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I'm trying to compute explained variance and standardized estimates. My structural model includes interactions of continuous latent variables predicting a continuous latent variable outcome. I'm using the formulas in Bengt's 2012 article "0.1. LATENT VARIABLE INTERACTIONS." In order to compute the variance of the endogenous latent variables, I need the variances of the latent exogenous variables, as well as the covariances of the latent exogenous variables that comprise the interaction term (which is created with the xwith command). In a previous post, I think that Linda mentioned that TECH3 produces output of the parameter estimates for the variances and covariances. Is this accurate? I am unable to figure out how to interpret the TECH3 output (using the TECH1 output as a guide) to identify the variances and covariances of the latent variables. How do I determine the variances and covariances of the exogenous latent variables when using the xwith command to create latent variable interactions? Thank you for your help with this! Luke 


You need TECH4. 


TECH4 is not available when running models with latent variable interactions that were created with the xwith command. *** WARNING in OUTPUT command TECH4 option is not available for TYPE=RANDOM. Request for TECH4 is ignored. 1 WARNING(S) FOUND IN THE INPUT INSTRUCTIONS Since TECH4 is available if I omit the latent variable interaction terms, can I use the variances and covariances of the exogenous latent variables from this TECH4 output to compute the variances of the endogenous latent variables in the SEM that includes the latent variable interactions? If not, how do I compute the variances of the endogenous latent variables in the SEM that includes the latent variable interactions? I have Bengt's 2012 article "0.1. LATENT VARIABLE INTERACTIONS," which has been referred to in other posts; however, I'm looking in particular to identify the values (e.g., variances and covariances) to include in the equations that Bengt provides in his article. Thanks, Luke 


In the example on page 3 corresponding to Figure 2, parameter values are given for a hypothetical example. The exogeneous factor is eta2, while eta1 and eta3 are endogenous factors. Following are the list of each parameter (their value) and under which heading you would find them in the output: variance of eta2 (=1): Variances residual variance of eta1 (=1): Residual variances residual variance of eta3 (=1): Residual variances slope for eta1 regressed on eta2 (=1); ON statements slope for eta3 regressed on eta1 (=0.5): ON statements slope for eta3 regressed on eta2 (=0.7): ON statements slope for eta3 regressed on eta1*eta2 (=0.4): ON statements From the above list of parameter values the variances and covariances are derived by usual rules as indicated on page 3  this needs to be done by the analyst. Hope this answers your question. 


Hi Bengt, I understand your example, and I have worked through the examples in your article using the values for the parameters that you supplied in your article. I am now trying to use the equations from your article to calculate explained variance and standardized estimates for my SEM. If TECH4 produces variances and covariances that I need to input into my equations to calculate the variance of the latent dependent variable, and since TECH4 is not available in Mplus output for SEMs that include latent variable interactions that are created with the xwith command, then where do I get the values for variances and covariances that I need to compute endogenous latent variable variances? A potential solution for which I'm seeking your advice is the following: run the SEM without latent variable interactions in order to obtain TECH4 output. Use the variance and covariance values from the TECH4 variance/covariance matrix for the equations in your article to calculate the variance of the endogenous latent variables. All other values needed for the equations that you provided in your article can be supplied by the Mplus output for the SEM that includes the latent variable interaction terms. I have worked this out in an Excel spreadsheet, which I can email you. 


To answer your first question, you get the variances and covariances for the endogenous latent variables by you yourself computing them using the general covariance algebra formulas of SEM that you have to explicate for your particular example. I do that on page 3 for my Figure 2 example. Regarding your potential solution, no you should not use TECH4 results from a model without the interaction because such a model does not correctly estimate the variances of the model with latent variable interaction. 


Thank you. I appreciate your help with this! 


I have conducted a level1 moderation using MSEM. I would like to plot and probe my latent variable interaction using Preacher's online interaction utility for multilevel models. http://www.quantpsy.org/interact/hlm2.htm Part of the information that I need to do this is the coefficient covariance between the coefficent of the focal predictor and the DV and the coefficient of the interaction and the DV. (Y 10, 30) I do not think that Mplus produces this coefficient covariance. Is that correct? If so, is there a way that it can be obtained or estimated? 


You can get what you want from TECH3 of the OUTPUT command. 

Chris Thomas posted on Wednesday, September 19, 2012  4:44 pm



Hi Bengt & Linda, I also have a question about formulae from Bengt's Latent Variable Interaction FAQ. Specifically, a version of Eq. 5 that accounts for control variables along with the two substantive variables composing the interaction. Let's assume there is one additional variable. There is already an eta3, so to be consistent with the existing equation and Figure 2, let's label this variable: eta4. The path coefficient is B4, and variance is V(eta4). To calculate V(eta3), I include a term for the path from eta4 to eta3; thus, to Eq. 5 I add "+ B4^2*V(eta4)" In addition, do I also need to repeat the 2*B1*B2*Cov(eta1,eta2) for each additional exogenous covariance? In the case of one control, cov(eta1,eta4) and cov(eta2,eta4). Thus, if my computational logic is correct, the extended version of Eq. 5 for one additional control variable would be: V(eta3) = B1^2*V(eta1) + B2^2*V(eta2) +2*B1*B2*Cov(eta1,eta2) + B3^2*V(eta1*eta2) + B4^2*V(eta4) + 2*B1*B4*Cov(eta1,eta4) + 2*B2*B4*Cov(eta2,eta4) + V(zeta3) As always, thanks for the invaluable assistance you provide through this website. 


The general approach that should be taken is to multiply the DVs righthand side expression with itself and take expectation of all those product terms. This is the variance of the DV when all the IVs have mean zero (center observed variables if they don't have mean zero). Then apply the rule in my FAQ note that the expectation of the product of 3 variables is zero (under normality) and use the expressions for products of 2 and 4 variables that my FAQ note gives. I think that will be a bit different than the formula you give. 


Dr. Muthen, I am attempting to adapt your formula from the FAQ section for calculating variance of twoway interactions to a threeway interaction between 2 latents (eta1 & eta 2) and 1 manifest variable (x1). The two latents are endogenous. I am using the following formulas to calculate variance of secondorder interactions: V_int12=(Veta1*Veta2)+(Cov_eta1_eta2^2); V_int13=(Veta1*Vx1)+(Cov_eta1_x1^2); V_int23=(Veta2*Vx1)+(Cov_eta2_x1^2); Would I be correct in using the following formula to calculate variance of the thirdorder interaction term? V_int123=( V_int12*Vx1); It is not clear to me how to derive the squared covariance of the 3 interaction components. Your thoughts on this matter would be appreciated. Thank you. 


I think you are using equations (9)(11) in my FAQ on interactions between 2 latent variables. You see there that the formula is derived via the expectation of 4thorder moments. With 3 variables interacting instead of 2, it seems that you would need the expectation of 6thorder moments. I don't have that expectation formula handy. 

Steve posted on Friday, July 05, 2013  4:56 am



Hello, I have a mediated structural model with a latent interaction on the outcomes (the interacting variables are endogenous). I am trying to obtain the variance for the interacting variables  but the output does not seem to provide variances for endogenous latent variables. The reason I am trying to do this is to compute "appropriate" standardized interaction effect (Wen, Marsh, & Hau, 2010). If this is not possible, I looked at breaking apart the model (just testing 'right side') in order to get this information. However, I'd rather not do this because I get very different results. This leads me to my final question: If I leave out the 'left side' of my model, I understand that Mplus is computing a different structure. However, can you explain why I get such different standardized coefficients as compared to my full mediated model (since 'left side' was not specified with any direct effects on 'right side'). Thank you very much. 


See the Latent Variable Interaction FAQ on the website. The only time breaking the model apart would not find different results is if the model fitted perfectly. 

Steve posted on Tuesday, July 09, 2013  3:12 am



Hello Linda, Thank you for your reply. I have reviewed the FAQ on Latent Variable Interaction  yet I'm afraid it is not clear to me how to calculate the variance of the dependent variables (mediators F4 and F5) from the Mplus output (as it is not provided). The relevant parts of my model are: F4xF5  F4 xwith F5; F4 on F1 F2 F3 Age Weight; F5 on F1 F2 F3 Age Weight; F6 on F4 F5 F4xF5 Age Weight; Would it be possible for you to tell me how I can calculate the variance for F4 and F5 from the Mplus output? Any help you could provide would be very much appreciated! Many thanks. 


The F4 and F5 variances don't involve latent variable interactions and they can be computed using MODEL CONSTRAINT where you express the variance using labels for the model parameters in the MODEL command. Using regular covariance algebra rules, that expression involves the slopes of their predictors, the predictor variances, and their covariances. 

Steve posted on Thursday, July 11, 2013  9:59 am



Dear Bengt, Thank you for your explanation and direction. I have reviewed the FAQ on Latent Variable Interaction and think that I now understand. If I may ask a quick question, can I confirm that the following is correct for a more simple example computing the variance of F3: F3 on F1 F2 V(F3) = [(slope for F3 on F1)^2] * V(F1) + [(slope for F3 on F2)^2] * V(F2) + (residual variance of F3) Many thanks. 


You forget the covariance term: 2* slope1*slope2*Cov(F1, F2) 

Steve posted on Friday, July 12, 2013  1:18 pm



Dear Bengt, thank you very much for your help. 

Tom Guy posted on Wednesday, November 13, 2013  7:56 am



Dear Bengt, I perform xwith interactions between exogenous latent variables. Is it possible to derive the covariances needed to do a spotlight analysis? 


I don't know what a spotlight analysis is. What I know about XWITH interactions is contained in the FAQ mentioned above. 

Tom Guy posted on Friday, November 15, 2013  1:30 am



Thank you for your answer. From my reading of the FAQ about xwith interactions, the covariance between two latent variables (that are both exogenous) cannot be determined by the equations (or am I wrong?). Can I find it somewhere in the Mplus output or calculate it somehow? Thanks in advance for your answer! 


The covariance between the exogenous latent variables is found in the regular output (labeled WITH). 


Hi a quick question regarding the equation (4) in the latent interaction note: Cov(xj xj) = V (xj) = E(x2j)  [E(xj)]2, where 2 is in effect square. I understand that [E(xj)]2 being squared latent mean, but I am not sure what E(x2j) is, and where I can locate it in the output of an latent interaction model. Thanks 


The output has parameter estimates for V(xj) and E(xj), where those are the latent variables in your case. E(x2j) is not a parameter. The point of (4) is that E(x2j) is obtained from V(xj) and E(xj). 


Hi, As far as I understand, V(xj) and E(xj) are not available when TYPE = RANDOM specified for a latent interaction model, because TECH can not be produced, and the model result only gives variance of reflect factor in my case. If one of the exogenous variables is measured by causal indicators (i.e. the latent variable is a formative construct), is it still possible to calculate its variance (based on (4)) and variance of the interaction term (based on (11))? I am wondering whether it is possible to ask for TECH4, so that I can get V(xj), but that is obviously without interaction terms in the model. That means that E(x2j) can be obtained by using DEFINE to square formative indicators in the model without interaction term. If E(x2j) and [E(xj)]2 are obtained, then V (xj) can be calculated. (11) can then be followed and also (16). Sorry it all gets a bit complicated, but this is the method I can work out so far since no output is readily available for this specific purpose. 


V(xj) and E(xj) are model parameters  and therefore printed in the output  when x is an exogenous factor (and not a formative factor). In other cases, V and E needs to be computed by Model Constraint using SEM formulas and my Interaction note. You can calculate the mean and variance of your formative factor using standard SEM formulas (see, e.g., Bollen's book). If you don't feel up to speed with such formulas yet, I would urge connecting with an experienced SEM person, or a statistics consultant. 

Andy posted on Wednesday, January 29, 2014  3:28 am



Dear Linda and Bengt, Regarding the following post: Bengt O. Muthen posted on Tuesday, May 29, 2012  1:42 pm I have question about Cov (eta1, eta2). What if there is no relationship between these two variables. 1. Do we need to correlate them if we want to test its interaction with eta3  and in that case use Cov (eta1, eta2) from output; 2. We don't have to correlate them, and we can use covariance that MPlus estimated between these two variables? Thank you. 


Maybe you are referring to Figure 1 on page 3 of the LV interaction pdf FAQ. If so, eta1 and eta2 don't need to be correlated. If you don't want them to correlate you just use fix their covariance at zero (otherwise the default is to covary them); in this case, Mplus will not estimate a covariance between them. 

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