SEM results
Data preparation
Most metadata are omitted, but duration (in seconds) is retained for analysis.
Reverse items
Item
Q6_4was reversed and recoded intoefficacy_small_letgo.Item
Q12_3was reversed and recoded intopost_eff_easy_offense.
Recoding
To produce more readable tables, I recoded some of the variables to their original meanings.
final data -
Survey data
We omit the fast respondents: those who answered faster than
CFA
First, we assessed the reliability of the measurement model for the latent variables.
| rowname | pj | trust | eff | vig |
|---|---|---|---|---|
| alpha | 0.9445 | 0.9277 | 0.8562 | 0.7734 |
| omega | 0.9446 | 0.9276 | 0.8559 | 0.7898 |
| omega2 | 0.9446 | 0.9276 | 0.8559 | 0.7898 |
| omega3 | 0.9446 | 0.9275 | 0.8555 | 0.7908 |
| avevar | 0.7089 | 0.8103 | 0.6645 | 0.5656 |
SEM -
pj = procedural justice
dj = distributive justice
trs = trust in the police
eff = police efficacy
vig = tendency to vigilancy
pr = previous exp score
lavaan 0.6.15 ended normally after 67 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 45
Used Total
Number of observations 465 551
Model Test User Model:
Standard Scaled
Test Statistic 331.957 266.062
Degrees of freedom 142 142
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.248
Yuan-Bentler correction (Mplus variant)
Model Test Baseline Model:
Test statistic 6730.294 5044.970
Degrees of freedom 170 170
P-value 0.000 0.000
Scaling correction factor 1.334
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.971 0.975
Tucker-Lewis Index (TLI) 0.965 0.970
Robust Comparative Fit Index (CFI) 0.976
Robust Tucker-Lewis Index (TLI) 0.972
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -8696.311 -8696.311
Scaling correction factor 1.362
for the MLR correction
Loglikelihood unrestricted model (H1) -8530.332 -8530.332
Scaling correction factor 1.275
for the MLR correction
Akaike (AIC) 17482.621 17482.621
Bayesian (BIC) 17669.013 17669.013
Sample-size adjusted Bayesian (SABIC) 17526.194 17526.194
Root Mean Square Error of Approximation:
RMSEA 0.054 0.043
90 Percent confidence interval - lower 0.046 0.036
90 Percent confidence interval - upper 0.061 0.050
P-value H_0: RMSEA <= 0.050 0.206 0.937
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.048
90 Percent confidence interval - lower 0.039
90 Percent confidence interval - upper 0.057
P-value H_0: Robust RMSEA <= 0.050 0.603
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.049 0.049
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Latent Variables:
Estimate Std.Err z-value P(>|z|)
pj =~
q5_1 1.000
q5_2 0.976 0.036 27.151 0.000
q5_3 1.009 0.041 24.505 0.000
q5_4 1.004 0.043 23.093 0.000
q5_5 0.958 0.041 23.599 0.000
q5_6 1.021 0.035 29.268 0.000
q5_7 0.998 0.037 27.176 0.000
dj =~
q8_1 1.000
trust =~
q7_1 1.000
q7_2 1.000 0.029 34.297 0.000
q7_3 0.994 0.035 28.784 0.000
eff =~
q6_1 1.000
q6_2 1.027 0.048 21.393 0.000
q6_3 1.016 0.050 20.212 0.000
vig =~
q9_1 1.000
q9_2 1.358 0.106 12.768 0.000
q9_3 0.843 0.083 10.202 0.000
Regressions:
Estimate Std.Err z-value P(>|z|)
vig ~
pj (h2) -0.288 0.149 -1.936 0.053
dj (h4) 0.001 0.076 0.018 0.985
trust (h1) -0.217 0.108 -2.020 0.043
eff (h3) 0.499 0.180 2.765 0.006
pre (h6) -0.042 0.040 -1.039 0.299
q1 (h5) 0.075 0.122 0.609 0.542
Covariances:
Estimate Std.Err z-value P(>|z|)
pj ~~
dj 0.767 0.056 13.740 0.000
trust 0.778 0.054 14.477 0.000
eff 0.694 0.053 13.099 0.000
dj ~~
trust 0.891 0.057 15.677 0.000
eff 0.738 0.055 13.309 0.000
trust ~~
eff 0.722 0.055 13.154 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
.q5_1 0.449 0.044 10.247 0.000
.q5_2 0.301 0.034 8.757 0.000
.q5_3 0.281 0.027 10.357 0.000
.q5_4 0.383 0.032 11.838 0.000
.q5_5 0.413 0.039 10.705 0.000
.q5_6 0.316 0.029 10.758 0.000
.q5_7 0.239 0.023 10.573 0.000
.q8_1 0.000
.q7_1 0.215 0.035 6.080 0.000
.q7_2 0.269 0.042 6.442 0.000
.q7_3 0.257 0.040 6.406 0.000
.q6_1 0.359 0.038 9.496 0.000
.q6_2 0.393 0.048 8.146 0.000
.q6_3 0.397 0.044 9.042 0.000
.q9_1 0.890 0.092 9.686 0.000
.q9_2 0.195 0.079 2.472 0.013
.q9_3 0.593 0.062 9.528 0.000
pj 0.823 0.068 12.115 0.000
dj 1.099 0.060 18.464 0.000
trust 1.033 0.067 15.362 0.000
eff 0.731 0.063 11.678 0.000
.vig 0.570 0.084 6.812 0.000
Defined Parameters:
Estimate Std.Err z-value P(>|z|)
H1 -0.217 0.108 -2.020 0.043
H2 -0.288 0.149 -1.936 0.053
H3 0.499 0.180 2.765 0.006
H4 0.001 0.076 0.018 0.985
H5 0.075 0.122 0.609 0.542
H6 -0.042 0.040 -1.039 0.299We fitted a structural equation model (SEM) to n = 465 observations using the lavaan R package (Rosseel 2012) as described in Figure XXX. We used the Robust Maximum Likelihood (MLR) estimation method, as the data deviate from normality. The goodness of fit test statistic was significant \(\chi^2(142) = 266.02, p < .001\), but common model fit indices suggested a good fit: CFI = .937, TLI = .923, RMSEA = .07 (90% CI : .064-.076).
https://www.researchgate.net/post/Is_it_necessary_that_in_model_fit_my_Chi-square_valuep-Value_must_be_non-significant_in_structure_equation_modeling_AMOS
Hyp 1: trust is negatively associated with vigilance: confirmed, (b = -0.219, p = .041)
Hyp 2: high PJ predicts low vig: confirmed (b = -.293, p = .047)
Hyp 3: high efficacy predicts low vig: REJECTED. Moreover, the estimate (b = .501, p = .005) suggests the other way around - higher perceived police efficacy is associated with MORE vigilantism.
Hyp 4: DJ predicts vig: rejected, (b = 0.000, p = .998).
Hyp 5: Previous victims will have higher vig. : no, (b = -.042, p = .299)
Hyp 6: The quality of police encounter moderates crime victims’ vigilantism : not supported (b = -.022, p = .401).
| label | est | se | z | pvalue |
|---|---|---|---|---|
| H1 | -0.2172 | 0.1075 | -2.02 | 0.0434 |
| H2 | -0.2876 | 0.1485 | -1.936 | 0.05281 |
| H3 | 0.4989 | 0.1805 | 2.765 | 0.005694 |
| H4 | 0.0014 | 0.07584 | 0.01846 | 0.9853 |
| H5 | 0.07464 | 0.1225 | 0.6094 | 0.5423 |
| H6 | -0.04191 | 0.04034 | -1.039 | 0.2989 |
SEM - experiment
lavaan 0.6.15 ended normally after 105 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 108
Used Total
Number of observations 317 421
Model Test User Model:
Standard Scaled
Test Statistic 1058.344 925.708
Degrees of freedom 453 453
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.143
Yuan-Bentler correction (Mplus variant)
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Latent Variables:
Estimate Std.Err z-value P(>|z|)
pj_pre =~
q5_1 1.000
q5_2 0.992 0.042 23.630 0.000
q5_3 0.995 0.046 21.587 0.000
q5_4 1.019 0.054 18.973 0.000
q5_5 0.925 0.051 18.234 0.000
q5_6 1.019 0.046 22.334 0.000
q5_7 0.982 0.048 20.529 0.000
dj_pre =~
q8_1 1.000
trust_pre =~
q7_1 1.000
q7_2 1.016 0.039 25.958 0.000
q7_3 0.995 0.051 19.688 0.000
eff_pre =~
q6_1 1.000
q6_2 1.020 0.061 16.814 0.000
q6_3 1.025 0.058 17.601 0.000
vig_pre =~
q9_1 1.000
q9_2 1.217 0.092 13.241 0.000
q9_3 0.862 0.104 8.301 0.000
pj_post =~
q11_1 1.000
q11_2 1.083 0.085 12.749 0.000
q11_3 1.395 0.104 13.367 0.000
q11_4 1.403 0.104 13.449 0.000
q11_5 1.273 0.106 12.005 0.000
q11_6 1.529 0.122 12.524 0.000
q11_7 1.544 0.124 12.474 0.000
eff_post =~
q12_1 1.000
q12_2 1.114 0.049 22.736 0.000
q12_3 0.239 0.093 2.576 0.010
trust_post =~
q13_1 1.000
fair =~
q14_1 1.000
vig_post =~
q15_1 1.000
q15_2 1.049 0.075 13.994 0.000
q15_3 0.941 0.107 8.763 0.000
q15_4 1.274 0.109 11.647 0.000
Covariances:
Estimate Std.Err z-value P(>|z|)
pj_pre ~~
dj_pre 0.736 0.065 11.242 0.000
trust_pre 0.708 0.062 11.496 0.000
eff_pre 0.661 0.063 10.447 0.000
vig_pre -0.028 0.053 -0.531 0.596
pj_post 0.225 0.048 4.650 0.000
eff_post 0.291 0.060 4.848 0.000
trust_post 0.409 0.070 5.858 0.000
fair 0.387 0.070 5.493 0.000
vig_post -0.027 0.054 -0.492 0.623
dj_pre ~~
trust_pre 0.822 0.061 13.370 0.000
eff_pre 0.702 0.063 11.103 0.000
vig_pre 0.009 0.059 0.153 0.878
pj_post 0.193 0.052 3.716 0.000
eff_post 0.324 0.064 5.056 0.000
trust_post 0.463 0.075 6.186 0.000
fair 0.421 0.074 5.719 0.000
vig_post -0.109 0.061 -1.784 0.074
trust_pre ~~
eff_pre 0.659 0.062 10.707 0.000
vig_pre -0.009 0.057 -0.162 0.871
pj_post 0.207 0.052 3.987 0.000
eff_post 0.293 0.061 4.804 0.000
trust_post 0.469 0.069 6.785 0.000
fair 0.396 0.068 5.786 0.000
vig_post -0.101 0.059 -1.701 0.089
eff_pre ~~
vig_pre 0.074 0.052 1.426 0.154
pj_post 0.215 0.047 4.619 0.000
eff_post 0.330 0.061 5.440 0.000
trust_post 0.429 0.069 6.201 0.000
fair 0.442 0.070 6.308 0.000
vig_post -0.035 0.056 -0.627 0.531
vig_pre ~~
pj_post 0.035 0.041 0.846 0.398
eff_post 0.097 0.055 1.769 0.077
trust_post -0.028 0.061 -0.462 0.644
fair 0.077 0.063 1.225 0.221
vig_post 0.357 0.071 4.995 0.000
pj_post ~~
eff_post 0.670 0.073 9.234 0.000
trust_post 0.694 0.076 9.088 0.000
fair 0.582 0.076 7.704 0.000
vig_post -0.103 0.049 -2.127 0.033
eff_post ~~
trust_post 0.913 0.083 11.025 0.000
fair 0.838 0.081 10.301 0.000
vig_post -0.094 0.062 -1.518 0.129
trust_post ~~
fair 1.004 0.089 11.223 0.000
vig_post -0.236 0.070 -3.393 0.001
fair ~~
vig_post -0.127 0.068 -1.872 0.061
Variances:
Estimate Std.Err z-value P(>|z|)
.q5_1 0.434 0.053 8.271 0.000
.q5_2 0.258 0.027 9.425 0.000
.q5_3 0.263 0.031 8.560 0.000
.q5_4 0.362 0.033 10.973 0.000
.q5_5 0.400 0.038 10.578 0.000
.q5_6 0.301 0.033 9.232 0.000
.q5_7 0.272 0.030 8.945 0.000
.q8_1 0.000
.q7_1 0.201 0.037 5.483 0.000
.q7_2 0.232 0.039 5.885 0.000
.q7_3 0.281 0.055 5.099 0.000
.q6_1 0.342 0.046 7.489 0.000
.q6_2 0.372 0.048 7.780 0.000
.q6_3 0.337 0.051 6.632 0.000
.q9_1 0.825 0.105 7.828 0.000
.q9_2 0.315 0.089 3.552 0.000
.q9_3 0.554 0.076 7.256 0.000
.q11_1 0.903 0.089 10.187 0.000
.q11_2 0.476 0.051 9.253 0.000
.q11_3 0.382 0.057 6.679 0.000
.q11_4 0.269 0.043 6.263 0.000
.q11_5 0.498 0.061 8.157 0.000
.q11_6 0.452 0.066 6.859 0.000
.q11_7 0.533 0.078 6.805 0.000
.q12_1 0.490 0.055 8.867 0.000
.q12_2 0.297 0.076 3.913 0.000
.q12_3 1.712 0.098 17.453 0.000
.q13_1 0.000
.q14_1 0.000
.q15_1 0.718 0.101 7.142 0.000
.q15_2 0.911 0.107 8.474 0.000
.q15_3 0.671 0.099 6.793 0.000
.q15_4 0.327 0.073 4.458 0.000
pj_pre 0.781 0.079 9.826 0.000
dj_pre 1.028 0.065 15.694 0.000
trust_pre 0.937 0.074 12.728 0.000
eff_pre 0.715 0.074 9.603 0.000
vig_pre 0.655 0.102 6.400 0.000
pj_post 0.610 0.095 6.417 0.000
eff_post 0.937 0.089 10.523 0.000
trust_post 1.339 0.087 15.318 0.000
fair 1.319 0.091 14.538 0.000
vig_post 0.824 0.124 6.643 0.000 pj_pre trust_pre eff_pre vig_pre pj_post eff_post vig_post
alpha 0.9421435 0.9222382 0.8632403 0.7696777 0.9335215 0.5813407 0.8446798
omega 0.9424672 0.9224658 0.8632685 0.7857340 0.9366787 0.6748849 0.8507725
omega2 0.9424672 0.9224658 0.8632685 0.7857340 0.9366787 0.6748849 0.8507725
omega3 0.9432340 0.9227713 0.8632405 0.7945334 0.9355912 0.6828780 0.8545203
avevar 0.7008055 0.7986372 0.6779158 0.5549926 0.6835056 0.4628598 0.5910267lavaan 0.6.15 ended normally after 67 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 45
Used Total
Number of observations 465 551
Model Test User Model:
Standard Scaled
Test Statistic 331.957 266.062
Degrees of freedom 142 142
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.248
Yuan-Bentler correction (Mplus variant)
Model Test Baseline Model:
Test statistic 6730.294 5044.970
Degrees of freedom 170 170
P-value 0.000 0.000
Scaling correction factor 1.334
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.971 0.975
Tucker-Lewis Index (TLI) 0.965 0.970
Robust Comparative Fit Index (CFI) 0.976
Robust Tucker-Lewis Index (TLI) 0.972
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -8696.311 -8696.311
Scaling correction factor 1.362
for the MLR correction
Loglikelihood unrestricted model (H1) -8530.332 -8530.332
Scaling correction factor 1.275
for the MLR correction
Akaike (AIC) 17482.621 17482.621
Bayesian (BIC) 17669.013 17669.013
Sample-size adjusted Bayesian (SABIC) 17526.194 17526.194
Root Mean Square Error of Approximation:
RMSEA 0.054 0.043
90 Percent confidence interval - lower 0.046 0.036
90 Percent confidence interval - upper 0.061 0.050
P-value H_0: RMSEA <= 0.050 0.206 0.937
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.048
90 Percent confidence interval - lower 0.039
90 Percent confidence interval - upper 0.057
P-value H_0: Robust RMSEA <= 0.050 0.603
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.049 0.049
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Latent Variables:
Estimate Std.Err z-value P(>|z|)
pj =~
q5_1 1.000
q5_2 0.976 0.036 27.151 0.000
q5_3 1.009 0.041 24.505 0.000
q5_4 1.004 0.043 23.093 0.000
q5_5 0.958 0.041 23.599 0.000
q5_6 1.021 0.035 29.268 0.000
q5_7 0.998 0.037 27.176 0.000
dj =~
q8_1 1.000
trust =~
q7_1 1.000
q7_2 1.000 0.029 34.297 0.000
q7_3 0.994 0.035 28.784 0.000
eff =~
q6_1 1.000
q6_2 1.027 0.048 21.393 0.000
q6_3 1.016 0.050 20.212 0.000
vig =~
q9_1 1.000
q9_2 1.358 0.106 12.768 0.000
q9_3 0.843 0.083 10.202 0.000
Regressions:
Estimate Std.Err z-value P(>|z|)
vig ~
pj (h2) -0.288 0.149 -1.936 0.053
dj (h4) 0.001 0.076 0.018 0.985
trust (h1) -0.217 0.108 -2.020 0.043
eff (h3) 0.499 0.180 2.765 0.006
pre (h6) -0.042 0.040 -1.039 0.299
q1 (h5) 0.075 0.122 0.609 0.542
Covariances:
Estimate Std.Err z-value P(>|z|)
pj ~~
dj 0.767 0.056 13.740 0.000
trust 0.778 0.054 14.477 0.000
eff 0.694 0.053 13.099 0.000
dj ~~
trust 0.891 0.057 15.677 0.000
eff 0.738 0.055 13.309 0.000
trust ~~
eff 0.722 0.055 13.154 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
.q5_1 0.449 0.044 10.247 0.000
.q5_2 0.301 0.034 8.757 0.000
.q5_3 0.281 0.027 10.357 0.000
.q5_4 0.383 0.032 11.838 0.000
.q5_5 0.413 0.039 10.705 0.000
.q5_6 0.316 0.029 10.758 0.000
.q5_7 0.239 0.023 10.573 0.000
.q8_1 0.000
.q7_1 0.215 0.035 6.080 0.000
.q7_2 0.269 0.042 6.442 0.000
.q7_3 0.257 0.040 6.406 0.000
.q6_1 0.359 0.038 9.496 0.000
.q6_2 0.393 0.048 8.146 0.000
.q6_3 0.397 0.044 9.042 0.000
.q9_1 0.890 0.092 9.686 0.000
.q9_2 0.195 0.079 2.472 0.013
.q9_3 0.593 0.062 9.528 0.000
pj 0.823 0.068 12.115 0.000
dj 1.099 0.060 18.464 0.000
trust 1.033 0.067 15.362 0.000
eff 0.731 0.063 11.678 0.000
.vig 0.570 0.084 6.812 0.000
Defined Parameters:
Estimate Std.Err z-value P(>|z|)
H1 -0.217 0.108 -2.020 0.043
H2 -0.288 0.149 -1.936 0.053
H3 0.499 0.180 2.765 0.006
H4 0.001 0.076 0.018 0.985
H5 0.075 0.122 0.609 0.542
H6 -0.042 0.040 -1.039 0.299lavaan 0.6.15 ended normally after 32 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 17
Used Total
Number of observations 358 421
Model Test User Model:
Standard Scaled
Test Statistic 139.996 134.233
Degrees of freedom 32 32
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.043
Yuan-Bentler correction (Mplus variant)
Model Test Baseline Model:
Test statistic 1081.702 906.116
Degrees of freedom 42 42
P-value 0.000 0.000
Scaling correction factor 1.194
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.896 0.882
Tucker-Lewis Index (TLI) 0.864 0.845
Robust Comparative Fit Index (CFI) 0.897
Robust Tucker-Lewis Index (TLI) 0.864
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -3558.087 -3558.087
Scaling correction factor 1.272
for the MLR correction
Loglikelihood unrestricted model (H1) -3488.089 -3488.089
Scaling correction factor 1.122
for the MLR correction
Akaike (AIC) 7150.173 7150.173
Bayesian (BIC) 7216.143 7216.143
Sample-size adjusted Bayesian (SABIC) 7162.210 7162.210
Root Mean Square Error of Approximation:
RMSEA 0.097 0.094
90 Percent confidence interval - lower 0.081 0.079
90 Percent confidence interval - upper 0.114 0.111
P-value H_0: RMSEA <= 0.050 0.000 0.000
P-value H_0: RMSEA >= 0.080 0.959 0.934
Robust RMSEA 0.096
90 Percent confidence interval - lower 0.080
90 Percent confidence interval - upper 0.114
P-value H_0: Robust RMSEA <= 0.050 0.000
P-value H_0: Robust RMSEA >= 0.080 0.949
Standardized Root Mean Square Residual:
SRMR 0.149 0.149
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Latent Variables:
Estimate Std.Err z-value P(>|z|)
vig_post =~
q15_1 1.000
q15_2 1.055 0.069 15.329 0.000
q15_3 0.957 0.096 9.963 0.000
q15_4 1.281 0.107 11.990 0.000
vig_pre =~
q9_1 1.000
q9_2 1.494 0.163 9.162 0.000
q9_3 0.892 0.095 9.346 0.000
Regressions:
Estimate Std.Err z-value P(>|z|)
vig_post ~
procj -0.185 0.145 -1.280 0.201
avail 0.018 0.143 0.128 0.898
procj:avail -0.082 0.205 -0.403 0.687
Variances:
Estimate Std.Err z-value P(>|z|)
.q15_1 0.711 0.092 7.762 0.000
.q15_2 0.890 0.103 8.666 0.000
.q15_3 0.678 0.092 7.369 0.000
.q15_4 0.336 0.091 3.712 0.000
.q9_1 0.944 0.108 8.776 0.000
.q9_2 0.068 0.109 0.623 0.533
.q9_3 0.660 0.078 8.460 0.000
.vig_post 0.807 0.116 6.983 0.000
vig_pre 0.549 0.101 5.450 0.000References
Rosseel, Yves. 2012. “lavaan: An R Package for Structural Equation Modeling.” Journal of Statistical Software 48 (2): 1–36. https://doi.org/10.18637/jss.v048.i02.