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 Jim Shaw posted on Saturday, February 10, 2007 - 3:25 pm
Dr. Muthen:

I am using Mplus 3.12 to estimate the parameters of a path model that includes continuous and binary observed dependent variables. I am using the WLSMV estimator and the bias-corrected bootstrap to derive standard errors and confidence limits for parameter estimates. I have two questions relating to the bootstrap confidence limits.

First, the Mplus User's Guide (version 3, p. 36) suggests that confidence limits may be derived for standardized estimates. However, the output includes confidence limits for the unstandardized coefficients only. Is there any way to obtain bias-corrected limits for the standardized coefficients? I believe that one could simply divide the unstandardized limits by the quotient of the unstandardized and standardized coefficients.

Second, I would like to know if there is any way to derive 90% CI limits for the coefficients. Mplus outputs the 95% and 99% limits by default. I could calculate the normal 90% limits by hand. However, I do not believe this is feasible for the percentile or bias-corrected percentile limits.

Thank you for your help.
 Linda K. Muthen posted on Sunday, February 11, 2007 - 2:00 pm
Confidence intervals are not given for standardized estimates. I'm not sure what part of page 36 suggests that. If you tell me, I will clarify it in the next version.

You can obtain standard errors for standardized coefficients from MODEL INDIRECT if the parameters of interest are regression coefficients and then use them for regular confidence intervals but not for bias-corrected confidence intervals. I know of no way to adjust bias-corrected confidence intervals from 95% or 99% to 90%.
 Jim Shaw posted on Thursday, February 15, 2007 - 12:31 pm
Dr. Muthen:

Thanks for the prompt response. In paragraph 3 on page 36 of the text I have, it states the following: "The CINTERVAL option is used to request confidence intervals for parameter estimates of the model, indirect effects, and standardized indirect effects." This statement is repeated in paragraph 2 on page 44.
 Jim Shaw posted on Thursday, February 15, 2007 - 12:38 pm
Dr. Muthen:

Is there any way to derive the standard errors for indirect effects when using the MLR estimator with integration? I am fitting a multinomial model to an outcome with 3 categories using the following code:

type = missing H1 ;
estimator = mlr ;
algorithm = integration ;
integration = montecarlo(1000) ;

dv#1 on A B C D E F G H;
dv#2 on A B C D E F G H;
C on A B E F G H ;
D on A B E F G H ;
A on E F G H ;
B on E F G H ;
A with B;
C with D;

While I can calculate the indirect effects by hand, I would like to be able to test their signficance. The 'bootstrap' and 'model indirect' options are not available with integration. For indirect effects involving 2 parameters, x and y, I believe I can calculate the variance as follows:

Var(x*y) = y^2*Var(x) + x^2*Var(y) + Var(x)*Var(y) + 2*x*y*Cov(xy) + Cov(xy)^2

However, I am unsure how to extrapolate this to an indirect effect involving 3 parameters, i.e., x*y*z.

Also, is it appropriate to calculate indirect effects when some of the estimates are linear regression coefficients and others are logit (or probit) coefficients?
 Linda K. Muthen posted on Thursday, February 15, 2007 - 5:41 pm
I think we planned on doing this but then didn't. We give the standard errors of the standardized coefficients so you can compute the confidence interval by hand.

You would have to compute standard errors by hand using the Delta method.

Yes, it is correct to use the product of both linear and probit regression coefficients.
 Jon Elhai posted on Thursday, February 15, 2007 - 6:49 pm
Dear Linda,
I was actually just asking a statistician this morning about one of these issues raised by Jim Shaw today on this listserv. I too asked if linear and logit coefficients could be multiplied for indirect effects, and was told that this isn't wise. Do you know of any sources on this matter? And do you have any suggestions on whether coefficients such as linear and poisson/negative binomial coefficients should be multiplied?
 Linda K. Muthen posted on Friday, February 16, 2007 - 8:12 am
I know of a source that is not yet published. I am waiting for permission to cite it. I don't think this would apply to Poisson or negative binomial coefficients.
 Linda K. Muthen posted on Friday, February 16, 2007 - 9:02 am
The following paper, which is under revision, is what supports the Mplus approach for the indirect effects that are a combination of linear and probit/logistic regression coefficients:

MacKinnon, D.P., Lockwood, C.M., Brown, C.H., and Hoffman, J.M. The intermediate endpoint effect in logistic and probit regression.
 Jim Shaw posted on Friday, February 16, 2007 - 5:50 pm
Dr. Muthen:

Using the delta method, I have been unable to reproduce the standard errors for indirect effects generated by Mplus with 'model indirect.' However, I have been able to replicate the results of studies in which the delta method was applied to Mplus output.

Yfantis and Flatman (YF; J. Chemometrics 1991; 5: 309) report the formula for the variance of an indirect effect involving three independent parameters (a, b, and c):

Var(a*b*c) = Var(a)*Var(b)*Var(c) + Var(a)*b^2*c^2 + Var(b)*a^2*c^2 + Var(c)*a^2*b^2 + Var(a)*Var(b)*c^2 + Var(a)*Var(c)*b^2 + Var(b)*Var(c)*a^2

This formula yields results that are closer to the Mplus estimates than those derived using the delta method. See below:

F1 = a*b*c
Mplus: 0.0159
Delta: 0.0345
YF: 0.0155

F2 = a*b*d
Mplus: 0.0053
Delta: 0.0219
YF: 0.0070

The difference between methods (YF vs. Mplus) does not appear to be related to round-off error.

The delta method is only asymptotically correct. Does a degree-of-freedom correction have to be applied to the parameter variances and covariances (output using the 'savedata' command) prior to performing the matrix calculations?
 Bengt O. Muthen posted on Friday, February 16, 2007 - 7:40 pm
When Mplus uses the Delta method, no assumption is made of uncorrelated parameter estimates. So for example with the indirect effect a*b, Mplus uses not only the variances for the a and b estimates but also their covariance.
 MAH posted on Wednesday, October 15, 2008 - 11:58 am
I am running a simple twin model of a continuous outcome. Estimator is ML. Are the confidence intervals also likelihood based?

 Bengt O. Muthen posted on Wednesday, October 15, 2008 - 5:51 pm
Yes, in the sense that the standard errors are ML SEs.
 Jack Noone posted on Sunday, April 10, 2011 - 10:43 pm
Hello. I am doing a test for indirect effects in a simple path analysis with categorical dependent variables. When I ask for bootstraped CIs I only get them for the raw STD estimate. Is it possible to get CIs centred around the STDYX estimate? Or is there a way of calculating these by hand?

I'm still using Mplus 4.21

 Linda K. Muthen posted on Monday, April 11, 2011 - 9:22 am
The is no option to increase what you obtain with the CINTERVAL option.
 EFried posted on Thursday, September 05, 2013 - 5:58 am
I have a question regarding bootstrapped CI in a SEM using ML estimator.

In m1 without bootstrapping:
y8 on x3:
estimate = 0.022
std etimate = 0.056
p = 0.001

In m2 with 10,000 bootstrap replications:
y8 on x3:
estimate = 0.022
std etimate = 0.056
p = 0.064
bootstrapped CI: 0.001 to 0.049 (lower and upper 2.5%).

Why is the p nonsignificant if the CI does not include 0?

Thank you
 Linda K. Muthen posted on Thursday, September 05, 2013 - 6:52 am
Please send the outputs and your license number to
 Noémie Bigras posted on Wednesday, August 24, 2016 - 8:26 am
Dear Dr Muthen,

I see in previous messages that standardized confidence intervals can't be provided when using bootstrap. This has been said in 2007 so I was wondering if this information is still accurate with the current version of Mplus. A reviewer was asking to provide standardized coefficients of indirect effects, but we reported unstandardized to make more sense with the unstandardized CIs. Is that correct?

Thank you very much,
 Bengt O. Muthen posted on Wednesday, August 24, 2016 - 5:33 pm
CIs for standardized effects are available in the current version 7.4.
 Bengt O. Muthen posted on Wednesday, August 24, 2016 - 5:40 pm
You see an example of this from our new book in Table 2.5 Part 2 at
 Noémie Bigras posted on Wednesday, September 14, 2016 - 4:28 pm
Then, should we now systematically report standardized CIs and coefficient for indirect effects now that they are available? It's because depending on which one I choose, results change. For example, standardized CIs range from -.09 to -.001 while the other, non standardized CIs range from -.05 to .000.

Thank you very much,
 Bengt O. Muthen posted on Wednesday, September 14, 2016 - 4:57 pm
If you prefer to report standardized effects you should use the CIs for those. Typically, CI covering zero or not agree. If they don't, try bootstrapping because then one or the other effect might have a non-normal distribution.

I would not regard -0.001 to be much different from 0.000.
 Eric Thibodeau posted on Friday, September 21, 2018 - 1:01 pm
I have two questions, one related to an earlier post in this thread.

So I'm running some longitudinal twin models. I'm asking MPlus to provide some additional parameters in model constraint to give me the A,C,E proportion of variance estimates. When I request confidence intervals, the CIs for those ACE estimates range from negative to positive and of course I want them to only be in the possible range of positive. Does bootstrapping those CIs fix that in an appropriate way?

Second question, is are MPlus CIs likelihood based as mentioned in this article?

many thanks!
 Bengt O. Muthen posted on Friday, September 21, 2018 - 5:44 pm
Q1: Yes, it can.

Q2: No. Bootstrapping is needed with ML. But you can also use Bayes to get non-symmetric CIs.
 Eric Thibodeau posted on Monday, September 24, 2018 - 8:35 am
Is the bootstrapping supposed to eliminate the presence of any negative ACE CIs?

So I'm running a bivariate (twin 1/twin2), multiple group (zygosity) linear growth twin model with individually-varying times of observation. The age definition variables require TYPE = RANDOM. Therefore, the bootstrapping requires ALGORITHM=INTEGRATION. However, ALGORITHM=INTEGRATION is not possible with multiple group models unless run as a mixture model with known classes.

I ran that known class growth mixture model and my CIs appear to still be symmetric. I have less out of bounds CIs for the ACE, proportion of variance, but there are still some that exist.

 Bengt O. Muthen posted on Monday, September 24, 2018 - 4:50 pm
If you like, you can send your output file to Support along with your license number. Point out which CIs you are focusing on.
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