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 Anonymous posted on Tuesday, May 04, 2004 - 4:14 pm
I'm trying to specify a growth model of my contiuous DV measured yearly from age 13 to age 22. I want to specify a piecewise model that captures linear growth from age 13-15, 16-19, and 20-22. Can you help me to determine how I should specify this model in mplus?
 Linda K. Muthen posted on Tuesday, May 04, 2004 - 5:53 pm
Example 6.11 shows how to set up a piecewise model for two parts. This can be generalized to three.
 Anonymous posted on Wednesday, May 05, 2004 - 2:01 pm
Thank you. Would you please take a look at this to make sure that I'm generalizing to 3 pieces correctly?

i BY y13-y22@1;
s1 BY y13@0 y14@1 y15@2 y16-y22@2;
s2 BY y13-y15@0 y16@0 Y17@1 y18@2 Y19@3 y20-y22@3;
s3 BY y13-y19@0 y20@0 y21@1 y22@2;

I'm confused about whether the y16-y22 for s1 should be @2 or @0. Likewise, should y20-y22 for s2 be @0 or @3? Thanks for your help.
 Linda K. Muthen posted on Wednesday, May 05, 2004 - 5:45 pm
The end point of one piece should be the beginning point for the next piece if they are sequential. In the user's guide example, the timescores are:


so the first piece ends at 2 for the third repeated measure. The second piece starts at the end of the first piece.

Your time scores are:

0 1 2 2 2 2 2 2 2 2
0 0 0 0 1 2 3 3 3 3
0 0 0 0 0 0 0 0 1 2

I'm not sure which measures are part of which piece given your time scores.
 Matthew McBee posted on Wednesday, August 30, 2006 - 10:11 am
I am creating a similar model and have a couple of questions.

I have seven timepoints. I am interested in understanding the effect of an intervention which would have occured between timepoints 6 and 7. The outcome is the percentage of students in a school that enter college.

Overall college enrollment rates have been slowly increasing in my state each year. I want to seperate this effect from the possible effect of the treatment. Should I specify my time scores as:

0 1 2 3 4 5 6
0 0 0 0 0 0 1


0 1 2 3 4 5 5
0 0 0 0 0 0 1 ?

Finally, will the model be identified given that the second slope will apply to only a single timepoint?
 Linda K. Muthen posted on Thursday, August 31, 2006 - 8:08 am
You are correct that the second piece would not be identified. I would use a single model and add a direct effect of the time 7 outcome on the dummy variable of intervention. So

y7 ON intervention;
 Geoffrey Hammond posted on Thursday, November 30, 2006 - 9:29 am

I have been recently trying to specify a piecewise paramterization. The observations are in months/10 and are at initiation, 3, 6, 12 ,18, 24, 30, and 36 months. I have been trying to specify two parts, with four observations each, one over the first year and the second over the remaining two years. Is the following correct?

i1 s1 Q1| pos1@0 pos2@0.3 pos3@0.6 pos4@1.2 pos5@1.8 pos6@1.8
pos7@1.8 pos8@1.8;

s2 Q2| pos1@0 pos2@0 pos3@0
pos4@0 pos5@0.6 pos6@1.2
pos7@1.8 pos8@2.4;

Thanks in advance
 Linda K. Muthen posted on Friday, December 01, 2006 - 10:01 am
Note that you need to mention the intercept growth factor in both parts of the growth model. Just use the same name. The way you have your second model specified, Mplus would interpret it as a linear model because it has two growth factors. In Chapter 16, there is a table that shows a piecewise growth model. I would specify the timescores for the piecewise growth model as follows.

Piece 1: 0 .5 1 2 2 2 2 2

Piece 2: 0 0 0 0 0 1 2 3
 Geoffrey Hammond posted on Sunday, December 03, 2006 - 5:38 pm
Thanks for the reply Linda.

Hovever for my second piece, shouldn't the order be:

Piece 1: 0 0.5 1 2 2 2 2 2

Piece 2: 0 0 0 0 1 2 3 4

In your proposed paramterization, I am not modelling the movement between 12 and 18 months?

Sorry, still a bit confused.
 Linda K. Muthen posted on Monday, December 04, 2006 - 8:28 am
Yes, the second piece should be as you state when there is one intercept growth factor.
 Dustin Pardini posted on Thursday, April 12, 2007 - 11:35 am
Hello Muthens,

I have some intervention data collected at pre, post 6, 12, 24, 36 month follow-ups. I want to compare two randomized intervention conditions on various externalizing outcomes. The problem is there is a large drop in externalizing problems from pre to post, then only modest linear change in the outcome across the follow-up assessments. A quadratic does not fit the data well. It is more like a piecewise model with the first slope consisting of change from pre-post, and the second consisting of linear change occuring after the intervention has been completed.

Because there are only two timepoints for the first piece the model it is not identified. I have considered leaving out the baseline assessment since random assignment makes the two treatment groups of interest equal prior to treatment. However, I was wondering if you had any suggestions for dealing with this problem.
 Bengt O. Muthen posted on Friday, April 13, 2007 - 9:11 am
One approach that I have used is to work with piecewise growth as you say, where the first piece has only a random intercept (centered at the baseline) but a fixed slope. This is then identified from 2 time points. The second piece can have its own intercept and slope which are both random. So part of the intervention effect is on the fixed slope of the first piece and part is on the random intercept and slope of the second piece.

I would not leave out the baseline assessment from the growth modeling.
 Dustin Pardini posted on Friday, April 13, 2007 - 11:01 am

I am having problems getting it to run. Is this how it would be specified for six equally spaced time points:

i1 s1 | ext1@0 ext2@1 ext3@1 ext4@1 ext5@1 ext6@1;

i2 s2 | ext1@0 ext2@0 ext3@1 ext4@2 ext5@4 ext6@6;

[ext1-ext6@0 i1 i2 s1 s2];

s1 with i1@0 s2@0 i2@0;

So would the interpretation for the first slope (which is fixed) be the the average pre-post intervention effect for all participants, and the second intercept would represent individual deviations from this average effect.
 Bengt O. Muthen posted on Friday, April 13, 2007 - 3:44 pm
No, when you have separate intercept factors, you work with 2 separate (but correlated) growth processes, e.g.:

i1 s1 | ext1@0 ext2@1;

i2 s2 | ext3@-3 ext4@-2 ext5@-1 ext6@0;

where you center i2 at the last time point. Nothing else needs to be said, except of course

s1 i2 s2 on tx;

Check Tech1 to make sure you get i1 correlated with i2, s2.
 Dustin Pardini posted on Sunday, April 15, 2007 - 7:08 am

Thanks. Just to clarify the regression should be

i1 i2 s2 on tx; Correct? Because S1 has a variance fixed to 0.

 Bengt O. Muthen posted on Sunday, April 15, 2007 - 10:31 am
You don't want to regress i1 on tx because i1 refers to the pre-intervention time point.

You can regress s1 on tx even though s1 has variance zero. The regression implies that the s1 mean is different for controls and tx subjects, so that describes the first drop that you mentioned.
 Katherine A. Johnson posted on Tuesday, June 19, 2007 - 2:54 pm
I am modeling change over time with 6 equally spaced time points each 2 years apart. My means are:

Would it be inappropriate to have a piecewise growth model with 4 pieces?

For example:
0 2 2 2 2 2
0 0 2 2 2 2
0 0 0 2 4 4
0 0 0 0 0 2

Thank you!
 Linda K. Muthen posted on Tuesday, June 19, 2007 - 5:47 pm
This would not be identified. We recommend at least three repeated measures for each piece.
 Gwen Marchand posted on Sunday, April 06, 2008 - 8:05 pm

I am using a piecewise growth model to describe late elementary school and the transition to middle school. The linear model for each piece estimates fine, but when I add a quadratic function, the model will not converge.

To ask for the quadratic function, I am adding a "q" as I would in a traditional growth model. Do I need to do something different? Here is an example of the syntax...

i s1 q1| g3fasxex@0 g3sasxex@1 4fasxex@2 g4sasxex@3 g5fasxex@4 g5sasxex@5 g6fasxex@5 g6sasxex@5 g7fasxex@5 g7sasxex@5;

i s2 q2|g3fasxex@0 g3sasxex@0 g4fasxex@0 g4sasxex@0 g5fasxex@0 g5sasxex@0 g6fasxex@1 g6sasxex@2 g7fasxex@3 g7sasxex@4;

If it helps, this data has planned missingness, so I have had to set the covariance coverage to 0 to account for no students having data in both the 3rd and the 7th grade.
 Gwen Marchand posted on Sunday, April 06, 2008 - 10:11 pm
A second question: if both slopes of a piecewise model appear similar, is there a way to test (through some kind of constraint) whether a single slope, rather than two slopes, fits the model just as well as two processes? Would you consider these models nested?

Thank you for any insight.

 Linda K. Muthen posted on Monday, April 07, 2008 - 9:10 am
If you have no data for certain timepoints, you reflect this through the time scores, for example, if you measured in years 1, 2, and 4, the time scores would be 0, 1, and 3. You don't do this through missing data.

Usually a piecewise model is used to break a growth curve that is not linear into linear pieces so I don't understand why you are adding a quadratic growth factor.

You can test if the slope of the two pieces are equal by chi-square difference testing of the two nested models or by using MODEL TEST.
 Bill Dudley posted on Thursday, July 03, 2008 - 9:26 am
I have longitudinal data on recovery from cancer treatment. There are six time points (0, 6, 12, 18, 24 and 36). The time = 0 is prior to treatment and then starting with 6 months we have the recovery phase. The means show a jump between T1 and t2 so it makes sense to fit a two piece model. The first linear piece has only two time points so I have looked at the notes below re fixing s1 variance and the s1, i covariance. I assume that in the second piece that the time 2 serves as the intercept for that curve. I also want to include covariates - age and treatment type (a binary variable).
Question1 - So I wonder if I should model an i2 as well as i.
Question2 - If s1 and i are fixed then it makes no sense to look for predictors of s1…. Correct?
Question 3 - I also can fit a nice quadratic curve to the data without the piecewise approach. I assume that although this fits, it is not valid given the nature of the data collection and the mean and that I should stick with the piecewise… Correct?
 Bengt O. Muthen posted on Thursday, July 03, 2008 - 4:45 pm
Q1 Yes

Q2 You can. Think of a gender dummy variable where s1 is different for males and females - that can be identified. But the residual variance for s1 then needs to still be zero.

Q3 If the second piece does not need a quadratic and if having only one piece that is quadratic fits the data well I would do one piece quadratic. My experience is that 2 pieces are needed - and they might have different covariate influence.
 Bill Dudley posted on Tuesday, July 08, 2008 - 6:58 am
Thanks for the reply to my post on July 3rd. Just a follow up if I might.

With regard to Q3 - are you saying that I should use a one piece quadratic model rather than a two piece or are you saying that in cases such as mine two piece models are preferred?
 Bengt O. Muthen posted on Tuesday, July 08, 2008 - 8:52 am
Use the 1-piece quadratic if it fits well. If it doesn't, use 2 pieces.
 Tim Stump posted on Friday, October 30, 2009 - 8:12 am
I'm specifying a linear growth model with 5 unequally-spaced time points--baseline and then followup at 2, 4, 8, and 12 wks from baseline.

I'd like to specify a piecewise model since most of the change in individuals occurs between baseline and 2 wks and then they don't exhibit as much change after that. Would the timescores in the following statements be correct:

i s1 | var0@0 var2@2 var4@2 var8@2 var12@2;
i s2 | var0@0 var2@0 var4@2 var8@6 var12@10;

I'm most unsure about the second slope factor.

 Bengt O. Muthen posted on Friday, October 30, 2009 - 8:41 am
This looks right. Note that the s1 variance cannot be identified from only 2 time points and will have to be fixed at zero.

You can also consider a more flexible model with 2 different intercepts:

i1 s1 | var@0 var2@2;
i2 s2 | var4@0 var8@4 var12@8;
 Tim Stump posted on Monday, November 02, 2009 - 6:53 am
Thanks for your reply to my post on 30oct2009. I have a follow-up question. By specifying s1@0, I'm setting the variance of the slope factor s1 to be zero. Can I still make comparisons on the mean of the slope factor s1? Say I have a binary covariate named group coded 0 or 1. If I add: i s1 s2 on group; to the model, does it still make sense to test for differences in the mean s1 across group even though the variance is contstrained to zero?
 Bengt O. Muthen posted on Monday, November 02, 2009 - 7:55 am
Yes, because it simply says that the s1 mean is different for the two groups.
 Jennifer Augustine posted on Monday, March 08, 2010 - 10:33 am

I would like to continue with this thread by asking some additional questions:

I would like to run a piecegrowth model. The reason I'd like to do this is because I have four time points, but a linear model does not fit the data. Specifically, there is a big change between time 1 and time 2. The change between time 2-4 is less dramatic. Substantively, time 1-2 represents the transition into elementary school and var is achievement. The data points are measured in two-year increments.

My model looks like this:

i1 s1 | var1@-1 var2@0 ;
s1@0 ;
i1 s2 | var2@0 var3@2 var4*4 ;

I have found I have to free the time 4 factor loading to get the model to run. The estimated time score for this model is 2.067.

Here are my questions:

First, should this var4 time score alarm me? If I run a gc from times 2-4 and free the time 4 loading, the model runs and has good model fit. The estimated time 4 factor loading for this model is 2.65. If I estimate a quadratic using times 1-4, I get an error message.

Secondly, I’d like to allow the intercepts to be different (specifying i1 and i2), but this model does not run. Could you suggest why this might be and how I can fix it?

Thank you very much for your help!
 Linda K. Muthen posted on Tuesday, March 09, 2010 - 10:43 am
For a piecewise model, you need to include all outcomes in both pieces. Your piecewise model is set up incorrectly. See Example 6.11.

If you want different intercepts, you could try:

i1 s1 |var1 var2;
i2 s2 |var3 var4;
s1@0 s2@0;

With only two measures, you cannot identify the variances of the slope growth factors. You can't include the same outcome in more than one growth model.
 Jennifer Augustine posted on Tuesday, March 09, 2010 - 12:38 pm

Thank you for your help. So, if I wanted to estimate a piecewise model with the same intercept for each piece (as above), my model should actually look like:

i1 s1 | var1@-1 var2@0 var3@0 var4@0;
i1 s2 | var1@0 var2@0 var3@2 var4*4 ;

Also, you mention that with only two measures, I cannot identify the variances of the slope growth factors. This is why I need to set s1@0, which I understand, but as I think about it, is beginning to seem problematic because I'd like to look at the covariance of s1 with s2. Moreover, I am assuming that everyone has the same s1, which is not the correct.

Are there other ways of dealing with this issue that you might suggest? Perhaps I could calculate a deviation score between t1 and t2, and use that to predict i1 and s2? As I mentioned in my previous post, I was considering a piecewise approach because there is considerable growth between t1 and t2, and more modest growth between t2-t4.
 Linda K. Muthen posted on Wednesday, March 10, 2010 - 9:46 am
If s1 is zero, the covariance between s1 and s2 is also zero.

See the Topic 3 course handout starting at Slide 106 where six ways to model non-linear growth are described. Maybe something there will help you.
 Dustin Pardini posted on Tuesday, April 13, 2010 - 9:07 am
Hello Muthens,

I would like to do a GMM identifying latent classes of individuals who have differential responses to treatment (e.g., responders, non-responders). I have intervention data collected at pre, post 6, 12, 24, 36 month follow-ups. When I initially modeled the data using a traditional growth curve approach, a quadratic growth did not fit the data well. Instead, I found that a two process model fit well with the first process consisting of change from pre-post, and the second process of linear change occuring after the intervention has been completed (see below).

Now that I am moving toward a GMM, I am not sure how to include both processes to define different latent classes across all assessment points. Is it possible to use both processes to define a latent trajectory classes.

i1 s1 | ext1@0 ext2@1;

i2 s2 | ext3@-3 ext4@-2 ext5@-1 ext6@0;
 Linda K. Muthen posted on Wednesday, April 14, 2010 - 7:56 am
A sequential process growth model can be used in GMM.
 Gregory Kirkner posted on Sunday, April 18, 2010 - 7:04 pm
I assume that "Piecewise Growth Modeling" is not an option when a dependent variable is measured at only three time points, correct?
 Linda K. Muthen posted on Sunday, April 18, 2010 - 8:26 pm
That is true.
 Jon Elhai posted on Thursday, May 19, 2011 - 6:04 pm
I'm not sure if piecewise growth modeling would be appropriate... But I have 4 timepoints in a growth model, each of which are equidistant (measured every 2 weeks). But for the scores over the four weeks, the slope looks unlike the slopes I usually see in the Mplus forum. In this example, the observed scores would take the shape of something like: 100 (time1), 50 (time2), 80 (time3), 65 (time4). Would a piecewise model be appropriate, or some transformation of the time scores?
 Linda K. Muthen posted on Friday, May 20, 2011 - 9:22 am
Four time points is not enough to identify a piecewise model. Perhaps the outcome is not suitable to a developmental model given the up and down nature of the data.
 Anne Chan  posted on Sunday, April 15, 2012 - 4:59 am

I specified two models. The first model is just a basic linear one-slope model:

i s| Y75@0 Y76@1 Y77@2 Y79@4 Y82@7;

The second one is a piecewise (two-pieces) model, constraining the slope 1 and slope 2 to be the same:

i s1| Y75@0 Y76@1 Y77@2 Y79@2 Y82@2;
i s2| Y75@0 Y76@0 Y77@0 Y79@2 Y82@5;

[s1] (p1);
[s2] (p2);

p1 = p2;

May I ask if they are basically the SAME model? Thanks a lot!
 Linda K. Muthen posted on Sunday, April 15, 2012 - 5:05 pm
The two models are the same with respect to the means but not the variances and covariances.
 Marina Milyavskaya posted on Monday, May 21, 2012 - 12:09 pm
I am trying to model data on students' well-being across the school year, and have 6 time points. What I am finding is that well-being decreases from September-December, then jumps back up in January, and decreases again throughout the second semester (January-April). My means are as follows:


I tried a piecewise model, with 3 pieces, one for each of the semesters, and then one for the change during the break (December-January), which I constrained to 0 as suggested above
i s1 | t1wb@0 t2wb@1 t3wb@2 t4wb@2 t5wb@2 t6wb@2;
i s2 | t1wb@0 t2wb@0 t3wb@0 t4wb@1 t5wb@1 t6wb@1;
i s3 | t1wb@0 t2wb@0 t3wb@0 t4wb@0 t5wb@1 t6wb@2;

Is this correct? The fit is not great...

thank you!
 Bengt O. Muthen posted on Monday, May 21, 2012 - 8:37 pm
How about having 2 intercepts so you easily capture the Xmas break jump:

i s1 | t1wb@0 t2wb@1 t3wb@2;
i2 s2 | t4wb@0 t5wb@1 t6wb@2;
 JMC posted on Wednesday, July 03, 2013 - 7:08 am
I am running into a similar problem as a post above. I have three time points (time 1-time 2 with traditional instruction and time 2-time 3 with an intervention). I wanted to compare the direction and rate of change for significance, but I don't think there are enough time points to do this. Is there any work around for this? I saw above a suggestion for setting up an intervention dummy variable, but I don't know how to set this up. Thank you.
 Bengt O. Muthen posted on Wednesday, July 03, 2013 - 3:05 pm
If the intervention happens between the second and third time points, you can have a dummy variable pointing to Y3.
 Kara Thompson posted on Thursday, September 05, 2013 - 9:15 am
Dear Drs. Muthen,

I am running multivariate piecewise growth models. I have been using a single intercept with 2 slopes, but someone has recommended I use 2 intercepts instead. Are there any advantages/disadvantages to using 2 intercepts rather than 1 intercept in a piecewise model?

 Linda K. Muthen posted on Thursday, September 05, 2013 - 10:33 am
Using two intercepts is not really a piecewise model. With two intercepts, you have two sequential processes. Your theory should guide your decision.
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