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 patrick sturgis posted on Thursday, January 11, 2007 - 11:50 am
I am estimating an LGC model in which the repeated measure (3 time points) is ordinal (6 categories). Using delta parameterization I am specifying the model
Bollen and Curran (2006 p240-242) describe, where the first two thresholds of the categorical repeated measure are constratined to 0/1 respectively and the remaining thresholds are estimated but constrained to equality over time.
This allows an estimate of the mean of the intercept and slope growth parameters for the underlying continuous variable (the default in MPlus constrains slope mean to zero). However, the plot function will permit only plots of the probability of membership of each category at each time point.

Is it possible to obtain a plot for the estiamted growth parameters (i.e. a single line)? thanks,

 Linda K. Muthen posted on Thursday, January 11, 2007 - 12:00 pm
The Mplus growth model default for categorical outcomes is to hold the thresholds equal across time, fix the intercept growth factor mean to zero, and estimate the slope growth factor mean. You can change this default parameterization to have the intercept growth factor mean free and fix a threshold, for example, the first threshold, to zero at each time point. You will obtain the same model fit. This is just a reparameterization of the model. In both cases you can produce the plot you wanted, using the estimated probit or logit mean values underlying the categorical responses. A plot of these is not automated in Mplus but is easily done by plotting the function i + x_t*s, where for linear probit/logit growth i is the intercept growth factor mean, x_t is the time score at time point t, and s is the slope growth factor mean.

The Bollen-Curran reference is on a somewhat different topic, where, in line with Joreskog, the latent response variable underlying the categorical item is given a mean and a variance at each time point by fixing 2 of the thresholds at each time point. Unlike the two parameterizations described above, the approach cannot be used for binary items.
 patrick sturgis posted on Thursday, January 11, 2007 - 12:39 pm
Dear Linda

thanks very much for your swift response. so, I can fit the plot in excel or similar using the simple method you describe. one additional question - how should the scale of the y axis be interpreted? that is, what is the scale of the underlying continuous variable? If it is the logit scale, can we interpret the units as being changes in the log odds of being in the next highest category or some such? Many thanks

 Bengt O. Muthen posted on Friday, January 12, 2007 - 9:47 am
Well, this calls for more words - which is one reason we plot the more down-to-earth probabilities instead.

With logit link the scale is in logits, which with ordered polytomous response is the log odds of being in the highest (or 2 highest, etc) category vs the lower one(s). So for example, with 3 categories (u=0,1,2) and linear logit growth

logit_t = i + s*x_t

where i and s are growth factors and x_t are the time scores, this time-specific logit translates into time- and category-specific logits by the usual proportional-odds model as

logit(P(u=2)) = -tau2 + logit_t
logit(P(u=2 or 1) = -tau1 +logit_t

where as usual logit(y) = log(y/(1-y). I don't think I have seen this type of presentation of the logit growth model sdo it may not be very commonplace - anyone?
 patrick sturgis posted on Monday, February 12, 2007 - 7:27 am
I am estimating a growth model over three time points with an ordered categorical variable as the outcome. I am using the knownclasses command. following is the model commands:
VARIABLE: NAMES ARE gender zscf0 zscf10 zsc26 zsc30 zscf16;
USEVARIABLES ARE zscf10-zsc30;


ANALYSIS: type=MIXTURE missing ;

i s | zscf10@0 zsc26@1.6 zsc30@2;
i s ;
i with s;
i s ;
i with s;

SERIES=zscf10 (10) zsc26 (26) zsc30 (30);

OUTPUT: standardized sampstat tech1;

when I run this model, I obtain the growth parameters in logit scale but no predicted probabilities (as I do when just using ML rather than mixture). How can I get Mplus to display these (I know they are there somewhere because I can plot them using the PLOT command). thanks,

 Linda K. Muthen posted on Monday, February 12, 2007 - 9:12 am
I'm having a hard time figuring the difference in your two runs. Can you send the two outputs, with and without probability values, and your license number to so I can see what you mean?
 patrick sturgis posted on Monday, June 18, 2007 - 2:30 am
Dear Bengt & Linda

I am having trouble with a reviewer who wants a precise account of the interpretation of the growth parameters I present in my paper where I have a linear growth model with 3 time points and the outcome is ordinal. My approach in the paper is to present the growth parameters without too much interpretation and to get the substantive implications across by moving pretty quickly to the predicted probability plots. I have formulated my statement of the interpretation of the growth parameters as follows:

"Where the ordinal regression approach is used, Beta_i in equation (1) (my addition - this is just the standard linear growth model notation) is interpreted as the expected change in the log of the odds that individual i is in category j, or higher, of the ordinal outcome y, for a unit change in t. A more intuitive expression of these model parameters can be obtained in the form of predicted probabilities of membership in each category of y, at each point in time".

I would be grateful if you could let me know if I have mis-represented your approach in this statement! best wishes,

 Linda K. Muthen posted on Monday, June 18, 2007 - 7:45 am
Your statement is correct if you used maximum likelihood estimation and the default logit link. Note that this is not a special Mplus approach. It is logistic regression applied to a random effect growth model for which there are several good references among them the book by Fitzmaurice et al.
 patrick sturgis posted on Thursday, June 21, 2007 - 8:43 am
Thanks Linda.
 Karin Landolt posted on Friday, July 06, 2007 - 9:17 am
Dear Linda and Bengt

I am estimating a latent growth model with categorical data. Some models require THETA parameterization.
Using different parameterizations, the slope parameter and the thresholds changed considerably (about three times higher with THETA). We thought that this was due to our data, but also using the data file of Example 6.4 (Users Guide), the slope was -0.407 with DELTA and 0.543 with THETA.
Could you please explain me why this difference occurs and how a high difference can be interpreted?

 Linda K. Muthen posted on Friday, July 06, 2007 - 10:31 am
The metric is not the same. This will affect that parameter estimates and standard errors but the ratio of the two should be close to the same. This is discussed in Web Note 4.
 John Woo posted on Tuesday, August 23, 2011 - 9:24 am
Hi Dr. Muthens,

When doing LCM with categorical outcome variable, it is mentioned that y* (latent continuous variable) is used to link y (categorical variable).

is y* = logit y?

That is, if y* = a + b*time, then is

logit y = a + b*time ?

Thank you..
 Bengt O. Muthen posted on Wednesday, August 24, 2011 - 12:25 pm
In logit/probit regression

y* = a + b*x + e,

where e is the residual. The logit is a + b*x, so no residual. The expected y* is the logit because E(e)=0.
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