Preparation

This tutorial expects:

• Version 8 or higher of Mplus.This tutorial was made using Mplus version 8_3.
• Basic knowledge of hypothesis testing
• Basic knowledge of correlation and regression
• Basic knowledge of Bayesian inference
• Basic knowledge of coding in Mplus

Example Data

The data we will be using for this exercise is based on a study about predicting PhD-delays (Van de Schoot, Yerkes, Mouw and Sonneveld 2013).The data can be downloaded here. Among many other questions, the researchers asked the Ph.D. recipients how long it took them to finish their Ph.D. thesis (n=333). It appeared that Ph.D. recipients took an average of 59.8 months (five years and four months) to complete their Ph.D. trajectory. The variable B3_difference_extra measures the difference between planned and actual project time in months (mean=9.97, minimum=-31, maximum=91, sd=14.43). For more information on the sample, instruments, methodology and research context we refer the interested reader to the paper.

For the current exercise we are interested in the question whether age (M = 31.7, SD = 6.86) of the Ph.D. recipients is related to a delay in their project.

The relation between completion time and age is expected to be non-linear. This might be due to that at a certain point in your life (i.e., mid thirties), family life takes up more of your time than when you are in your twenties or when you are older.

So, in our model the \(gap\) (B3_difference_extra) is the dependent variable and \(age\) (E22_Age) and \(age^2\)(E22_Age_Squared ) are the predictors. The data can be found in the file phd-delays_nonames.csv . (In Mplus the first row CANNOT have the variable names, these have already been deleted for you)

Preparation – Importing and Exploring Data

You can find the data in the file phd-delays_nonames.csv , which contains all variables that you need for this analysis. Although it is a .csv-file, you can directly load it into Mplus using the following syntax:

TITLE: Bayesian analysis summary
DATA: FILE IS phd-delays_nonames.csv;
VARIABLE: NAMES ARE diff child sex Age Age2;
USEVARIABLES ARE diff Age Age2;
OUTPUT: sampstat;

Once you loaded in your data, it is advisable to check whether your data import worked well. Therefore, first have a look at the summary statistics of your data. You can do this by looking at the sampstat ouput.

MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 Means
    DIFF               9.967      0.790     12.622      0.000
    AGE               31.676      0.375     84.433      0.000
    AGE2            1050.217     35.916     29.241      0.000

Regression – Default Priors

In this exercise you will investigate the impact of Ph.D. students’ \(age\) and \(age^2\) on the delay in their project time, which serves as the outcome variable using a regression analysis (note that we ignore assumption checking!).

As you know, Bayesian inference consists of combining a prior distribution with the likelihood obtained from the data. Specifying a prior distribution is one of the most crucial points in Bayesian inference and should be treated with your highest attention (for a quick refresher see e.g. Van de Schoot et al. 2017). In this tutorial, we will first rely on the default prior settings, thereby behaving a ‘naive’ Bayesians (which might not always be a good idea).

To run a multiple regression with Mplus, you first specify the model (after the MODEL line using the ON command for regression coefficients, and the [] command for the intercept). As default Mplus does not run a Bayesian analysis, so you would have to change the ESTIMATOR to BAYES under ANALYSIS in the input file and then look at the output under MODEL RESULTS.

TITLE: Bayesian analysis with default priors

DATA: FILE IS phd-delays_nonames.csv;

VARIABLE: NAMES ARE diff child sex Age Age2; ! All the variables in the dataset

USEVARIABLES ARE diff Age Age2; ! The variables we use in this analysis

ANALYSIS:
ESTIMATOR IS bayes; ! Specify that we want to use a Bayesian analysis
Bseed = 23082018; !specify a seed value for reproducing the results
CHAINS = 3; ! set the number of chains we want to use

MODEL: 
[diff] (intercept);       ! specify that we want an intercept
! this model would also work without this line, but this way it is possible to easily set a prior
diff ON Age (Beta_Age);   ! Regression coefficient 1. 
diff ON Age2(Beta_Age2);  ! Regression coefficient 2 
! You need to name these regression coefficients to later set priors
diff (e);                 ! Error variance


OUTPUT: tech8; ! Specify the output we would like

MODEL RESULTS

                                Posterior  One-Tailed         95% C.I.
                    Estimate       S.D.      P-Value   Lower 2.5%  Upper 2.5%  Significance

 DIFF       ON
    AGE                2.647       0.622      0.000       1.348       3.874      *
    AGE2              -0.026       0.007      0.000      -0.039      -0.013      *

 Intercepts
    DIFF             -46.578      12.898      0.000     -70.394     -21.095      *

 Residual Variances
    DIFF             198.117      16.978      0.000     171.316     232.769      *

The results that stem from a Bayesian analysis are genuinely different from those that are provided by a frequentist model.

     PRIORS FOR ALL PARAMETERS            PRIOR MEAN      PRIOR VARIANCE     PRIOR STD. DEV.

     Parameter 1~N(0.000,infinity)           0.0000            infinity            infinity
     Parameter 2~N(0.000,infinity)           0.0000            infinity            infinity
     Parameter 3~N(0.000,infinity)           0.0000            infinity            infinity
     Parameter 4~IG(-1.000,0.000)          infinity            infinity            infinity

Regression – User-specified Priors

In Mplus, you can also manually specify your prior distributions. Be aware that usually, this has to be done BEFORE peeking at the data, otherwise you are double-dipping (!). In theory, you can specify your prior knowledge using any kind of distribution you like. However, if your prior distribution does not follow the same parametric form as your likelihood, calculating the model can be computationally intense. Conjugate priors avoid this issue, as they take on a functional form that is suitable for the model that you are constructing. For your normal linear regression model, conjugacy is reached if the priors for your regression parameters are specified using normal distributions (the residual variance receives an inverse gamma distribution, which is neglected here). In Mplus, you are quite flexible in the specification of informative priors.

Let’s re-specify the regression model of the exercise above, using conjugate priors. We leave the priors for the intercept and the residual variance untouched for the moment. Regarding your regression parameters, you need to specify the hyperparameters of their normal distribution, which are the mean and the variance. The mean indicates which parameter value you deem most likely. The variance expresses how certain you are about that. We try 4 different prior specifications (for now we only look at the priors for the regression coefficients and ignore the intercept and the error term), for both the \(\beta_{age}\) regression coefficient, and the \(\beta_{age^2}\) coefficient.

First, we use the following prior specifications:

\(Age\) ~ N(3, 0.4)

\(Age^2\) N(0, 0.1)

In Mplus, the priors are set in the under the MODEL PRIORS command using the ~ sign and an N for a normal distribution

The priors are presented in code as follows:

MODEL: 
[diff] (intercept);       ! specify that we want an intercept
diff ON Age (Beta_Age);   ! Regression coefficient 1
diff ON Age2(Beta_Age2);  ! Regression coefficient 2 
diff (e);                 ! Error variance


MODEL PRIORS:
  Beta_Age ~ N(3, .4);
  Beta_Age2 ~ N(0, .1);

MODEL RESULTS

                                Posterior  One-Tailed         95% C.I.
                    Estimate       S.D.      P-Value   Lower 2.5%  Upper 2.5%  Significance

 DIFF       ON
    AGE                2.794       0.460      0.000       1.863       3.697      *
    AGE2              -0.027       0.005      0.000      -0.038      -0.017      *

 Intercepts
    DIFF             -50.109       9.537      0.000     -67.545     -30.948      *

 Residual Variances
    DIFF             198.178      16.947      0.000     171.160     232.840      *

MODEL PRIORS:
  Beta_Age ~ N(3, 1000);
  Beta_Age2 ~ N(0, 1000);

MODEL PRIORS:
  Beta_Age ~ N(20, .4);
  Beta_Age2 ~ N(0, .1);

MODEL PRIORS:
  Beta_Age ~ N(20, 1000);
  Beta_Age2 ~ N(0, 1000);