use loadings of
0, 1, *, *, *, * .
• It is necessary to have the 0 and 1 fixed but the 1 does not have to be second;
we could use 0, *, *, *, *,1.
You may ask how you could justify allowing some of the time loadings to be free if
there was a one month or one year difference between waves of data. The answer is
that developmental time may be different than chronological time.
Allowing these loadings to be free has an advantage over the quadratic in that it uses
fewer degrees of freedom but still allows for growth spurts.
This model is not nested under a quadratic, but you could think of a linear growth
model with fixed values for each year (0, 1, 2, 3, 4, 5, 6) being nested within the free
model that uses 0, 1, *, *, *, *. If the free model fits much better than the fixed linear
model, you might use this instead of the quadratic model.
Growth Curve and Related Models, Alan C. Acock 23

5 Working with Missing Values
Mplus has two ways of working with missing values. The simplest is to use full information
maximum likelihood estimation with missing values (FIML). This uses all available data and is
the default. For example, some adolescents were interviewed all six years but others may have
skipped one, two, or even more years. We use all available information with this approach. The
second approach is to utilize multiple imputations.
Growth Curve and Related Models, Alan C. Acock 24

• Multiple imputations should not be confused with single imputation available from SPSS if a
person purchases their missing values module and which gives incorrect standard errors.
• Multiple imputation involves
a. Imputing multiple datasets (usually 5-10) using appropriate procedures.
b. Estimating the model for each of these datasets, and
c. Then pooling the estimates and standard errors.
When the standard errors are pooled this way, they incorporate the variability across the 5-10
solutions and are thereby produced unbiased estimates of standard errors. Multiple imputations
can be done with:
• Norm, a freeware program that works for normally distributed, continuous variables and is
often used even on dichotomized variables.
• A Stata user has written a program called ice that is an implementation of the S-Plus/R
program called MICE, that has advantages over Norm. It does the imputation by using different
estimation models for outcome variables that are continuous, counts, or categorical. See
Royston (2005). SAS has similar capabilities.
• Mplus can read these multiple datasets, estimate the model for each dataset, and pool the
estimates and their standard errors.
We will not illustrate the multiple imputation approach because that involves working with other
programs to impute the datasets. However, the Mplus User’s Guide, discusses how you specify the
datasets in the Data: section.
6 Multiple Cohort Growth Model with Missing Waves
Major datasets often have multiple cohorts. NLSY97 has youth who were 12-18 in 1997. Seven
years later, they are 19-25. It is quite likely that many growth processes that involve going from
the age of 12 to the age of 19 are different than going from 19-25. For example, involvement in
minor crimes (petty theft, etc.) may increase from 12 to 19, but then decrease from there to 25.
Here is what we might have for our NLSY97 data (data inside tables are scores, person 1, born in
1985, in 1997 at age of 12 had a score of 3 on the outcome variable)
Score by survey year for a single case from each cohort
Survey Year
Individual Cohort 1997 1998 1999 2000 2001 2002 2003
1 1985 3 4 5 6 7 7 8
2 1985 2 4 3 5 6 7 7
Growth Curve and Related Models, Alan C. Acock 25

3 1984 4 5 6 7 6 6 5
4 1982 6 7 5 4 3 2 2
5 1982 5 5 6 4 2 2 1
We can rearrange this data
Data for first 5 cases
Case Cohort 12 13 14 15 16 17 18 19 20 21
1 1985 3 4 5 6 7 7 8 * * *
2 1985 2 4 3 5 6 7 7 * * *
3 1984 * 4 5 6 7 6 6 5 * *
4 1982 * * * 6 7 5 4 3 2 2
5 1982 * * * 5 5 6 4 2 2 1
• In this table the top row is the age at which the data was collected. To capture everybody we
would need to extend the table to HD25 because the youth who were 18 in 1997 are 25 seven
years latter.
• This table would have massive amounts of missing data, but the missingness would not be
related to other variables. It would be missing completely at random (MCAR).
• We could develop a growth curve that covered the full range from age 12 to age 25. We would
have 14 waves of data even though each participant was only measured 7 times. Each
participant would have data for a maximum of 7 of the years and have missing values for a
minimum of 7 years.
• We would want to estimate a growth model with a quadratic term and expect the linear slope to
be positive (growth from 12-18) and the quadratic term to be negative (decline from 18-25).
• Mplus has a special Analysis: type called MCOHORT. There is an example on the Mplus
WebPage and we will not cover it here. This is an extraordinary way to deal with missing
values.
Here is an example from data Muthén analyzed:
Growth Curve and Related Models, Alan C. Acock 26

7 Multiple group growth curves
Multiple group analysis using SEM is extremely flexible—some would say it is too flexible
because there are so many possibilities. We use gender for our grouping variable because we are
interested in the trend in BMI for girls compared to boys. We think of adolescent girls are more
concerned about their weight and therefore more likely to have a lower BMI than boys and to have
a flatter trajectory.
There are several ways of comparing a model across multiple groups.
One approach is to see if the same model fits each group, allowing all of the estimated parameters
to be different.
• Here we are saying that a linear growth model fits the data for both boys and girls, but
• We are not constraining girls and boys to have the same values on any of the parameters. They
may differ on the
- intercept mean
- slope mean
- intercept variance
- slope variance
- covariance of intercept and slope residuals
- residual errors
- covariance of the residual errors that may be specified.
Growth Curve and Related Models, Alan C. Acock 27

• We can then put increasing invariance constraints on the model.
a. At a minimum, we want to test whether the two groups have a different intercept (level) and
slope.
b. If this constraint is acceptable we can add additional constraints on the variances,
covariances, and error terms.
First, we will estimate the model simultaneously for girls and boys with no constraints on the
parameters. Here is the program with new commands highlighted:
Title:
bmi_growth_gender.inp
Data:
File is "C:\mplus examples\bmi_stata.dat" ;
Variable:
Names are
id grlprb_y boyprb_y grlprb_p boyprb_p male
race_eth bmi97 bmi98 bmi99 bmi00 bmi01 bmi02
bmi03 white black hispanic asian other;
Missing are all (-9999) ;
! usevariables keeps bmi variables and gender
Usevariables are male bmi97 bmi98 bmi99
bmi00 bmi01 bmi02 bmi03 ;
Grouping is male (0=female 1=male);
Model:
i s | bmi97@0 bmi98@1 bmi99@2 bmi00@3 bmi01@4
bmi02@5 bmi03@6;
Output:
Sampstat Mod(3.84) ;
Plot:
Type is Plot3;
Series = bmi97 bmi98 bmi99 bmi00 bmi01 bmi02 bmi03(*);
I’ve put the only changes we need to make in bold, underline.
• We have a binary variable, male, that is coded 0 for females and 1 for males.
• We add male to the list of variables we are using.
• We add a subcommand to the Variable: section that says we have a grouping variable,
names it, and defines what the values are so the output will be labeled nicely.
Growth Curve and Related Models, Alan C. Acock 28

• The command Grouping is male (0=female 1 = male); is going to give us a
separate set of estimates for the parameters for girls (labeled female) and boys (labeled
male).
• The estimation does both groups simultaneously.
Here is selected, annotated output:
SUMMARY OF ANALYSIS
Number of groups 2
Number of observations
Group FEMALE 859
Group MALE 909
The following shows that we have the same variables in the model
Number of dependent variables 7
Number of independent variables 0
Number of continuous latent variables 2
Observed dependent variables
Continuous
BMI97 BMI98 BMI99 BMI00 BMI01 BMI02
BMI03
Continuous latent variables
I S
Variables with special functions
Grouping variable MALE
SAMPLE STATISTICS
ESTIMATED SAMPLE STATISTICS FOR FEMALE
Means
BMI97 BMI98 BMI99 BMI00 BMI01
________ ________ ________ ________ ________
1 20.432 21.840 22.375 22.916 23.443
Means
BMI02 BMI03
________ ________
1 24.295 24.727
ESTIMATED SAMPLE STATISTICS FOR MALE
Means
BMI97 BMI98 BMI99 BMI00 BMI01
________ ________ ________ ________ ________
1 20.698 21.848 22.896 23.665 24.220
Means
BMI02 BMI03
________ ________
1 24.467 25.111
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Growth Curve and Related Models, Alan C. Acock 29

Value 411.966
Degrees of Freedom 46 was 23
P-Value 0.0000
Chi-Square Contributions From Each Group
FEMALE 150.775
MALE 261.191
Chi-Square Test of Model Fit for the Baseline Model
Value 11735.530
Degrees of Freedom 42
P-Value 0.0000
CFI/TLI
CFI 0.969
TLI 0.971
Loglikelihood
H0 Value -27639.607
H1 Value -27433.624
Information Criteria
Number of Free Parameters 24
Akaike (AIC) 55327.213
Bayesian (BIC) 55458.676
Sample-Size Adjusted BIC 55382.430
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.095
90 Percent C.I. 0.087 0.103
SRMR (Standardized Root Mean Square Residual)
Value 0.072
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
Group FEMALE
S WITH
I 0.522 0.103 5.050 0.000
Means
I 20.881 0.143 145.640 0.000
S 0.663 0.025 27.015 0.000
Variances
I 15.141 0.859 17.626 0.000
S 0.264 0.026 10.221 0.000
Residual Variances
BMI97 4.662 0.334 13.980 0.000
BMI98 3.368 0.242 13.940 0.000
BMI99 2.753 0.190 14.503 0.000
BMI00 5.154 0.308 16.718 0.000
BMI01 3.084 0.226 13.649 0.000
BMI02 13.344 0.769 17.360 0.000
BMI03 6.105 0.517 11.812 0.000
Group MALE
S WITH
I 0.278 0.102 2.719 0.007
Growth Curve and Related Models, Alan C. Acock 30

Means
I 21.180 0.139 152.166 0.000
S 0.732 0.024 30.661 0.000
Variances
I 14.911 0.824 18.094 0.000
S 0.254 0.025 10.292 0.000
Residual Variances
BMI97 6.693 0.417 16.066 0.000
BMI98 3.237 0.227 14.279 0.000
BMI99 3.671 0.223 16.487 0.000
BMI00 3.730 0.224 16.656 0.000
BMI01 2.489 0.185 13.434 0.000
BMI02 5.416 0.357 15.190 0.000
BMI03 10.857 0.676 16.063 0.000
Here is the graph of the two growth curves. It appears that the girls have a lower initial level and a
flatter rate of growth of BMI.
We should not rely on our visual inspection, but should explicitly test whether the girls and boys
have a significant difference in their intercept and their slope. We can re-estimate the model with
the intercept and slope invariant (or do it twice so we could have separate tests.) To do this we
make the following modifications to the model:
Notice that we added two lines to the Model: section,
Title: bmi_growth_gender_equal.inp
Data:
Growth Curve and Related Models, Alan C. Acock 31

Variable:
Model:
File is "C:\mplus examples\bmi_stata.dat" ;
Names are
id grlprb_y boyprb_y grlprb_p boyprb_p male
race_eth bmi97 bmi98 bmi99 bmi00 bmi01 bmi02
bmi03 white black hispanic asian other;
Missing are all (-9999) ;
Usevariables are male bmi97 bmi98 bmi99
bmi00 bmi01 bmi02 bmi03 ;
Grouping is male (0=female 1=male);
i s | bmi97@0 bmi98@1 bmi99@2 bmi00@3 bmi01@4
bmi02@5 bmi03@6;
Model:
[i] (1);
[s] (2);
Model male:
[i] (1);
[s] (2);
Output: Sampstat Mod(3.84) ;
Plot:
Type is Plot3;
Series = bmi97 bmi98 bmi99 bmi00 bmi01 bmi02
bmi03(*);
We kept the group subcommand and added the grouping variable, male. We added two Model:
subcommands under the Model: command. The first one refers to the first group.
• Since girls were coded 0 and boys were coded 1, the first group is girls. We put the name of
parameters in square brackets, [i] and [s].
• If we had called these initial and trend we would have typed [initial] and [trend].
We put an arbitrary number in parentheses after the parameter name. Thus, we put (1)
after [i].
In the second subcommand, Model male:, we put the name of the parameters followed by the
same numbers as they had in the first gorup. Thus, the intercept gets the number 1 for girls and
also gets the number 1 for boys. This tells Mplus these must be held equal. They are still
optimized, but with the constraint that they are equal.
• If we had typed [i] (2) under Model male: what would happen? We would have
constrained the boys intercept to be equal to the girls slope—not something we would want to
do.
Growth Curve and Related Models, Alan C. Acock 32

• If we had omitted the [s] (2) under Model male: What would happen. We would have
constrained both solutions to have the same intercept [i] (1), but not constrained them to
have the same slopes.
• The first Model: command is understood to be the group coded as zero on the male variable.
• These changes force the intercept to be equal in both groups because they are both assigned
parameter (1) and the slopes to be equal because they are both assigned a parameter (2).
• Any parameters with a (1) after them are equal in both groups as are any parameters with a
(2) after them in both groups.
• Notice that we have square brackets [ ] around the names of the intercept and slope.
When we run the revised program we obtain a chi-square that has two extra degrees of freedom
because of the two constraints.
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value 418.884
Degrees of Freedom 48
P-Value 0.0000
We had a chi-square(46) = 411.966 without these constraints. The
difference has a chi-square of 6.918 with 2 degrees of freedom.
Using Stata the significance is chi-square(2) = 6.918, p < .05
. display 1-chi2(2,6.918)
.03146121
Chi-Square Contributions From Each Group
The model fits females much better than it fits males:
FEMALE 154.530
MALE 264.353
Chi-Square Test of Model Fit for the Baseline Model
Value 11735.530
Degrees of Freedom 42
P-Value 0.0000
CFI/TLI
CFI 0.968
TLI 0.972
Loglikelihood
H0 Value -27643.065
Growth Curve and Related Models, Alan C. Acock 33

H1 Value -27433.624
Information Criteria
Number of Free Parameters 22
Akaike (AIC) 55330.131
Bayesian (BIC) 55450.638
Sample-Size Adjusted BIC 55380.746
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.093
90 Percent C.I. 0.085 0.102
SRMR (Standardized Root Mean Square Residual)
Value 0.079
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
Group FEMALE
S WITH
I 0.528 0.104 5.097 0.000
Means
I 21.046 0.100 210.324 0.000
S 0.700 0.017 40.814 0.000
Group MALE
S WITH
I 0.281 0.102 2.742 0.006
Means
I 21.046 0.100 210.324 0.000
S 0.700 0.017 40.814 0.000
We should also look at the variances and covariances.
• Although we can say there is a highly significant difference between the level and trend for
girls and boys, we need to be cautious because this difference of chi-square has the same
problem with a large sample size that the original chi-squares have.
• In fact, the measures of fit are hardly changed whether we constrain the intercept and slope to
be equal or not. Moreover, the visual difference in the graph is not dramatic.
We could also put other constraints on the two solutions such as equal variances and covariances,
and even equal residual error variances, but we will not.
Growth Curve and Related Models, Alan C. Acock 34

8 An Alternative to Multiple Group Analysis
An alternative way of doing this, where there are two groups, is to enter the grouping variable as a
predictor. This requires re-conceptualizing our model. We can think of the indicator variable
Male having a direct path to both the intercept and the slope. Because the indicator variable is
coded as 1 for male and 0 for female,
• If the path from Male to the Intercept is positive this means that boys have a higher initial
level on BMI.
• Similarly, if there is a positive path from Male to the Slope, this indicates that boys have a
steeper slope than girls on BMI. This direct effect actually represents an interaction between
the trajectory and gender.
• Such results would be consistent with our expectation that boys both start higher and gain more
fat than girls during adolescence.
• This approach does not let us test for other types of invariances such as the residual variances,
covariances, and error terms.
a. We are forcing these to be the same for both females and males; this may be unreasonable.
b. The random effect for the slope for boys, R2, may be greater or less than it is for girls. We
will not be able to evaluate this possibility with this approach.
The following figure shows these two paths. We are explaining why some people have a higher or
lower initial level and why some have a steeper or flatter slope by whether they are a girl or a boy.
We are predicting that boys have a higher initial level and a steeper slope.
Here is the figure:
Growth Curve and Related Models, Alan C. Acock 35

Here is the program:
Title: bmi_gender_alternatives.inp
bmi growth curve using gender as a single covariate.
This is an alternative to using gender as two groups.
Data: File is "c:\Mplus examples\bmi_stata.dat" ;
Variable: Names are
id grlprb_y boyprb_y grlprb_p boyprb_p male
race_eth bmi97 bmi98 bmi99 bmi00 bmi01 bmi02 bmi03
white black Hispanic asian other;
Missing are all (-9999) ;
Usevariables are male bmi97 bmi98 bmi99
bmi00 bmi01 bmi02 bmi03 ;
Model:
Growth Curve and Related Models, Alan C. Acock 36

i s | bmi97@0 bmi98@1 bmi99@2 bmi00@3 bmi01@4 bmi02@5 bmi03@6;
i on male;
s on male;
Output: Sampstat Mod(3.84) standardized;
Plot:
Type is Plot3;
Series = bmi97 bmi98 bmi99 bmi00 bmi01 bmi02
bmi03(*);
Here is selected, annotated output:
SUMMARY OF ANALYSIS
Number of groups
1
Number of observations
1771
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
We cannot compare this chi-square to the two group chi-square
because it is not a nested model.
Value 301.244
Degrees of Freedom 28
P-Value 0.0000
Chi-Square Test of Model Fit for the Baseline Model
Value 11544.530
Degrees of Freedom 28
P-Value 0.0000
CFI/TLI
CFI 0.976
TLI 0.976
Loglikelihood
H0 Value -29020.154
H1 Value -28869.532
Information Criteria
Number of Free Parameters 14
Akaike (AIC) 58068.308
Bayesian (BIC) 58145.018
Growth Curve and Related Models, Alan C. Acock 37

Sample-Size Adjusted BIC 58100.541
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.074
90 Percent C.I. 0.067 0.082
Probability RMSEA

BMI00 4.354 0.185 23.534 0.000
BMI01 2.834 0.149 18.972 0.000
BMI02 9.409 0.398 23.624 0.000
BMI03 8.626 0.424 20.360 0.000
I 15.045 0.597 25.198 0.000
S 0.253 0.018 14.175 0.000
R-SQUARE
Observed Two-Tailed
Variable Estimate S.E. Est./S.E. P-Value
BMI97 0.724 0.012 59.339 0.000
BMI98 0.832 0.009 93.431 0.000
BMI99 0.846 0.008 110.551 0.000
BMI00 0.820 0.008 99.975 0.000
BMI01 0.888 0.006 138.194 0.000
BMI02 0.731 0.011 65.511 0.000
BMI03 0.772 0.011 70.133 0.000
You can see why we rarely report the R-square for the intercept
and slope.
Latent Two-Tailed
Variable Estimate S.E. Est./S.E. P-Value
I 0.001 0.002 0.608 0.543
S 0.007 0.006 1.266 0.206
We see that the intercept is 20.385 and the slope is .625. How is gender related to this?
For girls the equation is:
Est. BMI = 20.911 + .656(Time) + .242(Male) + .086(Male)(Time)
20.911 + .656(Time) + .242(0) + .086(0)(Time)
= 20.911 + .656(Time)
For boys the equation is:
Est BMI = 20.911 + .656(Time) + .242(1) + .086(1)(Time)
= (20.911 + .242) + (.625 + .086)(Time)
Growth Curve and Related Models, Alan C. Acock 39

= 21.153 + .711(Time)
Where Time is coded as 0, 1, 2, 3, 4, 5, 6
Using these we estimate the BMI for girls is initially 20.911. By the seventh year when she is
18(Time = 6) her estimated BMI will be 20.385 + .656(6) or 24.847
Using these results, we estimate the BMI for boys is initially 21.153. By the seventh year it will be
21.153 + .711(6) or 25.419. Since a BMI of 25 is considered overweight, by the age of 18 we
estimate the average boy will be classified as overweight and the average girl is not far behind!
We could use the plots provided by Mplus, but if we wanted a nicer looking plot we could use
another program. I used Stata getting this graph.
The Stata command is (this is driven by a drop down menu)
twoway (connected Girls Age, lcolor(black) lpattern(dash) ///
lwidth(medthick)) (connected Boys Age, lcolor(black) ///
lpattern(solid) lwidth(medthick)), ///
ytitle(Body Mass Index) xtitle(Age of Adolescent) ///
caption(NLSY97 Data)
and the data is
+-----------------------+
| Age Girls Boys |
|-----------------------|
1. | 12 20.911 21.153 |
2. | 18 24.847 25.419 |
+-----------------------+
Body Mass Index for Adolescents
Comparison of Girls and Boys
Growth Curve and Related Models, Alan C. Acock 40

Limitations of this approach
• When we treat a categorical variable as a grouping variable and do multiple comparisons we
can test the equality of all the parameters.
• When we treat it as a predictor as in this example, we only test whether the intercept and
slope are different for the two groups (interaction). In this example we do not allow the
other parameters to be different for boys and girls and this might be a problem in some
applications.
9 Growth Curves with Time Invariant Covariates
An extension of having a single categorical predictor includes having a series of covariates that
explain variance in the intercept and slope. In this example we use what are known as time
invariant covariates. These are covariates that either remain constant (gender) or for which you
have a measure only at the start of the study. These are some times considered fixed effects since
their value cannot change from one wave to another. It is possible to add time varying covariates
as well.
Growth Curve and Related Models, Alan C. Acock 41

This has been called the Conditional Latent Trajectory Modeling (Curran & Hussong, 2003)
because your initial level and trajectory (slope) are conditional on other variables.
*The variable White (whites = 1; nonwhites = 0) compares Whites to the combination of African
American and Hispanic. Asian & Pacific Islander, and Other have been deleted from this analysis
because of small sample size.
In this figure we have two covariates.
• One is whether the adolescent
is white versus African
American or Hispanic and
• The other is a latent variable
reflecting the level of
emotional problems a youth
has. There are two indicators
of emotional problems, one
from a parent report,
boyprb_p, and the other from
a youth report, boyprb_y.
• The emotional problems are
problems as reported at age 12.
• A researcher may predict that
Whites have a lower initial BMI
(intercept) which persists during
adolescence, but the White
advantage does not increase (same
slope as nonwhites).
• Alternatively, a researcher may
predict that being White predicts a
lower initial BMI (intercept) and
less increase of the BMI (smaller
slope) during adolescence.
a. This suggests that minorities start with a disadvantage (high BMI) and
b. This disadvantaged gets even greater across adolescence.
Growth Curve and Related Models, Alan C. Acock 42

• A researcher may argue that emotional problems are associated with both higher initial BMI
(intercept) and a more rapid increase in BMI over time (slope).
• By including a covariate that is a latent variable itself, emotional problems, we will show how
these are handled by Mplus.
We estimated this model for boys only; girls were excluded.
The following is our Mplus program:
Title: bmi_timea.inp
bmi growth curve using race/ethnicity and emotional
problems as a second covariate. There are two indicators
of emotional problems.
Data: File is "c:\Mplus examples\bmi_stata.dat" ;
Variable: Names are
id grlprb_y boyprb_y grlprb_p boyprb_p male race_eth bmi97
bmi98 bmi99 bmi00 bmi01 bmi02 bmi03 white black hispanic
asian other;
Missing are all (-9999) ;
Usevariables are boyprb_y boyprb_p white bmi97 bmi98 bmi99
bmi00 bmi01 bmi02 bmi03 ;
Useobservations = male eq 1 and asian ne 1 and other ne 1;
Model: i s q| bmi97@0 bmi98@1 bmi99@2 bmi00@3 bmi01@4 bmi02@5
bmi03@6;
emot_prb by boyprb_p boyprb_y ;
i on white emot_prb;
s on white emot_prb;
q on white emot_prb;
Output: Sampstat Mod(3.84) standardized;
Plot:
Type is Plot3;
Series = bmi97 bmi98 bmi99 bmi00 bmi01 bmi02 bmi03(*);
I have highlighted the new lines in the Mplus program.
• The format of the Useobservations subcommand is similar to if or select used
with other programs.
• The Useobservations = male eq 1 and asian ne 1 and other ne 1;
restricts our sample to males (male eq 1). This is very handy when using the same
dataset for a variety of models where you want some models to only include selected
participants.
• We have dropped Asians and members of the “other” category. There are relatively few of
them in this sample dataset and they may have very different BMI trajectories. Also, the
meaning of the category “other” is ambiguous.
Growth Curve and Related Models, Alan C. Acock 43

• I added a quadratic term in the Model: command. I first estimated this model using just a
linear slope and the fit was not very good (results not shown here). Adding the quadratic
improved the fit.
• This example has a measurement model for a latent covariate, emot_prb. In other
programs this can involve complicated programming. Here it is done with the single line.
(You usually would like to have 3 and preferably 4 indicators of a latent variable.)
emot_prb by boyprb_p boyprb_y ;
• The by is a key word in Mplus for creating latent variables used in Confirmatory Factor
Analysis and SEM.
• On the right of the by are two observed variables. The boyprb_p is the report of parents
about the adolescent’s emotional problems. The boyprb_y is the youths own report.
• It is desirable to have three or more indicators of a latent variable, but we only have two
here so that will have to do.
• To the left of the by is the name we give to the latent variable, emot_prb. This new latent
variable did not appear in the list of variables we are using, but it is defined here.
• The “by” term
o fixes the first variable to the right as a reference indicator, boyprb_p, and assigns a
loading of 1 to it.
o It lets the loading of the second variable, boyprb_y, be estimated. It also creates
error/residual variances that are labeled e1 and e2 in the figure.
o The default is that these errors are uncorrelated.
o It is good practice to have the strongest indicator on the right of the “by” be the
reference indicator with a loading fixed at 1.0. You can run the model and if this does
not happen, you can re-run it, reversing the order of the items on the right of the
“by.”
• The next three new lines,
o i on white emot_prb;
o s on white emot_prb; and
o q on white emot_prb;
o Define the relationship between the covariates and the intercept and slope. These
represent interactions of each covariate with the intercept and slope.
o These are the γ1wi in the equation for HLM users.
o Mplus uses the on command to signify that a variable depends on another variable in
the structural part of the model. The by command is the key to understanding how
Growth Curve and Related Models, Alan C. Acock 44

Mplus sets up the measurement model and the on is the key to how Mplus sets up the
structural model.
There are many defaults. Mplus assumes there are residual variances and covariances for the
intercept and slopes. It fixes the intercepts at zero. It assumes the intercept and slope variances are
correlated.
Here is selected results:
Mplus VERSION 5.1
MUTHEN & MUTHEN
07/01/2008 8:01 PM
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value 88.824
Degrees of Freedom 34
P-Value 0.0000
Chi-Square Test of Model Fit for the Baseline Model
Value 5975.020
Degrees of Freedom 45
P-Value 0.0000
CFI/TLI
CFI 0.991
TLI 0.988
Loglikelihood
H0 Value -17221.918
H1 Value -17177.507
Information Criteria
Number of Free Parameters 29
Akaike (AIC) 34501.837
Bayesian (BIC) 34640.190
Sample-Size Adjusted BIC 34548.093
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.043
90 Percent C.I. 0.032 0.054
Probability RMSEA

Measurement of latent variable.
EMOT_PRB BY
BOYPRB_P 1.000 0.000 999.000 999.000
BOYPRB_Y 0.575 0.171 3.374 0.001
Emotional problems does not have a significant effect on the initial level
at age 12, but significantly increases the slope. Significant negative
effect on quadratic is a bit confusing.
I ON
EMOT_PRB 0.300 0.168 1.793 0.073
S ON
EMOT_PRB 0.212 0.089 2.370 0.018
Q ON
EMOT_PRB -0.037 0.015 -2.462 0.014
Whites have a significant advantage initially (intercept), but there is not
a significant compounding of this over time since White does not
significantly influence the slope or quadratic.
I ON
WHITE -1.030 0.292 -3.529 0.000
S ON
WHITE 0.130 0.138 0.941 0.346
Q ON
WHITE -0.030 0.023 -1.293 0.196
S WITH
I 0.701 0.329 2.128 0.033
Q WITH
I -0.101 0.053 -1.922 0.055
S -0.174 0.034 -5.081 0.000
WHITE WITH
EMOT_PRB -0.111 0.028 -4.013 0.000
Intercepts
BOYPRB_Y 2.108 0.052 40.712 0.000
BOYPRB_P 1.893 0.058 32.668 0.000
BMI97 0.000 0.000 999.000 999.000
BMI98 0.000 0.000 999.000 999.000
BMI99 0.000 0.000 999.000 999.000
BMI00 0.000 0.000 999.000 999.000
BMI01 0.000 0.000 999.000 999.000
BMI02 0.000 0.000 999.000 999.000
BMI03 0.000 0.000 999.000 999.000
I 21.279 0.210 101.495 0.000
S 1.171 0.100 11.719 0.000
Q -0.077 0.016 -4.649 0.000
The linear slope of 1.171 is huge when you project this over the six years.
The quadratic slope being negative indicates that there is some leveling off
in the increase in BMI.
Variances
Growth Curve and Related Models, Alan C. Acock 46

EMOT_PRB 1.485 0.455 3.262 0.001
Residual Variances
BOYPRB_Y 1.850 0.170 10.875 0.000
BOYPRB_P 1.203 0.442 2.722 0.006
BMI97 5.535 0.479 11.564 0.000
BMI98 3.276 0.227 14.402 0.000
BMI99 3.333 0.221 15.067 0.000
BMI00 3.182 0.218 14.586 0.000
BMI01 2.361 0.186 12.670 0.000
BMI02 5.225 0.357 14.654 0.000
BMI03 8.961 0.743 12.057 0.000
I 13.094 0.869 15.070 0.000
S 1.291 0.215 6.007 0.000
Q 0.030 0.006 5.092 0.000
STANDARDIZED MODEL RESULTS
STDYX Standardization
Two-Tailed
Estimate S.E. Est./S.E. P-Value
EMOT_PRB BY
BOYPRB_P 0.743 0.111 6.708 0.000
BOYPRB_Y 0.458 0.073 6.308 0.000
Notice the z-tests are slightly different. Most standard packages assume the
unstandardized test is the same.
I ON
EMOT_PRB 0.099 0.052 1.909 0.056
S ON
EMOT_PRB 0.222 0.080 2.775 0.006
Q ON
EMOT_PRB -0.253 0.087 -2.918 0.004
I ON
WHITE -0.140 0.039 -3.558 0.000
S ON
WHITE 0.056 0.059 0.942 0.346
Growth Curve and Related Models, Alan C. Acock 47

Q ON
WHITE -0.083 0.064 -1.293 0.196
S WITH
I 0.170 0.090 1.891 0.059
Q WITH
I -0.163 0.094 -1.731 0.083
S -0.891 0.021 -42.268 0.000
WHITE WITH
EMOT_PRB -0.183 0.049 -3.748 0.000
Residual Variances
BOYPRB_Y 0.790 0.067 11.864 0.000
BOYPRB_P 0.448 0.165 2.717 0.007
BMI97 0.290 0.025 11.460 0.000
BMI98 0.171 0.012 13.889 0.000
BMI99 0.151 0.011 13.794 0.000
BMI00 0.132 0.010 13.364 0.000
BMI01 0.096 0.008 11.571 0.000
BMI02 0.185 0.013 14.176 0.000
BMI03 0.269 0.022 12.260 0.000
I 0.966 0.016 62.173 0.000
S 0.952 0.034 27.829 0.000
Q 0.937 0.043 22.016 0.000
Unfortunately, we cannot get graphs when we have covariates. You could create these yourself by
substituting fix values for race and emotional problems.
Growth Curve and Related Models, Alan C. Acock 48

10 Mediational Models with Time Invariant Covariates
Sometimes all of the covariates are time invariant or at least measured at just the start of the study.
Curran and Hussong (2003) discuss a study of a latent growth curve on drinking problems with a
covariate of parental drinking. Parental drinking influences both the initial level and the rate of
growth of drinking problem behavior among adolescents. The question is whether some other
variables might mediate this relationship
• Parental monitoring
• Peer influence
Mplus allows us to estimate the direct and indirect effect of Parent Drinking on the Intercept and
Slope. It also provides a test of significance for these effects.
Growth Curve and Related Models, Alan C. Acock 49

11 Time Varying Covariates
We have illustrated time invariant covariates that are measured at time 1. It is possible to extend
this to include time varying covariates. Time varying covariates either are measured after the
process has started or have a value that changes (hours of nutrition education, level of program
fidelity). Although we will not show our output, we will illustrate the use of time varying
covariates in a figure. In this figure the time varying covariates, a21 to a24 might be
• Hours of nutrition education completed between waves. Independent of the overall growth
trajectory, η1, students who have several hours of nutrition education programming may have a
decrease in their BMI
• Physical education curriculum. A physical activity program might lead to reduced BMI.
Students who spend more time in this physical activity program might have a lower BMI
independent of the overall growth trend. Hours in physical education courses will vary from
year to year.
• This would be a good way to incorporate fidelity into a program evaluation.
This figure is borrowed from Muthén where he is examining growth in math performance over 4
years. The w vector contains x variables are covariates that directly influence the intercept, η0, or
slope, η1. The aij are number of math courses taken each year.
yit = repeated measures on the outcome (math achievement)
a1it = Time score (0, 1, 2, 3) as discussed previously
a2it = Time varying covariates
(# of math courses taken that year)
w = Vector of x covariates that
are time invariant and measured at
or before the first yit
In this example we might think of
the yi variables being measures of
conflict behavior where y1 is at age
17 and y4 is at age 25. We know
there is a general decline in
conflict behavior during this time
interval. Therefore, the slope η1 is
expected to be negative.
Growth Curve and Related Models, Alan C. Acock 50

Now suppose we also have a measure of alcohol abuse for each of the 4 waves (aij). We might
hypothesize that during a year in which an adolescent has a high score on alcohol abuse (say
number of days the person drinks 5 or more drinks in the last 30 days) that there will be an
elevated level of conflict behavior that cannot be explained by the general decline (negative
slope).
The negative slope reflects the general decline in conflict behavior by young adults as the move
from age 17 to age 25. The effect of aij on yi provides the additional explanation that those years
when there is a lot of drinking; there will be an elevated level of conflict that does not fit the
general decline.
If you want more, here are a few references
2. Basic growth curve modeling
a. Bollen, K. A., & Curran, P. J. (2006). Latent Curve Models: A Structural Equation
Perspective. Hoboken, NJ: Wiley.
b. Curran, F. J., & Hussong, A. M. (2003). The Use of latent Trajectory Models in
Psychopathology Research. Journal of Abnormal Psychology. 112:526-544. This is a
general introduction to growth curves that is accessible.
c. Duncan, T. E., Duncan, S. C., & Strycker, L. A. (2006). An Introduction to Latent
Variable Growth Curve Modeling: Concepts, Issues, and Applications (2 nd ed.).
Mahwah NJ: Lawrence Erlbaum. The second edition of a classic text on growth curve
modeling.
d. Kaplan, D. (2000). Chapter 8: Latent Growth Curve Modeling. In D. Kaplan,
Structural Equation Modeling: Foundations and Extensions (pp 149-170). Thousand
Oaks, CA: Sage. This is a short overview.
e. Wang, M. (2007). Profiling retirees in the retirement transition and adjustment
process: Examining the longitudinal change patterns of retirees' psychological wellbeing.
Journal of Applied Psychology, 92(2), 455-474. This is a nice example of
presenting results showing some graphs and tables.
3. Limited Outcome Variables: Binary and count variables
a. Muthén, B. (1996). Growth modeling with binary responses. In A. V. Eye & C.
Clogg (Eds.) Categorical Variables in Developmental Research: Methods of analysis
(pp 37-54). San Diego, CA: Academic Press.
Growth Curve and Related Models, Alan C. Acock 51

. Long, J. S., & Freese, J. (2006). Regression Models for Categorical Dependent
Variables Using Stata, 2 nd ed. Stata Press (www.stata-press.com). This provides the
most accessible and still rigorous treatment of how to use an interpret limited
dependent variables.
c. Rabe-Hesketh, S., & Skrondal, A. (2005). Multilevel and Longitudinal Modeling
Using Stata. Stata Press (www.stata-press.com). This discusses a free set of
commands that can be added to Stata that will do most of what Mplus can do and
some things Mplus cannot do. It is hard to use and very slow.
4. Growth mixture modeling
a. Muthén, B., & Muthén, L. K. (2000). Integrating person-centered and variablecentered
analysis: Growth mixture modeling with latent trajectory classes.
Alcoholism: Clinical and Experimental Research. 24:882-891.
This is an excellent and accessible conceptual introduction.
b. Muthén, B. (2001). Latent variable mixture modeling. In G. Marcoulides, & R.
Schumacker (Eds.) New Developments and Techniques in Structural Equation
Modeling (pp. 1-34). Mahwah, NJ: Lawrence Erlbaum.
c. Muthén, B., Brown, C. H., Booil, J., Khoo, S. Yang, C. Wang, C., Kellam, S., Carlin,
J., & Liao, J. (2002). General growth mixture modeling for randomized preventive
interventions. Biostatistics, 3:459-475
d. Muthén, B. Latent Variable analysis: Growth Mixture Modeling and Related
Techniques for Longitudinal Data. (2004) In D. Kaplan (ed.), Handbook of
quantitative methodology for the social sciences (pp. 345-368). Newbury Park, CA:
Sage Publications
e. Muthén, B., Brown, C. H., Booil Jo, K, M., Khoo, S., Yang, C. Wang, C., Kellam, S.,
Carlin, J., Liao, J. (2002). General growth mixture modeling for randomized
preventive interventions. Biostatistics. 3,4, pp. 459-475.
5. The web page for Mplus, www.statmodel.com , maintains a current set of references, many
as PDF files. These are organized by topic and some include data and the Mplus program.
Growth Curve and Related Models, Alan C. Acock 52