7/9/10  

Introduction Growth Curves Using Mplus 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

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 1

Alan C. Acock, July, 2010 Introduction to MPlus

1  

7/9/10  

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”

2

Alan C. Acock, July, 2010

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

based on a graphic interface

  It has the slowest introduction of new capabilities

3

Alan C. Acock, July, 2010

2  

7/9/10  

A brief history--Mplus  Version 6  Version 5  Version 4  Version 3  Version 2  Version 1

4

April 2010 November 2007 February 2006 March 2004 February 2001 November 1998

Alan C. Acock, July, 2010

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

5

Alan C. Acock, July, 2010

3  

7/9/10  

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) 6

Alan C. Acock, July, 2010

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

7

Alan C. Acock, July, 2010

4  

7/9/10  

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 8

Alan C. Acock, July, 2010

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 9

Alan C. Acock, July, 2010

5  

7/9/10  

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 10

Alan C. Acock, July, 2010

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) 11

Alan C. Acock, July, 2010

6  

7/9/10  

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

12

Alan C. Acock, July, 2010

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

13

Alan C. Acock, July, 2010

7  

7/9/10  

Growth trajectory of 10 randomly selected kids

14

Alan C. Acock, July, 2010

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 15

Alan C. Acock, July, 2010

8  

7/9/10  

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

16

Alan C. Acock, July, 2010

A figure for how MPlus conceptualizes a growth curve

17

Alan C. Acock, July, 2010

9  

7/9/10  

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) 18

Alan C. Acock, July, 2010

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

19

Alan C. Acock, July, 2010

10  

7/9/10  

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 20

Alan C. Acock, July, 2010

What’s in this figure? The Slope Growth Factor   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

21

Alan C. Acock, July, 2010

11  

7/9/10  

What’s in this figure? The Slope Growth Factor   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

22

Alan C. Acock, July, 2010

What’s in this figure? The Slope Growth Factor   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

23

Alan C. Acock, July, 2010

12  

7/9/10  

What’s in this figure? The Slope Growth Factor   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

24

Alan C. Acock, July, 2010

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 25

Alan C. Acock, July, 2010

13  

7/9/10  

What’s in this figure? Residual Variance & Random Effects   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 26

Alan C. Acock, July, 2010

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

27

Alan C. Acock, July, 2010

14  

7/9/10  

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

28

Alan C. Acock, July, 2010

The Mplus program for a linear growth curve

29

Alan C. Acock, July, 2010

15  

7/9/10  

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: 

! 30

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

Alan C. Acock, July, 2010

What’s new compared to day 1? 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. 31

Alan C. Acock, July, 2010

16  

7/9/10  

What’s new compared to day 1?   A few words make a picture because of defaults   Mplus assumes there is a random effect and makes RI & RS random    

   

32

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 !

Alan C. Acock, July, 2010

What’s new compared to day 1?   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 33

Alan C. Acock, July, 2010

17  

7/9/10  

Selected Growth Curve Output—Missing

34

Alan C. Acock, July, 2010

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 35

Alan C. Acock, July, 2010

18  

7/9/10  

Selected Growth Curve Output— Correlations

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

36

Alan C. Acock, July, 2010

Selected Growth Curve Output—Fit

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

37

Alan C. Acock, July, 2010

19  

7/9/10  

Selected Output—Estimates

Growth curve is: BMI’ = 21.035 + 0.701×Year! 38

Alan C. Acock, July, 2010

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

39

Alan C. Acock, July, 2010

20  

7/9/10  

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 40

Alan C. Acock, July, 2010

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

41

Alan C. Acock, July, 2010

21  

7/9/10  

Graphic results   Most who started high show an increase (one dropped)

42

Alan C. Acock, July, 2010

Graphic results   Growth curve versus actual means   GraphsView graphsSample & estimated means

43

Alan C. Acock, July, 2010

22  

7/9/10  

Quadratic Growth Curve

44

Alan C. Acock, July, 2010

Mplus program for quadratic curve

45

Alan C. Acock, July, 2010

23  

7/9/10  

Tests for Model Fit—quadratic vs. linear

46

Alan C. Acock, July, 2010

Quadratic—Selected Output

47

Alan C. Acock, July, 2010

24  

7/9/10  

Quadratic—Plot

There is a good fit. The sample means and the estimated means are very close 48

Alan C. Acock, July, 2010

Quadratic—Plot

There is still significance variance (random effect) for both the intercept and the slope among individual adolescents that needs to be explained 49

Alan C. Acock, July, 2010

25  

7/9/10  

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   3 variances   3 covariances

My1, My2, My3 Var(Y1),Var(Y2),Var(Y3) Cov(Y1,Y2), Cov(Y1,Y3), Cov(Y2,Y3)

  9 known statistics

50

Alan C. Acock, July, 2010

Counting parameters--Linear

51

Alan C. Acock, July, 2010

26  

7/9/10  

3 waves linear growth curve needs 8   3 waves of data: H0: model of simple growth curve   A   B   C   D   E   F,G,H   8

variance of intercept variance of slope covariance of intercept & slope Mean of intercept growth factor Mean of slope growth factor 3 error variances Total number of parameters you are estimating

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

52

Alan C. Acock, July, 2010

4 waves gives huge increase in information   3 waves of data: H1: model is called unrestricted   4 means   4 variances   6 covariances

My1, My2, My3, My4 Var(Y1),Var(Y2),Var(Y3),Var(Y4) 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

53

Alan C. Acock, July, 2010

27  

7/9/10  

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

54

Alan C. Acock, July, 2010

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 55

Alan C. Acock, July, 2010

28  

7/9/10  

Missing values—Auxiliary Variables   These variables are not part of your model but still serve two  

 

 

  56

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

Alan C. Acock, July, 2010

Full Information Maximum Likelihood with Auxiliary variables

57

Alan C. Acock, July, 2010

29  

7/9/10  

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

58

Alan C. Acock, July, 2010

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

59

Alan C. Acock, July, 2010

30  

7/9/10  

Multiple Cohort Extension   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

60

Alan C. Acock, July, 2010

Multiple Cohort Extension   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

61

Alan C. Acock, July, 2010

31  

7/9/10  

Multiple group comparisons   Suppose we are interested in differences between Males and                 62

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

Alan C. Acock, July, 2010

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

63

Alan C. Acock, July, 2010

32  

7/9/10  

Two group solution, same form only

64

Alan C. Acock, July, 2010

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 65

Alan C. Acock, July, 2010

33  

7/9/10  

Multiple Groups—Form only

66

Alan C. Acock, July, 2010

Multiple Groups—Form only

67

Alan C. Acock, July, 2010

34  

7/9/10  

Multiple groups: Different Intercept & Slope?

68

Alan C. Acock, July, 2010

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 69

Alan C. Acock, July, 2010

35  

7/9/10  

Multiple groups—equal I and s

70

Alan C. Acock, July, 2010

Multiple groups—equal I and s

71

Alan C. Acock, July, 2010

36  

7/9/10  

Multiple groups—equal I and s

72

Alan C. Acock, July, 2010

Multiple groups—equal I and s   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 73

Alan C. Acock, July, 2010

37  

7/9/10  

Alternative to Multiple Group Analysis

74

Alan C. Acock, July, 2010

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 75

Alan C. Acock, July, 2010

38  

7/9/10  

Alternative to Multiple Group Analysis

76

Alan C. Acock, July, 2010

Alternative to Multiple Group Analysis

77

Alan C. Acock, July, 2010

39  

7/9/10  

Alternative to Multiple Group Analysis

78

Alan C. Acock, July, 2010

Alternative to Multiple Group Analysis

79

Alan C. Acock, July, 2010

40  

7/9/10  

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)

80

Alan C. Acock, July, 2010

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 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)

81

Alan C. Acock, July, 2010

41  

7/9/10  

Alternative to Multiple Group Analysis— Graphic representation   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 82

Alan C. Acock, July, 2010

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. 83

Alan C. Acock, July, 2010

42  

7/9/10  

Alternative to Multiple Group Analysis— Graphic representation using Stata

84

Alan C. Acock, July, 2010

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 85

Alan C. Acock, July, 2010

43  

7/9/10  

Adding Time Invariant Covariates

86

Alan C. Acock, July, 2010

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 87

Alan C. Acock, July, 2010

44  

7/9/10  

Program for time invariant covariates

88

Alan C. Acock, July, 2010

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

89

Alan C. Acock, July, 2010

45  

7/9/10  

Program for time invariant covariates

90

Alan C. Acock, July, 2010

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

91

Alan C. Acock, July, 2010

46  

7/9/10  

Program for time invariant covariates— Mediation & Moderation

Note, the mediation modeled using the Model Indirect: subcommand 92

Alan C. Acock, July, 2010

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

93

Alan C. Acock, July, 2010

47  

7/9/10  

Time Varying Covariates   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

94

Alan C. Acock, July, 2010

Time Varying Covariates Performance in Treatment Group

0

1

2

3

4

5

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. 95

Alan C. Acock, July, 2010

48  

7/9/10  

Time Varying Covariates   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

96

Alan C. Acock, July, 2010

Time Varying Covariates   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

97

Alan C. Acock, July, 2010

49  

7/9/10  

Time Varying Covariates

98

Alan C. Acock, July, 2010

50