Growth Curves and the Study of Romantic Relationships Among Young Adults.
Alan C. Acock
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
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:
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.
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
phy_con y11 y12 y13 y14 y21 y22 y23 y24 par_con ;
Missing are
all (-9999) ;
usevariables are
y11-y24;
Model:
i1 s1 | y11@0 y12@1 y13@2 y14@3 ;
i2 s2 | y21@0 y22@1 y23@2 y24@4 ;
y11 y12 y13 y14 pwith y21 y22 y23 y24 ;Correlates
corresponding errors
s2 on i1;“on” for regress s2 on i1
s1 on i2;
i1 with s1;“with”, i1 covaries with s1
i2 with s2;
Output:
Sampstat standardized Mod(3.84);
Here is Selected Output:
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value 231.699
Degrees of Freedom 18
P-Value 0.0000
Chi-Square Test of Model Fit for the Baseline Model
Value 3645.607
Degrees of Freedom 28
P-Value 0.0000
CFI/TLI
CFI 0.941These are a bit low, cf .95
TLI 0.908Some still compare to .90
Loglikelihood
H0 Value -6093.514
H1 Value -5977.665
Information Criteria
Number of Free Parameters 26
Akaike (AIC) 12239.029
Bayesian (BIC) 12348.609
Sample-Size Adjusted BIC 12266.083
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.154Way over < .06
90 Percent C.I. 0.137 0.172
Probability RMSEA <= .05 0.000
SRMR (Standardized Root Mean Square Residual)
Value 0.046Okay
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
I1 |
Y11 1.000 0.000 999.000 999.000
Y12 1.000 0.000 999.000 999.000
Y13 1.000 0.000 999.000 999.000
Y14 1.000 0.000 999.000 999.000
S1 |
Y11 0.000 0.000 999.000 999.000
Y12 1.000 0.000 999.000 999.000
Y13 2.000 0.000 999.000 999.000
Y14 3.000 0.000 999.000 999.000
I2 |
Y21 1.000 0.000 999.000 999.000
Y22 1.000 0.000 999.000 999.000
Y23 1.000 0.000 999.000 999.000
Y24 1.000 0.000 999.000 999.000
S2 |
Y21 0.000 0.000 999.000 999.000
Y22 1.000 0.000 999.000 999.000
Y23 2.000 0.000 999.000 999.000
Y24 4.000 0.000 999.000 999.000
S2 ON
I1 0.062 0.016 3.777 0.000
S1 ON
I2 0.035 0.024 1.431 0.153
I1 WITH
S1 0.168 0.034 4.898 0.000
I2 WITH
S2 0.120 0.025 4.912 0.000
I1 0.430 0.085 5.068 0.000
S2 WITH
S1 -0.016 0.013 -1.199 0.231
Y11 WITH
Y21 0.168 0.046 3.630 0.000
Y12 WITH
Y22 0.159 0.029 5.466 0.000
Y13 WITH
Y23 0.184 0.039 4.768 0.000
Y14 WITH
Y24 0.123 0.066 1.856 0.063
Means
I1 2.177 0.064 33.819 0.000Men start higher, could
I2 1.830 0.058 31.292 0.000 test with equality
constraint
Intercepts
Y11 0.000 0.000 999.000 999.000
Y12 0.000 0.000 999.000 999.000
Y13 0.000 0.000 999.000 999.000
Y14 0.000 0.000 999.000 999.000
Y21 0.000 0.000 999.000 999.000
Y22 0.000 0.000 999.000 999.000
Y23 0.000 0.000 999.000 999.000
Y24 0.000 0.000 999.000 999.000
S1 1.935 0.050 38.387 0.000 Huge slope for men
S2 0.607 0.041 14.871 0.000 With parallel these
are Under Intercepts.
Variances
I1 1.637 0.130 12.608 0.000 Lots of variance left to
I2 1.277 0.104 12.313 0.000 Explain adding
covariates
Residual Variances
Y11 0.543 0.063 8.583 0.000
Y12 0.463 0.040 11.594 0.000
Y13 0.483 0.046 10.516 0.000
Y14 0.472 0.075 6.322 0.000
Y21 0.646 0.063 10.193 0.000
Y22 0.405 0.039 10.284 0.000
Y23 0.709 0.060 11.881 0.000
Y24 0.428 0.129 3.318 0.001
S1 0.175 0.020 8.795 0.000 Something to explain
S2 0.100 0.014 7.237 0.000 adding covariates
STANDARDIZED MODEL RESULTS
STDYX Standardization
Two-Tailed
Estimate S.E. Est./S.E. P-Value
S2 ON
I1 0.242 0.065 3.750 0.000
S1 ON
I2 0.094 0.066 1.431 0.152
I1 WITH
S1 0.314 0.070 4.455 0.000
I2 WITH
S2 0.336 0.079 4.245 0.000
I1 0.297 0.051 5.814 0.000
S2 WITH
S1 -0.118 0.103 -1.145 0.252
Y11 WITH
Y21 0.284 0.070 4.068 0.000
Y12 WITH
Y22 0.368 0.056 6.520 0.000
Y13 WITH
Y23 0.314 0.057 5.493 0.000
Y14 WITH
Y24 0.274 0.135 2.021 0.043
Interpretation
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.
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.
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.
- 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.
- What other covariates are not included:
- prior history of conflict in romantic relationships.
- Parent-child conflict when they were an adolescent
- Arrest history for crimes against persons
- 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 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 ;
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 regress the intercepts on par_con
i2 on par_con;
i1 with s1;
i2 with s2;
s1 on par_con;These do the same for the slopes
s2 on par_con;
Output:
Sampstat standardized Mod(all);
SAMPLE STATISTICS
ESTIMATED SAMPLE STATISTICS
Means
Y11 Y12 Y13 Y14 Y21
______
1 2.163 4.163 6.216 8.188 1.588
Means
Y22 Y23 Y24 PAR_CON
______
1 2.622 3.688 4.672 3.137
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value 15.688This does much better than
Degrees of Freedom 23model without the covariate
P-Value 0.8683
Chi-Square Test of Model Fit for the Baseline Model
Value 3877.730
Degrees of Freedom 36
P-Value 0.0000
CFI/TLI
CFI 1.000
TLI 1.003
Loglikelihood
H0 Value -6766.605
H1 Value -6758.761
Information Criteria
Number of Free Parameters 29
Akaike (AIC) 13591.209
Bayesian (BIC) 13713.433
Sample-Size Adjusted BIC 13621.385
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.000
90 Percent C.I. 0.000 0.020
Probability RMSEA <= .05 1.000
SRMR (Standardized Root Mean Square Residual)
Value 0.025
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
I1 |
Y11 1.000 0.000 999.000 999.000
Y12 1.000 0.000 999.000 999.000
Y13 1.000 0.000 999.000 999.000
Y14 1.000 0.000 999.000 999.000
S1 |
Y11 0.000 0.000 999.000 999.000
Y12 1.000 0.000 999.000 999.000
Y13 2.000 0.000 999.000 999.000
Y14 3.000 0.000 999.000 999.000
I2 |
Y21 1.000 0.000 999.000 999.000
Y22 1.000 0.000 999.000 999.000
Y23 1.000 0.000 999.000 999.000
Y24 1.000 0.000 999.000 999.000
S2 |
Y21 0.000 0.000 999.000 999.000
Y22 1.000 0.000 999.000 999.000
Y23 2.000 0.000 999.000 999.000
Y24 3.000 0.000 999.000 999.000
S1 ON
I2 -0.007 0.026 -0.256 0.798 Wrong way, but insignif.
S2 ON
I1 0.064 0.027 2.332 0.020 Might use stadardized
S1 ON
PAR_CON 0.117 0.017 6.920 0.000All of these are
sign. Might use standardized
S2 ON
PAR_CON 0.046 0.021 2.178 0.029
I1 ON
PAR_CON 0.464 0.038 12.233 0.000
I2 ON
PAR_CON 0.272 0.036 7.569 0.000
I1 WITH
S1 0.081 0.030 2.661 0.008
I2 WITH
S2 0.115 0.031 3.728 0.000
S2 WITH
S1 -0.025 0.015 -1.637 0.102
Y11 WITH
Y21 0.200 0.043 4.658 0.000
Y12 WITH
Y22 0.158 0.028 5.586 0.000
Y13 WITH
Y23 0.160 0.033 4.902 0.000
Y14 WITH
Y24 0.143 0.055 2.619 0.009
Intercepts
Y11 0.000 0.000 999.000 999.000
Y12 0.000 0.000 999.000 999.000
Y13 0.000 0.000 999.000 999.000
Y14 0.000 0.000 999.000 999.000
Y21 0.000 0.000 999.000 999.000
Y22 0.000 0.000 999.000 999.000
Y23 0.000 0.000 999.000 999.000
Y24 0.000 0.000 999.000 999.000
I1 0.709 0.131 5.410 0.000With covariates means
S1 1.656 0.056 29.428 0.000 for both I and S go
I2 0.740 0.124 5.956 0.000 under Intercepts.
S2 0.752 0.060 12.464 0.000
Residual Variances
Y11 0.558 0.063 8.857 0.000
Y12 0.469 0.040 11.731 0.000
Y13 0.472 0.044 10.622 0.000
Y14 0.488 0.073 6.698 0.000
Y21 0.488 0.057 8.558 0.000
Y22 0.415 0.036 11.596 0.000
Y23 0.443 0.044 10.109 0.000
Y24 0.561 0.078 7.228 0.000
I1 1.149 0.100 11.494 0.000
S1 0.146 0.018 8.096 0.000
I2 1.036 0.089 11.594 0.000
S2 0.179 0.020 8.759 0.000
STANDARDIZED MODEL RESULTS
STDYX Standardization
Two-Tailed
Estimate S.E. Est./S.E. P-Value
S1 ON
I2 -0.018 0.068 -0.257 0.797
S2 ON
I1 0.183 0.078 2.334 0.020
S1 ON
PAR_CON 0.409 0.055 7.425 0.000 compare these
S2 ON
PAR_CON 0.150 0.068 2.212 0.027
I1 ON
PAR_CON 0.533 0.037 14.252 0.000
I2 ON
PAR_CON 0.363 0.045 8.062 0.000
I1 WITH
S1 0.198 0.082 2.417 0.016
I2 WITH
S2 0.268 0.080 3.349 0.001
S2 WITH
S1 -0.155 0.099 -1.562 0.118
Y11 WITH
Y21 0.383 0.071 5.394 0.000
Y12 WITH
Y22 0.358 0.054 6.623 0.000
Y13 WITH
Y23 0.350 0.061 5.731 0.000
Y14 WITH
Y24 0.274 0.096 2.868 0.004
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?
- 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.
- Parental Conflict Physical Conflict
- Parental Conflict Intercept Physical Conflict
- Parental Conflict Slope Physical Conflict
- 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.
- Intercept for man Physical Conflict
- 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
- Slope for man Physical Conflict
- 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 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*, τ, 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:
A Continuous Latent Factor and a Binary Response Variable and Threshold
Rule:τ is the threshold,
where
U = 1 if Y*τ,
U = 0 if Y* ≤ τ
Another way of looking at this is:
A person with a low score on τ (tau) will have a low probability of endorsing the item.
A person with a high score on τ (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.
Analysis:
Estimator = ML;Time consuming-10 minutes; does Logistic regressions
Processors = 2;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 9
Number of independent variables 1
Number of continuous latent variables 4
Observed dependent variables
Continuous
Y11 Y12 Y13 Y14 Y21 Y22
Y23 Y24
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 0.742Gives distribution this way for categorical variables
Category 2 0.258
SAMPLE STATISTICS
ESTIMATED SAMPLE STATISTICS
Means
Y11 Y12 Y13 Y14 Y21
______
1 2.163 4.163 6.216 8.188 1.588
Means
Y22 Y23 Y24 PAR_CON
______
1 2.622 3.688 4.672 3.137
TESTS OF MODEL FIT
Loglikelihood
H0 Value -6134.428
Information Criteria
Number of Free Parameters 36
Akaike (AIC) 12340.856
Bayesian (BIC) 12492.582
Sample-Size Adjusted BIC 12378.315
(n* = (n + 2) / 24)
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
Two-Tailed
Estimate S.E. Est./S.E. P-Value
I1 |
Y11 1.000 0.000 999.000 999.000
Y12 1.000 0.000 999.000 999.000
Y13 1.000 0.000 999.000 999.000
Y14 1.000 0.000 999.000 999.000
S1 |
Y11 0.000 0.000 999.000 999.000
Y12 1.000 0.000 999.000 999.000
Y13 2.000 0.000 999.000 999.000
Y14 3.000 0.000 999.000 999.000
I2 |
Y21 1.000 0.000 999.000 999.000
Y22 1.000 0.000 999.000 999.000
Y23 1.000 0.000 999.000 999.000
Y24 1.000 0.000 999.000 999.000
S2 |
Y21 0.000 0.000 999.000 999.000
Y22 1.000 0.000 999.000 999.000
Y23 2.000 0.000 999.000 999.000
Y24 3.000 0.000 999.000 999.000
S1 ON
I2 -0.019 0.026 -0.707 0.479
S2 ON
I1 0.054 0.027 1.993 0.046
S1 ON
PAR_CON 0.121 0.017 7.124 0.000
S2 ON
PAR_CON 0.050 0.021 2.413 0.016
I1 ON
PAR_CON 0.463 0.038 12.126 0.000
I2 ON
PAR_CON 0.272 0.036 7.505 0.000
PHY_CON ON
S1 0.113 0.393 0.287 0.774
S2 0.445 0.357 1.249 0.212
I1 0.218 0.125 1.752 0.080
I2 0.208 0.131 1.581 0.114
PHY_CON ON
PAR_CON 0.153 0.096 1.590 0.112
I1 WITH
S1 0.074 0.030 2.445 0.014
I2 WITH
S2 0.105 0.031 3.399 0.001
I1 0.128 0.069 1.843 0.065
S2 WITH
S1 -0.019 0.015 -1.237 0.216
Y11 WITH
Y21 0.173 0.044 3.942 0.000
Y12 WITH
Y22 0.154 0.028 5.482 0.000
Y13 WITH
Y23 0.164 0.033 5.011 0.000
Y14 WITH
Y24 0.134 0.055 2.453 0.014
Intercepts
Y11 0.000 0.000 999.000 999.000
Y12 0.000 0.000 999.000 999.000
Y13 0.000 0.000 999.000 999.000
Y14 0.000 0.000 999.000 999.000
Y21 0.000 0.000 999.000 999.000
Y22 0.000 0.000 999.000 999.000
Y23 0.000 0.000 999.000 999.000
Y24 0.000 0.000 999.000 999.000
I1 0.710 0.132 5.380 0.000 Mean Intercept hard to
S1 1.664 0.056 29.588 0.000 interpret because I
I2 0.742 0.125 5.923 0.000 failed to center par_con
S2 0.758 0.060 12.594 0.000 .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 0.546 0.063 8.703 0.000
Y12 0.463 0.040 11.649 0.000
Y13 0.474 0.044 10.672 0.000
Y14 0.487 0.072 6.722 0.000
Y21 0.472 0.057 8.315 0.000
Y22 0.413 0.036 11.531 0.000
Y23 0.446 0.044 10.169 0.000
Y24 0.552 0.077 7.134 0.000
I1 1.179 0.102 11.542 0.000
S1 0.148 0.018 8.231 0.000
I2 1.063 0.091 11.650 0.000
S2 0.183 0.020 8.946 0.000
LOGISTIC REGRESSION ODDS RATIO RESULTS
PHY_CON ON
S1 1.119These have the usual limitations of odds ratios
S2 1.561when variables are on different scales. An odds
I1 1.244ratio of more than 1 uses odds ratio – 1, 1.165 – 1 =
I2 1.231.165 or 16.5%. For each unit change in par_con there
Is a 16.5% increase in the odds of physical conflict.
PHY_CON ONBoth females and males initial level have similar
PAR_CON 1.165effects, 24% for men and 23% for women. Significance
For these are above for the unstandardized
coefficients
STANDARDIZED MODEL RESULTS
STDYX Standardization
Two-Tailed
Estimate S.E. Est./S.E. P-Value
I1 |
Y11 0.866 0.017 52.214 0.000
Y12 0.786 0.018 44.834 0.000
Y13 0.678 0.021 32.208 0.000
Y14 0.582 0.022 26.330 0.000
S1 |
Y11 0.000 0.000 999.000 999.000
Y12 0.258 0.015 16.988 0.000
Y13 0.445 0.023 19.092 0.000
Y14 0.574 0.028 20.138 0.000
I2 |
Y21 0.849 0.019 44.843 0.000
Y22 0.755 0.020 38.596 0.000
Y23 0.629 0.022 28.533 0.000
Y24 0.521 0.022 23.816 0.000
S2 |
Y21 0.000 0.000 999.000 999.000
Y22 0.304 0.017 18.153 0.000
Y23 0.507 0.024 20.848 0.000
Y24 0.631 0.028 22.225 0.000
S1 ON
I2 -0.049 0.069 -0.709 0.478
S2 ON
I1 0.155 0.078 1.992 0.046
S1 ON
PAR_CON 0.418 0.054 7.674 0.000Standarized coefficents
Are straight forward for
S2 ONcontinous variables.
PAR_CON 0.164 0.067 2.455 0.014
I1 ON
PAR_CON 0.527 0.038 14.061 0.000
I2 ON
PAR_CON 0.358 0.045 7.976 0.000
PHY_CON ON
S1 0.024 0.085 0.287 0.774
S2 0.102 0.082 1.255 0.209
I1 0.144 0.081 1.773 0.076
I2 0.119 0.074 1.593 0.111
PHY_CON ON
PAR_CON 0.115 0.072 1.600 0.109
I1 WITH
S1 0.178 0.079 2.242 0.025
I2 WITH
S2 0.239 0.077 3.085 0.002
I1 0.114 0.060 1.907 0.056
S2 WITH
S1 -0.114 0.096 -1.198 0.231
Y11 WITH
Y21 0.342 0.076 4.519 0.000
Y12 WITH
Y22 0.353 0.054 6.506 0.000
Y13 WITH
Y23 0.357 0.061 5.879 0.000
Y14 WITH
Y24 0.258 0.097 2.668 0.008
Intercepts
Y11 0.000 0.000 999.000 999.000
Y12 0.000 0.000 999.000 999.000
Y13 0.000 0.000 999.000 999.000
Y14 0.000 0.000 999.000 999.000
Y21 0.000 0.000 999.000 999.000
Y22 0.000 0.000 999.000 999.000
Y23 0.000 0.000 999.000 999.000
Y24 0.000 0.000 999.000 999.000
I1 0.556 0.116 4.777 0.000
S1 3.964 0.288 13.749 0.000
I2 0.672 0.126 5.318 0.000
S2 1.703 0.178 9.584 0.000
Thresholds
PHY_CON$1 1.616 0.400 4.044 0.000
Residual Variances
Y11 0.250 0.029 8.720 0.000
Y12 0.175 0.016 10.941 0.000
Y13 0.133 0.013 10.087 0.000
Y14 0.101 0.016 6.501 0.000
Y21 0.279 0.032 8.672 0.000
Y22 0.193 0.018 10.963 0.000
Y23 0.145 0.015 9.800 0.000
Y24 0.123 0.017 7.030 0.000
I1 0.722 0.040 18.248 0.000
S1 0.837 0.040 20.735 0.000
I2 0.872 0.032 27.092 0.000
S2 0.922 0.031 29.719 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.
Two-Tailed
Estimate S.E. Est./S.E. P-Value
I1 |
Y11 0.866 0.017 52.214 0.000
Y12 0.786 0.018 44.834 0.000
Y13 0.678 0.021 32.208 0.000
Y14 0.582 0.022 26.330 0.000
S1 |
Y11 0.000 0.000 999.000 999.000
Y12 0.258 0.015 16.988 0.000
Y13 0.445 0.023 19.092 0.000
Y14 0.574 0.028 20.138 0.000
I2 |
Y21 0.849 0.019 44.843 0.000
Y22 0.755 0.020 38.596 0.000
Y23 0.629 0.022 28.533 0.000
Y24 0.521 0.022 23.816 0.000
S2 |
Y21 0.000 0.000 999.000 999.000
Y22 0.304 0.017 18.153 0.000
Y23 0.507 0.024 20.848 0.000
Y24 0.631 0.028 22.225 0.000
S1 ON
I2 -0.049 0.069 -0.709 0.478
S2 ON
I1 0.155 0.078 1.992 0.046
S1 ON
PAR_CON 0.287 0.037 7.832 0.000
S2 ON
PAR_CON 0.113 0.046 2.462 0.014
I1 ON
PAR_CON 0.362 0.025 14.722 0.000
I2 ON
PAR_CON 0.246 0.030 8.172 0.000
PHY_CON ON
S1 0.024 0.085 0.287 0.774
S2 0.102 0.082 1.255 0.209
I1 0.144 0.081 1.773 0.076
I2 0.119 0.074 1.593 0.111
PHY_CON ON
PAR_CON 0.079 0.049 1.602 0.109
I1 WITH
S1 0.178 0.079 2.242 0.025
I2 WITH
S2 0.239 0.077 3.085 0.002
I1 0.114 0.060 1.907 0.056
S2 WITH
S1 -0.114 0.096 -1.198 0.231
Y11 WITH
Y21 0.342 0.076 4.519 0.000
Y12 WITH
Y22 0.353 0.054 6.506 0.000
Y13 WITH
Y23 0.357 0.061 5.879 0.000
Y14 WITH
Y24 0.258 0.097 2.668 0.008
Intercepts
Y11 0.000 0.000 999.000 999.000
Y12 0.000 0.000 999.000 999.000
Y13 0.000 0.000 999.000 999.000
Y14 0.000 0.000 999.000 999.000
Y21 0.000 0.000 999.000 999.000
Y22 0.000 0.000 999.000 999.000
Y23 0.000 0.000 999.000 999.000
Y24 0.000 0.000 999.000 999.000
I1 0.556 0.116 4.777 0.000
S1 3.964 0.288 13.749 0.000
I2 0.672 0.126 5.318 0.000
S2 1.703 0.178 9.584 0.000
Thresholds
PHY_CON$1 1.616 0.400 4.044 0.000
Residual Variances
Y11 0.250 0.029 8.720 0.000
Y12 0.175 0.016 10.941 0.000
Y13 0.133 0.013 10.087 0.000
Y14 0.101 0.016 6.501 0.000
Y21 0.279 0.032 8.672 0.000
Y22 0.193 0.018 10.963 0.000
Y23 0.145 0.015 9.800 0.000
Y24 0.123 0.017 7.030 0.000
I1 0.722 0.040 18.248 0.000
S1 0.837 0.040 20.735 0.000
I2 0.872 0.032 27.092 0.000
S2 0.922 0.031 29.719 0.000
R-SQUARE
Observed Two-Tailed
Variable Estimate S.E. Est./S.E. P-Value
PHY_CON 0.122 0.038 3.229 0.001 Model fit ≠ sig. of R2
Y11 0.750 0.029 26.107 0.000
Y12 0.825 0.016 51.561 0.000
Y13 0.867 0.013 65.498 0.000
Y14 0.899 0.016 57.776 0.000
Y21 0.721 0.032 22.421 0.000
Y22 0.807 0.018 45.911 0.000
Y23 0.855 0.015 57.907 0.000
Y24 0.877 0.017 50.175 0.000
Latent Two-Tailed
Variable Estimate S.E. Est./S.E. P-Value
I1 0.278 0.040 7.031 0.000
S1 0.163 0.040 4.025 0.000
I2 0.128 0.032 3.988 0.000
S2 0.078 0.031 2.506 0.012
Beginning Time: 15:32:58
Ending Time: 15:42:43
Elapsed Time: 00:09:45
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.