introduction growth curves using mplus - onidpeople.oregonstate.edu/~acock/growth/2010 mplus...

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Alan C. Acock University Distinguished Professor of Family Studies &

Knudson Chair for Family Research & Policy Oregon State University

College of Health and Human Sciences Summer Workshop Series

July 2010

Introduction Growth Curves Using Mplus

A brief history  LISREL (Joreskog and Sorbom) was being

developed in the late 1960s and released commercially in the early 1970s  Originally relied on entering 8 matrices specifying

all the parameters that were being estimated or fixed at a certain value

 Today has a graphic interface that generates the commands from a path diagram

 Extremely capable alternative to Mplus

Alan C. Acock, July, 2010 Introduction to MPlus 1

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A brief history  EQS (Bentler) was developed much later and

replaced the matrices with writing out a separate equation for each relationship

  It now has a nice “Diagrammer”

Alan C. Acock, July, 2010 2

A brief history  AMOS (now an SPSS product) was developed

based on a graphic interface

  It has the slowest introduction of new capabilities


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A brief history--Mplus  Version 6 April 2010  Version 5 November 2007  Version 4 February 2006  Version 3 March 2004  Version 2 February 2001  Version 1 November 1998


A brief history--MPlus  Very rapid development  Late development allowed a non graphic

interface to be highly efficient  Destroys the idea that a picture is worth

1000 words  Develop statistical applications, not drawing


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A brief history--MPlus  MPlus was developed during the 1990s

when growth curves were being introduced to social and behavioral sciences

 Need separate drawing program, but this is best for publication quality  Omni Graffle (Mac) for most figures here  Office Visio or Open Office Draw (PC)


A Brief History--Mplus


  Bengt Muthén is the statistician   Linda Muthén is the language/interface/business   Several people have contributed programming   Economy of Scale idea is reversed   Microsoft has 40,000 programmers so it takes a long time to

make a useful change   Mplus has a couple programmers so it rapidly adds features   Many new features are added between versions

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Buying Mplus


 Greatly reduced Student prices  There are three modules (they apparently learned

this module idea from SPSS). You probably want all three

 There is an annual maintenance and this lets you  Get “free” support  Get “free” updates  I started with Mplus 3.0 and have only paid for

the annual maintenance fee ($175) since then

Resources for Learning  Barbara Byrne. (2010). Structural Equation

Modeling with MPlus: Basic Concepts, Applications, and Programming. (Was to be available July 1

 www.statmodel.com  Large, 752 page, User’s Manual as pdf file  Short courses on video  There are 8 of these, each is one day long  Download handouts to follow videos


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Resources for Learning  www.ats.ucla.edu/stat/seminars/  UCLA has several online examples and videos  We will utilize files from the Mplus manual for many

of our examples. These typically involve simulated data. Sometimes we well assign hypothetical variable names to make these somewhat realistic

 Kline, Rex. (2010). Principles & practices of structural equation modeling (3rd ed.). N.Y. Guilford


Introduction to the Concept


  Growth Curves are 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

  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 issue becomes explaining individual differences in trajectories (why some people go up, down, or stay the same)

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Introduction to the Concept


  We will introduce growth curves for multiple groups such as comparing women and men

  Time invariant and time variant covariates will be introduced   Mediation will be introduced   Develop your models incrementally

  Start with a simple grow curve  Add complexities one at a time

Sample data


  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 examined 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

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Growth trajectory of 10 randomly selected kids


Observations about these 10 kids


  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 adolescents are getting a higher 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

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Observations about these 10 kids


  Clearly the kids have different intercepts. We need to consider if there is significant random variation in the intercept

  It is less clear that the kids have different slopes. We can test to see if there is a significant random variation in the slope

  The intercept and the slope might be correlated. Those who start with a low BMI might not increase as much as those who start with a high BMI

A figure for how MPlus conceptualizes a growth curve


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What’s in this figure?


  This figure is much simpler than it first appears. The key variables are the two latent variables labeled the Intercept growth factor and the Slope growth factor

  The Intercept Growth Factor   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.

 When covariates are added, especially when a zero value on covariates is rare and covariates are not centered (household income)

What’s in this figure? The Intercept Growth Factor


  A straight line may over or underestimate any one mean including the initial mean

  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 or subtracted

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What’s in this figure? The Slope Growth Factor


  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

  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  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



  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

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  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 and can fit better

  Compared to a quadratic for a curve, this approach doesn’t require a monotonic function



  You might use 0, 1.5, 2.0, 4.0, etc. to match the time differences for the measurements

  Many national surveys take 6 months to administer so the baseline might be in January or it might be in June

  A person might be measured at month 6, month 13, & month 30 (last at first wave, first at second wave, last at third wave)

  Another person might be interviewed at month 1, 13, & 25

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  Might use month rather than year for coding the wave   First person would be coded 5, 12, & 29   Second person would be coded 0, 12, 24

  Mplus has a feature that allows each participant to have a different interval which is important when the time between waves varies

  TSCORE!  This Mplus feature is underutilized

What’s in this figure? Residual Variance & Random Effects


  The individual variation around the Intercept and Slope are represented in Figure 1 by the RI and Rs.   RI is the variance in the intercept around the mean slope   RS is the variance in the slope around the mean slope   These are what statisticians call the random effects

  The value of the Intercept Growth Factor & the Slope Growth Factor is really what statisticians call a fixed effect—what would happen if every body was the same

  We expect substantial variance in both of these as some individuals have a higher or lower starting BMI (intercept) and some individuals will increase (or decrease) their BMI at a different rate (slope) than the average growth rate

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  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. Clearly there is a random effect for the intercept growth factor

  Some children have a BMI that increases and others do not. Perhaps there is a random effect for the slope growth factor

  The variances, RI and R2 are critical if we are going to explore more complex models with covariates (e.g., an intervention, gender, psychological problems, race, household income, physical activity, fidelity of implementation)

  These covariates might explain why some individuals have a steeper or less steep initial level and growth rate than the average



  What’s this about a random intercept model and a random coefficients (slope) model

  A random intercept model would fix the variance of RS at 0.0 and let the variance of RI be free

  A random coefficients model would let both RS & RI be free

  We can estimate the model with both free   Estimate the model with just RI free   Estimate the model with neither free

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What’s in a figure? Random Errors


  The ei are random error terms   We are usually fitting some functional form of a growth rate

such as linear of quadratic   Some years may move above or below the growth trajectory   A lot of error is eliminated when we have a random

coefficient model since some people may be systematically going up/down quicker than the fixed effect

The Mplus program for a linear growth curve


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What’s new compared to day 1?


  Usevariables are: subcommand to only include the BMI variables since we are doing a growth curve for these variables

  We drop the Analysis: section if we had 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

  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;!



i s | bmi97@0 bmi98@1 bmi99@2!! !bmi00@3 bmi01@4 bmi02@5 bmi03@6;!

  The i represents the intercept. We could enter intercept, start, initial, level, whatever

  The s represents the slope. We could enter slope, growth, increase, change, whatever

  A few words make a picture because of defaults   Linda assume the constant of 1.0 for each year to represent the

intercept   The slope is defined by fixing the paths from the slope to bmi97 at

0.0, the path from the slope at bmi98 at 1.0, etc.

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  A few words make a picture because of defaults   Mplus assumes there is a random effect and makes RI & RS random

effects   Mplus assumes the intercept and slope covary, i.e., cov(RI,RS) is

free to be estimated   The intercepts for BMI97 – BMI 2003 are all assumed to

be zero. There is an implicit command [[email protected]] where the [] refer to the intercepts

  Mplus assumes the means for the intercept and slope are free (fixed effects)

  The errors, e97 – e03, are assumed free and uncorrelated. To correlate e97 with e98 we would add a command e97 with e98 !



  A big change from day 1 is to ask for a plot of the growth trajectory. This is initiated by Plot:

  The type of plot we want for a growth trajectory is Plot3, Type is plot3;!

 Mplus must know what to plot so we enter the variable used for our waves to define the series

  The asterisk at the end of the series (*) is used to tell Mplus we are using the same values for the waves that we listed in the Model command, 0 – 5

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Selected Growth Curve Output—Missing


Selected Growth Curve Output—Means


You can see a steady increase in the BMI between 1997 and 2003 The rate of increase seems to be leveling off

Increases by 1.3 units between 1997 and 1998, by about 0.8 units between 1998 and 1999, but this gradually decreases to under 0.6 units between 2002 and 2003. A quadratic might work with a decreasing rate of increase over time Biggest problem for any definite functional form is the big increase between 1997 and 1998

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Selected Growth Curve Output—Correlations


Correlations get weaker as waves get farther apart. BMI97 & BMI03 a bit high. BMI01 & BMI02 a bit high.

Selected Growth Curve Output—Fit


Correlations get weaker as waves get farther apart. BMI97 & BMI03 a bit high. BMI01 & BMI02 a bit high.

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Selected Output—Estimates


Growth curve is: BMI’ = 21.035 + 0.701×Year!

Selected Output—Res. Var & Errors


RI = 15.051, the SD = 3.88. This is significant. About 95% of the kids have an intercept between 15.051 ± 2×3.88 or 7.29 and 22.81. This is fitting the intercept and not the actual mean at 1997 because we are using a linear growth curve. The initial mean was much higher. RS = 0.255, the SD = 0.50. About 95% have an increase in their BMI of 0.70 ± 2×0.50 or between 0.20 BMI units per year and 1.20 BMI units. Both seem like subsantial random effects

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Selected Output—Modification Indices


Rarely one to free the intercept loadings of 1.1 even though that has a huge M.I. The M.I. = 37.34 for BMI03 with BMI97 suggests we are not fitting at both ends. A quadratic might be needed. We do NOT automatically free parameters that have a large M.I. Could improve fit with covariates

Graphic results


  Click on Graphs Observed individual values   Set seed, random order, number of curves (20)

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Graphic results


  Most who started high show an increase (one dropped)

Graphic results


  Growth curve versus actual means   GraphsView graphsSample & estimated means

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Quadratic Growth Curve


Mplus program for quadratic curve


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Tests for Model Fit—quadratic vs. linear


Quadratic—Selected Output


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Quadratic—Plot


There is a good fit. The sample means and the estimated means are very close

Quadratic—Plot


There is still significance variance (random effect) for both the intercept and the slope among individual adolescents that needs to be explained

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They say I need 3 waves, why have more: Information available with 3 waves


  3 waves of data: H1: model is called unrestricted   3 means My1, My2, My3   3 variances Var(Y1), Var(Y2), Var(Y3)   3 covariances Cov(Y1,Y2), Cov(Y1,Y3), Cov(Y2,Y3)   9 known statistics

Counting parameters--Linear


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3 waves linear growth curve needs 8


  3 waves of data: H0: model of simple growth curve  A variance of intercept   B variance of slope  C covariance of intercept & slope  D Mean of intercept growth factor   E Mean of slope growth factor   F,G,H 3 error variances   8 Total number of parameters you are estimating

 With 3 waves we have 9 – 8 = 1 degree of freedom

4 waves gives huge increase in information


  3 waves of data: H1: model is called unrestricted   4 means My1, My2, My3, My4   4 variances Var(Y1), Var(Y2), Var(Y3), Var(Y4)   6 covariances Cov(Y1,Y2), Cov(Y1,Y3), Cov(Y1,Y4)

Cov(Y2,Y3), Cov(Y2,Y4), Cov(Y3, Y4)   14 known statistics

 With 4 waves we are still estimating 8 parameters so we have 14 – 8 = 6 degree of freedom

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Alternative to Quadratic


  Instead of using 0,1,2,3,4,5,6 for linear component waves and 0,1,4,9,16,25,36 for quadratic component waves

  Could use 0, 1, *, *, *, *!  Could use 0, *, *, *, *, 1   If using months, e.g., 12, 24, 36 and want quadratic,

need to rescale, 36 square is 1,296. May divide by 10, e.g., 1.2, 2.4, 3.6 to keep scale of quadratic reasonable

  These are not nested under quadratic so can’t test, can us BIC

Missing values


  Missing completely at random (MCAR)   Missing at random (MAR)   John is measured at waves 1, 2, 5   Sue (a part of refresh sample) is measured at waves 2, 3, 4, 5   Antoine is measured a just wave 1   Juan is measured at waves 1,2,3,4   Use all available information to estimate mean/variance for

each wave (Antoine gives information about wave 1)   Use all available information to estimate covariances (John &

Sue do this for some, but not all covariances

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Missing values—Auxiliary Variables


  These variables are not part of your model but still serve two purposes

  First they help predict the score on a missing value, e.g., education may not be relevant to your model, but it would help predict income

  Second, they help meet the MAR assumption by explaining who does and who does not have a missing value. Gender is an example as men are more likely to skip items.

  Observations have missing values on a random basis (MAR)—AFTER you control for all variables in your model and all auxiliary variables

  Mplus has a simple way to add auxiliary variables

Full Information Maximum Likelihood with Auxiliary variables


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Full Information Maximum Likelihood with Multiple Imputation


  No values are imputed for missing values under the full information maximum likelihood approach, just means, variances, and covariances are analyzed

  Although we will not show it here, Mplus can use multiple imputation   It can use multiple imputation dataset from other packages

(Stata, SAS, etc.)   It can do the multiple imputation itself starting in version 6

  There is much debate about the best solution to missing values but it usually doesn’t matter and they are asymptotically equivalent

Multiple Cohort Extension


  Many national surveys will have multiple cohorts. For example, the National Longitudinal Study of Youth, 1997 began in 1997 with kids who were 12 to 18.

  In year two these kids were 13 to 19, in year three they were 14 to 21, and in year four they were 15 to 22

  With just four waves of data we can represent a growth curve for people from 12 to 21

  Perhaps a study was discontinued at this point, but 3-years later they did it again   The follow-up survey would have the people 18 to 25   Using five waves this way we would have data from 12 to 25—

transition from pre-adolescence to adulthood

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  Mplus uses the command, Data cohort to rearrange your data

  We will not illustrate how this is done here but will show this in Stata on day 3



  Instead of organizing the data by age, it is organized by birth cohort

  There is massive missing values. In their example there are 90 possible observations, but 50 of these are missing

  This is MCAR unless there is a cohort effect such as people born in 1963 have a different set of historical circumstances than people born in 1964 or 1965

  Muthén did this with a few waves of data to describe growth curves for drinking behavior from adolescence to mid 30s

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Multiple group comparisons


  Suppose we are interested in differences between Males and Females in their growth on BMI

  We could treat these as subgroups and simultaneously estimate the model in both groups—Many comparisons:

  Do they have the same intercept growth factor   Same slope growth factor   Same variance for intercept growth factor   Same variance for slope growth factor   Same covariance of intercept and slope residuals   Same residual errors   Same covariance of residual errors

Multiple group comparisons


  At the least we can test whether they have the same intercept growth factor and same slope growth factor (fixed effects comparison)

  The problem with this approach to multiple group comparison is that it quickly becomes overwhelming

  We will check the intercept and slope invariance here   First, we estimate the model without any constraints (as

we’ve done before, however   We will estimate it simultaneously for two groups

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Two group solution, same form only


Two group solution, same form only


  We add our categorical variable to the Usevariables command. This is binary, but could have more categories

  We tell Mplus how the data is grouped and add labels for each set of outcome

grouping is male (0 = female 1 = male);!

  This simultaneously estimates the model separately for males and females and reports two sets of results with no invariance constraints

  You do NOT need to sort data to do this

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Multiple Groups—Form only


Multiple Groups—Form only


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Multiple groups: Different Intercept & Slope?


Multiple groups: Different Intercept & Slope? There are two model statements


  Fist is an overall and describes the group with the lowest score on the grouping variable (females were 0; males were 1).

  Second models the next higher score on the grouping variable, males.   Don’t list things that stay the same   Only list things that are different or   That you constrain to be equal

  Mplus puts the same number whenever it wants to force coefficients to be equal. When Mplus puts [] around a latent variable, it is referring to the mean. When Mplus puts [] around an observed variable it is referring to the intercept

  The intercept and slope are forced to be the same in the two groups

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Multiple groups—equal I and s




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  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

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Alternative to Multiple Group Analysis




  Enter the grouping variable as a predictor   Males have a higher intercept and a steeper slope

  Positive slope from male to intercept growth factor means males have a higher intercept

  Positive slope from male to slope growth factor means males have a steeper slope

  Limitations  This imposes restrictions that everything else is equal for

females and males   Same error terms, same residual variances, same covariance of

residual variances   If those differences are important need the two-group approach

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Alternative to Multiple Group Analysis—Graphic representation


  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)



  We see that the intercept is 20.385 and the slope is .625. How is gender related to this?

  For boys the equation is: Est. BMI = 20.911 + .656(Time) + .242(Male) + .086(Male)(Time)

= 20.911 + .656(Time) + .242(1) + .086(1)(Time) = (20.911 + .242) + (.625 + .086)(Time)

= 21.153 + .711(Time)

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  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

Alternative to Multiple Group Analysis—Graphic representation using Stata


Note, When we treat the grouping variable as a predictor, as in this example, we only test whether the intercept and slope are different for the two groups. This is an interaction between gender and growth trajectory. We do not allow the other parameters to be different and do not test whether this is reasonable or not.

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Alternative to Multiple Group Analysis—Graphic representation using Stata


Growth Curves with Time Invariant Covariates


  A covariate is time invariant if it is constant for the duration of the growth curve (gender, race, condition, etc.)

  Our alternative approach using gender illustrates this   It has been called conditional latent trajectory modeling

(Curran & Hussong, 2003)—the intercept growth factor and slope growth factor are conditioned on other variables

  You could call this moderated growth trajectories with gender serving as the moderator

  You could call this an interaction between gender and the growth trajectory

  Here is a slightly more complex example

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Adding Time Invariant Covariates


Adding Time Invariant Covariates


  Here we add two covariates  White is coded 1 for white and 0 for nonwhites   Emotional problems is a latent variable that was measured in

1997 (not any other year)   There is a youth self-report of emotional problems   There is a parent report of the youth’s emotional problems

  Both covariates influence the intercept, linear slope and quadratic slope

  The time invariant covariates moderate (interact with) the linear slope growth factor and the quadratic slope growth factor

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Program for time invariant covariates




  The Useobservations command restricts the sample to males who are not Asian nor Other on race/ethnicity

  Emotional problems is a latent variable

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Mediation & Moderation with Time Invariant Covariates


  We are predicting the growth in drinking problems among adolescents

  The amount of parental drinking may contribute to this   Parental monitoring may contribute   Peer influence may contribute   These variables MODERATE (interact with) the trajectory. The rate of

change varies depending on these three variables

  Parental drinking may be mediated by monitoring and peers   If parents monitor their adolescent and there are positive peers

then the effect of parental drinking may be greatly diminished

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Program for time invariant covariates—Mediation & Moderation


Note, the mediation modeled using the Model Indirect: subcommand

Time Varying Covariates


  Time varying covariates can be very important   We might have a randomized trial with a treatment and

control group   The trial might involve a four year intervention   The treatment group should have a more positive trajectory

than the control group   The fidelity of the program may vary from year to year

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  If the treatment is having a positive effect (positive trajectory), there might be annual variations around this trajectory, depending on the fidelity with which the program was implemented each year

  Years of high fidelity might push the outcome variable above the overall trajectory;

  Years of low fidelity might do the opposite   These annual perturbations may be explainable by

corresponding variations in program fidelity



0 1 2 3 4 5

Performance in Treatment Group

Baseline (0) was normal level of implementation, Year 2 (1) was a high level of implementation, but year 3 (2) and year 4 (3) there was a staff problem that resulted in a lower level of implementation. This was corrected in year 4 and year 5 as okay. These variations in fidelity of implementation pushed the score each year above or below the overall trajectory.

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  The error terms are correlated for adjacent years   The W represents a vector of time invariant covariates that

moderate the intercept and the slope   Intervention vs. control  Gender

  The Ai variables represent the time varying covariates   Fidelity of implementation   Peer group influence

  The time varying covariates directly influence the Yi scores



  Might study decline in drinking behavior from age 22 to age 27

  Slope is negative   Time varying covariates might include:   Drinking Behavior of Peers   Work status (student to employed full time)   Number of children   Percent of friends who have children

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introduction growth curves using mplus - onidpeople.oregonstate.edu/~acock/growth/2010 mplus...

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