Jenn Tein @ ASU has a chapter that should be relevant:
Tein, J.-Y., & MacKinnon, D. P. (2003). Estimating Mediated Effects with Survival Data. H. Yanai, A. O. Rikkyo, K. Shigemasu, Y. Kano, J. J. Meulman (Eds.) New Developments on Psychometrics (pp. 405-412). Tokyo, Japan: Springer-Verlag Tokyo Inc.
I'd contact her or Dave MacKinnon directly if you have trouble tracking down the book.......
Steffi posted on Thursday, June 22, 2006 - 2:29 pm
I am afraid my problem is very basic but I cannot come up with a good explanation.
Running a DTSA with four binary indicators (x1-x4), and two continuous predictors (y1 and y2) the model quickly converges (Mixture Missing) and yields reasonable results. The relevant Model statement is:
The predictors y1 and y2 are correlated by default in the Mplus analysis. Just like in regression analysis, these correlations are not part of the model's parameter set. You should not mention y1 with y2.
If you really want the mediation model that you have problems with, contact support.
Steffi posted on Thursday, June 22, 2006 - 11:26 pm
thank you for your fast reply! Indeed, the mediation is what I am actually after.
But out of curiosity: other than in "standard" CFA I assume it is not possible to constrain corr(y1, y2) to zero?
I think you are asking if in a mixture analysis you can constrain the correlation of two variables to zero if they are predictors of a continuous latent variable that in turn influences categorical observed variables. That is possible in Mplus. If you have problems doing so, please send your input, output, data, and license number to email@example.com.
I have a follow up question on Stan Hong, Lois Gelfand and my earlier post:
I want to test a mediation effect in discrete time survival analysis (X->M->Y) with X = continuous predictor, M = continuous mediator and Y = vector of binary event indicators (proportional hazard odds assumption). I have censored observations, and from my reading of the literature, the best approach would be the product of coefficients method (a*b, where a is a standard regression coefficient and b is a logit coefficient) using bootstrapped SEs. With a logit link function and ML, am I correct that such a test is not (yet) possible in Mplus?
The next best approach is probably to compute SE using the Delta Method as recommended by Linda somewhere, but this also needs to be done by hand, correct?
Before tackling this task, I want to make sure that there is no "better" alternative (such as a probit-link function with WLSMV as indicated in another post, etc. but I am very hesitant to adopt such an approach since I have never seen it before in a DTSA context). What would you recommend me and can you provide me with any good references for mediation effects with dichotomous outcomes (I am aware of MacKinnons message from Dec. 04 and many of his excellent papers, but I wonder to which extent these findings can be generalized to DTSA with censored observations)?
I want to do discrete-time hazard model. My data set is in long format---observations at different ages are nested within individuals, which are nested in cluster (census tracts). Can I specify id of individuals as "idvariable" and census tracts as "cluster" to do three level model? Or do I have to restructure the data so that each age interval becomes one variable and to take a multivariate approach at the lower level?
Also, can I introduce frailty to account for unobserved heterogeneity for individuals without restructuring the data?
I am trying to use the "Model Constraint: New" command described by Bengt above on July 26, 2006 4:25 pm to obtain the Delta method for estimating the standard error of a mediation effect in a cox regression model.
However, I get a fatal error message: "*** FATAL ERROR Internal Error Code: PR1004 - Parameter restriction split problem. An internal error has occurred. Please contact us about the error, providing both the input and data files if possible."
Am I doing something very wrong here? My syntax is below:
The problem is that one variable is continuous and the other time censored. This is not allowed in MODEL CONSTRAINT. If you are creating an indirect effect, I am not sure that the product can be used here. You can ask for TECH3 and compute the standard error of the product yourself.
HI Linda, Thanks for your reply on Nov 8. Could I please confirm with you about not being sure if the product can be used in the example syntax I posted on Nov 8? I would ideally like to estimate the indirect effect of RGRLEV on P_123Y (survival time). However, this would mean multiplying the estimates of the regression of FAMPROB6 on RGRLEV, and the cox regression of P_123Y on RGRLEV. As these are different kinds of regression analyses, perhaps I cannot multiply the paths together to obtain an indirect effect- is this correct?
Is there any way around this? I suppose I could forego the cox regression model and estimate the logistic model regressing I_123Y (survival or not) on FAMPROB6 - and then multiply the path from RGRLEV to FAMPROB6 and then FAMPROB6 to I_123Y to obtain the indirect effect of RGRLEV on I_123Y.
I have two questions concerning discrete-time mediated survival analysis. Our data set is in long format (observations nested within individuals). The predictor is dichotomous, mediators are continuous, dependent variable is an event indicator (no missing data).
1. Our model is a survival analysis with a preceding panel regression model. Is it correct to use TYPE=COMPLEX in conjunction with the CLUSTER option to account for the nonindependence of observations (due to the long format)? Or are there any alternatives you would suggest in our case (any literature suggestions are appreciated)?
2. Is it true that indirect effects can only be estimated with a probit regression in our case? Is there any alternative model specification based on the standard logistic link function?
1. It sounds like you have longitudinal data ("panel regression model" followed by a subsequent survival model. That seems to be best handled by letting the panel part be in wide, not long form (single-level analysis) since the survival modeling is in wide form (so first the columns with the panel outcomes for the panel time points, followed by columns for the even indicators).
2. You can consider the indirect effect with f as end point as in regular linear regression since f (using the Mplus UG notation) is continuous.
We can see that your suggestion is a viable alternative. But: We are mainly interested in the effects of time-varying mediators on the outcome. If we use a wide data setup, we get several effects (as many as there are panel waves, in our case 8) on the respective event indicators. We feel this is quite cumbersome to depict/interpret.
Would it be correct to impose an equality contraint on the various effects per (time-varying) covariate? In this case there would be only one effect per time-varying covariate. (Still, if the model fits worsens after the constraint, we continue to have the "cumbersome" multiple effects, don't we?)
We are pretty sure the relationship is not invariant across time. But if this is the case, the results become quite technical/complicated for a non-methodological journal article, which we are trying to avoid... Besides, isn't the equality constraint just what the "usual" time-discrete event history analysis with time-varying covariates does (unless you include interactions with times of measurement)?
One last question: Would you consider the "long format" approach (every row in the data refers to an episode of observation) incorrect, even if nonindependence of observations is controlled for by adjusting standard errors (by using the Mplus CLUSTER command)? In the "classic" (i.e., non-SEM) literature, time-discrete event history analysis is usually done in long format (this is why we figured we could run our analysis that way).
I am doing a discrete time survival analysis by following models described in the Muthén and Maysen's article, and the Mplus user’s guide. However, I am experiencing some difficulties.
1) I have 10 time points but for the first two time points the binary U’s indicators have only 0 values. As the current version of Mplus doesn’t allow a categorical indicator with only one value, I thought of doing the analysis without the first two time points. That strategy can however result in bias in the estimation of the hazard because of the suppression of information related to time interval. Is that reasoning correct? If yes, is there any way to evaluate that bias? If the bias is negligible I can carry on with the discrete time survival analysis especially because the continuous one doesn’t really suit my problematical.
2) In the Muthén and Maysen's article they mentioned imposing specific structure to the logit baseline hazard. So, I was wandering how to test linear trend on the logit baseline hazard via MODEL CONSTRAINT.
I am looking forward to hearing your answer. Thank you for your invaluable help.
I am planning to do a discrete time survival analysis and had a few questions I was hoping you could answer:
1) How should the data be coded for people who have missing assessments but then come back into the study? For example, say I have six assessments and someone drops out at time 3, but then comes back into the study at time 5 and reports the event. Should it be coded:
0 0 999 999 1 999 (999=missing)
if this hypothetical person did not report the event at time 5 or 6 would it be coded?
0 0 999 999 0 0
2) How does one handle unequal spacing between time points in these models?
1. In the Muthen/Masyn implementation of discrete-time survival analysis in Mplus, only non-repeatable events such as onset of drug use are considered. Because of that, I think the coding for your two cases would be
0 0 0 0 1 999
0 0 0 0 0 0
Missing has a different meaning in this coding scheme.
2. The spacing does not matter unless you put a growth model on the thresholds.
Thanks for the reply. Since the people have missing information for some assessments it seems problematic to assume that the event has not occurred in for these phases (i.e., 0). I thought there was a way to take periodic missingness into account in these models. If I understand correctly dropout prior to an event occurring would be handled by coding 999 at the timepoint where dropout occurred, but handling periodic missingness as I outlined previously seems more tricky.
I have a dataset where the units under observation are subsidiaries of multi-national companies. In a period of 20 years new subsidiaries are setup that enter into the dataset. How should I code for the initial periods when they were absent because they were not born at all? They are not really missing values.
Since you posted under discrete-time survival, I assume you observe some event such as failure. Perhaps you should consider as your survival time variable, the number of years between being born and the event?
Yes, we do observe a failure and that is the event of interest, not the founding/entry. As of now I am not considering time-variant covariates (country level contextual factors) but the idea was to incorporate them at a later point of time. If I didn't have these I could have potentially coded each subject from its first year of entry. Even if I did this it would still bias any unobserved heterogeneity for the particular calendar year, for ex: there could have been a bad economic condition during a particular year in a particular country. So I just reasoned that I will code it as missing since it is not part of the risk set in any case.
If I correctly understood your suggestion - I can incorporate age as a time-variant covariate. I am just wondering if there would be too much of a correlation between this and the dependent variables (u) and wouldn't that overshadow the effects of my other variables?
We think age should not be the covariate - it should be the dependent survival variable. Instead we would recommend that the economic conditions that you are talking about should be a time-varying covariate - that can be done as in the http://statmodel.com/download/lilyFinalReportV6.pdf
Discrete time survival may be a good approach here that can also accommodate time-varying covariates easily.
Yes, discrete time survival is the most appropriate approach for more than one reason for my analysis.
If I have age as the survival time dependent I cannot have macro-economic conditions as a simple time-varying covariate. Let me illustrate with an example. In a particular country consider subsidiary 1 as existing between 2006 and 2010. Subsidiary 2 exists from 2004 to 2008. I can have 7 dummy binary variables for age 1 to 7 but neither of them reached age 6. I now have 7 values for macro-economic condition. The macro-economic condition of year 2008 was fatal for one and not the other but with a simple time-varying covariate it appears that the fifth year (2008) was fatal for both.
If I correctly understood why you provided the particular link, what I can do is incorporate the existence of the subsidiary in a given year and year based time-varying variables in the latent part. This will incorporate the calendar year based effect on the survivability, through the class membership. Am I correct in interpreting what you said?
Now if you also add discrete time survival - the value 5 will also be split into binaries age1, age2,...,age7 age1, age2,...,age7, x1, x2, ..., x7 0,0,0,0,0,999,999 ,0,0,1,0,0,0,0 0,0,0,0,1,999,999 ,0,0,1,0,0,0,0
Dear all, we are doing a discrete-time survival analysis in which we examine whether the effect of one time-invariant predictor (proportionality assumed and confirmed) on event occurences is mediated by a time-varying predictor (proportionality NOT confirmed). The basic model looks like this:
! c'-path event_T1 on t_iv (1); .. event_Tn on t_iv (1);
! b-paths event_T1 on t_v1 (b1); .. event_Tn on t_vn (bn);
! a-paths t_v1 on t_iv (a1); .. t_vn on t_iv (an);
To test mediation, we use the model constraint approach and everything works fine. However, given that the time-varying mediator is non-proportional we get n mediation effects (in our study n = 8); this makes sense but to facilitate the interpretation and reporting of the results, I was wondering whether there is a procedure or an approach to average the n effects? Or would you recommend to compute the model with proportionality assumed for the time-varying predictor?
Stefano posted on Friday, January 04, 2013 - 6:33 am
I am running a discrete time survival analysis with a time-invariant predictor and considering a mediator (proportionality NOT confirmed). Based on the last note on this thread, I would like to obtain the average of the n indirect effects, but I didn´t manage to write the correct code.
I am interested in modeling student permanence in higher education, over a 4 years period using a discrete-time survival model. In that sense the event is repeatable every year, which it seems to be not plausible to model in Mplus. However, if I change the event to be the opposite, i.e. to dropout from higher education, it can be modeled.
If I estimate the discrete-time survival model of higher education dropout. I wonder if I can still interpret these results in relation to student permanence in higher education. Assuming that the effects on permanence would be the opposite of the findings for the dropout model.
Or, is there a way around to estimate a repeatable event model in Mplus?
I would appreciate any suggestion to model student permanence.
Have a look at the paper on our website under Papers, Survival Analysis:
Masyn, K. E. (2009). Discrete-time survival factor mixture analysis for low-frequency recurrent event histories. Research in Human Development, 6, 165-194. download paper contact first author show abstract
If you estimate the hazard of dropout, you can get back to survival estimates of dropout (i.e., the probability of "permanence" at time t). Consider that in classical survival analysis, modeling the hazard rates of death allow you obtain estimates of, say, the median survival time. In my understanding of your research question, modeling the hazard of dropout will allow you to estimate permanence (survival) rates across time.
With only a four year time period, you could also consider using a simple Markov chain model (in Mplus this would be specified as a latent transition analysis with a single, perfect, indicator latent class model for each year). With this specification, you could extend to a constrained mixed Markov chain model (e.g., "mover-stayer" model).
Thank you Katherine and Bengt for your kind responses, they are very helpful.
Katherine, my research is focused on investigating successful higher education permanence for first-generation students in higher education. Therefore, my interest on explaining permanence rather than dropout for this group of students.
I wonder if I can ask you for a couple of references to your suggestions.
I am not sure how to interpret the estimated coefficients of the dropout model in relation to the survival (permanence) of the students. However, I understand that survival rates can be derived from the hazard rates.
Can you point to a reference of Markov models for a similar case to student permanence?
I am helping a postdoc to fit a discrete-time survival model with mediators. We are using WLSMV with the THETA parameterization in order to obtain global fit stats and also refitted it with BAYES as a sensitivity analysis. The form of the model is:
Exposures + covariates -> mediators set 1
Mediators set 1 -> mediators set 2
Exposures + covariates + mediators set 1 + mediators set 2 -> outcome
We also correlate residuals of mediators at each mediation stage and bring x-variables with missing data into the analysis.
Coauthors are struggling with interpreting the probit coefficients, so I was wondering:
1. Would it be appropriate in your opinion to multiply the probit coefficients by 1.7 and then exponentiate them to obtain an approximate hazard ratio interpretation? (I realize this would be only approximate).
2. Alternatively, could we use the instructions on pp. 552-553 of the Mplus UG to compute predicted probabilities at various levels of an exposure and then compute their differences using MODEL CONSTRAINT? If so, would it be sufficient to use predictors of the outcome in a single equation as in the UG example or is the expression for the probability more complex due to the presence of mediators, correlated residuals, and/or bringing x-variables with missing data into the model?
WLSMV should not be used for discrete-time survival because it doesn't handle the missing data correctly. Use ML or Bayes. As opposed to WLSMV and Bayes, ML offers logit but may require many dimensions of integration.
Thank you for this feedback. So noted. We used a person-period data structure as described in Paul Allison's SAS logistic regression book. We then used WLSMV in order to obtain robust results because some variables in the analysis are continuous, but non-normal.
We could use MLR, but would then need to compute the indirect effects by hand, which we were hoping to avoid. That does have the appeal of using the logit link, as you pointed out, and the event is somewhat rare, so perhaps that would be best statistically. Bootstrap with numerical integration with many points of integration could be quite time-consuming.
What about WLSMV + multiple imputation? Would that be OK?
In any event, with WLSMV + MI or Bayes, would it be possible to compute approximate hazard ratios using exp(1.7*probit coefficient)?
For MLR or Bayes (or WLSMV+MI if that combination is OK), would it be possible to compute predicted probabilities as described in (2) of my original post?
I assume the discrete-time part is the "outcome" that you mention; perhaps it is the "f" of UG ex 6.19. And you might be interested in indirect and direct effects of the exposure on f. Given MLR's heavy numerical integration demands in your case, I would use Bayes. Perhaps a 1.7 (or 1.8) multiplication is reasonable - I haven't tried that for hazards.
Regarding your idea in (2), I assume you are thinking about the probabilities of "u1-u4" in ex 6.19 terms. Those are influenced by the distribution of f, so you need to get the mean and variance of f as a function of the mediators at different exposure values (which is without complications).
Thank you so much for this discussion and your guidance. Mplus Discussion is such a wonderful resource for enhancing the rigor of our research projects.
From our discussion here, it sounds like we should go with our Bayes-based results. I appreciate your opinion on the idea of converting the probit coefficients from the Bayesian analysis to hazard ratios.
For the idea of generating predicted probabilities, we're not actually using the UG Ex. 6.19 approach (though we tried it to verify we got very similar results to our approach and we did). We are using the discrete time survival approach described by Paul Allison in his SAS logistic book on pp. 235-236, "Each individual's survival history is broken down into a set of discrete time units that are treated as distinct observations. After pooling these observations, the next step is to estimate a binary regression model predicting whether an event did or did not occur in each time unit." So, the outcome (in our case the outcome is contracting HSV-2) is treated as an observed 0/1 variable with one record for each discrete time period the person was observed (i.e., a person-period data structure).
I'm not sure whether that would change any of the recommendations and conclusions you offered above?
I see, so you want P(u=1) for each of your u's (using notation of ex 6.19). That can be expressed in Model Constraint using the PHI function for the standard normal distribution function of probit. But note that the residual variance of the underlying y* conditional on the exposures and covariates is not 1 as in regular probit, but you have to add to 1 the residual variance expressions due to the 2 continuous mediators. A similar exercise is carried out in section 8.1.1 of our RMA book for the case of a single cont's mediator and a binary outcome.
As a follow-up to this helpful thread, we wound up using ESTIMATOR = BAYES and mediation assessed via the traditional mediation approach with MODEL INDIRECT. We used the approach of multiplying the probit coefficients by 1.7 and exponentiating them to obtain approximate ORs (HRs). A statistical co-author on the resulting manuscript has raised a question regarding interpretation of the model estimates. He asks: “How can one interpret the regression coefficients in a model with exposure X, continuous mediator M and binary outcome Y? Specifically, can the coefficients in the regression of X on M be interpreted as standard linear regression coefficients (change in M per one unit change in X)?” To those questions, I would also ask how the indirect effect is interpreted - as a linear regression of X on Y* through M or as the probit regression of X on Y through M? We are trying to figure out which coefficients to multiply by 1.7 and expotentiate to get approximate odds ratios and which to leave untransformed as linear regression betas. I hope this all makes sense. Thanks very much in advance for your help and best wishes,
Yes, X to M is a standard linear regression. I would think you want the influence of X on Y through M as expressed by the probit regression, that is, taking into account the M residual variance in addition to the probit residual variance of 1 - just like in Section 8.1.1. in our book.
I am still struggling to understand this. Thanks for bearing with me. So, for the X->M->Y relationship, is the distinction that the product of X->M and M->Y coefficients in the classic mediation approach is really the product of X->M and M->Y*, where Y* is the continuous underlying latent representation of Y? And thus if one wants the effect on Y rather than the effect on Y*, it is not as simple as computing an approximate OR as exp(1.7*indirect effect coefficient)?
I have been using X->M->Y as a simplified example. Our mediation model has multiple covariates and mediators, with some mediators being binary and at least one other being continuous. The variables are in an X -> M1 mediators -> M2 mediators -> Y configuration. What would you advise in this type of situation to maximize interpretability of results?
For Q3, conceptually the answer is yes, though in our application Y is a single person-period 0/1 variable indicating HSV-2 incident infection. The data structure follows the structure recommended in Paul Allison's SAS survival regression book in the chapter on discrete time survival. I.e., "long" format with repeated observations for each period per person.
It's hard to say what the best approach is here - it is a bit of a research topic for several reasons and I don't feel I can answer well without doing research on it. First, you have a binary Y and actually several binary Y's so mediation effects need special counterfactual formulas. Second, you have sequential mediation which needs special counterfactual formulas or be done using marginal structural modeling (as referred to in the "causality" literature). And when one takes the simpler WLSMV and Bayes approaches of using a probit link and considering Y* instead of Y to keep the regressions linear, the issue of how to get back odds ratio effects via 1.7 scaling is untried as well especially for discrete-time survival purposes. I wish I could be more helpful. Perhaps SEMNET has suggestions - Cam McIntosh is now actively discussing causality matters.
Thanks so much, Bengt, on behalf of my coauthors and me. After some discussion, we have decided that we will report the Mplus output directly as linear regressions among the continuous Y's and the categorical Y*'s and explain in the data analysis description in the methods that the results should be interpreted as linear regressions for the Y's and probit regressions for the Y*'s. Does that sound correct? If it is, do you have a favorite way to describe the regression of a Y* variable onto an X? E.g., "For every one unit change in X, the latent variable representing Y changes by <coefficient>"? Also, if you have favorite citations for references that discuss interpreting probit coefficients for Y* variables, those would be much appreciated. Thanks again for everything,
You consider a linear regression for both M as DV and for Y* as DV (we'll get to Y later). And, yes: "For every one unit change in X, the latent variable representing Y changes by <coefficient>" And you want to say how much that Y* change is in Y* SDs (so use STDYX). But I would also relate that Y* change to the corresponding Prob(Y=1) change. The probability change due to 1 X unit of change is different dependent on where on the Y* scale you are - I discuss this in Section 5.2.6 of the RMA book.
Thank you, Bengt. This is very interesting and I'm learning a lot. I truly appreciate you making the time to walk me through these steps and hope that our discussion benefits others who might find themselves in a similar situation.
I had one follow-up question to your helpful post above: To relate the Y* change to the corresponding Prob(Y=1) change, I'm wondering how to do that. Would it be permissible to follow the approach shown in the RMA book on p. 232-233 (generating the predicted probabilities and plotting them)? I ask because our situation diverges from the example shown in the RMA book in two ways: 1) We're using BAYES with estimation of Y* rather than ML with estimation of Y as in the RMA coal miner example on p. 233 and 2) We're working with a complex path mediation model rather than a regular regression model. (Our previous discussions above about the challenges of computing predicting probabilities for Y given the presence of mediators prompted this question.) It would be fantastic if we could present predicted probability plots as I imagine at least some of our more epi-oriented coauthors and readers would appreciate a linking of the Y* results to the Prob(Y=1).
Sorry - one more clarifying question, if I may: For the recommendation to standardize, I can see how STDYX would be the best choice for quantifying associations among two continuous variables (including continuous underlying latent representations of M* and Y* and any continuous X's). If, however, X is binary, then I assume I'd want to substitute STDY for that association, correct?
Here is some more detail on the Y* mean and variance and the connection with P(Y=1):
Look at section 8.1.1 in our RMA book, formulas (8.1) - (8.6). This is with probit so relevant for Bayes. This is for one mediator but it can be generalized to several ones like you have. Note the reduced form expression (8.3) and the fact that the V(Y* | X) variance is not 1 as in simple probit regression (where c=1), but also has a term due to the mediator residual variance (you will have more than one such term due to more than one mediator). Do the same reduced-form exercise for your case. Note the probability expression in (8.6) which can be computed using the PHI function in Model Constraint. With a binary X and one binary Y, you can even formulate an odds ratio based on the 2 Y probabilities and the 2 X values. This OR won't be constant with respect to control (covariate) values as in logistic regression but the ORs can nevertheless be computed - and then plotted in Model Constraint's PLOT-LOOP statements. See also Section 5.2.10 in the book about the fact that probit can also give estimated ORs. Hope this helps.