3 Example

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

library(modelbpp)
head(dat_serial_4_weak)
#>             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 0.6-17
#> 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 0.6.17 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           : 9
#> Number of model(s) converged        : 9
#> Number of model(s) passed post.check: 9
#> 
#> The models (sorted by BPP):
#>             model_df df_diff Prior     BIC   BPP Cumulative
#> original           2       0 0.111 464.320 0.253      0.253
#> drop: y~x          3      -1 0.111 464.340 0.251      0.504
#> add: m1~~y         1       1 0.111 465.900 0.115      0.619
#> add: y~m1          1       1 0.111 465.900 0.115      0.733
#> drop: m2~m1        3      -1 0.111 465.953 0.112      0.845
#> add: m1~y          1       1 0.111 466.142 0.102      0.947
#> add: m2~x          1       1 0.111 468.575 0.030      0.977
#> drop: m1~x         3      -1 0.111 469.403 0.020      0.997
#> drop: y~m2         3      -1 0.111 473.291 0.003      1.000
#> 
#> 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.
#> - Cumulative: Cumulative BIC posterior probability.

The total number of models examined, including the original model, is 9.

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 (0.251)

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           : 9
#> Number of model(s) converged        : 9
#> Number of model(s) passed post.check: 9
#> 
#> The models (sorted by BPP):
#>             model_df df_diff Prior     BIC   BPP Cumulative
#> original           2       0 0.500 464.320 0.730      0.730
#> drop: y~x          3      -1 0.062 464.340 0.090      0.821
#> add: m1~~y         1       1 0.062 465.900 0.041      0.862
#> add: y~m1          1       1 0.062 465.900 0.041      0.904
#> drop: m2~m1        3      -1 0.062 465.953 0.040      0.944
#> add: m1~y          1       1 0.062 466.142 0.037      0.981
#> add: m2~x          1       1 0.062 468.575 0.011      0.992
#> drop: m1~x         3      -1 0.062 469.403 0.007      0.999
#> drop: y~m2         3      -1 0.062 473.291 0.001      1.000
#> 
#> 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.
#> - Cumulative: Cumulative BIC posterior probability.

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 Advanced Options

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_add = 2,
                      df_change_drop = 2)

This is the printout. By default, when there are more than 10 models, only the top 10 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           : 18
#> Number of model(s) converged        : 18
#> Number of model(s) passed post.check: 18
#> 
#> The models (sorted by BPP):
#>                  model_df df_diff Prior     BIC   BPP Cumulative
#> original                2       0 0.056 464.320 0.214      0.214
#> drop: y~x               3      -1 0.056 464.340 0.212      0.426
#> add: m1~~y              1       1 0.056 465.900 0.097      0.523
#> add: y~m1               1       1 0.056 465.900 0.097      0.620
#> drop: m2~m1             3      -1 0.056 465.953 0.095      0.714
#> drop: m2~m1;y~x         4      -2 0.056 465.973 0.094      0.808
#> add: m1~y               1       1 0.056 466.142 0.086      0.894
#> add: m2~x               1       1 0.056 468.575 0.025      0.920
#> drop: m1~x              3      -1 0.056 469.403 0.017      0.936
#> drop: m1~x;y~x          4      -2 0.056 469.423 0.017      0.953
#> add: m1~y;m2~x          0       2 0.056 470.154 0.012      0.965
#> add: m1~~y;m2~x         0       2 0.056 470.154 0.012      0.976
#> add: m2~x;y~m1          0       2 0.056 470.154 0.012      0.988
#> drop: m1~x;m2~m1        4      -2 0.056 471.036 0.007      0.995
#> drop: y~m2              3      -1 0.056 473.291 0.002      0.998
#> drop: y~m2;y~x          4      -2 0.056 474.819 0.001      0.999
#> drop: m2~m1;y~m2        4      -2 0.056 474.924 0.001      1.000
#> drop: m1~x;y~m2         4      -2 0.056 478.374 0.000      1.000
#> 
#> 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.
#> - Cumulative: Cumulative BIC posterior probability.

The number of models examined, including the original model, is 18.

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.

4.2 Excluding Some Parameters From the Search

When calling model_set(), users can specify parameters that must be excluded from the list to be added (must_not_add), or must not be dropped (must_not_drop).

For example, it may not make sense To add the error covariance between m1 and y (m1~~y). We can exclude it when calling model_set()

out1_no_cov <- model_set(fit1,
                         must_not_add = "m1~~y")

This is the output:

out1_no_cov
#> 
#> Call:
#> model_set(sem_out = fit1, must_not_add = "m1~~y")
#> 
#> Number of model(s) fitted           : 8
#> Number of model(s) converged        : 8
#> Number of model(s) passed post.check: 8
#> 
#> The models (sorted by BPP):
#>             model_df df_diff Prior     BIC   BPP Cumulative
#> original           2       0 0.125 464.320 0.286      0.286
#> drop: y~x          3      -1 0.125 464.340 0.283      0.569
#> add: y~m1          1       1 0.125 465.900 0.130      0.699
#> drop: m2~m1        3      -1 0.125 465.953 0.126      0.825
#> add: m1~y          1       1 0.125 466.142 0.115      0.940
#> add: m2~x          1       1 0.125 468.575 0.034      0.974
#> drop: m1~x         3      -1 0.125 469.403 0.023      0.997
#> drop: y~m2         3      -1 0.125 473.291 0.003      1.000
#> 
#> 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.
#> - Cumulative: Cumulative BIC posterior probability.

The number of models reduced to 8.

This is the plot:

graph1_no_cov <- model_graph(out1_no_cov)
plot(graph1_no_cov)

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           : 9
#> Number of model(s) converged        : 9
#> Number of model(s) passed post.check: 9
#> 
#> The models (sorted by BPP):
#>             model_df df_diff Prior     BIC   BPP Cumulative
#> original           2       0 0.500 464.320 0.730      0.730
#> drop: y~x          3      -1 0.062 464.340 0.090      0.821
#> add: m1~~y         1       1 0.062 465.900 0.041      0.862
#> add: y~m1          1       1 0.062 465.900 0.041      0.904
#> drop: m2~m1        3      -1 0.062 465.953 0.040      0.944
#> add: m1~y          1       1 0.062 466.142 0.037      0.981
#> add: m2~x          1       1 0.062 468.575 0.011      0.992
#> drop: m1~x         3      -1 0.062 469.403 0.007      0.999
#> drop: y~m2         3      -1 0.062 473.291 0.001      1.000
#> 
#> 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.
#> - Cumulative: Cumulative BIC posterior probability.

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.

Get Started

Shu Fai Cheung

2024-02-20

1 Introduction

2 Workflow

3 Example

3.1 Step 1: Fit the Original Model

3.2 Step 2: Call `model_set()`

3.3 Step 3: Examine the Results

3.4 Repeat Step 2 with User Prior

4 Advanced Options

4.1 More Neighboring Models

4.2 Excluding Some Parameters From the Search

4.3 Models With Constraints

4.4 Recompute BPPs Without Refitting the Models

4.5 Many Neighboring Models

4.6 More Options

5 Further Information

References

Get Started

Shu Fai Cheung

2024-02-20

1 Introduction

2 Workflow

3 Example

3.1 Step 1: Fit the Original Model

3.2 Step 2: Call model_set()

3.3 Step 3: Examine the Results

3.4 Repeat Step 2 with User Prior

4 Advanced Options

4.1 More Neighboring Models

4.2 Excluding Some Parameters From the Search

4.3 Models With Constraints

4.4 Recompute BPPs Without Refitting the Models

4.5 Many Neighboring Models

4.6 More Options

5 Further Information

References

3.2 Step 2: Call `model_set()`