You have to carry the minus sign for all the x terms, not just the first.
Also the y=3 probability is incorrect - the last term should be dropped. Think of a normal curve where you are interested in the area between 2 thresholds - only the lower and the higher limit for the category are involved.
Can I "use" the standardised probit coefficients & threshfolds to calculate probabilities? Do you expect them the probabilties calculated from the standardised coefficients to differ from those calculated from the unstandardied coefficients?
Yes, unless you have residual variances for the indicators that are different from unity. So it depends on whether you have covariates in the model and whether you have a multiple-group situation. See the handout of Topic 2 for how to compute probit-based probabilities when you have factors and covariates.
Thank you for that, I found the handout topic 2 very helpful. However, I am not considering the probability effects of the covariates on the indicators but the probability effect of the factor on the indicators. My question is, do I also have to include the covariates in the computation of the factor probability effects on the indicators?
If the covariates influence the factor and not the indicators directly you don't have to include the covariates in the computation. You just have to know what the mean and variance of the factor is, either conditional on a certain covariate value or marginally.
Web Note 4 gives the answer in formula (7) which shows that you need to involve the residual variance theta. Theta is not a free parameter but is computed as a remainder to add up to unit latent response variable variance when there are no covariates in the model. This means that Delta in equation (10) is one which then gives you theta as a function of lambda and psi. Then you take the square root of that theta value and use it to divide your F arguments.
Thank you very much for this. Should I then use the inverse of the square root of the theta to compute both marginal factor effects on the indicators and model estimated probability proportion for the categorical indicator ? Can I also use the formulas in Webnote 4 to compute the factor probability effects on the reference indicator whose loading is fixed to one (to scale the factor; note that I am using the theta not delta parameterization)? & how come MPLUS computes unstandardised thresholds for this reference indicator if, according to Kamata & Bauer (2008:138), such thresholds should be 0? thank you. ramzi
Yes, but with the Theta parameterization the latent response variable variance is not 1. Instead, you should work with theta=1. Note also that with covariates you follow the formulas of theTopic 2 item bias segment.
There are different ways to parameterize the model. I have not read the reference you give, but it might be that they fix the threshold to zero for the indicator with lambda fixed at 1 because they instead free the factor mean. These are equivalent parameterizations in terms of model fit. And as far as I can see none is preferable from an interpretation point of view.
If I understand correctly then, the formula I posted on the 23rd November to compute the probability effect of a standard deviation increase in the factor on the categorical indicator __ F(t-B*MEAN)-F(t-B*(MEAN+1SD)) __ should be the correct one to use with covariates and the theta parametrisation.
I am however still not sure how to interpret this probability. It should not be the probability of changing thresholds at the mean of the factor because the other threshold, threshold2, is not included in the argument?
Regarding your first question, yes this is the correct formula if your covariates do not have direct effects on the item and you use the Theta parameterization. This is seen in the Topic 2 handout I pointed to - see page 163, eqns (46) and (47). You just have to drop x and note that theta=1.
Regarding your second question, your formula concerns the probability of being below the threshold which with a binary item implies the probability of observing 0 (not 1). It sounds like you consider a polytomous item. With say 3 categories you have 3 different probabilities to consider. Those probs are computed in line with eqns (18)-(20) of Technical Appendix 1 on our web site, where the x is your factor.
Sarah Ryan posted on Wednesday, September 28, 2011 - 12:50 pm
I have a model with 4 control covariates, 4 LV's, and 2 x's predicting an observed y with 5 levels. I am using WLSMV.
The latent factors, but not any of their indicators, are regressed on the observed covariates. The observed y is regressed on the observed covariates. After reading the posts above, I'm still not clear on whether I need to include the covariates (ie use formulas 46 and 47 of Topic 2 handout) in the prediction of probabilities for y. I think I probably do, but can you confirm or disconfirm?
Yes, you do need to include the covariates - as is done in those formulas with x. Unless, you have centered (subtracted their means) those covariates and want to compute the probability at their means (which is zero for the centered version).
Sarah Ryan posted on Thursday, September 29, 2011 - 4:48 pm
Okay, so in calculating y probabilities do I include only those covariates that share a significant association with the outcome (given covariate associations with all other predictors)?
Also, I would use this equation, then (with zero values for latent factor means), yes?
P (uij=1|eta*ij,xi) = 1– F[(tj – lambda*j_eta*i - kj_xi)*(1/sqrt(theta))]
(with theta being the y* residual variance in the standardized output).
Are you aware of any resources that can guide me in expanding this for a five-level outcome? I'm having trouble, and I'm not sure if it is because I'm not expanding the equation correctly or if my output is problematic. My thresholds are 1.51, 2.13, 2.19, and 2.96 and the y* residual variance is .45 (ie 1/sqrt(.45)=1.36), so I'm getting either very large or very small probabilities.
You should use all covariates included in the model, unless you consider a simplified model.
You get the factor mean value from Tech4 - it might not be zero, depending on the covariates influencing it.
See Appendix 1 of the V2 appendix to get the expression for the probabilities of an outcome u with more than 2 categories.
Sarah Ryan posted on Friday, September 30, 2011 - 5:14 pm
Once again, thank you. With this information, I have predicted probabilities and the world makes sense again!
Jak posted on Thursday, January 19, 2012 - 7:54 am
1) When I use (multilevel categorical) MLR estimation with link = probit, is it correct that the variance of the underlying latent response variables is 1?
2) I am trying to obtain the univariate probabilities for the categorical variables from the estimated thresholds. I thought a (first) threshold of -1.28 should match an observed proportion of .10 in the first category. But this seems incorrect, I hope you can help me out.
I have estimated a probit model and have a question about the interpretation of the coefficients. I have noticed in other sources that authors recommend calculating marginal effects to make the probit coefficients more interpretable. However, this is not the method recommended in Chapter 14 of the Mplus user’s manual. Is there something about the way that the Mplus program estimates the coefficients that affects the translation of the probit coefficients? Why does the Mplus manual recommend calculating the predicted probabilities directly from the probit equation rather than from marginal effects? Thanks in advance for any help you can provide.
Marginal effects are certainly also of interest, but are not provided automatically by Mplus. For definitions, see e.g. the book by Long (1997), section 3.7.4. The marginal effect of one covariate varies as a function of other covariate values so it isn't a simple description. I think you can express them in Mplus using the new LOOP and PLOT options of MODEL CONSTRAINT. This way you would also get confidence intervals.
db40 posted on Monday, September 01, 2014 - 9:32 am
Dear Dr. Muthen,
I am trying to calculate probit probabilities for a mediation model and I am wondering if this is correct.
Starting with X > M
Prob (M = 1 | physical abuse = 1)= F (- threshold + unstandardised est * 1) = F (-1.682) = a probability of 0.046284.
Threshold = M1$1 1.995 (add the minus in front) X1 (physical abuse = YES =1 ) = 0.313
I am using SEM with probit regression, since my outcome is a binary variable, and I would like to know if I am calculating the probabilities properly. I have six equation as you can see below, and I would like to estimate probabilities for U4 (binary outcome)
F1 BY U1-U3; Y1 ON Y2 Y3; Y4 ON F1 Y1 Y2 Y3; F1 ON Y1 Y2 Y3; Y5 ON F1 Y1 Y2 Y3 Y4; U4 ON F1 Y1 Y2 Y3 Y4 Y5;
I have the following questions:
1) Is it necessary to use only the last equation to estimate probabilities of U4 or is it necessary to substitute the variables in the last equation for the equations that predict those variables? For example, in the second equation Y1 is regressed on Y2 and Y3. Thus, should I substitute the Y1 in the last equation for Y2 and Y3? 2) The F1 is a continuous latent variable and its indicators are ordinal variables (5 categories). By default the Mplus estimate the latent variable with mean 0 and SD equal 1, doesn’t it? However, when I use the TECH4 option the mean of F1 is not 0. Thus, I don’t know in which value I need fix the latent variable to calculate the probabilities.
With regard the mean of latent variable I saw that when I use the TECH4 option when I created the latent variable without the role SEM model the mean is zero. However, when I use TECH4 option with all the equations from the SEM the mean change. Is this normal? Can I really consider the mean equal zero in this case?
I am using SEM with probit regression and I would like to estimate predict probabilities. I am using the equation below that I found in Mplus guide. However, I do not know where and how the residual variance need to be incorporated in this equation. I did not found this in Mplus guide. P (u = 1 | x) = F (-t + b*x +b2*x2...) Another question: The “R-Squared + residual variance” need to be equal 1, doesn't it? I do not know what is happening, because when I use categorical variable the sum of these two estimates is higher than 1. You can see part of my output below. The order of the columns is: variable, R-Square and residual variance Variable R-SQUARE Residual Variance A_DM: 0.217 0.847 ESC_CAT5: 0.821 0.246 RENDAF: 0.745 0.339 CLASSE5 : 0.783 0.293 AF : 0.116 0.954
There are no free residual variances estimated. Or, viewed a different way, they are fixed at 1 in probit and already incorporated in the parameters. The standardized solution will give R-square+resvar=1. The stand sol. gives resvars as "remainders", making the variance add up to 1.
Thank you very much for your response! Now I understood.
I have one more question. I would like to plot graphs with probabilities of an event according with our main latent variable (X). For doing that I will fix X in 5 differents points (-2SD, -1SD, mean, +1SD, +2SD) and the others variables in the model will be fixed in their mean. Thus I will have 5 different probabilities and it will be possible to plot a graph. However, doing that, I am estimating only the direct effect of X on Y, but I have also interesting in the total effect of X on Y and I would like to know if there is some way to estimate the probabilities that will reflect the total effect instead of only direct effect.
I have one main explicative continuos variable, 6 mediators (1 binary and 5 continuous variables, including one latent variable) and a binary outcome. You can see below my equations. I would like to estimate probabilities of U4 (total and direct effect) according with F1. The direct effect of F1 on U4, I saw that I can estimate using the equation P (u = 1 | x) = F (-t + b*x) with the estimates of this equation: U4 ON F1 Y1 Y2 Y3 Y4 Y5 Y6. But I also want to estimate probabilies to reflect the total effect of F1 on U4 and I do not know how to do that.
F1 BY U1-U3; U4 ON F1 Y1 Y2 Y3 Y4 Y5 Y6; Y1 ON Y2 Y3; Y2 ON F1 Y1; Y3 ON Y1 Y2 Y5; Y4 ON F1 Y1 Y2 Y5 Y6; Y5 ON F1 Y1 Y2 Y3 Y4; Y6 ON F1 Y3 Y4; F1 ON Y1 Y2 Y3;
Binary mediator and binary outcome both in principle call for the counterfactually-defined effects discussed in the paper on our website
Muthén, B. & Asparouhov, T. (2015). Causal effects in mediation modeling: An introduction with applications to latent variables. Structural Equation Modeling: A Multidisciplinary Journal, 22(1), 12-23. DOI:10.1080/10705511.2014.935843
This shows how to get the effects expressed in probabilities. But you have many complications in your model including reciprocal interaction (Y1 ON Y2 and Y2 ON Y1), so it would be hard to apply the above approach. You would need extensive statistical consulting.
The best you can do is to instead of focusing on the DV u4 you have to focus on u4*, the underlying continuous latent response variable for u4. The effects are then handled by WLSMV Model Indirect.
I try to figure out if the LOOP and PLOT options that are referred to in the Muthén et al. book on regression (2016, p. 232) are applicable when I want to compute probabilities from an SEM with a binary outcome (probit). To be more specific: Two LVs (LV1 and LV2) predict the outcome. LV2 is a mediator. I use the following statement: Model constraint: Loop(LV1,-1sd,+1sd,0.1); Plot(propens); propens=phi(-t1+b1*hedo); where "1sd" refers to sqrrt(var_LV1). Since the mean of the LV is zero, does this result in a plot of probabilities 1 SD below and above the latent variable mean with the influence of LV2 kept zero? Thank you.
I have a model in which a binary response variable is regressed on a continuous latent factor using the WLSMV estimator. I can show the magnitude of the association using the ICC plot that Mplus provides showing the estimated probability of scoring a 0 and a 1 on the binary response variable against levels of the latent factor.
My question is whether it is possible to get the standard error of the estimated probabilities in Mplus. This would allow me to plot estimated probabilities with confidence intervals. The manual has the formula for calculating probit probabilities but not standard errors.
Riva Denny posted on Monday, January 08, 2018 - 10:35 am
Hi Dr. Muthen,
I am looking for clarification on which equation to use to interpret my probit SEM results as probabilities. I have located two equations in Mplus materials and am hoping you can point me in the right direction regarding which one to use for my model and any additional articles or documentation that would aid my understanding.
There is the equation in the User Guide on page 440: P (u = 1 | x) = F (a + b*x) = F (-t + b*x)
And the equation in the Topic 2 Handout on slide 163 (equation 6/7) that amounts to: P = 1 - probability ((threshold - z)/sqrt(theta))
I understand that the real difference between these equations is the theta term in the second equation. In my reading I have not been able to discern the context in which one equation should be used vs. the other.
My model is recursive and has a binary outcome and 5 mediators (3 continuous, one of which is latent, 1 binary, and one with 5 categories; there are some correlations specified between the continuous mediators). I am using the WLSMV estimator with the Theta parameterization, and TYPE=COMPLEX with a 4 group cluster variable (but no cluster-level variables).
Which equation should I use given my model specification and why?
I greatly appreciate any assistance you are able to provide.
The mediation model implies a more elaborate formula than for the regression model you refer to. This is shown for one mediator on pages 309-310 in our book Regression and Mediation Analysis using Mplus. The expression involves the residual variance for the mediator.
Riva Denny posted on Monday, February 19, 2018 - 11:50 am
Hi Dr. Muthen,
Thank you for your January 8 response. I have gotten a copy of your book and found the pages you referenced (and others) very helpful! I do have a couple follow-up questions though.
1) When applying the probability formula on page 310 to a model with multiple mediators (in my case 4 (I have simplified my model since my last post): 3 latent continuous (with categorical predictors) and 1 observed categorical variable, would the variance portion of the equation be operationalized as: V(y*|x) = V(Beta51*error_y1 + Beta52*error_eta2 + Beta53*error_eta3 + Beta54*error_eta4 + error_y5) = Beta51^2*sigma^2_y1 + Beta52^2*sigma^2_eta2 + Beta53^2*sigma^2_eta3 + Beta54^2*sigma^2_eta4 + c where the beta terms are the respective mediator coefficients and the sigma^2 terms are the standardized residual variance values for the respective mediating variables and c=1?
2) Also, when substituting values into my E(y*|x) term, is it correct to assume that the intercept for the categorical mediator is zero as it is for the other latent mediators in the model, since with the WLSMV estimator I am actually using the latent form of the categorical mediator?
Thank you! ~Riva
Riva Denny posted on Monday, February 19, 2018 - 11:52 am
Relatedly, I would like to confirm the nature of my coefficients when using the WLSMV estimator. I understand from my reading in your book and on this forum that the WLSMV estimator works with the latent continuous variable underlying the categorical response variables (for both the mediator and the final outcome variable). It also appears from other forum responses that in models with a mixture of continuous and categorical outcomes that the coefficients are linear when the dependent variable is continuous and probit when the dependent variable is categorical, and that this is still the case when using the WLSMV estimator.
Is this true, and if so why? If latent variables are being used for my categorical mediator and binary outcome variables why would the coefficients in their equations be different from those in the equations for the mediators that I have specified as latent?
Page 310 refers to a single mediator - with several mediators, the formula is more complex, involving the covariance matrix for the residuals of the mediators. This is shown in equation (4) of the article:
Nguyen, T.Q., Webb-Vargas, Y., Koning, I.K. & Stuart, E.A. (2016). Causal mediation analysis with a binary outcome and multiple continuous or ordinal mediators: Simulations and application to an alcohol intervention. Structural Equation Modeling: A Multidisciplinary Journal, 23:3, 368-383 DOI: 10.1080/10705511.2015.1062730
This means that you have to do matrix multiplication operations in say R to get the probability expression.
Regarding your second post, for categorical DVs, WLSMV considers linear regressions among their corresponding latent continuous response variables. So the coefficients are linear regression coefficients. But if you consider the regression of the observed categorical DV on some predictors, the coefficients are probit coefficients. So the same coefficients have different meaning depending on what you consider as the DV.
Note also that indirect effect for the observed binary outcome and/or binary mediator that you consider has special (counterfactual) formulas as you see in Chapter 8. WLSMV does not give those effects but instead considers the effect on and via the latent response variables.