Reliability‑Corrected Single‑Item SEM
relcorr_single_item.RdReplace (some of) the first‑order latent variables in a lavaan measurement model by **single composite indicators whose error variances are fixed from Cronbach's \(\alpha\). The function returns a modified lavaan model syntax together with an augmented data set that contains the newly created composite variables, so that you can fit the full SEM in a single step.
Arguments
- syntax
A character string containing lavaan model syntax. Must at least include the measurement relations (
=~).- data
A
data.frame,tibbleor object coercible to a data frame that holds the raw observed indicators.- choose
*Optional.* Character vector with the names of the latent variables you wish to replace by single indicators. Defaults to **all** first‑order latent variables in `syntax`.
Value
An object of S3 class `modsem_relcorr` (a named list) with elements:
- `syntax`
Modified lavaan syntax string.
- `data`
Data frame with additional composite indicator columns.
- `parTable`
Parameter table corresponding to `syntax`.
- `reliability`
Named numeric vector of reliabilities (one per latent variable).
- `AVE`
Named numeric vector with Average Variance Extracted values.
- `lVs`
Character vector of latent variables that were corrected.
- `single.items`
Character vector with names for the generated reliability corrected items
Details
The resulting object can be fed directly into modsem or lavaan::sem
by supplying syntax = $syntax and data = $data.
Examples
# \dontrun{
tpb_uk <- "
# Outer Model (Based on Hagger et al., 2007)
ATT =~ att3 + att2 + att1 + att4
SN =~ sn4 + sn2 + sn3 + sn1
PBC =~ pbc2 + pbc1 + pbc3 + pbc4
INT =~ int2 + int1 + int3 + int4
BEH =~ beh3 + beh2 + beh1 + beh4
# Inner Model (Based on Steinmetz et al., 2011)
INT ~ ATT + SN + PBC
BEH ~ INT + PBC
BEH ~ INT:PBC
"
corrected <- relcorr_single_item(syntax = tpb_uk, data = TPB_UK)
print(corrected)
#> Average Variance Extracted:
#> ATT: 0.773
#> SN: 0.759
#> PBC: 0.765
#> INT: 0.790
#> BEH: 0.769
#>
#> Chronbach's Alpha:
#> ATT: 0.941
#> SN: 0.917
#> PBC: 0.931
#> INT: 0.984
#> BEH: 0.942
#>
#> Generated Syntax:
#> INT ~ ATT
#> INT ~ SN
#> INT ~ PBC
#> BEH ~ INT
#> BEH ~ PBC
#> BEH ~ INT:PBC
#> ATT =~ 1*composite_ATT_
#> SN =~ 1*composite_SN_
#> PBC =~ 1*composite_PBC_
#> INT =~ 1*composite_INT_
#> BEH =~ 1*composite_BEH_
#> composite_ATT_ ~~ 0.230703252252833*composite_ATT_
#> composite_SN_ ~~ 0.219569422798239*composite_SN_
#> composite_PBC_ ~~ 0.269459429965901*composite_PBC_
#> composite_INT_ ~~ 0.0783705643995773*composite_INT_
#> composite_BEH_ ~~ 0.276556964349188*composite_BEH_
#>
#> Generated Items:
#> 'data.frame': 1169 obs. of 5 variables:
#> $ composite_ATT_: num 5.5 5 2 5.5 4.5 1 1 1 4.25 6.25 ...
#> $ composite_SN_ : num 6.5 4.75 3.5 6.5 5.25 3 2.25 4 4.5 6 ...
#> $ composite_PBC_: num 4 4.5 1.75 5.75 5.5 1 1 1 5 5.75 ...
#> $ composite_INT_: num 4 4.5 1 5.25 5.25 1 1 1 5 5 ...
#> $ composite_BEH_: num 1 3.75 1.25 2.75 5.25 1 1 1 2.5 5.5 ...
syntax <- corrected$syntax
data <- corrected$data
est_dca <- modsem(syntax, data = data, method = "dblcent")
est_lms <- modsem(syntax, data = data, method="lms", nodes=32)
summary(est_lms)
#>
#> modsem (version 1.0.11):
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of observations 1169
#> Number of iterations 49
#> Loglikelihood -7276.79
#> Akaike (AIC) 14591.57
#> Bayesian (BIC) 14687.79
#>
#> Numerical Integration:
#> Points of integration (per dim) 32
#> Dimensions 1
#> Total points of integration 32
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -9352.47
#> Akaike (AIC) 18740.95
#> Bayesian (BIC) 18832.1
#> Chi-square 34.76
#> Degrees of Freedom (Chi-square) 2
#> P-value (Chi-square) 0.000
#> RMSEA 0.118
#>
#> Comparative Fit to H0 (LRT test):
#> Loglikelihood change 2075.69
#> Difference test (D) 4151.38
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> INT 0.865
#> BEH 0.940
#> R-Squared Baseline Model (H0):
#> INT 0.856
#> BEH 0.869
#> R-Squared Change (H1 - H0):
#> INT 0.009
#> BEH 0.071
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information observed
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> PBC =~
#> cmpst_PBC_ 1.000
#> ATT =~
#> cmpst_ATT_ 1.000
#> SN =~
#> compst_SN_ 1.000
#> INT =~
#> cmpst_INT_ 1.000
#> BEH =~
#> cmpst_BEH_ 1.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> INT ~
#> PBC 1.054 0.053 19.825 0.000
#> ATT 0.000 0.051 -0.008 0.993
#> SN 0.026 0.030 0.871 0.384
#> BEH ~
#> PBC 0.719 0.043 16.802 0.000
#> INT 0.301 0.036 8.372 0.000
#> PBC:INT 0.150 0.008 18.283 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .cmpst_PBC_ 3.956 0.031 128.919 0.000
#> .cmpst_ATT_ 3.926 0.038 104.339 0.000
#> .compst_SN_ 4.466 0.038 117.695 0.000
#> .cmpst_INT_ 3.851 0.038 100.233 0.000
#> .cmpst_BEH_ 2.660 0.051 52.364 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> PBC ~~
#> ATT 3.516 0.091 38.798 0.000
#> SN 2.296 0.086 26.562 0.000
#> ATT ~~
#> SN 2.187 0.091 24.055 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .cmpst_PBC_ 0.269
#> .cmpst_ATT_ 0.231
#> .compst_SN_ 0.220
#> .cmpst_INT_ 0.078
#> .cmpst_BEH_ 0.277
#> PBC 3.905 0.094 41.382 0.000
#> ATT 3.916 0.127 30.785 0.000
#> SN 2.527 0.104 24.215 0.000
#> .INT 0.695 0.044 15.773 0.000
#> .BEH 0.328 0.032 10.347 0.000
#>
# }