# 1 Introduction

This article illustrates how to use model_set() and other functions from the package modelbpp to:

• Fit a set of neighboring models, each has one more or one less degree of freedom than the original fitted model.

• Compute the BIC posterior probability (BPP), for each model (Wu et al., 2020).

• Use BPP to assess to what extent each model is supported by the data, compared to all other models under consideration.

# 2 Workflow

1. Fit an SEM model, the original model, as usual in lavaan.

2. Call model_set() on the output from Step 1. It will automatically do the following:

• Enumerate the neighboring models of the original model.

• Fit all the models and compute their BIC posterior probabilities (BPPs).

3. Examine the results by:

• printing the output of model_set(), or

• generating a graph using model_graph().

# 3 Example

This is a sample dataset, dat_serial_4_weak, with four variables:

library(modelbpp)
#>             x         m1         m2          y
#> 1  0.09107195  1.2632493  0.7823926  0.3252093
#> 2 -1.96063838 -0.7745526  0.2002867 -0.6379673
#> 3  0.20184014  0.2238152  0.2374072  0.4205998
#> 4 -2.25521708  0.1185732 -0.1727878  0.9320889
#> 5 -0.15321350  0.1509888  1.1251386  0.6892537
#> 6 -2.00640303 -0.1595208 -0.1553136 -0.2364792

## 3.1 Step 1: Fit the Original Model

Fit this original model, a serial mediation model, with one direct path, from x to y:

library(lavaan)
#> This is lavaan 0.6-18
#> lavaan is FREE software! Please report any bugs.
mod1 <-
"
m1 ~ x
m2 ~ m1
y ~ m2 + x
"
fit1 <- sem(mod1, dat_serial_4_weak)

This the summary:

summary(fit1,
fit.measures = TRUE)
#> lavaan 0.6-18 ended normally after 1 iteration
#>
#>   Estimator                                         ML
#>   Optimization method                           NLMINB
#>   Number of model parameters                         7
#>
#>   Number of observations                           100
#>
#> Model Test User Model:
#>
#>   Test statistic                                 3.376
#>   Degrees of freedom                                 2
#>   P-value (Chi-square)                           0.185
#>
#> Model Test Baseline Model:
#>
#>   Test statistic                                39.011
#>   Degrees of freedom                                 6
#>   P-value                                        0.000
#>
#> User Model versus Baseline Model:
#>
#>   Comparative Fit Index (CFI)                    0.958
#>   Tucker-Lewis Index (TLI)                       0.875
#>
#> Loglikelihood and Information Criteria:
#>
#>   Loglikelihood user model (H0)               -216.042
#>   Loglikelihood unrestricted model (H1)       -214.354
#>
#>   Akaike (AIC)                                 446.084
#>   Bayesian (BIC)                               464.320
#>   Sample-size adjusted Bayesian (SABIC)        442.212
#>
#> Root Mean Square Error of Approximation:
#>
#>   RMSEA                                          0.083
#>   90 Percent confidence interval - lower         0.000
#>   90 Percent confidence interval - upper         0.232
#>   P-value H_0: RMSEA <= 0.050                    0.261
#>   P-value H_0: RMSEA >= 0.080                    0.628
#>
#> Standardized Root Mean Square Residual:
#>
#>   SRMR                                           0.050
#>
#> Parameter Estimates:
#>
#>   Standard errors                             Standard
#>   Information                                 Expected
#>   Information saturated (h1) model          Structured
#>
#> Regressions:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   m1 ~
#>     x                 0.187    0.059    3.189    0.001
#>   m2 ~
#>     m1                0.231    0.091    2.537    0.011
#>   y ~
#>     m2                0.341    0.089    3.835    0.000
#>     x                 0.113    0.052    2.188    0.029
#>
#> Variances:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>    .m1                0.278    0.039    7.071    0.000
#>    .m2                0.254    0.036    7.071    0.000
#>    .y                 0.213    0.030    7.071    0.000

The fit is acceptable, though the RMSEA is marginal (CFI = 0.958, RMSEA = 0.083).

## 3.2 Step 2: Call model_set()

Use model_set() to find the neighboring models differ from the target model by one on model degrees of freedom, fit them, and compute the BPPs:

out1 <- model_set(fit1)

## 3.3 Step 3: Examine the Results

To examine the results, just print the output:

out1
#>
#> Call:
#> model_set(sem_out = fit1)
#>
#> Number of model(s) fitted           : 7
#> Number of model(s) converged        : 7
#> Number of model(s) passed post.check: 7
#>
#> The models (sorted by BPP):
#>             model_df df_diff Prior     BIC   BPP   cfi rmsea
#> original           2       0 0.143 464.320 0.323 0.958 0.083
#> drop: y~x          3      -1 0.143 464.340 0.320 0.848 0.129
#> add: y~m1          1       1 0.143 465.900 0.147 1.000 0.000
#> drop: m2~m1        3      -1 0.143 465.953 0.143 0.800 0.148
#> add: m2~x          1       1 0.143 468.575 0.038 0.939 0.142
#> drop: m1~x         3      -1 0.143 469.403 0.025 0.695 0.183
#> drop: y~m2         3      -1 0.143 473.291 0.004 0.577 0.216
#>
#> Note:
#> - BIC: Bayesian Information Criterion.
#> - BPP: BIC posterior probability.
#> - model_df: Model degrees of freedom.
#> - df_diff: Difference in df compared to the original/target model.
#> - To show cumulative BPPs, call print() with 'cumulative_bpp = TRUE'.
#> - At least one model has fixed.x = TRUE. The models are not checked for
#>   equivalence.
#> - Since Version 0.1.3.5, the default values of exclude_feedback and
#>   exclude_xy_cov changed to TRUE. Set them to FALSE to reproduce
#>   results from previous versions.

The total number of models examined, including the original model, is 7.

(Note: The total number of models was 9 in previous version. Please refer to the Note in the printout for the changes.)

The BIC posterior probabilities (BPPs) suggest that the original model is indeed the most probable model based on BPP. However, the model with the direct path dropped, drop: y~x, only has slightly lower BPP (0.320)

This suggests that, with equal prior probabilities (Wu et al., 2020), the support for the model with the direct and without the direct path have similar support from the data based on BPP.

Alternatively, we can use model_graph() to visualize the BPPs and model relations graphically:

graph1 <- model_graph(out1)
plot(graph1)

Each node (circle) represents one model. The larger the BPP, the larger the node.

The arrow points from a simpler model (a model with larger model df) to a more complicated model (a model with smaller model df). If two models are connected by an arrow, then one model can be formed from another model by adding or removing one free parameter (e.g., adding or removing one path).

## 3.4 Repeat Step 2 with User Prior

In real studies, not all models are equally probable before having data (i.e., not all models have equal prior probabilities). A researcher fits the original model because

• its prior probability is higher than other models, at least other neighboring models (otherwise, it is not worthy of collecting data assess thi original model), but

• the prior probability is not so high to eliminate the need for collecting data to see how much it is supported by data.

Suppose we decide that the prior probability of the original model is .50: probable, but still needs data to decide whether it is empirically supported

This can be done by setting prior_sem_out to the desired prior probability when calling model_set():

out1_prior <- model_set(fit1,
prior_sem_out = .50)

The prior probabilities of all other models are equal. Therefore, with nine models and the prior of the target model being .50, the prior probability of the other eight model is (1 - .50) / 8 or .0625.

This is the printout:

out1_prior
#>
#> Call:
#> model_set(sem_out = fit1, prior_sem_out = 0.5)
#>
#> Number of model(s) fitted           : 7
#> Number of model(s) converged        : 7
#> Number of model(s) passed post.check: 7
#>
#> The models (sorted by BPP):
#>             model_df df_diff Prior     BIC   BPP   cfi rmsea
#> original           2       0 0.500 464.320 0.741 0.958 0.083
#> drop: y~x          3      -1 0.083 464.340 0.122 0.848 0.129
#> add: y~m1          1       1 0.083 465.900 0.056 1.000 0.000
#> drop: m2~m1        3      -1 0.083 465.953 0.055 0.800 0.148
#> add: m2~x          1       1 0.083 468.575 0.015 0.939 0.142
#> drop: m1~x         3      -1 0.083 469.403 0.010 0.695 0.183
#> drop: y~m2         3      -1 0.083 473.291 0.001 0.577 0.216
#>
#> Note:
#> - BIC: Bayesian Information Criterion.
#> - BPP: BIC posterior probability.
#> - model_df: Model degrees of freedom.
#> - df_diff: Difference in df compared to the original/target model.
#> - To show cumulative BPPs, call print() with 'cumulative_bpp = TRUE'.
#> - At least one model has fixed.x = TRUE. The models are not checked for
#>   equivalence.
#> - Since Version 0.1.3.5, the default values of exclude_feedback and
#>   exclude_xy_cov changed to TRUE. Set them to FALSE to reproduce
#>   results from previous versions.

If the prior of the target is set to .50, then, taking into account both the prior probabilities and the data, the target model is strongly supported by the data.

This is the output of model_graph():

graph1_prior <- model_graph(out1_prior)
plot(graph1_prior)

## 4.1 More Neighboring Models

If desired, we can enumerate models “farther away” from the target model. For example, we can set the maximum difference in model df to 2, to include models having two more or two less df than the original model:

out1_df2 <- model_set(fit1,
df_change_drop = 2)

This is the printout. By default, when there are more than 20 models, only the top 20 models on BPP will be printed:

out1_df2
#>
#> Call:
#> model_set(sem_out = fit1, df_change_add = 2, df_change_drop = 2)
#>
#> Number of model(s) fitted           : 14
#> Number of model(s) converged        : 14
#> Number of model(s) passed post.check: 14
#>
#> The models (sorted by BPP):
#>                  model_df df_diff Prior     BIC   BPP   cfi rmsea
#> original                2       0 0.071 464.320 0.269 0.958 0.083
#> drop: y~x               3      -1 0.071 464.340 0.267 0.848 0.129
#> add: y~m1               1       1 0.071 465.900 0.122 1.000 0.000
#> drop: m2~m1             3      -1 0.071 465.953 0.119 0.800 0.148
#> drop: m2~m1;y~x         4      -2 0.071 465.973 0.118 0.690 0.160
#> add: m2~x               1       1 0.071 468.575 0.032 0.939 0.142
#> drop: m1~x              3      -1 0.071 469.403 0.021 0.695 0.183
#> drop: m1~x;y~x          4      -2 0.071 469.423 0.021 0.585 0.185
#> add: m2~x;y~m1          0       2 0.071 470.154 0.015 1.000 0.000
#> drop: m1~x;m2~m1        4      -2 0.071 471.036 0.009 0.536 0.196
#> drop: y~m2              3      -1 0.071 473.291 0.003 0.577 0.216
#> drop: y~m2;y~x          4      -2 0.071 474.819 0.001 0.422 0.218
#> drop: m2~m1;y~m2        4      -2 0.071 474.924 0.001 0.419 0.219
#> drop: m1~x;y~m2         4      -2 0.071 478.374 0.000 0.314 0.238
#>
#> Note:
#> - BIC: Bayesian Information Criterion.
#> - BPP: BIC posterior probability.
#> - model_df: Model degrees of freedom.
#> - df_diff: Difference in df compared to the original/target model.
#> - To show cumulative BPPs, call print() with 'cumulative_bpp = TRUE'.
#> - At least one model has fixed.x = TRUE. The models are not checked for
#>   equivalence.
#> - Since Version 0.1.3.5, the default values of exclude_feedback and
#>   exclude_xy_cov changed to TRUE. Set them to FALSE to reproduce
#>   results from previous versions.

The number of models examined, including the original model, is 14.

This is the output of model_graph():

graph1_df2 <- model_graph(out1_df2,
node_label_size = .75)
plot(graph1_df2)

Note: Due to the number of nodes, node_label_size is used to reduce the size of the labels.

## 4.3 Models With Constraints

If the original model has equality constraints, they will be included in the search for neighboring models, by default. That is, removing one equality constraint between two models is considered as a model with an increase of 1 df.

## 4.4 Recompute BPPs Without Refitting the Models

Users can examine the impact of the prior probability of the original model without refitting the models, by using the output of model_set() as the input, using the model_set_out argument:

out1_new_prior <- model_set(model_set_out = out1,
prior_sem_out = .50)

The results are identical to calling model_set() with the original lavaan output as the input:

out1_new_prior
#>
#> Call:
#> model_set(model_set_out = out1, prior_sem_out = 0.5)
#>
#> Number of model(s) fitted           : 7
#> Number of model(s) converged        : 7
#> Number of model(s) passed post.check: 7
#>
#> The models (sorted by BPP):
#>             model_df df_diff Prior     BIC   BPP   cfi rmsea
#> original           2       0 0.500 464.320 0.741 0.958 0.083
#> drop: y~x          3      -1 0.083 464.340 0.122 0.848 0.129
#> add: y~m1          1       1 0.083 465.900 0.056 1.000 0.000
#> drop: m2~m1        3      -1 0.083 465.953 0.055 0.800 0.148
#> add: m2~x          1       1 0.083 468.575 0.015 0.939 0.142
#> drop: m1~x         3      -1 0.083 469.403 0.010 0.695 0.183
#> drop: y~m2         3      -1 0.083 473.291 0.001 0.577 0.216
#>
#> Note:
#> - BIC: Bayesian Information Criterion.
#> - BPP: BIC posterior probability.
#> - model_df: Model degrees of freedom.
#> - df_diff: Difference in df compared to the original/target model.
#> - To show cumulative BPPs, call print() with 'cumulative_bpp = TRUE'.
#> - At least one model has fixed.x = TRUE. The models are not checked for
#>   equivalence.
#> - Since Version 0.1.3.5, the default values of exclude_feedback and
#>   exclude_xy_cov changed to TRUE. Set them to FALSE to reproduce
#>   results from previous versions.

## 4.5 Many Neighboring Models

When a model has a lot of free parameters, the number of neighboring models can be large and it will take a long time to fit all of them. Users can enable parallel processing by setting parallel to TRUE when calling model_set().

## 4.6 More Options

Please refer to the help page of model_set() for options available.

# 5 Further Information

For further information on other functions, please refer to their help pages.

# References

Wu, H., Cheung, S. F., & Leung, S. O. (2020). Simple use of BIC to assess model selection uncertainty: An illustration using mediation and moderation models. Multivariate Behavioral Research, 55(1), 1–16. https://doi.org/10.1080/00273171.2019.1574546