Latent Growth Curve Analysis

Daniel Boduszek

Department of Behavioural and Social Sciences

University of Huddersfield, UK

d.boduszek@hud.ac.uk

Introduction to Latent Growth Analysis

Amos procedure

Interpretation of results

Outline

Latent Growth Curve Analysis – a way to explain change Measurement has to be taken at 3 or more times With two data points, a straight line will fit perfectly, every

time, because two points determine a line There is nothing to test (no degrees of freedom - no data that

could disprove the straight line)

2 statistical approaches for studying change (LGA) Structural equation modelling of latent growth curves (our

focus!!!) Multilevel modelling (hierarchical modelling)

Introduction

Change

Linear – going up or down in a straight direction

Non-linear - going up rapidly and then leveling off

Introduction

Two parameters determine a straight line

One of them is Intercept

It is the value at the start of the process.

Sometimes call it a constant, because it is what we start

with and the standard from which change is measured

Growth curve—the intercept

The intercept for the range of data is 10 (mean). The 13-year-olds have an average delinquency score of 10

The intercept is the value of the outcome when the growth curve begins

Growth curve—the intercept

Slope – parameter which tells us much the curve grows over time

Between13 to 17 years old the mean delinquency score jumps from 10 to 70

The slope is 15 (average)

Growth curve—the slope

Latent Growth Model

Delinquency Measured at Age 13, 14, 15, and 16

Latent Growth Model

We assume our measures are not perfectly reliable

Variety of symbols are used for measurement error including e, ε and δ.

Latent Growth Model

There might be more than one slope for complex nonlinear models.

Latent Growth Model

The four lines from the intercept to the four y’s are all fixed at a constant value of 1

Since all four values are fixed at the same value, this is the constant level of delinquency, if there were no growth

Latent Growth Model

The values for the lines are fixed at 0, 1, 2, and 3 respectively

0 means zero growth – the initial level

Fixing the values from the slope is how you identify model growth

Latent Growth Model

Using values of 1, 2, and 3 results in a linear growth curve, since there is a 1-year difference between each measure

The y1 is delinquency at the age of 13, the y2 is one year later at 14 (so its value is 1), the y3 is one more year latter at 15 (so its value is 1 + 1 = 2), and y4 is one more year later at 16 (so its value is 1 + 1 + 1 = 3)

Suppose we had no data for when the adolescents were 14 we would use 0, 2, and 3

Latent Growth Model

If the model should have a curve, we would have two slopes.

One would be fixed at 0, 1, 2, and 3 as in Figure 13. The Second slope would have the four paths fixed at 0, 1, 4, and 9, the square of the corresponding the linear paths

Latent Growth Model

For both the slope and intercept there is a mean and a variance.

Mean Intercept: Where does the

average person start? Slope: What is the average

rate of change?

Variance Intercept: How much do

individuals differ in where they start?

Slope: How much do individuals differ in their rates of change: “Different slopes for different folks.”

Latent Growth Model

Correlation of variances

If positive covariance It means that individuals

who have higher intercepts also have higher slopes and individuals with lower intercepts have lower slopes

E.g., an adolescent who starts out with a high initial level of delinquency will become increasingly more delinquent than others

Latent Growth Model

Latent Growth Model with predictors

Empirical Example with Two Growth Curves

Amos procedure for LGM

Data used in this example:

268 offenders from high security prison

Criminal identity measured at 3 time points (2009, 2010, 2011)

To launch the Amos on your computer go to Start

All Programs

IBM SPSS Statistics

IBM SPSS Amos 20

Amos Graphics

Amos is a part of IBM SPSS software thus it can read the SPSS file without converting.

To load the data go to File and Data Files.

The Data Files dialog box then opens.

Click on File Name and navigate to the location where the data file is stored.

By default, Amos looks for an SPSS file.

Choose Plugins ---> Growth Curve Model.

Enter the number of measures when you are prompted for the number of time points.

In this example, you would enter 3 for the number of time points because you have data for criminal identity at 3 distinct points in time

Choose View ---> Analysis Properties ---> Estimation tab.

Check the Estimate Means and Intercepts check box.

Choose Output tab ---> Standardized regression weights

Click on View and select Variables in Data Set

Click on (and hold) your first observed variable (CrIdent) and move it to the observed box (X1)

Then CrIdent2 to X2 and CrIdent3 to X3

Right-click on each of the 3 observed variable boxes one at a time and select Object Properties.

Click on the Parameters tab to fix the intercept parameter; fix each observed variable's intercept value to 0

Specify regression weights for each parameter. All of the Intercept paths get a regression weight of 1 (automatically done ). The slope gets a weight relative to the time point for each measure (in this example 0, 1, 2).

Move the pointer onto the arrow that leads from the slope to the Criminal Identity variable so it is highlighted

Right click and select Object Properties from the menu

Specify a value of 0 for the regression weight

Keep the Object Properties window open while you move to the next arrows and apply the appropriate weights to the slope

You can create a Figure Caption that displays important information about your model. This will help you identify important changes as you test different models.

Select the Diagram option from the Main Menu

Select the Figure Caption option and left click above your path diagram

In the Caption window type the following commands:

chi square = \cmin

df = \df

p = \p

It is now time to fit the model.

Choose the Analysis tab from the Main Menu

Select the Calculate Estimates option

Result (Default model) Minimum was achieved Chi-square = 15.263 Degrees of freedom = 3 Probability level = .002

Before interpreting the results on the model diagram, you should first verify that the model fits the data well on an overall basis.

The chi-square test of overall model fit was statistically significant (chi-square = 15.263, df= 3, p = .002). Model does not fit the data Warning: Chi-square is

sensitive to large sample (more than 200)

Check CFI and TLI in Baseline

Comparison (values above .95 indicate good model fit) Good fit!

After you determine that the model fit the data acceptably, you may interpret the parameter estimates.

The mean intercept value of 18.81 indicates that the average starting amount of criminal social identity was 18.81 units.

The mean slope value was 7.34 (the average growth in criminal social identity).

Criminal social identity is expected to increase by 7.34 each studied time period, beginning with an average score of 18.81

The means and variances were statistically significant when tested with the null hypothesis

Substantively, the finding that the variances of the intercepts is statistically significant suggests that there is non-trivial variation in the amount of criminal social identity at the initial time point.

Significant variation in slope values, indicating that criminal social identity may be quite varied over time.

Interestingly, the amount of criminal social identity at the initial time of measurement appeared to be unrelated to changes over time, as illustrated by the non-significant correlation of .08 between the slopes and intercepts.