Growth Curves and the Study of Romantic Relationships
Among Young Adults.
Alan C. Acock
[email protected]
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 Curves For Couple Data
With couple data we need to identify a pair of parallel growth curves. The
following figure is a representation of what we are doing:
Dyadic Growth Curve Presentation—Alan C. Acock
1
This figure is a straightforward extension of our simple linear growth curve.
 The y11 to y14 are the four waves for the male member in the couple. 
 The y21 to y24 are the corresponding scores for the female member of
the couple. These are, of course, distinguishable pairs and this model
would not work this way for same sex couples. We could put equality
constraints so that the path from s1  y14 = s2  y24, etc., if
we have non-distinguishable pairs.
 We have an intercept and slope for both the males and the females and
these would be identified the same way as we did with the male only
growth curve. 

Dyadic Growth Curve Presentation—Alan C. Acock
2
What is new?
 The corresponding errors, e11 e21, e12-e21, etc (not show
explicitly in figure but represented by unlabeled arrows going to year
y) are logically correlated. 
 Anything that could cause error at wave 0 for males is likely there for
the female as well. For example, they may have shared a financial
crisis, or some other event shared at that time that makes them
especially prone to conflict or prone to being pleasant. 
 This non-random error needs to be correlated to take it “out” of the
growth trajectory.
 The initial level or intercept for both of them may be very different as
would happen if he engaged in more verbal conflict than she did, but
across our 500 couples we would expect some correlation. The curved
arrow between the intercepts represents this.
 The same argument applies to the slopes.
 In conventional regression models we assume the intercept and slope
are uncorrelated. Here we explicitly allow them to be correlated, i1 –
s1 and i2 – s2. It is often the case that individuals who start much
higher or much lower than the mean initial level have different
trajectories. 
 We also have a direct effect going from his intercept to her slope and
from her intercept to his slope. We expect that couples where the man
has a high initial level of verbal aggression will have the woman show
a steeper increase in her level of aggression, and vice versa.
Here is the Mplus Program (Control Statements):
Title: parallel_growth.inp
Data:
File is monte1.dat ;
Variable:
Names are
Dyadic Growth Curve Presentation—Alan C. Acock
3
phy_con y11 y12 y13 y14
Missing are
all (-9999) ;
usevariables are
y11-y24;
Model:
i1 s1 | y11@0 y12@1 y13@2
i2 s2 | y21@0 y22@1 y23@2
y11 y12 y13 y14 pwith y21
y21 y22 y23 y24 par_con ;
y14@3 ;
y24@4 ;
y22 y23 y24 ;Correlates
corresponding errors
“on” for regress s2 on i1
s2 on i1;
s1 on i2;
i1 with s1;
“with”,
i2 with s2;
Output:
Sampstat standardized Mod(3.84);
i1 covaries with s1
Here is Selected Output:
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value
Degrees of Freedom
P-Value
231.699
18
0.0000
Chi-Square Test of Model Fit for the Baseline Model
Value
Degrees of Freedom
P-Value
3645.607
28
0.0000
CFI/TLI
CFI
0.941
TLI
0.908
These are a bit low, cf .95
Some still compare to .90
Loglikelihood
H0 Value
H1 Value
Dyadic Growth Curve Presentation—Alan C. Acock
-6093.514
-5977.665
4
Information Criteria
Number of Free Parameters
Akaike (AIC)
Bayesian (BIC)
Sample-Size Adjusted BIC
(n* = (n + 2) / 24)
26
12239.029
12348.609
12266.083
RMSEA (Root Mean Square Error Of Approximation)
Estimate
90 Percent C.I.
Probability RMSEA <= .05
0.154
0.137
0.000
Way over < .06
0.172
SRMR (Standardized Root Mean Square Residual)
Value
0.046
Okay
MODEL RESULTS
I1
S.E.
Est./S.E.
1.000
1.000
1.000
1.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
0.000
1.000
2.000
3.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
1.000
1.000
1.000
1.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
0.000
1.000
2.000
4.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
0.062
0.016
3.777
0.000
|
Y11
Y12
Y13
Y14
S1
|
Y11
Y12
Y13
Y14
I2
|
Y21
Y22
Y23
Y24
S2
|
Y21
Y22
Y23
Y24
S2
ON
I1
S1
Two-Tailed
P-Value
Estimate
ON
Dyadic Growth Curve Presentation—Alan C. Acock
5
I2
0.035
0.024
1.431
0.153
0.168
0.034
4.898
0.000
0.120
0.430
0.025
0.085
4.912
5.068
0.000
0.000
-0.016
0.013
-1.199
0.231
0.168
0.046
3.630
0.000
0.159
0.029
5.466
0.000
0.184
0.039
4.768
0.000
0.123
0.066
1.856
0.063
Means
I1
I2
2.177
1.830
0.064
0.058
33.819
31.292
0.000
0.000
Intercepts
Y11
Y12
Y13
Y14
Y21
Y22
Y23
Y24
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
S1
1.935
0.050
38.387
0.000
S2
0.607
0.041
14.871
Huge slope for men
0.000 With parallel these
are Under Intercepts.
Variances
I1
1.637
0.130
12.608
0.000
I2
1.277
0.104
12.313
0.000
0.543
0.063
8.583
0.000
I1
WITH
S1
I2
WITH
S2
I1
S2
WITH
S1
Y11
WITH
Y21
Y12
WITH
Y22
Y13
WITH
Y23
Y14
WITH
Y24
Residual Variances
Y11
Dyadic Growth Curve Presentation—Alan C. Acock
Men start higher, could
test with equality
constraint
Lots of variance left to
Explain adding
covariates
6
Y12
Y13
Y14
Y21
Y22
Y23
Y24
S1
0.463
0.483
0.472
0.646
0.405
0.709
0.428
0.175
0.040
0.046
0.075
0.063
0.039
0.060
0.129
0.020
11.594
10.516
6.322
10.193
10.284
11.881
3.318
8.795
0.000
0.000
0.000
0.000
0.000
0.000
0.001
0.000
S2
0.100
0.014
7.237
0.000
Estimate
S.E.
Est./S.E.
0.242
0.065
3.750
0.000
0.094
0.066
1.431
0.152
0.314
0.070
4.455
0.000
0.336
0.297
0.079
0.051
4.245
5.814
0.000
0.000
-0.118
0.103
-1.145
0.252
0.284
0.070
4.068
0.000
0.368
0.056
6.520
0.000
0.314
0.057
5.493
0.000
0.274
0.135
2.021
0.043
Something to explain
adding covariates
STANDARDIZED MODEL RESULTS
STDYX Standardization
S2
Two-Tailed
P-Value
ON
I1
S1
ON
I2
I1
WITH
S1
I2
WITH
S2
I1
S2
WITH
S1
Y11
WITH
Y21
Y12
WITH
Y22
Y13
WITH
Y23
Y14
WITH
Y24
Interpretation
Dyadic Growth Curve Presentation—Alan C. Acock
7
The parallel growth curve is a much more complicated model than the single
growth curve.
 Where the single growth curve for men fit the data almost perfectly, the
parallel growth curve has a Chi-square(18) = 231.70, p < .001 indicating
it fails to fit the data perfectly. 
 Both the CFI = .94 and the TLI = .91 are at the lower end of a good fit.
 The RMSEA = .15 is evidence of a poor fit, but the Standardized Root
Mean Square Residual, SRMR = 0.046 indicates a good fit. 
 These are, at best, mixed results. Let’s interpret the model assuming that
these criteria justify doing so.
The male member of the couple has an initial level of 2.18 which is higher
than the initial level for women of 1.83.
 Both are highly significant, p < .001. 
 We could constrain these to be equal and compare the models to see if
they differ significantly. 
 We also could interpret these with real data in terms of effect size by
considering the standard deviation for verbal conflict of men and the
standard deviation for verbal conflict of women.
Not only do men appear to have higher initial verbal conflict, during the four
weeks the couples were followed, the men have a steeper slope, i.e, they
have an increasing gap with them becoming more hostile. The slope for the
men is 1.94 compared to 0.61 for the women. Both are statistically
significant. As with the initial level, we might put equality constraints on
these to test if they are significantly different from each other.
Men who have higher initial conflict have a direct positive effect on the
growth rate of women. The direct effect is 0.062, p < .001. There is a similar
but somewhat weaker direct effect of the initial conflict of women on the
growth rate of men, 0.035, p ns.
Dyadic Growth Curve Presentation—Alan C. Acock
8
Rather than relying on Mplus for graphics, you could write out the equation
and use Stata or Excel to do a very nice graph of the parallel growth
trajectories. Men start higher and go up more steeply.
We could say that to some extent “birds of a feather flock together” because
the initial levels of men and women in couples are correlated. Those men
who bring higher conflict to a relationship are attached to women who also
bring higher initial conflict. Here you might report the fully standardized
coefficient since it is the simple correlation ri1-s1 = 0.31, p < .001. But, there
is no significant correlation between how quickly he increases his level of
conflict and how quickly she does the same (I missed something generating
the simulated data here).
A Time Invariant Covariate to Explain the Growth
Trajectories
The next step is to add covariates that may be able to explain these
trajectories. There are two types of covariates, time invariant and time
varying. Here we will only consider one time invariant covariate that I have
labeled parental conflict. It would make much more sense to have two of
these, one for her parents’ conflict and the other for his parents’ conflict, but
to keep it simple and since it is only simulated data anyway, we have just
one variable called parental conflict and assume they both have the same
score on this variable.
Time invariant covariates are predictors that do not very across the duration
of the panel.
 Examples include variables such as gender, ethnicity, etc. 
Dyadic Growth Curve Presentation—Alan C. Acock
9



Some variables such as education may be treated as time invariant with
some populations, but not others. Young adults are often still in school
and their level of education could change across a 4 year panel.
Time varying covariates normally predict the score at a particular wave
and might explain why people did better at one wave than another—
perhaps because program fidelity was especially high at one wave.
Another example would be work related stress that could vary across
waves and might explain why a participant would deviate from the
overall growth trajectory at a particular wave. Examples of time varying
covariates are in my other material at oregonstate.edu/~acock/growth.
Time varying covariates are predictors that may vary from wave to wave.
 If you have an intervention and there are 4 waves of data, the fidelity of
implementation could vary from one wave to another. 
 With young adults, education could vary across waves.
 Time invariant covariates can directly predict the intercept and slope as
well as some distal outcome variable. 
What predicts the initial level and the rate of growth in verbal conflict across
for waves of a romantic relationship? We have used parental conflict.
1. The assumption is that those study participants who were exposed to
high levels of parental conflict will have a higher level of initial verbal
conflict in an intimate relationship plus they will have a steeper slope.
2. What other covariates are not included:
a. prior history of conflict in romantic relationships.
b. Parent-child conflict when they were an adolescent
c. Arrest history for crimes against persons
d. History of drug abuse
When we only include a single predictor we have misspecified our model. A
properly specified model includes all relevant predictors. No model is going
Dyadic Growth Curve Presentation—Alan C. Acock
10
to be specified perfectly because we never know that we have all relevant
predictors. We need to be sensitive to misspecification because our
predictor, parental conflict, may have a different effect when other time
invariant covariates are included.
INPUT INSTRUCTIONS
Title: parallel_growth_extendeda.inp
Data:
File is monte1.dat ;
Variable:
Names are
phy_con y11 y12 y13 y14 y21 y22 y23 y24 par_con ;
Missing are
all (-9999) ;
usevariables are
y11-y24 par_con ;
Model:
i1 s1 | y11@0 y12@1 y13@2 y14@3 ;
Dyadic Growth Curve Presentation—Alan C. Acock
11
i2 s2 | y21@0 y22@1 y23@2 y24@3 ;
y11 y12 y13 y14 pwith y21 y22 y23 y24 ;
s1 on i2;
s2 on i1;
i1 on par_con;
These
i2 on par_con;
i1 with s1;
i2 with s2;
s1 on par_con;
These
s2 on par_con;
regress the intercepts on par_con
do the same for the slopes
Output:
Sampstat standardized Mod(all);
SAMPLE STATISTICS
ESTIMATED SAMPLE STATISTICS
Means
Y11
Y12
________
________
1
2.163
4.163
1
Means
Y22
________
2.622
Y23
________
3.688
Y13
________
6.216
Y14
________
8.188
Y24
________
4.672
PAR_CON
________
3.137
Y21
________
1.588
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value
15.688
This does much better than
Degrees of Freedom
P-Value
23
0.8683
model without the covariate
Chi-Square Test of Model Fit for the Baseline Model
Value
Degrees of Freedom
P-Value
3877.730
36
0.0000
CFI/TLI
CFI
TLI
1.000
1.003
Loglikelihood
H0 Value
H1 Value
Dyadic Growth Curve Presentation—Alan C. Acock
-6766.605
-6758.761
12
Information Criteria
Number of Free Parameters
Akaike (AIC)
Bayesian (BIC)
Sample-Size Adjusted BIC
(n* = (n + 2) / 24)
29
13591.209
13713.433
13621.385
RMSEA (Root Mean Square Error Of Approximation)
Estimate
90 Percent C.I.
Probability RMSEA <= .05
0.000
0.000
1.000
0.020
SRMR (Standardized Root Mean Square Residual)
Value
0.025
MODEL RESULTS
I1
S.E.
Est./S.E.
1.000
1.000
1.000
1.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
0.000
1.000
2.000
3.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
1.000
1.000
1.000
1.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
0.000
1.000
2.000
3.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
-0.007
0.026
-0.256
0.798
|
Y11
Y12
Y13
Y14
S1
|
Y11
Y12
Y13
Y14
I2
|
Y21
Y22
Y23
Y24
S2
|
Y21
Y22
Y23
Y24
S1
ON
I2
S2
Two-Tailed
P-Value
Estimate
Wrong way, but insignif.
ON
Dyadic Growth Curve Presentation—Alan C. Acock
13
0.020 Might use stadardized
I1
0.064
0.027
2.332
ON
PAR_CON
0.117
0.017
6.920
ON
PAR_CON
0.046
0.021
2.178
0.029
ON
PAR_CON
0.464
0.038
12.233
0.000
ON
PAR_CON
0.272
0.036
7.569
0.000
0.081
0.030
2.661
0.008
0.115
0.031
3.728
0.000
-0.025
0.015
-1.637
0.102
0.200
0.043
4.658
0.000
0.158
0.028
5.586
0.000
0.160
0.033
4.902
0.000
Y24
0.143
0.055
2.619
0.009
Intercepts
Y11
Y12
Y13
Y14
Y21
Y22
Y23
Y24
I1
S1
I2
S2
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.709
1.656
0.740
0.752
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.131
0.056
0.124
0.060
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
5.410
29.428
5.956
12.464
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
0.000
0.000
0.000
0.000
Residual Variances
Y11
0.558
0.063
8.857
0.000
S1
S2
I1
I2
I1
All of these are
sign. Might use standardized
0.000
WITH
S1
I2
WITH
S2
S2
WITH
S1
Y11
WITH
Y21
Y12
WITH
Y22
Y13
WITH
Y23
Y14
WITH
Dyadic Growth Curve Presentation—Alan C. Acock
With covariates means
for both I and S go
under Intercepts.
14
Y12
Y13
Y14
Y21
Y22
Y23
Y24
I1
S1
I2
S2
0.469
0.472
0.488
0.488
0.415
0.443
0.561
1.149
0.146
1.036
0.179
0.040
0.044
0.073
0.057
0.036
0.044
0.078
0.100
0.018
0.089
0.020
11.731
10.622
6.698
8.558
11.596
10.109
7.228
11.494
8.096
11.594
8.759
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Estimate
S.E.
Est./S.E.
-0.018
0.068
-0.257
0.797
I1
0.183
0.078
2.334
0.020
ON
PAR_CON
0.409
0.055
7.425
0.000
ON
PAR_CON
0.150
0.068
2.212
0.027
ON
PAR_CON
0.533
0.037
14.252
0.000
ON
PAR_CON
0.363
0.045
8.062
0.000
0.198
0.082
2.417
0.016
0.268
0.080
3.349
0.001
-0.155
0.099
-1.562
0.118
0.383
0.071
5.394
0.000
STANDARDIZED MODEL RESULTS
STDYX Standardization
S1
ON
I2
S2
S1
S2
I1
I2
ON
I1
compare these
WITH
S1
I2
WITH
S2
S2
WITH
S1
Y11
WITH
Y21
Y12
Two-Tailed
P-Value
WITH
Dyadic Growth Curve Presentation—Alan C. Acock
15
Y22
Y13
0.358
0.054
6.623
0.000
0.350
0.061
5.731
0.000
0.274
0.096
2.868
0.004
WITH
Y23
Y14
WITH
Y24
5 Adding a Categorical Distal Outcome
After you are satisfied that you have included the appropriate predictors of
the intercept and slope, you are ready to predict some distal outcome. The
outcome is distal in that it’s measurement should be after the last wave,
although if it is at the last wave that might be acceptable. It is something that
your growth process produces. For this example, I’ve selected whether there
was any physical aggression in the relationship at some particular point.
What would explain this?
1.
Antecedent time invariant covariates. We would expect people who
have been exposed to more conflict in the relationship between their
parents would be more likely to exibit physical aggression toward their
partner. We could make a similar argument about several other time
invariant covariates we might want to include in an actu al study.
a. Parental Conflict  Physical Conflict
b. Parental Conflict  Intercept  Physical Conflict
c. Parental Conflict  Slope  Physical Conflict
2.
The initial level of verbal agression for both the man and the woman in
the relationship. People who come into a relationship with a high level
of verbal conflict from the start, are more likely to become physically
agressive rather than just verbabaly aggressive.
i. Intercept for man  Physical Conflict
Dyadic Growth Curve Presentation—Alan C. Acock
16
ii.
iii.
iv.
3.
Intercept for woman Physical Conflict
Intercept for man  Slope for Woman  Physical Conflict
Intercept for woman  Slope for Man  Physical Conflict
Slope (trajectory) of verbal conflict for both the man and woman would
influence their adoption of physical oflict
i. Slope for man  Physical Conflict
ii. Slope for woman  Physical Conflict
Here is our Model:
Mplus’ ability to work with categorical and count variables is a powerful
feature. This has been underutilized, I think, because people do not know
Dyadic Growth Curve Presentation—Alan C. Acock
17
how to interpret the results and the way Mplus presents them is not
altogether clear.
Mplus, by default does a Weighted Least Squares estimate for these models,
but can do a full Maximum Liklihood estimate if told to. This does greatly
increase the time. This model took about 8 minutes on my MacBook Pro,
but many models that are more complicated can take a day or more to
converge. The default is probably good until you get a reasonable model
going, and then do the maximum likelihood for that model.
Here is the underlying logic Mplus uses for the binary outcome. It says there
is actually a latent variable, Y*. If you are above some threshold on Y*, ,
then you will go into the higher category and if you are below that threshold
you will go in the lower category. Where U is the binary variable we can
graph this as:
Dyadic Growth Curve Presentation—Alan C. Acock
18
A Continuous Latent Factor and a Binary Response Variable and
Threshold
Rule: is the threshold,
where
U = 1 if Y* > ,
U = 0 if Y* ≤ 
Another way of looking at this is:
Dyadic Growth Curve Presentation—Alan C. Acock
19


A person with a low score on  (tau) will have a low probability of
endorsing the item.
A person with a high score on  (tau) will have a high probability of
endorsing the item.
Mplus VERSION 5.2
MUTHEN & MUTHEN
01/14/2009
3:32 PM
Title: parallel_growth_extendedb.inp
Data:
File is monte1.dat ;
Variable:
Names are
phy_con y11 y12 y13 y14 y21 y22 y23 y24 par_con ;
Missing are
all (-9999) ;
usevariables are
phy_con y11-y24 par_con ;
Categorical is phy_con ;
Mplus makes it binary if 2 values, multinomial if
More than 2 values; Counts also possible.
Dyadic Growth Curve Presentation—Alan C. Acock
20
Analysis:
Estimator = ML;
Processors = 2;
Time consuming-10 minutes; does Logistic regressions
Makes a big difference—I want 8 processors
☺
Model:
i1 s1 | y11@0 y12@1 y13@2 y14@3 ;
i2 s2 | y21@0 y22@1 y23@2 y24@3 ;
y11 y12 y13 y14 pwith y21 y22 y23 y24 ;
s1 on i2;
s2 on i1;
i1 on par_con;
i2 on par_con;
i1 with s1;
i2 with s2;
i2 with i1;
s2 with s1;
s1 on par_con;
s2 on par_con;
phy_con on s1 s2 i1 i2 par_con;
Output:
Sampstat standardized ;
Number of dependent variables
Number of independent variables
Number of continuous latent variables
9
1
4
Observed dependent variables
Continuous
Y11
Y23
Y12
Y24
Y13
Y14
Y21
Y22
Binary and ordered categorical (ordinal)
PHY_CON
Observed independent variables
PAR_CON
Continuous latent variables
I1
S1
I2
S2
Estimator
ML
SUMMARY OF CATEGORICAL DATA PROPORTIONS
PHY_CON
Category 1
Category 2
0.742
0.258
Gives distribution this way for categorical variables
Dyadic Growth Curve Presentation—Alan C. Acock
21
SAMPLE STATISTICS
ESTIMATED SAMPLE STATISTICS
1
Means
Y11
________
2.163
Y12
________
4.163
Y13
________
6.216
Y14
________
8.188
1
Means
Y22
________
2.622
Y23
________
3.688
Y24
________
4.672
PAR_CON
________
3.137
Y21
________
1.588
TESTS OF MODEL FIT
Loglikelihood
H0 Value
-6134.428
Information Criteria
Number of Free Parameters
Akaike (AIC)
Bayesian (BIC)
Sample-Size Adjusted BIC
(n* = (n + 2) / 24)
36
12340.856
12492.582
12378.315
There is no Chi-square or usual fit measures. The AIC, BIC can be used to
compare models (say dropping direct effects of covariates on distal outcome).
MODEL RESULTS
I1
Two-Tailed
P-Value
Estimate
S.E.
Est./S.E.
1.000
1.000
1.000
1.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
0.000
1.000
2.000
3.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
|
Y11
Y12
Y13
Y14
S1
|
Y11
Y12
Y13
Y14
Dyadic Growth Curve Presentation—Alan C. Acock
22
I2
|
Y21
Y22
Y23
Y24
S2
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
0.000
1.000
2.000
3.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
-0.019
0.026
-0.707
0.479
0.054
0.027
1.993
0.046
0.121
0.017
7.124
0.000
0.050
0.021
2.413
0.016
0.463
0.038
12.126
0.000
0.272
0.036
7.505
0.000
0.113
0.445
0.218
0.208
0.393
0.357
0.125
0.131
0.287
1.249
1.752
1.581
0.774
0.212
0.080
0.114
0.153
0.096
1.590
0.112
0.074
0.030
2.445
0.014
0.105
0.128
0.031
0.069
3.399
1.843
0.001
0.065
-0.019
0.015
-1.237
0.216
0.173
0.044
3.942
0.000
|
Y21
Y22
Y23
Y24
S1
ON
I2
S2
ON
I1
S1
ON
PAR_CON
S2
ON
PAR_CON
I1
ON
PAR_CON
I2
ON
PAR_CON
PHY_CON
S1
S2
I1
I2
ON
PHY_CON
ON
PAR_CON
I1
WITH
S1
I2
WITH
S2
I1
S2
WITH
S1
Y11
WITH
Y21
Y12
1.000
1.000
1.000
1.000
WITH
Dyadic Growth Curve Presentation—Alan C. Acock
23
Y22
0.154
0.028
5.482
0.000
0.164
0.033
5.011
0.000
Y24
0.134
0.055
2.453
0.014
Intercepts
Y11
Y12
Y13
Y14
Y21
Y22
Y23
Y24
I1
S1
I2
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.710
1.664
0.742
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.132
0.056
0.125
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
5.380
29.588
5.923
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
0.000
0.000
0.000
0.758
0.060
12.594
0.000
Y13
WITH
Y23
Y14
WITH
S2
Mean Intercept hard to
interpret because I
failed to center par_con
.71 would be score at
Start IF you scored 0 on
Par_con.
Thresholds
PHY_CON$1
3.128
0.801
3.907
0.000
Residual Variances
Y11
Y12
Y13
Y14
Y21
Y22
Y23
Y24
I1
S1
I2
S2
0.546
0.463
0.474
0.487
0.472
0.413
0.446
0.552
1.179
0.148
1.063
0.183
0.063
0.040
0.044
0.072
0.057
0.036
0.044
0.077
0.102
0.018
0.091
0.020
8.703
11.649
10.672
6.722
8.315
11.531
10.169
7.134
11.542
8.231
11.650
8.946
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
LOGISTIC REGRESSION ODDS RATIO RESULTS
PHY_CON
S1
S2
I1
I2
ON
1.119
1.561
1.244
1.231
These have the usual limitations of odds ratios
when variables are on different scales. An odds
ratio of more than 1 uses odds ratio – 1, 1.165 – 1 =
.165 or 16.5%. For each unit change in par_con there
Is a 16.5% increase in the odds of physical conflict.
Dyadic Growth Curve Presentation—Alan C. Acock
24
PHY_CON
ON
PAR_CON
1.165
Both females and males initial level have similar
effects, 24% for men and 23% for women. Significance
For these are above for the unstandardized
coefficients
STANDARDIZED MODEL RESULTS
STDYX Standardization
I1
Two-Tailed
P-Value
Estimate
S.E.
Est./S.E.
0.866
0.786
0.678
0.582
0.017
0.018
0.021
0.022
52.214
44.834
32.208
26.330
0.000
0.000
0.000
0.000
0.000
0.258
0.445
0.574
0.000
0.015
0.023
0.028
999.000
16.988
19.092
20.138
999.000
0.000
0.000
0.000
0.849
0.755
0.629
0.521
0.019
0.020
0.022
0.022
44.843
38.596
28.533
23.816
0.000
0.000
0.000
0.000
0.000
0.304
0.507
0.631
0.000
0.017
0.024
0.028
999.000
18.153
20.848
22.225
999.000
0.000
0.000
0.000
-0.049
0.069
-0.709
0.478
0.155
0.078
1.992
0.046
0.418
0.054
7.674
0.000
0.164
0.067
2.455
0.014
0.527
0.038
14.061
0.000
|
Y11
Y12
Y13
Y14
S1
|
Y11
Y12
Y13
Y14
I2
|
Y21
Y22
Y23
Y24
S2
|
Y21
Y22
Y23
Y24
S1
ON
I2
S2
ON
I1
S1
ON
PAR_CON
S2
ON
PAR_CON
I1
Standarized coefficents
Are straight forward for
continous variables.
ON
PAR_CON
Dyadic Growth Curve Presentation—Alan C. Acock
25
I2
ON
PAR_CON
0.358
0.045
7.976
0.000
0.024
0.102
0.144
0.119
0.085
0.082
0.081
0.074
0.287
1.255
1.773
1.593
0.774
0.209
0.076
0.111
0.115
0.072
1.600
0.109
0.178
0.079
2.242
0.025
0.239
0.114
0.077
0.060
3.085
1.907
0.002
0.056
-0.114
0.096
-1.198
0.231
0.342
0.076
4.519
0.000
0.353
0.054
6.506
0.000
0.357
0.061
5.879
0.000
Y24
0.258
0.097
2.668
0.008
Intercepts
Y11
Y12
Y13
Y14
Y21
Y22
Y23
Y24
I1
S1
I2
S2
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.556
3.964
0.672
1.703
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.116
0.288
0.126
0.178
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
4.777
13.749
5.318
9.584
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
0.000
0.000
0.000
0.000
Thresholds
PHY_CON$1
1.616
0.400
4.044
0.000
Residual Variances
Y11
Y12
0.250
0.175
0.029
0.016
8.720
10.941
0.000
0.000
PHY_CON
S1
S2
I1
I2
ON
PHY_CON
ON
PAR_CON
I1
WITH
S1
I2
WITH
S2
I1
S2
WITH
S1
Y11
WITH
Y21
Y12
WITH
Y22
Y13
WITH
Y23
Y14
WITH
Dyadic Growth Curve Presentation—Alan C. Acock
26
Y13
Y14
Y21
Y22
Y23
Y24
I1
S1
I2
S2
0.133
0.101
0.279
0.193
0.145
0.123
0.722
0.837
0.872
0.922
0.013
0.016
0.032
0.018
0.015
0.017
0.040
0.040
0.032
0.031
10.087
6.501
8.672
10.963
9.800
7.030
18.248
20.735
27.092
29.719
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
STDY Standardization
These are what I would interpret IF I had a binary predictor
and a continous outcome variable. For example if we had a
binary variable for attends church (0,1)  intercept with a
standardized on Y of .3, this would mean that those who say
they attend church are .3 standard deviations higher on the
initial level.
I1
Two-Tailed
P-Value
Estimate
S.E.
Est./S.E.
0.866
0.786
0.678
0.582
0.017
0.018
0.021
0.022
52.214
44.834
32.208
26.330
0.000
0.000
0.000
0.000
0.000
0.258
0.445
0.574
0.000
0.015
0.023
0.028
999.000
16.988
19.092
20.138
999.000
0.000
0.000
0.000
0.849
0.755
0.629
0.521
0.019
0.020
0.022
0.022
44.843
38.596
28.533
23.816
0.000
0.000
0.000
0.000
0.000
0.304
0.507
0.631
0.000
0.017
0.024
0.028
999.000
18.153
20.848
22.225
999.000
0.000
0.000
0.000
-0.049
0.069
-0.709
0.478
|
Y11
Y12
Y13
Y14
S1
|
Y11
Y12
Y13
Y14
I2
|
Y21
Y22
Y23
Y24
S2
|
Y21
Y22
Y23
Y24
S1
ON
I2
Dyadic Growth Curve Presentation—Alan C. Acock
27
S2
ON
I1
0.155
0.078
1.992
0.046
0.287
0.037
7.832
0.000
0.113
0.046
2.462
0.014
0.362
0.025
14.722
0.000
0.246
0.030
8.172
0.000
0.024
0.102
0.144
0.119
0.085
0.082
0.081
0.074
0.287
1.255
1.773
1.593
0.774
0.209
0.076
0.111
0.079
0.049
1.602
0.109
0.178
0.079
2.242
0.025
0.239
0.114
0.077
0.060
3.085
1.907
0.002
0.056
-0.114
0.096
-1.198
0.231
0.342
0.076
4.519
0.000
0.353
0.054
6.506
0.000
0.357
0.061
5.879
0.000
Y24
0.258
0.097
2.668
0.008
Intercepts
Y11
Y12
Y13
Y14
Y21
Y22
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
S1
ON
PAR_CON
S2
ON
PAR_CON
I1
ON
PAR_CON
I2
ON
PAR_CON
PHY_CON
S1
S2
I1
I2
ON
PHY_CON
ON
PAR_CON
I1
WITH
S1
I2
WITH
S2
I1
S2
WITH
S1
Y11
WITH
Y21
Y12
WITH
Y22
Y13
WITH
Y23
Y14
WITH
Dyadic Growth Curve Presentation—Alan C. Acock
28
Y23
Y24
I1
S1
I2
S2
0.000
0.000
0.556
3.964
0.672
1.703
0.000
0.000
0.116
0.288
0.126
0.178
999.000
999.000
4.777
13.749
5.318
9.584
999.000
999.000
0.000
0.000
0.000
0.000
Thresholds
PHY_CON$1
1.616
0.400
4.044
0.000
Residual Variances
Y11
Y12
Y13
Y14
Y21
Y22
Y23
Y24
I1
S1
I2
S2
0.250
0.175
0.133
0.101
0.279
0.193
0.145
0.123
0.722
0.837
0.872
0.922
0.029
0.016
0.013
0.016
0.032
0.018
0.015
0.017
0.040
0.040
0.032
0.031
8.720
10.941
10.087
6.501
8.672
10.963
9.800
7.030
18.248
20.735
27.092
29.719
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Estimate
S.E.
Est./S.E.
0.122
0.750
0.825
0.867
0.899
0.721
0.807
0.855
0.877
0.038
0.029
0.016
0.013
0.016
0.032
0.018
0.015
0.017
3.229
26.107
51.561
65.498
57.776
22.421
45.911
57.907
50.175
Estimate
S.E.
Est./S.E.
0.278
0.163
0.128
0.078
0.040
0.040
0.032
0.031
7.031
4.025
3.988
2.506
R-SQUARE
Observed
Variable
PHY_CON
Y11
Y12
Y13
Y14
Y21
Y22
Y23
Y24
Latent
Variable
I1
S1
I2
S2
Two-Tailed
P-Value
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Model fit ≠ sig. of R2
Two-Tailed
P-Value
0.000
0.000
0.000
0.012
Beginning Time: 15:32:58
Ending Time: 15:42:43
Elapsed Time: 00:09:45
Dyadic Growth Curve Presentation—Alan C. Acock
29
So Where Are We?
MPlus is an excellent tool for working with dyadic data to model parallel growth processes. It is
especially useful for distinguishable pairs. It can be used for growth processes that involve
continuous variables, binary variables, or count variables.
Incorporating covariates to explain variation in the growth trajectories across your sample of
dyads is straightforward. We can also have distal outcomes and examine direct and indirect
effects.
7 References
Bollen, K. A., & Curran, P. J. (2006). Latent Curve Models: A Structural Equation
Perspective. Hoboken, NJ: Wiley.
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.
Duncan, T. E., Duncan, S. C., & Strycker, L. A. (2006). An Introduction to Latent Variable
Growth Curve Modeling: Concepts, Issues, and Applications (2nd ed.). Mahwah NJ: Lawrence
Erlbaum. The second edition of a classic text on growth curve modeling.
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.
Long, J. S., & Freese, J. (2006). Regression Models for Categorical Dependent Variables
Using Stata, 2nd 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.
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.
Dyadic Growth Curve Presentation—Alan C. Acock
30
Muthén, B., & Muthén, L. K. (2000). Integrating person-centered and variable-centered
analysis: Growth mixture modeling with latent trajectory classes. Alcoholism: Clinical and
Experimental Research. 24:882-891. This is an excellent and accessible conceptual introduction.
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.
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
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
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.
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.
Wang, M. (2007). Profiling retirees in the retirement transition and adjustment process:
Examining the longitudinal change patterns of retirees' psychological well-being. Journal of
Applied Psychology, 92(2), 455-474. This is a nice example of presenting results showing some
graphs and tables.
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.
My web page for this topic is oregonstate.edu/~acock/growth.
Dyadic Growth Curve Presentation—Alan C. Acock
31