# An Introduction to Factored Regression Models with Blimp

## Abstract

**:**

## 1. Introduction

`mdmb`,

`smcfcs`,

`JointAI`packages [12,13,14].

## 2. Factored Regression Modeling Framework

#### 2.1. Illustrating Factored Regression with Moderation

#### 2.2. Imputation of Missing Observations

## 3. Mediation Models as Factored Regressions

#### Mediated Latent Variable Model as Factored Regressions

## 4. Fitting Factored Regressions with Blimp

`# Read in and set up data`

**DATA:**pain.dat; # Read Data in

**VARIABLES:**# List Variable Names`id txgrp male age edugroup workhrs`

`exercise pain severity anxiety stress`

`control depress interfere disability`

`dep1:dep7 interf1:interf6 disab1:disab6;`

**ORDINAL:**severity male; # Specify ordinal data

**MISSING:**999; # Missing data code`#`” for single-line comments. Second, a semicolon (

`;`) terminates a statement. The

`DATA`command specifies the file path to the data set are reading in. This data set can be tab, space, or comma-delimited. In the above example, we have the

`pain.dat`file in the same folder as the input script, so we do not need to specify the full file path. Next, the

`VARIABLES`command specifies the names and order of the variables in the data set. Blimp also allows these names to appear in the first row of the data set, but then the

`VARIABLES`command must not be used. As a shortcut, Blimp allows specifying multiple variables using a colon. For example,

`dep1:dep7`is replaced by “

`dep1 dep2 dep3 dep4 dep5 dep6 dep7`”. The

`ORDINAL`command above specifies that the variable

`severity`and

`male`are both ordinal and that we want to model them via a probit specification. Blimp will automatically use the data to determine the number of categories and fit the appropriate ordered probit model. The

`MISSING`command specifies the numerical value that represents a missing value in the data set.

#### 4.1. Fitting a Single Mediator Model

`depress`) mediates the effect between pain severity (

`severity`) and the psychosocial disability construct (

`disability`) while controlling for biological sex (

`male`). We provide the path diagram for the model of interest in Figure 3 with the mean structure excluded. To construct the factor regression, we first factor the joint distribution for

`disability`and

`depress`conditional on

`severity`and

`male`.

`depress`. These three paths match the same three variables that appear right of the pipe (i.e., ∣). Similarly, we have two paths pointing to

`depress`and the same two variables that appear right of the pipe above. These conditional distributions correspond to the two regression equations we expect from a mediation analysis, and we specify the models via the following Blimp syntax.

`# Specify the single mediator model`

**MODEL:**`# Single-Mediator model controlling for biological sex`

`disability ∼ [email protected] severity male;`

`depress ∼ [email protected] male;`

**FIXED:**male; # Specify no distribution for male

**PARAMETERS:**# Post compute the mediated effect`indirect = apath * bpath;`

`MODEL`command signifies that we would like to specify our factored models. We list the first factorization,

`disability`conditional on

`depress`,

`severity`, and

`male`, on the first line. A tilde (∼) replaces the pipe in the factorization to specify the appropriate regression model. The

`@`syntax denotes that we want the regression slope between

`disability`and

`depress`to be labeled as the

`bpath`. The second line after the

`MODEL`command specifies the second factorization,

`depress`conditional on

`severity`, and

`male`and has the

`apath`label. Both of these labeled paths are indicated in Figure 3. Note, we did not specify any regressions for

`male`or

`severity`. By default, Blimp will specify a partially factored model for all predictors (i.e., who are never left of a tilde). Since

`male`is complete, there is no need to include any distributional assumptions about it, and we can indicate this using the

`FIXED`command. Thus, Blimp will by default will estimate the regression of

`severity`on

`male`for us. Alternatively, we could explicitly specify this model by including “

`severity ∼ male`” in the

`MODEL`command.

`PARAMETERS`command allows us to specify quantities of interest. As we will see below, one of the advantages of using simulation methods to estimate the models is that we can easily take the sampled values and create quantities of interest. These quantities will have all the same summarizes as any other parameter (e.g., point estimate, uncertainty, and interval estimates). In this example, we are calculating the mediated effect, saved as the parameter called

`indirect`, by multiplying the

`apath`times the

`bpath`.

`# Specify the MCMC sampler parameters`

**SEED:**398721; # Set a prng seed

**BURN:**1000; # Set number of burn iterations

**ITERATIONS:**10000; # Set number of post-burn iterations

**CHAINS:**4; # Specify number independent of chains`SEED`command is an arbitrary positive integer used to replicate the results of the pseudo-random number generator. While many programs will have a default value if not specified, Blimp purposely requires one to be specified to ensure replicability. Second, the

`BURN`command specifies the number of warm-up iterations the MCMC sampler runs. These iterations will be discarded and not summarized for the parameter summaries. We use the burn-in iterations to ensure convergence, that is, properly sampling from the posterior distributions for the parameters and imputations. We will discuss this more when we look at the output below. Thirdly, the

`ITERATIONS`command requests the total number of post-burn iterations we want to be drawn and summarized as part of the MCMC estimation procedure. Fourthly, the

`CHAINS`command specifies how many independent MCMC processes we want to run. In our example, we are requesting four to be run simultaneously with random starting values. Each chain will be run on a separate processor, allowing us better to utilize the computational power of a modern computer.

#### 4.1.1. Adding Auxiliary Variables to the Model

`anxiety`), stress (

`stress`), and perceived control over pain (

`control`).

`anxiety`,

`stress`, and

`control`into the model as auxiliary variables, we must not substantively change the meaning of the analysis model. We accomplish this by modeling the joint distribution of the auxiliary variables conditional on the predictors and outcomes. Just as we factored the analysis model, we take the joint distribution of the auxiliary variables conditional on the four variables from the analysis (two predictors and two outcomes) and factorize it into the following three conditional distributions.

`MODEL`command with the following one that includes the factored auxiliary variables.

`# Specify the single mediator model`

**MODEL:**`# Single-Mediator model controlling for biological sex`

`disability ∼ [email protected] severity male;`

`depress ∼ [email protected] male;`

`# Model for the Auxiliary Variables`

`anxiety ∼ stress control disability depress severity male;`

`stress ∼ control disability depress severity male;`

`control ∼ disability depress severity male;`

`disability`,

`depress`, and

`severity`), we explicitly include the conditional distributions into the factorization without changing the meaning of our original analysis model’s densities. In other words, when we draw imputations on the missing variables, these factored regression densities will be a part of the sampling step. To illustrate, the distribution of

`depress`conditional on all other variables is proportional to the product of five densities.

`# Specify auxiliary variable model with one line`

`anxiety stress control ∼ disability depress severity male;`

#### 4.1.2. Output from Single Mediator Model

`OPTIONS: labels`will also print out a table displaying the numbers. As a general rule of thumb, we expect the MCMC sampler to converge once all PSR statistics are below approximately $1.05$ to $1.10$ [25], and the table indicates that the algorithm quickly achieved this within the 1000 burn-in iterations requested.

`PARAMETERS`command, and parameters that are used in the unspecified default models (referred to as a “Predictor Model” in Blimp).

`severity`. The

`PREDICTORS`section indicates that we have fixed

`male`(i.e., made no distributional assumptions about the complete predictor) and estimated an ordinal probit model for the binary

`severity`variable. The subsequent output then lists all five models that we specified in the syntax and the one generated parameter,

`indirect`.

`MODELS`

`[1] anxiety ∼ Intercept stress control disability depress severity male`

`[2] control ∼ Intercept disability depress severity male`

`[3] depress ∼ Intercept [email protected] male`

`[4] disability ∼ Intercept [email protected] severity male`

`[5] stress ∼ Intercept control disability depress severity male`

`GENERATED PARAMETERS`

`[1] indirect = apath*bpath`

`Intercept`. This section serves as an overview of the regression equations and reflects the printing order of the models. We can see the three auxiliary models (i.e., regressions for

`anxiety`,

`control`, and

`stress`) and two analysis models with the labeled paths. Finally, there is also a section showing how the generated parameters were computed—i.e., the

`indirect`parameter was computed by multiplying the

`apath`and

`bpath`parameters.

`OUTCOME MODEL ESTIMATES:`

`Summaries based on 10,000 iterations using 4 chains.`

`disability`model’s output.

`N_EFF`). For those unfamiliar with results from a Bayesian analysis, heuristically, we can think of the posterior median and standard deviation as analogous to the point estimate and standard error. Similarly, the 95% posterior interval is comparable to a confidence interval. The PSR is the same PSR we discussed earlier but now computed on all post-burn-in summaries. The effective sample size is a crude approximation of the “effective number of independent simulation draws” ([25] p. 286) for each parameter. Typically speaking, these will be lower than the actual number of samples because of autocorrelation in the MCMC simulation procedure, and it is recommended that more iterations are needed if the effective sample size is less than ten per chain (e.g., less than 40 in our example; [25] p. 287). The model’s output is sectioned into four main categories. The first two sections are the variance parameters and regression coefficients from the model. The next section is the standardized solutions for the regression coefficients. The final section provides the variance explained by the regression coefficients (i.e., ${R}^{2}$) and the residual variance. For example, our regression coefficients explained about 20% of the variance, and we are 95% confident the value lies between $0.12$ and $0.29$.

`indirect`quantity.

`severity`regressed on

`male`. This output also prints out the same posterior summaries but is not of substantive interest. Rather, the model serves to produce imputations for the incomplete predictor,

`severity`.

#### 4.2. Single Mediator Model with a Moderator

`male`) moderates the A and B paths of the mediation model. The path diagram in Figure 4 adds these two additional paths (labeled

`amod`and

`bmod`in the diagram) with the arrows pointing to the labeled A and B paths. Estimating this model in Blimp is a straightforward extension from the previous example. Notably, the factorization that we discussed in the previous example remains unchanged. What does change is the form of the two substantive models; that is, the models now include the products between

`male`and

`severity`or

`depress`. Below we provide the syntax to extend the mediation model to include the moderated A and B paths.

`# Specify the mediation with moderated paths`

**MODEL:**`# Single-Mediator model with male moderating a and b paths`

`disability ∼ [email protected] severity male depress*[email protected];`

`depress ∼ [email protected] male severity*[email protected];`

`# Specify auxiliary variable model with one line`

`anxiety stress control ∼ disability depress severity male;`

**FIXED:**male; # Specify no distribution for male

**PARAMETERS:**# Post compute the mediated effect`indirect.female = (apath + (amod * 0)) * (bpath + (bmod * 0));`

`indirect.male = (apath + (amod * 1)) * (bpath + (bmod * 1));`

`indirect.diff = indirect.female - indirect.male;`

`male`by

`depress`interaction into the regression and labeled the parameter

`bmod`. Similarly, we have included the

`male`by

`severity`interaction and labeled the parameter

`amod`. Importantly, with these two products added to our regression models, the missing observations in both

`depress`and

`severity`will now be imputed by taking into account the hypothesized nonlinear relationship. Said differently, the likelihoods in the factorizations will directly include the interaction when drawing imputations for missing observations. We have opted to label each moderated path to compute the indirect effects for both males and females. Just like the previous example, we use the

`PARAMETERS`command to post compute the quantities after the sampler estimates the model. The first two lines of the

`PARAMETERS`command computes the indirect effect for females and males, respectively. The third line illustrates that in Blimp, we can use these computed values to calculate the difference in indirect effects between the two groups.

`# Specify Simple command to obtain`

`# conditional regressions`

**SIMPLE:**`severity | male;`

`depress | male;`

**CENTER:**severity depress; # Center variables`SIMPLE`command (shown above) can be added on to the script to compute the conditional effect of

`severity`or

`depress`given

`male`equals zero (i.e., females) and one (i.e., males). The variable to the left of the vertical bar is the focal variable, and to the right is the moderator. In our example, because we have specified

`male`as ordinal, Blimp will produce the conditional intercept and slope for each value of the variable. Finally, in line with a typical interaction analysis, we have centered both

`severity`and

`depress`using the

`CENTER`command. Note, by using the

`CENTER`command, Blimp uses the Bayesian estimated mean to center for both variables. This approach allows us to fully capture the mean estimates’ uncertainty and is especially important when the variables are incomplete.

#### Single Mediator Model with a Moderator Output

`severity`and

`depress`when being used as a predictor.

`CENTERED PREDICTORS`

`Grand Mean Centered: severity depress`

`MODELS`

`[1] anxiety ∼ Intercept stress control disability depress severity male`

`[2] control ∼ Intercept disability depress severity male`

`[3] depress ∼ Intercept [email protected] male severity*[email protected]`

`[4] disability ∼ Intercept [email protected] severity male depress*[email protected]`

`[5] stress ∼ Intercept control disability depress severity male`

`GENERATED PARAMETERS`

`[1] indirect.female = (apath+(amod*0))*(bpath+(bmod*0))`

`[2] indirect.male = (apath+(amod*1))*(bpath+(bmod*1))`

`[3] indirect.diff = indirect.female-indirect.male`

`disability`model again, we present a truncated output table with the standardized coefficients section removed (i.e., where the vertical ellipsis are).

`depress`and

`severity`are centered at their overall means for this model; thus, substantively speaking, the interpretation of the intercept is an adjusted mean for the female’s group. In addition, we now have the

`depress`by

`male`interaction, which resulted in an approximately incremental 2% gain in variance explained when compared to the previous model.

`disability`model, Blimp prints an additional table that provides the conditional effects that we requested with the

`SIMPLE`command.

`disability`predicted by

`severity`only differ in the intercept because there is no

`male`by

`severity`interaction in this regression. The second set of equations,

`disability`predicted by

`depress`, are the intercepts and slopes holding all other predictors constant at zero (i.e., their means). As with all the generated quantities, the conditional slopes also include 95% posterior intervals that give us a sense of the uncertainty around the parameter. Comparing the female and male slopes, we can see that the interval does not include the other posterior median, which would suggest the slope differences are meaningful and most likely not due to sampling variability.

#### 4.3. Adding Latent Variables to the Mediation Model

`depress`and

`disability`. Figure 5 is the path diagram for this model, where the latent factor for depression is normally distributed with seven items as indicators (

`dep1`to

`dep7`; represented by a set of ellipses in the path diagram), and the latent factor for disability is normally distributed with six items as indicators (

`disab1`to

`disab6`). As with our previous path diagrams, we have excluded the mean structure from the diagram. In addition, we have fixed the first item for both factors to one for identification.

`depress`and

`disability`with their latent variables, ${\eta}_{\mathrm{dep}}$ and ${\eta}_{\mathrm{disab}}$. In addition to the structural model, we now must also factor out the measurement model. Multiplying that factorization to the structural model gives us the full functional notation.

`ORDINAL`command; therefore, in line with traditional ordinal factor analysis, their regression models will follow an ordered probit specification. Using the factorization that we discussed above, we can specify our model via the following Blimp model syntax.

`# Declare latent variables`

**LATENT:**eta_dep eta_disab;`# Single-Mediator model with male moderating a and b paths`

**MODEL:**`# Structural Models`

`eta_disab ∼ [email protected] severity male eta_dep*[email protected];`

`eta_dep ∼ [email protected] male severity*[email protected];`

`# Measurement Models`

`dep1 ∼ [email protected]; dep2 ∼ eta_dep; dep3 ∼ eta_dep;`

`dep4 ∼ eta_dep; dep5 ∼ eta_dep; dep6 ∼ eta_dep;`

`dep7 ∼ eta_dep;`

`disab1 ∼ [email protected]; disab2 ∼ eta_disab; disab3 ∼ eta_disab;`

`disab4 ∼ eta_disab; disab5 ∼ eta_disab; disab6 ∼ eta_disab;`

`LATENT`command specifies two new latent variables to add to our data set,

`eta_dep`and

`eta_disab`. By declaring these variables, Blimp will allow us to use them in the

`MODEL`command. As discussed previously, these variables have every observation missing, and each iteration, Blimp will produce imputations via data augmentation according to the model we specified. Mapping onto how we generally conceptualize SEM, we have broken the model syntax down into the structural and measurement parts. The structural part maps onto the same form we specified earlier for the single mediator model. The only difference is replacing the manifest scale scores with their respective latent variables,

`eta_dep`and

`eta_disab`. By default, Blimp excludes the intercept for any latent variable; thus, fixing it to zero for identification. Turning to the measurement model, we specify the regression equations for the two latent variables. In line with standard SEM conventions, we fix the first loading to one (i.e.,

`disab1`and

`dep1`) by using the

`@`symbol followed by a one. While this syntax matches both the factorization and how Blimp conceptualizes the model, the syntax also allows specifying measurement models concisely using a right-pointing arrow (

`->`).

`# Compact syntax to specify measurement models.`

`eta_dep -> dep1:dep7;`

`eta_disab -> disab1:disab6;`

`:`) between the names to list all variables names between

`dep1`to

`dep7`and

`disab1`to

`disab6`. Second, when using a right-pointing arrow (

`->`) to predict variables using a factor, by default, Blimp will fix the first variable’s loading to one. Additionally, note that using this syntax still requires the latent variable to be specified via the

`LATENT`command.

`anxiety`,

`stress`, and

`control`as auxiliary variables. In line with the previous example, we predict every auxiliary variable as a function of all the other manifest variables in our model. As discussed, this will allow for the imputations on the missing analysis variables to account for the relationship to the auxiliary variables.

**MODEL:**`# Specify auxiliary variable model with one line`

`anxiety stress control ∼ disab1:disab6 dep1:dep7 severity male;`

**FIXED:**male; # Specify no distribution for male

**PARAMETERS:**# Post compute the mediated effect`indirect.female = (apath + (amod * 0)) * (bpath + (bmod * 0));`

`indirect.male = (apath + (amod * 1)) * (bpath + (bmod * 1));`

`indirect.diff = indirect.female - indirect.male;`

**SIMPLE:**# Specify conditional regressions`severity | male;`

`eta_dep | male;`

**CENTER:**severity; # Center variables`male`is fixed with no distribution and post compute the mediated effect for both groups and the difference between the two effects. Finally, we center

`severity`and request for the conditional regression effects for our two focal predictors given the moderator,

`male`.

#### Latent Single Mediator Model with a Moderator Output

`dep2`regression model.

`eta_dep`, and the standardized coefficient is analogous to the standardized solution in any confirmatory factor analysis. In addition, because there is only one predictor, the proportion of variance explained by the coefficients is equivalent to the estimate of

`dep2`’s reliability under a factor analytic approach. As with other values, Blimp provides the posterior interval for this measure, characterizing the precision in the reliability coefficient.

`eta_dep`by manifest

`male`interaction, and the imputations on

`eta_dep`are drawn in accordance to the nonlinearity. When comparing the above results to the previous example, the scaling of the regression slopes has changed because of the latent variable. Despite this, we can see that we are explaining about 10% more variance with the latent variable ($0.31$ versus $0.21$ with manifest scale score), and this is one of the advantages of incorporating a model to account for the measurement error.

## 5. Conclusions

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Name | Definition | Missing % | Range |
---|---|---|---|

male | Biological Sex Dummy Code | 0.0 | 0 or 1 |

severity | Severe pain dummy code | 7.3 | 0 or 1 |

depress | Depression composite score | 13.5 | 7 to 28 |

disability | Psychosocial disability composite | 9.1 | 10 to 34 |

dep1 | Couldn’t experience any positive feelings at all | 4.7 | 1 to 4 |

dep2 | Difficult to work up the initiative to do things | 2.2 | 1 to 4 |

dep3 | I felt that I had nothing to look forward to | 1.8 | 1 to 4 |

dep4 | I felt down-hearted and blue | 1.5 | 1 to 4 |

dep5 | Unable to become enthusiastic about anything | 2.2 | 1 to 4 |

dep6 | I felt I wasn’t worth much as a person | 4.0 | 1 to 4 |

dep7 | I felt that life was meaningless | 2.9 | 1 to 4 |

disab1 | I isolate myself as much as I can from the family | 3.3 | 1 to 6 |

disab2 | I am doing fewer social activities | 4.7 | 1 to 6 |

disab3 | I sometimes behave as if I were confused | 3.6 | 1 to 6 |

disab4 | I laugh or cry suddenly | 3.6 | 1 to 6 |

disab5 | I act irritable and impatient with myself | 4.7 | 1 to 6 |

disab6 | I do not speak clearly when I am under stress | 3.6 | 1 to 6 |

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Keller, B.T. An Introduction to Factored Regression Models with Blimp. *Psych* **2022**, *4*, 10-37.
https://doi.org/10.3390/psych4010002

**AMA Style**

Keller BT. An Introduction to Factored Regression Models with Blimp. *Psych*. 2022; 4(1):10-37.
https://doi.org/10.3390/psych4010002

**Chicago/Turabian Style**

Keller, Brian Tinnell. 2022. "An Introduction to Factored Regression Models with Blimp" *Psych* 4, no. 1: 10-37.
https://doi.org/10.3390/psych4010002