Example of SEM in the ‘Real World’

These are data from a clinical interview of personality pathology in a large mixed group of participants assessed in Pittsburgh (PI: Paul Pilkonis). We have 0/1/2 (absent, subthreshold, present) for diagnostic criteria of 10 DSM-IV personality disorders.

load("pdsymptoms_for_sem.RData")
psych::describe(allPD)

	vars	n	mean	sd	median	trimmed	mad	min	max	range	skew	kurtosis	se
PTNUM	1	445	4616.052	2363.311	5082	4707.655	2987.4	1004	9011	8007	-0.297	-1.389	112.032
Par_1	2	445	0.151	0.411	0	0.039	0.0	0	2	2	2.777	7.341	0.019
Par_2	3	445	0.231	0.518	0	0.109	0.0	0	2	2	2.178	3.822	0.025
Par_3	4	445	0.178	0.458	0	0.059	0.0	0	2	2	2.603	6.130	0.022
Par_4	5	445	0.256	0.559	0	0.120	0.0	0	2	2	2.081	3.199	0.027
Par_5	6	445	0.294	0.590	0	0.157	0.0	0	2	2	1.854	2.239	0.028
Par_6	7	445	0.171	0.472	0	0.039	0.0	0	2	2	2.796	6.974	0.022
Par_7	8	445	0.052	0.268	0	0.000	0.0	0	2	2	5.625	33.143	0.013
Schizoid_1	9	445	0.029	0.181	0	0.000	0.0	0	2	2	6.660	48.514	0.009
Schizoid_2	10	445	0.130	0.426	0	0.000	0.0	0	2	2	3.368	10.653	0.020
Schizoid_3	11	445	0.117	0.402	0	0.000	0.0	0	2	2	3.587	12.361	0.019
Schizoid_4	12	445	0.025	0.169	0	0.000	0.0	0	2	2	7.431	60.895	0.008
Schizoid_5	13	445	0.198	0.546	0	0.036	0.0	0	2	2	2.642	5.527	0.026
Schizoid_6	14	445	0.013	0.115	0	0.000	0.0	0	1	1	8.408	68.856	0.005
Schizoid_7	15	445	0.045	0.228	0	0.000	0.0	0	2	2	5.462	32.307	0.011
Schizotypal_1	16	445	0.027	0.162	0	0.000	0.0	0	1	1	5.821	31.953	0.008
Schizotypal_2	17	445	0.065	0.273	0	0.000	0.0	0	2	2	4.459	21.091	0.013
Schizotypal_3	18	445	0.054	0.245	0	0.000	0.0	0	2	2	4.815	24.773	0.012
Schizotypal_4	19	445	0.029	0.205	0	0.000	0.0	0	2	2	7.682	62.769	0.010
Schizotypal_5	20	445	0.025	0.155	0	0.000	0.0	0	1	1	6.101	35.307	0.007
Schizotypal_6	21	445	0.022	0.188	0	0.000	0.0	0	2	2	9.037	85.172	0.009
Schizotypal_7	22	445	0.031	0.210	0	0.000	0.0	0	2	2	7.325	57.318	0.010
Schizotypal_8	23	445	0.036	0.239	0	0.000	0.0	0	2	2	7.091	51.540	0.011
Schizotypal_9	24	445	0.013	0.134	0	0.000	0.0	0	2	2	11.018	133.771	0.006
Histrionic_1	25	445	0.178	0.477	0	0.048	0.0	0	2	2	2.709	6.499	0.023
Histrionic_2	26	445	0.193	0.487	0	0.067	0.0	0	2	2	2.521	5.517	0.023
Histrionic_3	27	445	0.292	0.608	0	0.140	0.0	0	2	2	1.913	2.325	0.029
Histrionic_4	28	445	0.164	0.453	0	0.039	0.0	0	2	2	2.814	7.250	0.021
Histrionic_5	29	445	0.124	0.392	0	0.003	0.0	0	2	2	3.317	10.782	0.019
Histrionic_6	30	445	0.261	0.577	0	0.115	0.0	0	2	2	2.090	3.118	0.027
Histrionic_7	31	445	0.157	0.463	0	0.022	0.0	0	2	2	2.985	8.043	0.022
Histrionic_8	32	445	0.155	0.431	0	0.036	0.0	0	2	2	2.846	7.608	0.020
Narcissistic_1	33	445	0.207	0.518	0	0.070	0.0	0	2	2	2.467	5.035	0.025
Narcissistic_2	34	445	0.218	0.528	0	0.081	0.0	0	2	2	2.370	4.543	0.025
Narcissistic_3	35	445	0.166	0.469	0	0.034	0.0	0	2	2	2.857	7.312	0.022
Narcissistic_4	36	445	0.238	0.551	0	0.098	0.0	0	2	2	2.227	3.802	0.026
Narcissistic_5	37	445	0.375	0.668	0	0.221	0.0	0	2	2	1.522	0.886	0.032
Narcissistic_6	38	445	0.227	0.537	0	0.090	0.0	0	2	2	2.300	4.191	0.025
Narcissistic_7	39	445	0.348	0.639	0	0.199	0.0	0	2	2	1.615	1.274	0.030
Narcissistic_8	40	445	0.290	0.603	0	0.140	0.0	0	2	2	1.920	2.374	0.029
Narcissistic_9	41	445	0.283	0.590	0	0.140	0.0	0	2	2	1.939	2.518	0.028
Antisocial_1	42	445	0.285	0.631	0	0.112	0.0	0	2	2	1.998	2.455	0.030
Antisocial_2	43	445	0.110	0.402	0	0.000	0.0	0	2	2	3.773	13.574	0.019
Antisocial_3	44	445	0.065	0.281	0	0.000	0.0	0	2	2	4.650	22.912	0.013
Antisocial_4	45	445	0.130	0.431	0	0.000	0.0	0	2	2	3.392	10.721	0.020
Antisocial_5	46	445	0.184	0.513	0	0.036	0.0	0	2	2	2.738	6.284	0.024
Antisocial_6	47	445	0.274	0.627	0	0.098	0.0	0	2	2	2.072	2.722	0.030
Antisocial_7	48	445	0.213	0.534	0	0.070	0.0	0	2	2	2.439	4.782	0.025
Borderline_1	49	445	0.258	0.572	0	0.115	0.0	0	2	2	2.097	3.175	0.027
Borderline_2	50	445	0.443	0.697	0	0.305	0.0	0	2	2	1.259	0.159	0.033
Borderline_3	51	445	0.400	0.692	0	0.252	0.0	0	2	2	1.437	0.569	0.033
Borderline_4	52	445	0.526	0.770	0	0.409	0.0	0	2	2	1.038	-0.533	0.036
Borderline_5	53	445	0.270	0.607	0	0.106	0.0	0	2	2	2.078	2.874	0.029
Borderline_6	54	445	0.528	0.781	0	0.412	0.0	0	2	2	1.039	-0.575	0.037
Borderline_7	55	445	0.553	0.791	0	0.443	0.0	0	2	2	0.970	-0.719	0.038
Borderline_8	56	445	0.488	0.746	0	0.361	0.0	0	2	2	1.146	-0.259	0.035
Borderline_9	57	445	0.160	0.469	0	0.022	0.0	0	2	2	2.972	7.900	0.022
Avoidant_1	58	445	0.281	0.581	0	0.143	0.0	0	2	2	1.934	2.552	0.028
Avoidant_2	59	445	0.263	0.601	0	0.101	0.0	0	2	2	2.120	3.061	0.028
Avoidant_3	60	445	0.288	0.591	0	0.146	0.0	0	2	2	1.909	2.408	0.028
Avoidant_4	61	445	0.467	0.752	0	0.336	0.0	0	2	2	1.222	-0.138	0.036
Avoidant_5	62	445	0.429	0.709	0	0.289	0.0	0	2	2	1.330	0.258	0.034
Avoidant_6	63	445	0.535	0.760	0	0.420	0.0	0	2	2	1.002	-0.556	0.036
Avoidant_7	64	445	0.299	0.642	0	0.126	0.0	0	2	2	1.922	2.146	0.030
Dependent_1	65	445	0.146	0.434	0	0.020	0.0	0	2	2	3.059	8.768	0.021
Dependent_2	66	445	0.337	0.660	0	0.174	0.0	0	2	2	1.714	1.435	0.031
Dependent_3	67	445	0.213	0.486	0	0.101	0.0	0	2	2	2.229	4.208	0.023
Dependent_4	68	445	0.117	0.379	0	0.000	0.0	0	2	2	3.409	11.521	0.018
Dependent_5	69	445	0.144	0.432	0	0.017	0.0	0	2	2	3.096	9.001	0.020
Dependent_6	70	445	0.231	0.527	0	0.104	0.0	0	2	2	2.214	3.917	0.025
Dependent_7	71	445	0.384	0.693	0	0.232	0.0	0	2	2	1.511	0.742	0.033
Dependent_8	72	445	0.276	0.591	0	0.129	0.0	0	2	2	1.994	2.696	0.028
OC_1	73	445	0.292	0.597	0	0.148	0.0	0	2	2	1.888	2.311	0.028
OC_2	74	445	0.213	0.486	0	0.101	0.0	0	2	2	2.229	4.208	0.023
OC_3	75	445	0.326	0.622	0	0.179	0.0	0	2	2	1.716	1.642	0.029
OC_4	76	445	0.160	0.424	0	0.048	0.0	0	2	2	2.692	6.801	0.020
OC_5	77	445	0.227	0.545	0	0.084	0.0	0	2	2	2.325	4.227	0.026
OC_6	78	445	0.339	0.633	0	0.190	0.0	0	2	2	1.656	1.415	0.030
OC_7	79	445	0.054	0.245	0	0.000	0.0	0	2	2	4.815	24.773	0.012
OC_8	80	445	0.636	0.821	0	0.546	0.0	0	2	2	0.756	-1.101	0.039
GAS_Sx	81	267	57.408	11.531	55	55.916	10.4	40	99	59	1.144	0.848	0.706
GAS_Work	82	267	56.292	13.278	51	54.591	8.9	41	99	58	1.008	-0.057	0.813
GAS_Soc	83	267	57.438	10.998	55	56.270	8.9	41	99	58	1.007	0.617	0.673
SEV	84	418	61.900	13.789	65	63.393	13.3	10	85	75	-0.923	0.319	0.674
CON_PD*	85	418	1.646	0.479	2	1.682	0.0	1	2	1	-0.608	-1.634	0.023
totPD	86	445	17.616	11.699	16	16.669	11.9	0	55	55	0.684	-0.105	0.555
con_pd_num	87	418	0.646	0.479	1	0.682	0.0	0	1	1	-0.608	-1.634	0.023

bpd_only_syntax <- '
bpd =~ Borderline_6 + Borderline_1 + Borderline_2 + Borderline_3 + Borderline_4 + Borderline_5 + 
  Borderline_7 + Borderline_8 + Borderline_9
'

bpd_model <- sem(bpd_only_syntax, estimator="MLR", allPD)
summary(bpd_model, fit.measures=TRUE, standardized=TRUE)

## lavaan 0.6-3 ended normally after 26 iterations
## 
##   Optimization method                           NLMINB
##   Number of free parameters                         18
## 
##   Number of observations                           445
## 
##   Estimator                                         ML      Robust
##   Model Fit Test Statistic                     179.349     109.106
##   Degrees of freedom                                27          27
##   P-value (Chi-square)                           0.000       0.000
##   Scaling correction factor                                  1.644
##     for the Yuan-Bentler correction (Mplus variant)
## 
## Model test baseline model:
## 
##   Minimum Function Test Statistic             1328.671     787.319
##   Degrees of freedom                                36          36
##   P-value                                        0.000       0.000
## 
## User model versus baseline model:
## 
##   Comparative Fit Index (CFI)                    0.882       0.891
##   Tucker-Lewis Index (TLI)                       0.843       0.854
## 
##   Robust Comparative Fit Index (CFI)                         0.894
##   Robust Tucker-Lewis Index (TLI)                            0.858
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -3510.958   -3510.958
##   Scaling correction factor                                  1.764
##     for the MLR correction
##   Loglikelihood unrestricted model (H1)      -3421.283   -3421.283
##   Scaling correction factor                                  1.692
##     for the MLR correction
## 
##   Number of free parameters                         18          18
##   Akaike (AIC)                                7057.916    7057.916
##   Bayesian (BIC)                              7131.681    7131.681
##   Sample-size adjusted Bayesian (BIC)         7074.557    7074.557
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.113       0.083
##   90 Percent Confidence Interval          0.097  0.129       0.070  0.095
##   P-value RMSEA <= 0.05                          0.000       0.000
## 
##   Robust RMSEA                                               0.106
##   90 Percent Confidence Interval                             0.086  0.127
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.062       0.062
## 
## Parameter Estimates:
## 
##   Information                                 Observed
##   Observed information based on                Hessian
##   Standard Errors                   Robust.huber.white
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   bpd =~                                                                
##     Borderline_6      1.000                               0.569    0.729
##     Borderline_1      0.445    0.080    5.566    0.000    0.253    0.443
##     Borderline_2      0.850    0.076   11.139    0.000    0.483    0.694
##     Borderline_3      0.710    0.089    7.970    0.000    0.404    0.584
##     Borderline_4      0.947    0.083   11.388    0.000    0.539    0.701
##     Borderline_5      0.721    0.076    9.513    0.000    0.410    0.677
##     Borderline_7      0.881    0.078   11.344    0.000    0.501    0.634
##     Borderline_8      0.777    0.075   10.426    0.000    0.442    0.593
##     Borderline_9      0.395    0.056    7.096    0.000    0.225    0.480
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Borderline_6      0.285    0.032    8.806    0.000    0.285    0.468
##    .Borderline_1      0.262    0.026    9.983    0.000    0.262    0.804
##    .Borderline_2      0.251    0.025    9.935    0.000    0.251    0.518
##    .Borderline_3      0.315    0.030   10.628    0.000    0.315    0.659
##    .Borderline_4      0.301    0.032    9.322    0.000    0.301    0.509
##    .Borderline_5      0.199    0.024    8.322    0.000    0.199    0.542
##    .Borderline_7      0.374    0.035   10.719    0.000    0.374    0.598
##    .Borderline_8      0.360    0.032   11.195    0.000    0.360    0.648
##    .Borderline_9      0.169    0.024    7.158    0.000    0.169    0.770
##     bpd               0.324    0.040    8.112    0.000    1.000    1.000

semPaths_default(bpd_model)

modificationindices(bpd_model, minimum.value = 5)

	lhs	op	rhs	mi	epc	sepc.lv	sepc.all	sepc.nox
20	Borderline_6	~~	Borderline_1	8.70	-0.044	-0.044	-0.161	-0.161
26	Borderline_6	~~	Borderline_8	15.97	0.073	0.073	0.228	0.228
29	Borderline_1	~~	Borderline_3	56.57	0.110	0.110	0.383	0.383
34	Borderline_1	~~	Borderline_9	21.58	-0.049	-0.049	-0.231	-0.231
38	Borderline_2	~~	Borderline_7	8.44	-0.050	-0.050	-0.164	-0.164
44	Borderline_3	~~	Borderline_8	35.88	-0.106	-0.106	-0.316	-0.316
49	Borderline_4	~~	Borderline_9	5.93	-0.030	-0.030	-0.131	-0.131
52	Borderline_5	~~	Borderline_9	13.72	0.036	0.036	0.197	0.197

bpd_mod_syntax <- '
bpd =~ Borderline_6 + Borderline_1 + Borderline_2 + Borderline_3 + Borderline_4 + Borderline_5 + Borderline_7 + Borderline_8 + Borderline_9

Borderline_1 ~~ Borderline_3
'

bpd_model_mod <- sem(bpd_mod_syntax, estimator="MLR", allPD)
resid(bpd_model_mod, "cor")

## $type
## [1] "cor.bollen"
## 
## $cov
##              Brdr_6 Brdr_1 Brdr_2 Brdr_3 Brdr_4 Brdr_5 Brdr_7 Brdr_8
## Borderline_6  0.000                                                 
## Borderline_1 -0.047  0.000                                          
## Borderline_2 -0.023  0.058  0.000                                   
## Borderline_3 -0.031  0.000  0.071  0.000                            
## Borderline_4  0.003  0.073 -0.018  0.064  0.000                     
## Borderline_5 -0.034  0.014  0.046 -0.023 -0.026  0.000              
## Borderline_7  0.022  0.022 -0.063  0.020  0.004  0.012  0.000       
## Borderline_8  0.072 -0.047  0.032 -0.156  0.029 -0.031 -0.014  0.000
## Borderline_9  0.016 -0.148 -0.038  0.045 -0.068  0.093  0.034 -0.036
##              Brdr_9
## Borderline_6       
## Borderline_1       
## Borderline_2       
## Borderline_3       
## Borderline_4       
## Borderline_5       
## Borderline_7       
## Borderline_8       
## Borderline_9  0.000

The maximum-likelihood approach assumes that indicators are normally distributed. But here, we have trichotmous data, which are unlikely to satisfy this assumption. The typical approach would be to estimate the model with an explicit categorical threshold structure that captures the distance between adjacent ordinal indicators.

bpd_model_cat <- sem(bpd_only_syntax, estimator="WLSMV", allPD, ordered = paste0("Borderline_", 1:9))
summary(bpd_model_cat, fit.measures=TRUE)

## lavaan 0.6-3 ended normally after 21 iterations
## 
##   Optimization method                           NLMINB
##   Number of free parameters                         27
## 
##   Number of observations                           445
## 
##   Estimator                                       DWLS      Robust
##   Model Fit Test Statistic                      72.056     108.554
##   Degrees of freedom                                27          27
##   P-value (Chi-square)                           0.000       0.000
##   Scaling correction factor                                  0.691
##   Shift parameter                                            4.238
##     for simple second-order correction (Mplus variant)
## 
## Model test baseline model:
## 
##   Minimum Function Test Statistic             3882.434    2611.156
##   Degrees of freedom                                36          36
##   P-value                                        0.000       0.000
## 
## User model versus baseline model:
## 
##   Comparative Fit Index (CFI)                    0.988       0.968
##   Tucker-Lewis Index (TLI)                       0.984       0.958
## 
##   Robust Comparative Fit Index (CFI)                            NA
##   Robust Tucker-Lewis Index (TLI)                               NA
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.061       0.082
##   90 Percent Confidence Interval          0.044  0.079       0.067  0.099
##   P-value RMSEA <= 0.05                          0.129       0.001
## 
##   Robust RMSEA                                                  NA
##   90 Percent Confidence Interval                                NA     NA
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.085       0.085
## 
## Parameter Estimates:
## 
##   Information                                 Expected
##   Information saturated (h1) model        Unstructured
##   Standard Errors                           Robust.sem
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   bpd =~                                              
##     Borderline_6      1.000                           
##     Borderline_1      0.745    0.068   11.013    0.000
##     Borderline_2      0.936    0.051   18.323    0.000
##     Borderline_3      0.886    0.058   15.255    0.000
##     Borderline_4      0.950    0.055   17.325    0.000
##     Borderline_5      1.006    0.056   17.873    0.000
##     Borderline_7      0.883    0.058   15.295    0.000
##     Borderline_8      0.829    0.058   14.172    0.000
##     Borderline_9      0.840    0.059   14.349    0.000
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .Borderline_6      0.000                           
##    .Borderline_1      0.000                           
##    .Borderline_2      0.000                           
##    .Borderline_3      0.000                           
##    .Borderline_4      0.000                           
##    .Borderline_5      0.000                           
##    .Borderline_7      0.000                           
##    .Borderline_8      0.000                           
##    .Borderline_9      0.000                           
##     bpd               0.000                           
## 
## Thresholds:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     Borderlin_6|t1    0.390    0.061    6.376    0.000
##     Borderlin_6|t2    0.916    0.070   13.182    0.000
##     Borderlin_1|t1    0.874    0.069   12.757    0.000
##     Borderlin_1|t2    1.495    0.091   16.390    0.000
##     Borderlin_2|t1    0.458    0.062    7.406    0.000
##     Borderlin_2|t2    1.179    0.077   15.268    0.000
##     Borderlin_3|t1    0.580    0.063    9.171    0.000
##     Borderlin_3|t2    1.179    0.077   15.268    0.000
##     Borderlin_4|t1    0.372    0.061    6.094    0.000
##     Borderlin_4|t2    0.951    0.070   13.516    0.000
##     Borderlin_5|t1    0.899    0.069   13.013    0.000
##     Borderlin_5|t2    1.370    0.085   16.126    0.000
##     Borderlin_7|t1    0.348    0.061    5.718    0.000
##     Borderlin_7|t2    0.882    0.069   12.843    0.000
##     Borderlin_8|t1    0.427    0.061    6.938    0.000
##     Borderlin_8|t2    1.024    0.072   14.162    0.000
##     Borderlin_9|t1    1.191    0.078   15.335    0.000
##     Borderlin_9|t2    1.720    0.106   16.289    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .Borderline_6      0.301                           
##    .Borderline_1      0.612                           
##    .Borderline_2      0.387                           
##    .Borderline_3      0.451                           
##    .Borderline_4      0.369                           
##    .Borderline_5      0.292                           
##    .Borderline_7      0.454                           
##    .Borderline_8      0.519                           
##    .Borderline_9      0.507                           
##     bpd               0.699    0.053   13.154    0.000
## 
## Scales y*:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     Borderline_6      1.000                           
##     Borderline_1      1.000                           
##     Borderline_2      1.000                           
##     Borderline_3      1.000                           
##     Borderline_4      1.000                           
##     Borderline_5      1.000                           
##     Borderline_7      1.000                           
##     Borderline_8      1.000                           
##     Borderline_9      1.000

We still see some evidence of misfit.

modificationIndices(bpd_model_cat, minimum.value = 5)

	lhs	op	rhs	mi	epc	sepc.lv	sepc.all	sepc.nox
66	Borderline_1	~~	Borderline_3	34.53	0.338	0.338	0.644	0.644
70	Borderline_1	~~	Borderline_8	5.07	-0.177	-0.177	-0.313	-0.313
71	Borderline_1	~~	Borderline_9	10.22	-0.351	-0.351	-0.631	-0.631
81	Borderline_3	~~	Borderline_8	13.50	-0.280	-0.280	-0.579	-0.579

The model suggests freeing conditional independence for some items. Here are the diagnostic criteria from DSM-IV

frantic efforts to avoid real or imagined abandonment.
a pattern of unstable and intense interpersonal relationships characterised by alternating between extremes of idealization and devaluation.
identity disturbance: markedly and persistently unstable self-image or sense of self.
impulsivity in at least two areas that are potentially self-damaging (e.g. spending, sex, substance abuse, reckless driving, binge eating). This does not include suicidal or self-harming behaviour.
recurrent suicidal behaviour, gestures, or threats, or self-mutilating behaviour.
affective instability due to a marked reactivity of mood - intense feelings that can last from a few hours to a few days.
chronic feelings of emptiness.
inappropriate intense anger or difficulty controlling anger.
transient, stress-related paranoid ideas or severe dissociative symptoms.

Example of SEM in the ‘Real World’

Michael Hallquist

26 Mar 2019