Today we are learning how to use panel data to assess reverse causality, temporal ordering, and reciprocal relationships, implementing some cross-lagged panel models. We will again reshape data between wide and long formats, fit the traditional cross-lagged panel model, and try different model specifications.
The relationship between policing and crime is a tricky one. Some researchers argue that heavy-handed policing can contribute to increase crime via legal cynicism and strain. However, more police resources—including use of force—are often deployed in high-crime areas. Here, we examine evidence from Chicago. We use publicly available data on violent crimes (i.e., homicides and robberies) and complaints about police misuse of force across all 343 Chicago neighbourhoods between 1991 and 2016. We will use cross-lagged panel models to examine this relationship.
The chicago.Rdata file contains data on 343 Chicago
neighbourhood clusters, and 25 years (1991 through 2016). The
data.frame includes the following variables:
| Name | Description |
|---|---|
NC_NUM |
an ID variable indicating Chicago neighbourhood clusters |
year |
the year, ranging from 1991 to 2016 |
complaints |
(logged) number of complaints about police misuse of force in Chicago |
violent |
(logged) number of violent crimes reported |
population_density |
(logged) population density (measured in 1990) |
FAC_disadv |
factor scores indicating concentrated disadvantage (measured in 1990) |
Please download the chicago.RData data file from the website. Download the
file and store it in a sensible folder in your computer (preferably the
folder of your R Project). Then, we can use the load()
function to import the dataset into R.
load("data/chicago.RData") # adjust accordinglyQuestion 1
Is the dataset set up in a wide format or in a long format?
# check the first rows of the dataset
head(chicago)## NC_NUM year complaints violent population_density FAC_disadv
## 1 1 1991 2.197225 4.356709 9.708923 0.7612026
## 2 2 1991 1.386294 4.553877 9.280131 -0.6690897
## 3 3 1991 2.564949 4.189655 9.413296 -1.4276494
## 4 4 1991 1.098612 4.521789 9.763880 -1.0023245
## 5 5 1991 1.791759 4.406719 9.582598 -0.5795160
## 6 6 1991 0.000000 4.477337 9.392984 -0.3819274Question 2
To fit a cross-lagged panel model, we need to have a wide dataset.
Let’s use the pivot_wider() function again to reshape it
wide. To avoid tedious coding in the next question, let’s filter the
dataset so that it includes data from 2011 to 2014 only.
# load tidyverse
library(tidyverse)
# filter and reshape the dataset
chicago_wide <- chicago %>%
filter(year > 2010 & year < 2015) %>%
pivot_wider(names_from = year,
values_from = -c(NC_NUM, year, FAC_disadv, population_density))
# check columns of the wide dataset
names(chicago_wide)## [1] "NC_NUM" "population_density" "FAC_disadv"
## [4] "complaints_2011" "complaints_2012" "complaints_2013"
## [7] "complaints_2014" "violent_2011" "violent_2012"
## [10] "violent_2013" "violent_2014"Question 3 Using the sem() function
from the lavaan() package, let’s fit our first cross-lagged
panel model assessing the reciprocal relationship between complaints and
violent crime. Let’s start with autoregressive and cross-lagged
parameters constrained to equality. What is the average cross-lagged
effect of crime on police use-of-force? And what is the average
cross-lagged effect of police use-of-force on crime?
# load the lavaan package
library(lavaan)
# specify the model
clpm_model <- '
violent_2014 ~ a * violent_2013 + b * complaints_2013
violent_2013 ~ a * violent_2012 + b * complaints_2012
violent_2012 ~ a * violent_2011 + b * complaints_2011
complaints_2014 ~ c * complaints_2013 + d * violent_2013
complaints_2013 ~ c * complaints_2012 + d * violent_2012
complaints_2012 ~ c * complaints_2011 + d * violent_2011
violent_2011 ~~ complaints_2011
violent_2012 ~~ complaints_2012
violent_2013 ~~ complaints_2013
violent_2014 ~~ complaints_2014
'
# fit the CLPM
clpm_fit <- sem(clpm_model, estimator = "ML", data = chicago_wide)
# summary results
summary(clpm_fit, fit.measures = TRUE, standardized = TRUE)## lavaan 0.6-19 ended normally after 30 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 24
## Number of equality constraints 8
##
## Number of observations 342
##
## Model Test User Model:
##
## Test statistic 244.647
## Degrees of freedom 20
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 2544.730
## Degrees of freedom 28
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.911
## Tucker-Lewis Index (TLI) 0.875
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2368.411
## Loglikelihood unrestricted model (H1) -2246.088
##
## Akaike (AIC) 4768.822
## Bayesian (BIC) 4830.179
## Sample-size adjusted Bayesian (SABIC) 4779.424
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.181
## 90 Percent confidence interval - lower 0.161
## 90 Percent confidence interval - upper 0.202
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.060
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## violent_2014 ~
## vilnt_2013 (a) 0.887 0.014 64.357 0.000 0.887 0.889
## cmpln_2013 (b) 0.054 0.015 3.719 0.000 0.054 0.049
## violent_2013 ~
## vilnt_2012 (a) 0.887 0.014 64.357 0.000 0.887 0.882
## cmpln_2012 (b) 0.054 0.015 3.719 0.000 0.054 0.052
## violent_2012 ~
## vilnt_2011 (a) 0.887 0.014 64.357 0.000 0.887 0.905
## cmpln_2011 (b) 0.054 0.015 3.719 0.000 0.054 0.054
## complaints_2014 ~
## cmpln_2013 (c) 0.260 0.030 8.637 0.000 0.260 0.248
## vilnt_2013 (d) 0.286 0.028 10.036 0.000 0.286 0.304
## complaints_2013 ~
## cmpln_2012 (c) 0.260 0.030 8.637 0.000 0.260 0.277
## vilnt_2012 (d) 0.286 0.028 10.036 0.000 0.286 0.317
## complaints_2012 ~
## cmpln_2011 (c) 0.260 0.030 8.637 0.000 0.260 0.268
## vilnt_2011 (d) 0.286 0.028 10.036 0.000 0.286 0.303
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## violent_2011 ~~
## complants_2011 0.392 0.049 8.043 0.000 0.392 0.483
## .violent_2012 ~~
## .complants_2012 0.051 0.013 3.835 0.000 0.051 0.212
## .violent_2013 ~~
## .complants_2013 0.008 0.014 0.576 0.565 0.008 0.031
## .violent_2014 ~~
## .complants_2014 0.041 0.015 2.719 0.007 0.041 0.149
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .violent_2014 0.137 0.010 13.077 0.000 0.137 0.171
## .violent_2013 0.141 0.011 13.077 0.000 0.141 0.175
## .violent_2012 0.104 0.008 13.077 0.000 0.104 0.130
## .complants_2014 0.557 0.043 13.077 0.000 0.557 0.782
## .complants_2013 0.480 0.037 13.077 0.000 0.480 0.738
## .complants_2012 0.563 0.043 13.077 0.000 0.563 0.759
## violent_2011 0.833 0.064 13.077 0.000 0.833 1.000
## complants_2011 0.790 0.060 13.077 0.000 0.790 1.000Question 3 Let’s fit the same model, but now not constraining autoregressive and cross-lagged parameters to equality. What happens with the model?
# load the lavaan package
library(lavaan)
# specify the model
clpm_model_unconstrained <- '
violent_2014 ~ violent_2013 + complaints_2013
violent_2013 ~ violent_2012 + complaints_2012
violent_2012 ~ violent_2011 + complaints_2011
complaints_2014 ~ complaints_2013 + violent_2013
complaints_2013 ~ complaints_2012 + violent_2012
complaints_2012 ~ complaints_2011 + violent_2011
violent_2011 ~~ complaints_2011
violent_2012 ~~ complaints_2012
violent_2013 ~~ complaints_2013
violent_2014 ~~ complaints_2014
'
# fit the CLPM
clpm_unconstrained.fit <- sem(clpm_model_unconstrained, estimator = "ML", data = chicago_wide)
# summary results
summary(clpm_unconstrained.fit, fit.measures = TRUE, standardized = TRUE)## lavaan 0.6-19 ended normally after 31 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 24
##
## Number of observations 342
##
## Model Test User Model:
##
## Test statistic 228.397
## Degrees of freedom 12
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 2544.730
## Degrees of freedom 28
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.914
## Tucker-Lewis Index (TLI) 0.799
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2360.286
## Loglikelihood unrestricted model (H1) -2246.088
##
## Akaike (AIC) 4768.573
## Bayesian (BIC) 4860.608
## Sample-size adjusted Bayesian (SABIC) 4784.475
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.230
## 90 Percent confidence interval - lower 0.204
## 90 Percent confidence interval - upper 0.256
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.059
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## violent_2014 ~
## violent_2013 0.867 0.024 35.660 0.000 0.867 0.878
## complants_2013 0.081 0.027 2.982 0.003 0.081 0.073
## violent_2013 ~
## violent_2012 0.950 0.026 36.848 0.000 0.950 0.925
## complants_2012 -0.026 0.026 -0.990 0.322 -0.026 -0.025
## violent_2012 ~
## violent_2011 0.857 0.022 39.463 0.000 0.857 0.883
## complants_2011 0.093 0.022 4.154 0.000 0.093 0.093
## complaints_2014 ~
## complants_2013 0.225 0.055 4.098 0.000 0.225 0.219
## violent_2013 0.279 0.049 5.697 0.000 0.279 0.305
## complaints_2013 ~
## complants_2012 0.255 0.049 5.190 0.000 0.255 0.274
## violent_2012 0.302 0.048 6.258 0.000 0.302 0.330
## complaints_2012 ~
## complants_2011 0.314 0.052 6.030 0.000 0.314 0.321
## violent_2011 0.257 0.051 5.073 0.000 0.257 0.270
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## violent_2011 ~~
## complants_2011 0.392 0.049 8.043 0.000 0.392 0.483
## .violent_2012 ~~
## .complants_2012 0.050 0.013 3.766 0.000 0.050 0.208
## .violent_2013 ~~
## .complants_2013 0.008 0.014 0.548 0.583 0.008 0.030
## .violent_2014 ~~
## .complants_2014 0.041 0.015 2.749 0.006 0.041 0.150
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .violent_2014 0.137 0.010 13.077 0.000 0.137 0.169
## .violent_2013 0.137 0.010 13.077 0.000 0.137 0.166
## .violent_2012 0.103 0.008 13.077 0.000 0.103 0.131
## .complants_2014 0.556 0.043 13.077 0.000 0.556 0.802
## .complants_2013 0.480 0.037 13.077 0.000 0.480 0.729
## .complants_2012 0.562 0.043 13.077 0.000 0.562 0.741
## violent_2011 0.833 0.064 13.077 0.000 0.833 1.000
## complants_2011 0.790 0.060 13.077 0.000 0.790 1.000Question 4 Pick your preferred model (i.e.,
constraining autoregressive and cross-lagged paramters to equality or
not). Now, let’s add two time-constant control variables to the
cross-lagged panel model: population_density and
FAC_disadv. Interpret the estimated coefficients.
# specify the model
clpm_model_covariates <- '
violent_2014 ~ a * violent_2013 + b * complaints_2013
violent_2013 ~ a * violent_2012 + b * complaints_2012
violent_2012 ~ a * violent_2011 + b * complaints_2011
complaints_2014 ~ c * complaints_2013 + d * violent_2013
complaints_2013 ~ c * complaints_2012 + d * violent_2012
complaints_2012 ~ c * complaints_2011 + d * violent_2011
violent_2011 ~~ complaints_2011
violent_2012 ~~ complaints_2012
violent_2013 ~~ complaints_2013
violent_2014 ~~ complaints_2014
complaints_2011 ~ population_density + FAC_disadv
violent_2011 ~ population_density + FAC_disadv
'
# fit the CLPM
clpm_fit_cov <- sem(clpm_model_covariates, estimator = "ML", data = chicago_wide)
# summary results
summary(clpm_fit_cov, fit.measures = TRUE, standardized = TRUE)## lavaan 0.6-19 ended normally after 29 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 28
## Number of equality constraints 8
##
## Number of observations 342
##
## Model Test User Model:
##
## Test statistic 283.231
## Degrees of freedom 32
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 2637.815
## Degrees of freedom 44
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.903
## Tucker-Lewis Index (TLI) 0.867
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2341.161
## Loglikelihood unrestricted model (H1) -2199.545
##
## Akaike (AIC) 4722.322
## Bayesian (BIC) 4799.018
## Sample-size adjusted Bayesian (SABIC) 4735.573
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.152
## 90 Percent confidence interval - lower 0.136
## 90 Percent confidence interval - upper 0.168
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.068
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## violent_2014 ~
## vilnt_2013 (a) 0.887 0.014 64.357 0.000 0.887 0.889
## cmpln_2013 (b) 0.054 0.015 3.719 0.000 0.054 0.049
## violent_2013 ~
## vilnt_2012 (a) 0.887 0.014 64.357 0.000 0.887 0.882
## cmpln_2012 (b) 0.054 0.015 3.719 0.000 0.054 0.052
## violent_2012 ~
## vilnt_2011 (a) 0.887 0.014 64.357 0.000 0.887 0.905
## cmpln_2011 (b) 0.054 0.015 3.719 0.000 0.054 0.054
## complaints_2014 ~
## cmpln_2013 (c) 0.260 0.030 8.637 0.000 0.260 0.248
## vilnt_2013 (d) 0.286 0.028 10.036 0.000 0.286 0.304
## complaints_2013 ~
## cmpln_2012 (c) 0.260 0.030 8.637 0.000 0.260 0.277
## vilnt_2012 (d) 0.286 0.028 10.036 0.000 0.286 0.317
## complaints_2012 ~
## cmpln_2011 (c) 0.260 0.030 8.637 0.000 0.260 0.268
## vilnt_2011 (d) 0.286 0.028 10.036 0.000 0.286 0.303
## complaints_2011 ~
## ppltn_dnst -0.202 0.075 -2.708 0.007 -0.202 -0.141
## FAC_disadv 0.206 0.042 4.956 0.000 0.206 0.257
## violent_2011 ~
## ppltn_dnst 0.117 0.076 1.536 0.124 0.117 0.079
## FAC_disadv 0.244 0.042 5.750 0.000 0.244 0.296
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .complaints_2011 ~~
## .violent_2011 0.340 0.044 7.734 0.000 0.340 0.460
## .violent_2012 ~~
## .complants_2012 0.051 0.013 3.835 0.000 0.051 0.212
## .violent_2013 ~~
## .complants_2013 0.008 0.014 0.576 0.565 0.008 0.031
## .violent_2014 ~~
## .complants_2014 0.041 0.015 2.719 0.007 0.041 0.149
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .violent_2014 0.137 0.010 13.077 0.000 0.137 0.171
## .violent_2013 0.141 0.011 13.077 0.000 0.141 0.175
## .violent_2012 0.104 0.008 13.077 0.000 0.104 0.130
## .complants_2014 0.557 0.043 13.077 0.000 0.557 0.782
## .complants_2013 0.480 0.037 13.077 0.000 0.480 0.738
## .complants_2012 0.563 0.043 13.077 0.000 0.563 0.759
## .complants_2011 0.726 0.056 13.077 0.000 0.726 0.919
## .violent_2011 0.752 0.058 13.077 0.000 0.752 0.903Question 5 Now, let’s add contemporaneous as paths as well as cross-lags.
# specify the model
clpm_model_covariates_contemporaneous <- '
violent_2014 ~ a * violent_2013 + b * complaints_2013 + e*complaints_2014
violent_2013 ~ a * violent_2012 + b * complaints_2012 + e*complaints_2013
violent_2012 ~ a * violent_2011 + b * complaints_2011 + e*complaints_2012
complaints_2014 ~ c * complaints_2013 + d * violent_2013 + f*violent_2014
complaints_2013 ~ c * complaints_2012 + d * violent_2012 + f*violent_2013
complaints_2012 ~ c * complaints_2011 + d * violent_2011 + f*violent_2012
violent_2011 ~~ complaints_2011
violent_2012 ~~ complaints_2012
violent_2013 ~~ complaints_2013
violent_2014 ~~ complaints_2014
complaints_2011 ~ population_density + FAC_disadv
violent_2011 ~ population_density + FAC_disadv
'
# fit the CLPM
clpm_fit_cov_cont <- sem(clpm_model_covariates_contemporaneous, estimator = "ML", data = chicago_wide)
# summary results
summary(clpm_fit_cov_cont, fit.measures = TRUE, standardized = TRUE)## lavaan 0.6-19 ended normally after 33 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 34
## Number of equality constraints 12
##
## Number of observations 342
##
## Model Test User Model:
##
## Test statistic 283.231
## Degrees of freedom 30
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 2637.815
## Degrees of freedom 44
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.902
## Tucker-Lewis Index (TLI) 0.857
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2341.161
## Loglikelihood unrestricted model (H1) -2199.545
##
## Akaike (AIC) 4726.322
## Bayesian (BIC) 4810.687
## Sample-size adjusted Bayesian (SABIC) 4740.898
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.157
## 90 Percent confidence interval - lower 0.141
## 90 Percent confidence interval - upper 0.174
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.068
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## violent_2014 ~
## vilnt_2013 (a) 0.874 0.013 65.912 0.000 0.874 0.876
## cmpln_2013 (b) 0.042 0.014 3.015 0.003 0.042 0.038
## cmpln_2014 (e) 0.044 0.004 10.773 0.000 0.044 0.042
## violent_2013 ~
## vilnt_2012 (a) 0.874 0.013 65.912 0.000 0.874 0.870
## cmpln_2012 (b) 0.042 0.014 3.015 0.003 0.042 0.041
## cmpln_2013 (e) 0.044 0.004 10.773 0.000 0.044 0.040
## violent_2012 ~
## vilnt_2011 (a) 0.874 0.013 65.912 0.000 0.874 0.893
## cmpln_2011 (b) 0.042 0.014 3.015 0.003 0.042 0.042
## cmpln_2012 (e) 0.044 0.004 10.773 0.000 0.044 0.043
## complaints_2014 ~
## cmpln_2013 (c) 0.244 0.030 8.125 0.000 0.244 0.233
## vilnt_2013 (d) 0.038 0.016 2.324 0.020 0.038 0.040
## vilnt_2014 (f) 0.279 0.013 20.703 0.000 0.279 0.297
## complaints_2013 ~
## cmpln_2012 (c) 0.244 0.030 8.125 0.000 0.244 0.261
## vilnt_2012 (d) 0.038 0.016 2.324 0.020 0.038 0.042
## vilnt_2013 (f) 0.279 0.013 20.703 0.000 0.279 0.311
## complaints_2012 ~
## cmpln_2011 (c) 0.244 0.030 8.125 0.000 0.244 0.252
## vilnt_2011 (d) 0.038 0.016 2.324 0.020 0.038 0.040
## vilnt_2012 (f) 0.279 0.013 20.703 0.000 0.279 0.290
## complaints_2011 ~
## ppltn_dnst -0.202 0.075 -2.708 0.007 -0.202 -0.141
## FAC_disadv 0.206 0.042 4.956 0.000 0.206 0.257
## violent_2011 ~
## ppltn_dnst 0.117 0.076 1.536 0.124 0.117 0.079
## FAC_disadv 0.244 0.042 5.750 0.000 0.244 0.296
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .complaints_2011 ~~
## .violent_2011 0.340 0.044 7.734 0.000 0.340 0.460
## .violent_2012 ~~
## .complants_2012 -0.002 0.012 -0.173 0.863 -0.002 -0.009
## .violent_2013 ~~
## .complants_2013 -0.052 0.014 -3.795 0.000 -0.052 -0.200
## .violent_2014 ~~
## .complants_2014 -0.021 0.014 -1.545 0.122 -0.021 -0.079
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .violent_2014 0.135 0.010 13.070 0.000 0.135 0.168
## .violent_2013 0.141 0.011 13.092 0.000 0.141 0.175
## .violent_2012 0.101 0.008 13.078 0.000 0.101 0.126
## .complants_2014 0.545 0.042 13.077 0.000 0.545 0.764
## .complants_2013 0.486 0.037 13.085 0.000 0.486 0.748
## .complants_2012 0.543 0.042 13.080 0.000 0.543 0.731
## .complants_2011 0.726 0.056 13.077 0.000 0.726 0.919
## .violent_2011 0.752 0.058 13.077 0.000 0.752 0.903Question 6 As we discussed, the traditional
cross-lagged panel model does not properly handle unobserved
heterogeneity. Let’s fit Allison et al.’s dynamic panel model with fixed
effects. We can use the dpm() function from the
dpm package.
This function requires the dataset to be in a long format. Coding is
simple, so we can use all 25 years of data! We just need to tell
R that this is a panel dataset using the
panel_data() function.
# load the dpm package
library(dpm)
# treat the dataset as a panel dataset
chicago.panel <- chicago %>%
mutate(wave = year - 1990) %>%
panel_data(id = NC_NUM,
wave = wave)The dpm() function is very simple. With a long dataset
defined by a panel_data() function, we can simply:
# template to fit a dpm() model
my_dpm <- dpm(dependent_variable ~ pre(lag(independent_variable)) + z | w + v, # time-varying covariate z | time-constant covariates w and v
data = my_data, error.inv = T, information = "observed", missing = "ML",
y.free = F, x.free = F # constraining autoregressive and cross-lagged panel models to equality
)The DPM approach does not permit modelling reciprocal relationships. So, let’s start modelling the effects of complaints about police misuse of force on violent crime while accounting for reverse causality.
# fit a DPM with FEs
dpm_violent <- dpm(violent ~ pre(lag(complaints)),
data = chicago.panel, error.inv = T, information = "observed", missing = "ML")
# print results
summary(dpm_violent)## MODEL INFO:
## Dependent variable: violent
## Total observations: 342
## Complete observations: 342
## Time periods: 2 - 26
##
## MODEL FIT:
## 𝛘²(669) = 2430.269
## RMSEA = 0.088, 90% CI [0.084, 0.092]
## p(RMSEA < .05) = 0
## SRMR = 0.045
##
## | | Est. | S.E. | z val. | p |
## |:-----------------------|------:|------:|-------:|------:|
## | complaints (t - 1) | 0.020 | 0.005 | 3.922 | 0.000 |
## | violent (t - 1) | 0.539 | 0.010 | 52.503 | 0.000 |
##
## Model converged after 402 iterationsQuestion 7 Now, let’s add two time-constant control
variables to the cross-lagged panel model:
population_density and FAC_disadv.
# fit a DPM with FEs
dpm_violent_cov <- dpm(violent ~ pre(lag(complaints)) | FAC_disadv + population_density,
data = chicago.panel, error.inv = T, information = "observed", missing = "ML")
# print results
summary(dpm_violent_cov)## MODEL INFO:
## Dependent variable: violent
## Total observations: 342
## Complete observations: 342
## Time periods: 2 - 26
##
## MODEL FIT:
## 𝛘²(717) = 2558.631
## RMSEA = 0.087, 90% CI [0.083, 0.09]
## p(RMSEA < .05) = 0
## SRMR = 0.047
##
## | | Est. | S.E. | z val. | p |
## |:-----------------------|------:|------:|-------:|------:|
## | complaints (t - 1) | 0.021 | 0.005 | 4.050 | 0.000 |
## | FAC_disadv | 0.129 | 0.017 | 7.662 | 0.000 |
## | population_density | 0.089 | 0.030 | 3.001 | 0.003 |
## | violent (t - 1) | 0.539 | 0.010 | 52.492 | 0.000 |
##
## Model converged after 509 iterationsQuestion 8 Let’s now fit Hamaker et al.’s Random
Intercepts Cross-Lagged Panel Model. Unfortunately, the RI-CLPM is not
fully implemented in any packages—so we need to rely on
lavaan and manually code the model.
RICLPM <- '
# Create between components (random intercepts)
RI_complaints =~ 1*complaints_2011 + 1*complaints_2012 + 1*complaints_2013 + 1*complaints_2014
RI_violent =~ 1*violent_2011 + 1*violent_2012 + 1*violent_2013 + 1*violent_2014
# Create within-person centered variables
wcomplaints_2011 =~ 1*complaints_2011
wcomplaints_2012 =~ 1*complaints_2012
wcomplaints_2013 =~ 1*complaints_2013
wcomplaints_2014 =~ 1*complaints_2014
wviolent_2011 =~ 1*violent_2011
wviolent_2012 =~ 1*violent_2012
wviolent_2013 =~ 1*violent_2013
wviolent_2014 =~ 1*violent_2014
# Estimate lagged effects between within-person centered variables
wcomplaints_2012 ~ wcomplaints_2011 + wviolent_2011
wviolent_2012 ~ wcomplaints_2011 + wviolent_2011
wcomplaints_2013 ~ wcomplaints_2012 + wviolent_2012
wviolent_2013 ~ wcomplaints_2012 + wviolent_2012
wcomplaints_2014 ~ wcomplaints_2013 + wviolent_2013
wviolent_2014 ~ wcomplaints_2013 + wviolent_2013
# Estimate covariance between within-person centered variables at first wave
wcomplaints_2011 ~~ wviolent_2011 # Covariance
# Estimate covariances between residuals of within-person centered variables
wcomplaints_2012 ~~ wviolent_2012
wcomplaints_2013 ~~ wviolent_2013
wcomplaints_2014 ~~ wviolent_2014
# Estimate variance and covariance of random intercepts
RI_complaints ~~ RI_complaints
RI_violent ~~ RI_violent
RI_complaints ~~ RI_violent
# Estimate (residual) variance of within-person centered variables
wcomplaints_2011 ~~ wcomplaints_2011 # Variances
wviolent_2011 ~~ wviolent_2011
wcomplaints_2012 ~~ wcomplaints_2012 # Residual variances
wviolent_2012 ~~ wviolent_2012
wcomplaints_2013 ~~ wcomplaints_2013
wviolent_2013 ~~ wviolent_2013
wcomplaints_2014 ~~ wcomplaints_2014
wviolent_2014 ~~ wviolent_2014
'
RICLPM_nocovs <- lavaan(RICLPM,
data = chicago_wide,
missing = 'ML',
meanstructure = T,
int.ov.free = T)
summary(RICLPM_nocovs, standardized = T)## lavaan 0.6-19 ended normally after 63 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 35
##
## Number of observations 342
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 21.526
## Degrees of freedom 9
## P-value (Chi-square) 0.011
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## RI_complaints =~
## complants_2011 1.000 0.515 0.597
## complants_2012 1.000 0.515 0.602
## complants_2013 1.000 0.515 0.613
## complants_2014 1.000 0.515 0.609
## RI_violent =~
## violent_2011 1.000 0.862 0.953
## violent_2012 1.000 0.862 0.955
## violent_2013 1.000 0.862 0.950
## violent_2014 1.000 0.862 0.955
## wcomplaints_2011 =~
## complants_2011 1.000 0.692 0.802
## wcomplaints_2012 =~
## complants_2012 1.000 0.683 0.798
## wcomplaints_2013 =~
## complants_2013 1.000 0.663 0.790
## wcomplaints_2014 =~
## complants_2014 1.000 0.670 0.793
## wviolent_2011 =~
## violent_2011 1.000 0.273 0.302
## wviolent_2012 =~
## violent_2012 1.000 0.267 0.296
## wviolent_2013 =~
## violent_2013 1.000 0.285 0.314
## wviolent_2014 =~
## violent_2014 1.000 0.268 0.297
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## wcomplaints_2012 ~
## wcomplnts_2011 0.102 0.069 1.476 0.140 0.104 0.104
## wviolent_2011 -0.447 0.289 -1.550 0.121 -0.179 -0.179
## wviolent_2012 ~
## wcomplnts_2011 0.050 0.027 1.849 0.064 0.129 0.129
## wviolent_2011 0.194 0.157 1.233 0.218 0.198 0.198
## wcomplaints_2013 ~
## wcomplnts_2012 0.103 0.068 1.517 0.129 0.106 0.106
## wviolent_2012 0.524 0.213 2.459 0.014 0.211 0.211
## wviolent_2013 ~
## wcomplnts_2012 -0.077 0.034 -2.277 0.023 -0.185 -0.185
## wviolent_2012 0.090 0.120 0.753 0.451 0.085 0.085
## wcomplaints_2014 ~
## wcomplnts_2013 0.018 0.075 0.237 0.812 0.018 0.018
## wviolent_2013 -0.155 0.205 -0.754 0.451 -0.066 -0.066
## wviolent_2014 ~
## wcomplnts_2013 0.058 0.036 1.617 0.106 0.144 0.144
## wviolent_2013 0.011 0.171 0.066 0.947 0.012 0.012
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## wcomplaints_2011 ~~
## wviolent_2011 0.003 0.015 0.223 0.824 0.018 0.018
## .wcomplaints_2012 ~~
## .wviolent_2012 0.009 0.018 0.500 0.617 0.053 0.053
## .wcomplaints_2013 ~~
## .wviolent_2013 0.019 0.015 1.264 0.206 0.106 0.106
## .wcomplaints_2014 ~~
## .wviolent_2014 0.004 0.015 0.238 0.812 0.021 0.021
## RI_complaints ~~
## RI_violent 0.347 0.035 9.788 0.000 0.781 0.781
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .complants_2011 1.137 0.047 24.377 0.000 1.137 1.318
## .complants_2012 0.945 0.046 20.426 0.000 0.945 1.105
## .complants_2013 0.826 0.045 18.188 0.000 0.826 0.984
## .complants_2014 0.717 0.046 15.675 0.000 0.717 0.848
## .violent_2011 3.408 0.049 69.683 0.000 3.408 3.768
## .violent_2012 3.386 0.049 69.378 0.000 3.386 3.752
## .violent_2013 3.257 0.049 66.327 0.000 3.257 3.587
## .violent_2014 3.070 0.049 62.876 0.000 3.070 3.400
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## RI_complaints 0.265 0.033 8.080 0.000 1.000 1.000
## RI_violent 0.743 0.059 12.701 0.000 1.000 1.000
## wcomplnts_2011 0.479 0.045 10.746 0.000 1.000 1.000
## wviolent_2011 0.074 0.014 5.143 0.000 1.000 1.000
## .wcomplnts_2012 0.447 0.048 9.366 0.000 0.958 0.958
## .wviolent_2012 0.067 0.012 5.624 0.000 0.943 0.943
## .wcomplnts_2013 0.415 0.039 10.583 0.000 0.943 0.943
## .wviolent_2013 0.078 0.015 5.058 0.000 0.959 0.959
## .wcomplnts_2014 0.447 0.043 10.425 0.000 0.996 0.996
## .wviolent_2014 0.070 0.015 4.602 0.000 0.979 0.979
## .complants_2011 0.000 0.000 0.000
## .complants_2012 0.000 0.000 0.000
## .complants_2013 0.000 0.000 0.000
## .complants_2014 0.000 0.000 0.000
## .violent_2011 0.000 0.000 0.000
## .violent_2012 0.000 0.000 0.000
## .violent_2013 0.000 0.000 0.000
## .violent_2014 0.000 0.000 0.000Question 7 Repeat the procedures above to model the effects of violent crime on complaints about police misuse of force accounting for reverse causality.