lavaanif (!require(lavaan)) install.packages("lavaan")Loading required package: lavaan
This is lavaan 0.5-23.1097
lavaan is BETA software! Please report any bugs.library(lavaan)Example data: 634 older adults (age 80-100) self-reporting on 7 items assessing the Instrumental Activities of Daily Living (IADL) as follows:
Two versions of a response format were available:
Binary -> 0 = “needs help”, 1 = “does not need help”
Categorical -> 0 = “can’t do it”, 1=”big problems”, 2=”some problems”, 3=”no problems”
Higher scores indicate greater function. We will look at each response format in turn.
We will be using the categorical items in this example.
The data are in a text file named adl.dat originally used in Mplus (so no column names were included at the top of the file). The file contains more items than we will use, so we select only items above from the whole file.
#read in data file (Mplus file format so having to label columns)
adlData = read.table(file = "adl2.dat", header = FALSE, na.strings = ".", col.names = c("case", "female", paste0("dpa", 1:14), paste0("dia", 1:7), paste0("cpa", 1:14), paste0("cia", 1:7)))
#select Situations items and PersonID variables
iadlDataInit = adlData[c(paste0("cia", 1:7), "female")]
#remove cases with all items missing
removeCases = which(apply(X = iadlDataInit, MARGIN = 1, FUN = function (x){ length(which(is.na(x)))}) == 7)
iadlData = iadlDataInit[-removeCases,]
#recode group variable
iadlData$sex = NA_character_
iadlData$sex[which(iadlData$female==1)] = "female"
iadlData$sex[which(iadlData$female==0)] = "male"Here everything is estimated separately across groups except for parameters needed to be constrained for identification.
configuralSyntax = "
#============================================================================================================
#Factor loadings (AKA slopes/discriminations) all freely estimated in both groups with label for each group
IADL =~ c(L1M, L1F)*cia1 + c(L2M, L2F)*cia2 + c(L3M, L3F)*cia3 +
c(L4M, L4F)*cia4 + c(L5M, L5F)*cia5 + c(L6M, L6F)*cia6 + c(L7M, L7F)*cia7
#============================================================================================================
#Item Thresholds all freely estimated in both groups with label for each group
cia1|c(I1T1M, I1T1F)*t1; cia1|c(I1T2M, I1T2F)*t2; cia1|c(I1T3M, I1T3F)*t3;
cia2|c(I2T1M, I2T1F)*t1; cia2|c(I2T2M, I2T2F)*t2; cia2|c(I2T3M, I2T3F)*t3;
cia3|c(I3T1M, I3T1F)*t1; cia3|c(I3T2M, I3T2F)*t2; cia3|c(I3T3M, I3T3F)*t3;
cia4|c(I4T1M, I4T1F)*t1; cia4|c(I4T2M, I4T2F)*t2; cia4|c(I4T3M, I4T3F)*t3;
cia5|c(I5T1M, I5T1F)*t1; cia5|c(I5T2M, I5T2F)*t2; cia5|c(I5T3M, I5T3F)*t3;
cia6|c(I6T1M, I6T1F)*t1; cia6|c(I6T2M, I6T2F)*t2; cia6|c(I6T3M, I6T3F)*t3;
cia7|c(I7T1M, I7T1F)*t1; cia7|c(I7T2M, I7T2F)*t2; cia7|c(I7T3M, I7T3F)*t3;
#============================================================================================================
#Redidual variances all fixed to one for each group
cia1 ~~ c(1, 1)*cia1; cia2 ~~ c(1, 1)*cia2; cia3 ~~ c(1, 1)*cia3;
cia4 ~~ c(1, 1)*cia4; cia5 ~~ c(1, 1)*cia5; cia6 ~~ c(1, 1)*cia6; cia7 ~~ c(1, 1)*cia7
#============================================================================================================
#Factor variance fixed to 1 for identification in each group
IADL ~~ c(1,1)*IADL
#============================================================================================================
#Factor mean fixed to zero for identification in each group
IADL ~ c(0,0)*0
#============================================================================================================
"
configuralEstimates = lavaan(model = configuralSyntax, data = iadlData,
ordered = c("cia1", "cia2", "cia3", "cia4", "cia5", "cia6", "cia7"), std.lv = TRUE,
parameterization="theta", group = "sex")lavaan WARNING: 40 bivariate tables have empty cells; to see them, use:
lavInspect(fit, "zero.cell.tables")summary(configuralEstimates, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 249 iterations
Used Total
Number of observations per group
male 163 214
female 383 420
Estimator DWLS Robust
Minimum Function Test Statistic 26.447 64.705
Degrees of freedom 28 28
P-value (Chi-square) 0.549 0.000
Scaling correction factor 0.451
Shift parameter for each group:
male 1.795
female 4.217
for simple second-order correction (Mplus variant)
Chi-square for each group:
male 8.363 20.354
female 18.084 44.351
Model test baseline model:
Minimum Function Test Statistic 69203.021 35439.603
Degrees of freedom 42 42
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 1.000 0.999
Tucker-Lewis Index (TLI) 1.000 0.998
Robust Comparative Fit Index (CFI) NA
Robust Tucker-Lewis Index (TLI) NA
Root Mean Square Error of Approximation:
RMSEA 0.000 0.069
90 Percent Confidence Interval 0.000 0.044 0.047 0.092
P-value RMSEA <= 0.05 0.979 0.072
Robust RMSEA NA
90 Percent Confidence Interval NA NA
Standardized Root Mean Square Residual:
SRMR 0.025 0.025
Weighted Root Mean Square Residual:
WRMR 0.794 0.794
Parameter Estimates:
Information Expected
Standard Errors Robust.sem
Group 1 [male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL =~
cia1 (L1M) 4.973 0.883 5.633 0.000 4.973 0.980
cia2 (L2M) 3.749 0.825 4.546 0.000 3.749 0.966
cia3 (L3M) 3.436 0.538 6.383 0.000 3.436 0.960
cia4 (L4M) 4.693 0.983 4.774 0.000 4.693 0.978
cia5 (L5M) 2.997 0.525 5.705 0.000 2.997 0.949
cia6 (L6M) 3.342 0.854 3.914 0.000 3.342 0.958
cia7 (L7M) 1.116 0.238 4.686 0.000 1.116 0.745
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL 0.000 0.000 0.000
.cia1 0.000 0.000 0.000
.cia2 0.000 0.000 0.000
.cia3 0.000 0.000 0.000
.cia4 0.000 0.000 0.000
.cia5 0.000 0.000 0.000
.cia6 0.000 0.000 0.000
.cia7 0.000 0.000 0.000
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1|t1 (I1T1) -6.378 1.093 -5.837 0.000 -6.378 -1.257
cia1|t2 (I1T2) -4.683 0.844 -5.551 0.000 -4.683 -0.923
cia1|t3 (I1T3) -1.061 0.589 -1.802 0.072 -1.061 -0.209
cia2|t1 (I2T1) -5.013 0.939 -5.341 0.000 -5.013 -1.292
cia2|t2 (I2T2) -4.626 0.897 -5.159 0.000 -4.626 -1.192
cia2|t3 (I2T3) -2.827 0.727 -3.889 0.000 -2.827 -0.729
cia3|t1 (I3T1) -4.381 0.602 -7.272 0.000 -4.381 -1.224
cia3|t2 (I3T2) -3.389 0.579 -5.857 0.000 -3.389 -0.947
cia3|t3 (I3T3) -0.918 0.403 -2.280 0.023 -0.918 -0.257
cia4|t1 (I4T1) -5.722 1.075 -5.320 0.000 -5.722 -1.192
cia4|t2 (I4T2) -4.905 0.923 -5.314 0.000 -4.905 -1.022
cia4|t3 (I4T3) -2.679 0.730 -3.671 0.000 -2.679 -0.558
cia5|t1 (I5T1) -5.043 0.750 -6.724 0.000 -5.043 -1.596
cia5|t2 (I5T2) -3.312 0.556 -5.959 0.000 -3.312 -1.048
cia5|t3 (I5T3) -1.764 0.441 -3.997 0.000 -1.764 -0.558
cia6|t1 (I6T1) -5.770 1.270 -4.544 0.000 -5.770 -1.654
cia6|t2 (I6T2) -4.635 1.056 -4.388 0.000 -4.635 -1.328
cia6|t3 (I6T3) -3.140 0.854 -3.676 0.000 -3.140 -0.900
cia7|t1 (I7T1) -3.128 0.424 -7.374 0.000 -3.128 -2.088
cia7|t2 (I7T2) -2.573 0.324 -7.953 0.000 -2.573 -1.717
cia7|t3 (I7T3) -1.884 0.265 -7.120 0.000 -1.884 -1.257
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.cia1 1.000 1.000 0.039
.cia2 1.000 1.000 0.066
.cia3 1.000 1.000 0.078
.cia4 1.000 1.000 0.043
.cia5 1.000 1.000 0.100
.cia6 1.000 1.000 0.082
.cia7 1.000 1.000 0.446
IADL 1.000 1.000 1.000
Scales y*:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1 0.197 0.197 1.000
cia2 0.258 0.258 1.000
cia3 0.279 0.279 1.000
cia4 0.208 0.208 1.000
cia5 0.317 0.317 1.000
cia6 0.287 0.287 1.000
cia7 0.667 0.667 1.000
R-Square:
Estimate
cia1 0.961
cia2 0.934
cia3 0.922
cia4 0.957
cia5 0.900
cia6 0.918
cia7 0.554
Group 2 [female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL =~
cia1 (L1F) 3.029 0.309 9.790 0.000 3.029 0.950
cia2 (L2F) 3.465 0.506 6.845 0.000 3.465 0.961
cia3 (L3F) 5.205 1.101 4.728 0.000 5.205 0.982
cia4 (L4F) 3.204 0.317 10.108 0.000 3.204 0.955
cia5 (L5F) 2.271 0.216 10.512 0.000 2.271 0.915
cia6 (L6F) 1.736 0.175 9.941 0.000 1.736 0.867
cia7 (L7F) 1.011 0.157 6.452 0.000 1.011 0.711
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL 0.000 0.000 0.000
.cia1 0.000 0.000 0.000
.cia2 0.000 0.000 0.000
.cia3 0.000 0.000 0.000
.cia4 0.000 0.000 0.000
.cia5 0.000 0.000 0.000
.cia6 0.000 0.000 0.000
.cia7 0.000 0.000 0.000
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1|t1 (I1T1) -4.575 0.407 -11.248 0.000 -4.575 -1.434
cia1|t2 (I1T2) -3.358 0.345 -9.731 0.000 -3.358 -1.053
cia1|t3 (I1T3) -0.856 0.232 -3.690 0.000 -0.856 -0.268
cia2|t1 (I2T1) -5.605 0.687 -8.154 0.000 -5.605 -1.554
cia2|t2 (I2T2) -4.748 0.619 -7.670 0.000 -4.748 -1.317
cia2|t3 (I2T3) -2.955 0.496 -5.957 0.000 -2.955 -0.819
cia3|t1 (I3T1) -7.802 1.453 -5.371 0.000 -7.802 -1.472
cia3|t2 (I3T2) -6.737 1.308 -5.151 0.000 -6.737 -1.271
cia3|t3 (I3T3) -4.441 1.026 -4.329 0.000 -4.441 -0.838
cia4|t1 (I4T1) -4.526 0.387 -11.698 0.000 -4.526 -1.348
cia4|t2 (I4T2) -2.844 0.309 -9.202 0.000 -2.844 -0.847
cia4|t3 (I4T3) -0.742 0.240 -3.090 0.002 -0.742 -0.221
cia5|t1 (I5T1) -3.970 0.322 -12.338 0.000 -3.970 -1.600
cia5|t2 (I5T2) -1.944 0.211 -9.215 0.000 -1.944 -0.783
cia5|t3 (I5T3) -0.285 0.165 -1.729 0.084 -0.285 -0.115
cia6|t1 (I6T1) -3.303 0.249 -13.270 0.000 -3.303 -1.649
cia6|t2 (I6T2) -2.227 0.196 -11.351 0.000 -2.227 -1.112
cia6|t3 (I6T3) -1.250 0.168 -7.441 0.000 -1.250 -0.624
cia7|t1 (I7T1) -3.437 0.343 -10.006 0.000 -3.437 -2.417
cia7|t2 (I7T2) -2.761 0.243 -11.340 0.000 -2.761 -1.941
cia7|t3 (I7T3) -1.747 0.168 -10.369 0.000 -1.747 -1.228
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.cia1 1.000 1.000 0.098
.cia2 1.000 1.000 0.077
.cia3 1.000 1.000 0.036
.cia4 1.000 1.000 0.089
.cia5 1.000 1.000 0.162
.cia6 1.000 1.000 0.249
.cia7 1.000 1.000 0.494
IADL 1.000 1.000 1.000
Scales y*:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1 0.314 0.314 1.000
cia2 0.277 0.277 1.000
cia3 0.189 0.189 1.000
cia4 0.298 0.298 1.000
cia5 0.403 0.403 1.000
cia6 0.499 0.499 1.000
cia7 0.703 0.703 1.000
R-Square:
Estimate
cia1 0.902
cia2 0.923
cia3 0.964
cia4 0.911
cia5 0.838
cia6 0.751
cia7 0.506Here the IFA factor loadings are held equal across groups but the factor variance is freed.
metricSyntax1 = "
#============================================================================================================
#Factor loadings (AKA slopes/discriminations) all constriaind across both groups with same label for both groups *****
IADL =~ c(L1, L1)*cia1 + c(L2, L2)*cia2 + c(L3, L3)*cia3 +
c(L4, L4)*cia4 + c(L5, L5)*cia5 + c(L6, L6)*cia6 + c(L7, L7)*cia7
#============================================================================================================
#Item Thresholds all freely estimated in both groups with label for each group
cia1|c(I1T1M, I1T1F)*t1; cia1|c(I1T2M, I1T2F)*t2; cia1|c(I1T3M, I1T3F)*t3;
cia2|c(I2T1M, I2T1F)*t1; cia2|c(I2T2M, I2T2F)*t2; cia2|c(I2T3M, I2T3F)*t3;
cia3|c(I3T1M, I3T1F)*t1; cia3|c(I3T2M, I3T2F)*t2; cia3|c(I3T3M, I3T3F)*t3;
cia4|c(I4T1M, I4T1F)*t1; cia4|c(I4T2M, I4T2F)*t2; cia4|c(I4T3M, I4T3F)*t3;
cia5|c(I5T1M, I5T1F)*t1; cia5|c(I5T2M, I5T2F)*t2; cia5|c(I5T3M, I5T3F)*t3;
cia6|c(I6T1M, I6T1F)*t1; cia6|c(I6T2M, I6T2F)*t2; cia6|c(I6T3M, I6T3F)*t3;
cia7|c(I7T1M, I7T1F)*t1; cia7|c(I7T2M, I7T2F)*t2; cia7|c(I7T3M, I7T3F)*t3;
#============================================================================================================
#Redidual variances all fixed to one for each group
cia1 ~~ c(1, 1)*cia1; cia2 ~~ c(1, 1)*cia2; cia3 ~~ c(1, 1)*cia3;
cia4 ~~ c(1, 1)*cia4; cia5 ~~ c(1, 1)*cia5; cia6 ~~ c(1, 1)*cia6; cia7 ~~ c(1, 1)*cia7
#============================================================================================================
#Factor variance fixed to 1 for identification in group1; estimated for group 2
IADL ~~ c(1,NA)*IADL
#============================================================================================================
#Factor mean fixed to zero for identification in each group
IADL ~ c(0,0)*0
#============================================================================================================
"
metricEstimates1 = lavaan(model = metricSyntax1, data = iadlData, ordered = c("cia1", "cia2", "cia3", "cia4", "cia5", "cia6", "cia7"),
parameterization="theta", group = "sex")lavaan WARNING: 40 bivariate tables have empty cells; to see them, use:
lavInspect(fit, "zero.cell.tables")summary(metricEstimates1, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 210 iterations
Used Total
Number of observations per group
male 163 214
female 383 420
Estimator DWLS Robust
Minimum Function Test Statistic 42.967 63.802
Degrees of freedom 34 34
P-value (Chi-square) 0.139 0.001
Scaling correction factor 0.792
Shift parameter for each group:
male 2.851
female 6.700
for simple second-order correction (Mplus variant)
Chi-square for each group:
male 20.789 29.099
female 22.178 34.702
Model test baseline model:
Minimum Function Test Statistic 69203.021 35439.603
Degrees of freedom 42 42
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 1.000 0.999
Tucker-Lewis Index (TLI) 1.000 0.999
Robust Comparative Fit Index (CFI) NA
Robust Tucker-Lewis Index (TLI) NA
Root Mean Square Error of Approximation:
RMSEA 0.031 0.057
90 Percent Confidence Interval 0.000 0.057 0.035 0.078
P-value RMSEA <= 0.05 0.874 0.282
Robust RMSEA NA
90 Percent Confidence Interval NA NA
Standardized Root Mean Square Residual:
SRMR 0.031 0.031
Weighted Root Mean Square Residual:
WRMR 1.011 1.011
Parameter Estimates:
Information Expected
Standard Errors Robust.sem
Group 1 [male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL =~
cia1 (L1) 4.054 0.463 8.747 0.000 4.054 0.971
cia2 (L2) 4.658 0.757 6.153 0.000 4.658 0.978
cia3 (L3) 5.637 0.931 6.052 0.000 5.637 0.985
cia4 (L4) 4.415 0.556 7.940 0.000 4.415 0.975
cia5 (L5) 3.011 0.370 8.145 0.000 3.011 0.949
cia6 (L6) 2.571 0.316 8.147 0.000 2.571 0.932
cia7 (L7) 1.266 0.191 6.614 0.000 1.266 0.785
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL 0.000 0.000 0.000
.cia1 0.000 0.000 0.000
.cia2 0.000 0.000 0.000
.cia3 0.000 0.000 0.000
.cia4 0.000 0.000 0.000
.cia5 0.000 0.000 0.000
.cia6 0.000 0.000 0.000
.cia7 0.000 0.000 0.000
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1|t1 (I1T1) -5.250 0.542 -9.687 0.000 -5.250 -1.257
cia1|t2 (I1T2) -3.855 0.539 -7.152 0.000 -3.855 -0.923
cia1|t3 (I1T3) -0.873 0.444 -1.965 0.049 -0.873 -0.209
cia2|t1 (I2T1) -6.156 0.909 -6.772 0.000 -6.156 -1.292
cia2|t2 (I2T2) -5.681 0.879 -6.464 0.000 -5.681 -1.192
cia2|t3 (I2T3) -3.471 0.728 -4.769 0.000 -3.471 -0.729
cia3|t1 (I3T1) -7.009 1.048 -6.687 0.000 -7.009 -1.224
cia3|t2 (I3T2) -5.423 0.976 -5.558 0.000 -5.423 -0.947
cia3|t3 (I3T3) -1.469 0.648 -2.265 0.024 -1.469 -0.257
cia4|t1 (I4T1) -5.397 0.651 -8.295 0.000 -5.397 -1.192
cia4|t2 (I4T2) -4.626 0.633 -7.309 0.000 -4.626 -1.022
cia4|t3 (I4T3) -2.527 0.560 -4.514 0.000 -2.527 -0.558
cia5|t1 (I5T1) -5.064 0.566 -8.949 0.000 -5.064 -1.596
cia5|t2 (I5T2) -3.326 0.451 -7.379 0.000 -3.326 -1.048
cia5|t3 (I5T3) -1.771 0.380 -4.659 0.000 -1.771 -0.558
cia6|t1 (I6T1) -4.562 0.495 -9.225 0.000 -4.562 -1.654
cia6|t2 (I6T2) -3.664 0.409 -8.960 0.000 -3.664 -1.328
cia6|t3 (I6T3) -2.483 0.385 -6.455 0.000 -2.483 -0.900
cia7|t1 (I7T1) -3.368 0.399 -8.434 0.000 -3.368 -2.088
cia7|t2 (I7T2) -2.771 0.297 -9.329 0.000 -2.771 -1.717
cia7|t3 (I7T3) -2.029 0.248 -8.180 0.000 -2.029 -1.257
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.cia1 1.000 1.000 0.057
.cia2 1.000 1.000 0.044
.cia3 1.000 1.000 0.031
.cia4 1.000 1.000 0.049
.cia5 1.000 1.000 0.099
.cia6 1.000 1.000 0.131
.cia7 1.000 1.000 0.384
IADL 1.000 1.000 1.000
Scales y*:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1 0.239 0.239 1.000
cia2 0.210 0.210 1.000
cia3 0.175 0.175 1.000
cia4 0.221 0.221 1.000
cia5 0.315 0.315 1.000
cia6 0.363 0.363 1.000
cia7 0.620 0.620 1.000
R-Square:
Estimate
cia1 0.943
cia2 0.956
cia3 0.969
cia4 0.951
cia5 0.901
cia6 0.869
cia7 0.616
Group 2 [female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL =~
cia1 (L1) 4.054 0.463 8.747 0.000 3.033 0.950
cia2 (L2) 4.658 0.757 6.153 0.000 3.485 0.961
cia3 (L3) 5.637 0.931 6.052 0.000 4.218 0.973
cia4 (L4) 4.415 0.556 7.940 0.000 3.303 0.957
cia5 (L5) 3.011 0.370 8.145 0.000 2.253 0.914
cia6 (L6) 2.571 0.316 8.147 0.000 1.923 0.887
cia7 (L7) 1.266 0.191 6.614 0.000 0.947 0.688
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL 0.000 0.000 0.000
.cia1 0.000 0.000 0.000
.cia2 0.000 0.000 0.000
.cia3 0.000 0.000 0.000
.cia4 0.000 0.000 0.000
.cia5 0.000 0.000 0.000
.cia6 0.000 0.000 0.000
.cia7 0.000 0.000 0.000
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1|t1 (I1T1) -4.581 0.370 -12.366 0.000 -4.581 -1.434
cia1|t2 (I1T2) -3.362 0.320 -10.519 0.000 -3.362 -1.053
cia1|t3 (I1T3) -0.857 0.228 -3.759 0.000 -0.857 -0.268
cia2|t1 (I2T1) -5.636 0.633 -8.903 0.000 -5.636 -1.554
cia2|t2 (I2T2) -4.774 0.574 -8.321 0.000 -4.774 -1.317
cia2|t3 (I2T3) -2.971 0.468 -6.351 0.000 -2.971 -0.819
cia3|t1 (I3T1) -6.381 0.741 -8.608 0.000 -6.381 -1.472
cia3|t2 (I3T2) -5.510 0.687 -8.025 0.000 -5.510 -1.271
cia3|t3 (I3T3) -3.632 0.581 -6.250 0.000 -3.632 -0.838
cia4|t1 (I4T1) -4.653 0.392 -11.875 0.000 -4.653 -1.348
cia4|t2 (I4T2) -2.924 0.318 -9.209 0.000 -2.924 -0.847
cia4|t3 (I4T3) -0.763 0.246 -3.106 0.002 -0.763 -0.221
cia5|t1 (I5T1) -3.943 0.307 -12.857 0.000 -3.943 -1.600
cia5|t2 (I5T2) -1.931 0.205 -9.412 0.000 -1.931 -0.783
cia5|t3 (I5T3) -0.283 0.163 -1.733 0.083 -0.283 -0.115
cia6|t1 (I6T1) -3.574 0.289 -12.379 0.000 -3.574 -1.649
cia6|t2 (I6T2) -2.410 0.222 -10.844 0.000 -2.410 -1.112
cia6|t3 (I6T3) -1.353 0.185 -7.304 0.000 -1.353 -0.624
cia7|t1 (I7T1) -3.329 0.308 -10.791 0.000 -3.329 -2.417
cia7|t2 (I7T2) -2.674 0.212 -12.596 0.000 -2.674 -1.941
cia7|t3 (I7T3) -1.692 0.147 -11.548 0.000 -1.692 -1.228
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.cia1 1.000 1.000 0.098
.cia2 1.000 1.000 0.076
.cia3 1.000 1.000 0.053
.cia4 1.000 1.000 0.084
.cia5 1.000 1.000 0.165
.cia6 1.000 1.000 0.213
.cia7 1.000 1.000 0.527
IADL 0.560 0.138 4.048 0.000 1.000 1.000
Scales y*:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1 0.313 0.313 1.000
cia2 0.276 0.276 1.000
cia3 0.231 0.231 1.000
cia4 0.290 0.290 1.000
cia5 0.406 0.406 1.000
cia6 0.461 0.461 1.000
cia7 0.726 0.726 1.000
R-Square:
Estimate
cia1 0.902
cia2 0.924
cia3 0.947
cia4 0.916
cia5 0.835
cia6 0.787
cia7 0.473To check if the metric invariance model fits as well as the configural model, we can use the anova function:
anova(configuralEstimates, metricEstimates1)the condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedScaled Chi Square Difference Test (method = "satorra.2000")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
configuralEstimates 28 26.447
metricEstimates1 34 42.967 2.0956 1.8595 0.3211Here, disregard the error text. The p-value is large, so we will claim metric invariance holds and move onto scalar invariance.
thresholdSyntax1 = "
#============================================================================================================
#Factor loadings (AKA slopes/discriminations) from final metric model
IADL =~ c(L1, L1)*cia1 + c(L2, L2)*cia2 + c(L3, L3)*cia3 +
c(L4, L4)*cia4 + c(L5, L5)*cia5 + c(L6, L6)*cia6 + c(L7, L7)*cia7
#============================================================================================================
#Item Thresholds all constrained to be equal across groups except non-invariant item ****
cia1|c(I1T1, I1T1)*t1; cia1|c(I1T2, I1T2)*t2; cia1|c(I1T3, I1T3)*t3;
cia2|c(I2T1, I2T1)*t1; cia2|c(I2T2, I2T2)*t2; cia2|c(I2T3, I2T3)*t3;
cia3|c(I3T1, I3T1)*t1; cia3|c(I3T2, I3T2)*t2; cia3|c(I3T3, I3T3)*t3;
cia4|c(I4T1, I4T1)*t1; cia4|c(I4T2, I4T2)*t2; cia4|c(I4T3, I4T3)*t3;
cia5|c(I5T1, I5T1)*t1; cia5|c(I5T2, I5T2)*t2; cia5|c(I5T3, I5T3)*t3;
cia6|c(I6T1, I6T1)*t1; cia6|c(I6T2, I6T2)*t2; cia6|c(I6T3, I6T3)*t3;
cia7|c(I7T1, I7T1)*t1; cia7|c(I7T2, I7T2)*t2; cia7|c(I7T3, I7T3)*t3;
#============================================================================================================
#Redidual variances all fixed to one for each group
cia1 ~~ c(1, 1)*cia1; cia2 ~~ c(1, 1)*cia2; cia3 ~~ c(1, 1)*cia3;
cia4 ~~ c(1, 1)*cia4; cia5 ~~ c(1, 1)*cia5; cia6 ~~ c(1, 1)*cia6; cia7 ~~ c(1, 1)*cia7
#============================================================================================================
#Factor variance fixed to 1 for identification in males; estimated for females
IADL ~~ c(1,NA)*IADL
#============================================================================================================
#Factor mean fixed to zero for identification in males; estimated for females ****
IADL ~ c(0,NA)*0
#============================================================================================================
"
thresholdEstimates1 = lavaan(model = thresholdSyntax1, data = iadlData,
ordered = c("cia1", "cia2", "cia3", "cia4", "cia5", "cia6", "cia7"),
parameterization="theta", group = "sex")lavaan WARNING: 40 bivariate tables have empty cells; to see them, use:
lavInspect(fit, "zero.cell.tables")summary(thresholdEstimates1, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 161 iterations
Used Total
Number of observations per group
male 163 214
female 383 420
Estimator DWLS Robust
Minimum Function Test Statistic 97.897 151.935
Degrees of freedom 54 54
P-value (Chi-square) 0.000 0.000
Scaling correction factor 0.721
Shift parameter for each group:
male 4.850
female 11.396
for simple second-order correction (Mplus variant)
Chi-square for each group:
male 64.842 94.724
female 33.055 57.211
Model test baseline model:
Minimum Function Test Statistic 69203.021 35439.603
Degrees of freedom 42 42
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.999 0.997
Tucker-Lewis Index (TLI) 1.000 0.998
Robust Comparative Fit Index (CFI) NA
Robust Tucker-Lewis Index (TLI) NA
Root Mean Square Error of Approximation:
RMSEA 0.055 0.082
90 Percent Confidence Interval 0.037 0.072 0.066 0.097
P-value RMSEA <= 0.05 0.311 0.000
Robust RMSEA NA
90 Percent Confidence Interval NA NA
Standardized Root Mean Square Residual:
SRMR 0.031 0.031
Weighted Root Mean Square Residual:
WRMR 1.527 1.527
Parameter Estimates:
Information Expected
Standard Errors Robust.sem
Group 1 [male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL =~
cia1 (L1) 4.093 0.471 8.687 0.000 4.093 0.971
cia2 (L2) 4.715 0.762 6.190 0.000 4.715 0.978
cia3 (L3) 5.798 0.967 5.998 0.000 5.798 0.985
cia4 (L4) 4.465 0.558 8.007 0.000 4.465 0.976
cia5 (L5) 3.044 0.373 8.152 0.000 3.044 0.950
cia6 (L6) 2.579 0.316 8.155 0.000 2.579 0.932
cia7 (L7) 1.283 0.193 6.630 0.000 1.283 0.789
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL 0.000 0.000 0.000
.cia1 0.000 0.000 0.000
.cia2 0.000 0.000 0.000
.cia3 0.000 0.000 0.000
.cia4 0.000 0.000 0.000
.cia5 0.000 0.000 0.000
.cia6 0.000 0.000 0.000
.cia7 0.000 0.000 0.000
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1|t1 (I1T1) -5.386 0.520 -10.358 0.000 -5.386 -1.278
cia1|t2 (I1T2) -4.134 0.493 -8.392 0.000 -4.134 -0.981
cia1|t3 (I1T3) -1.545 0.444 -3.478 0.001 -1.545 -0.367
cia2|t1 (I2T1) -6.512 0.872 -7.467 0.000 -6.512 -1.351
cia2|t2 (I2T2) -5.744 0.820 -7.002 0.000 -5.744 -1.192
cia2|t3 (I2T3) -3.859 0.715 -5.400 0.000 -3.859 -0.801
cia3|t1 (I3T1) -7.560 1.036 -7.295 0.000 -7.560 -1.285
cia3|t2 (I3T2) -6.499 0.982 -6.618 0.000 -6.499 -1.105
cia3|t3 (I3T3) -4.110 0.836 -4.918 0.000 -4.110 -0.699
cia4|t1 (I4T1) -5.543 0.590 -9.392 0.000 -5.543 -1.211
cia4|t2 (I4T2) -3.990 0.535 -7.454 0.000 -3.990 -0.872
cia4|t3 (I4T3) -1.846 0.498 -3.708 0.000 -1.846 -0.403
cia5|t1 (I5T1) -4.674 0.425 -11.002 0.000 -4.674 -1.459
cia5|t2 (I5T2) -2.696 0.362 -7.455 0.000 -2.696 -0.841
cia5|t3 (I5T3) -1.076 0.331 -3.248 0.001 -1.076 -0.336
cia6|t1 (I6T1) -4.178 0.386 -10.838 0.000 -4.178 -1.511
cia6|t2 (I6T2) -3.054 0.342 -8.930 0.000 -3.054 -1.104
cia6|t3 (I6T3) -1.987 0.320 -6.213 0.000 -1.987 -0.719
cia7|t1 (I7T1) -3.521 0.308 -11.434 0.000 -3.521 -2.165
cia7|t2 (I7T2) -2.895 0.244 -11.876 0.000 -2.895 -1.780
cia7|t3 (I7T3) -1.978 0.196 -10.093 0.000 -1.978 -1.216
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.cia1 1.000 1.000 0.056
.cia2 1.000 1.000 0.043
.cia3 1.000 1.000 0.029
.cia4 1.000 1.000 0.048
.cia5 1.000 1.000 0.097
.cia6 1.000 1.000 0.131
.cia7 1.000 1.000 0.378
IADL 1.000 1.000 1.000
Scales y*:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1 0.237 0.237 1.000
cia2 0.207 0.207 1.000
cia3 0.170 0.170 1.000
cia4 0.219 0.219 1.000
cia5 0.312 0.312 1.000
cia6 0.362 0.362 1.000
cia7 0.615 0.615 1.000
R-Square:
Estimate
cia1 0.944
cia2 0.957
cia3 0.971
cia4 0.952
cia5 0.903
cia6 0.869
cia7 0.622
Group 2 [female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL =~
cia1 (L1) 4.093 0.471 8.687 0.000 3.016 0.949
cia2 (L2) 4.715 0.762 6.190 0.000 3.474 0.961
cia3 (L3) 5.798 0.967 5.998 0.000 4.272 0.974
cia4 (L4) 4.465 0.558 8.007 0.000 3.290 0.957
cia5 (L5) 3.044 0.373 8.152 0.000 2.243 0.913
cia6 (L6) 2.579 0.316 8.155 0.000 1.900 0.885
cia7 (L7) 1.283 0.193 6.630 0.000 0.945 0.687
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL -0.209 0.107 -1.942 0.052 -0.283 -0.283
.cia1 0.000 0.000 0.000
.cia2 0.000 0.000 0.000
.cia3 0.000 0.000 0.000
.cia4 0.000 0.000 0.000
.cia5 0.000 0.000 0.000
.cia6 0.000 0.000 0.000
.cia7 0.000 0.000 0.000
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1|t1 (I1T1) -5.386 0.520 -10.358 0.000 -5.386 -1.695
cia1|t2 (I1T2) -4.134 0.493 -8.392 0.000 -4.134 -1.301
cia1|t3 (I1T3) -1.545 0.444 -3.478 0.001 -1.545 -0.486
cia2|t1 (I2T1) -6.512 0.872 -7.467 0.000 -6.512 -1.801
cia2|t2 (I2T2) -5.744 0.820 -7.002 0.000 -5.744 -1.589
cia2|t3 (I2T3) -3.859 0.715 -5.400 0.000 -3.859 -1.067
cia3|t1 (I3T1) -7.560 1.036 -7.295 0.000 -7.560 -1.723
cia3|t2 (I3T2) -6.499 0.982 -6.618 0.000 -6.499 -1.481
cia3|t3 (I3T3) -4.110 0.836 -4.918 0.000 -4.110 -0.937
cia4|t1 (I4T1) -5.543 0.590 -9.392 0.000 -5.543 -1.612
cia4|t2 (I4T2) -3.990 0.535 -7.454 0.000 -3.990 -1.160
cia4|t3 (I4T3) -1.846 0.498 -3.708 0.000 -1.846 -0.537
cia5|t1 (I5T1) -4.674 0.425 -11.002 0.000 -4.674 -1.903
cia5|t2 (I5T2) -2.696 0.362 -7.455 0.000 -2.696 -1.098
cia5|t3 (I5T3) -1.076 0.331 -3.248 0.001 -1.076 -0.438
cia6|t1 (I6T1) -4.178 0.386 -10.838 0.000 -4.178 -1.946
cia6|t2 (I6T2) -3.054 0.342 -8.930 0.000 -3.054 -1.422
cia6|t3 (I6T3) -1.987 0.320 -6.213 0.000 -1.987 -0.926
cia7|t1 (I7T1) -3.521 0.308 -11.434 0.000 -3.521 -2.559
cia7|t2 (I7T2) -2.895 0.244 -11.876 0.000 -2.895 -2.104
cia7|t3 (I7T3) -1.978 0.196 -10.093 0.000 -1.978 -1.437
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.cia1 1.000 1.000 0.099
.cia2 1.000 1.000 0.077
.cia3 1.000 1.000 0.052
.cia4 1.000 1.000 0.085
.cia5 1.000 1.000 0.166
.cia6 1.000 1.000 0.217
.cia7 1.000 1.000 0.528
IADL 0.543 0.132 4.101 0.000 1.000 1.000
Scales y*:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1 0.315 0.315 1.000
cia2 0.277 0.277 1.000
cia3 0.228 0.228 1.000
cia4 0.291 0.291 1.000
cia5 0.407 0.407 1.000
cia6 0.466 0.466 1.000
cia7 0.727 0.727 1.000
R-Square:
Estimate
cia1 0.901
cia2 0.923
cia3 0.948
cia4 0.915
cia5 0.834
cia6 0.783
cia7 0.472anova(metricEstimates1, thresholdEstimates1)the condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedScaled Chi Square Difference Test (method = "satorra.2000")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
metricEstimates1 34 42.967
thresholdEstimates1 54 97.897 8.8105 2.2088 0.01532 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1anova(configuralEstimates, thresholdEstimates1)the condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedScaled Chi Square Difference Test (method = "satorra.2000")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
configuralEstimates 28 26.447
thresholdEstimates1 54 97.897 4.7991 2.3969 0.1253The difference tests showed the scalar model fit significantly no worse than the configural model but significantly worse than the metric invariance model, which is an odd finding. That said, we look at the modification indices to see if there are any parameters to free.
thresholdMI1 = modificationindices(thresholdEstimates1, free.remove = FALSE, maximum.number = 10000)
#restrict output to only factor loading parameters
thresholdMI1a = thresholdMI1[which(thresholdMI1$op == "|"),]
thresholdMI1a[order(-thresholdMI1a$mi),] lhs op rhs block mi mi.scaled epc sepc.lv sepc.all sepc.nox
16 cia3 | t3 1 19.549 27.096 2.585 2.585 0.439 0.439
67 cia3 | t3 2 5.856 8.117 -0.774 -0.774 -0.176 -0.176
22 cia5 | t3 1 4.522 6.267 -0.708 -0.708 -0.221 -0.221
21 cia5 | t2 1 2.910 4.033 -0.659 -0.659 -0.206 -0.206
24 cia6 | t2 1 2.642 3.662 -0.617 -0.617 -0.223 -0.223
10 cia1 | t3 1 2.502 3.468 0.659 0.659 0.157 0.157
25 cia6 | t3 1 2.500 3.465 -0.499 -0.499 -0.180 -0.180
19 cia4 | t3 1 2.193 3.039 -0.704 -0.704 -0.154 -0.154
15 cia3 | t2 1 1.820 2.523 0.921 0.921 0.156 0.156
18 cia4 | t2 1 1.563 2.167 -0.682 -0.682 -0.149 -0.149
73 cia5 | t3 2 1.013 1.404 0.159 0.159 0.065 0.065
23 cia6 | t1 1 0.730 1.012 -0.393 -0.393 -0.142 -0.142
20 cia5 | t1 1 0.727 1.008 -0.438 -0.438 -0.137 -0.137
61 cia1 | t3 2 0.612 0.848 -0.161 -0.161 -0.051 -0.051
72 cia5 | t2 2 0.605 0.839 0.137 0.137 0.056 0.056
66 cia3 | t2 2 0.568 0.788 -0.287 -0.287 -0.066 -0.066
75 cia6 | t2 2 0.552 0.764 0.129 0.129 0.060 0.060
76 cia6 | t3 2 0.548 0.759 0.109 0.109 0.051 0.051
70 cia4 | t3 2 0.479 0.663 0.154 0.154 0.045 0.045
13 cia2 | t3 1 0.437 0.606 0.345 0.345 0.072 0.072
69 cia4 | t2 2 0.332 0.460 0.145 0.145 0.042 0.042
9 cia1 | t2 1 0.250 0.347 0.242 0.242 0.058 0.058
14 cia3 | t1 1 0.214 0.296 0.354 0.354 0.060 0.060
11 cia2 | t1 1 0.189 0.262 0.282 0.282 0.059 0.059
74 cia6 | t1 2 0.186 0.258 0.100 0.100 0.047 0.047
71 cia5 | t1 2 0.183 0.253 0.110 0.110 0.045 0.045
27 cia7 | t2 1 0.128 0.178 0.101 0.101 0.062 0.062
64 cia2 | t3 2 0.110 0.153 -0.087 -0.087 -0.024 -0.024
26 cia7 | t1 1 0.107 0.149 0.124 0.124 0.076 0.076
28 cia7 | t3 1 0.097 0.135 -0.067 -0.067 -0.041 -0.041
60 cia1 | t2 2 0.067 0.092 -0.065 -0.065 -0.020 -0.020
65 cia3 | t1 2 0.066 0.091 -0.109 -0.109 -0.025 -0.025
77 cia7 | t1 2 0.062 0.086 -0.072 -0.072 -0.052 -0.052
62 cia2 | t1 2 0.061 0.084 -0.091 -0.091 -0.025 -0.025
78 cia7 | t2 2 0.055 0.076 -0.043 -0.043 -0.031 -0.031
79 cia7 | t3 2 0.029 0.040 0.020 0.020 0.014 0.014
8 cia1 | t1 1 0.025 0.034 0.087 0.087 0.021 0.021
17 cia4 | t1 1 0.022 0.031 0.087 0.087 0.019 0.019
59 cia1 | t1 2 0.007 0.010 -0.026 -0.026 -0.008 -0.008
68 cia4 | t1 2 0.006 0.009 -0.024 -0.024 -0.007 -0.007
12 cia2 | t2 1 0.000 0.000 -0.003 -0.003 -0.001 -0.001
63 cia2 | t2 2 0.000 0.000 0.001 0.001 0.000 0.000We see an issue with several items, but we will add one parameter at a time, starting with the third threshold for item 3.
thresholdSyntax2 = "
#============================================================================================================
#Factor loadings (AKA slopes/discriminations) from final metric model
IADL =~ c(L1, L1)*cia1 + c(L2, L2)*cia2 + c(L3, L3)*cia3 +
c(L4, L4)*cia4 + c(L5, L5)*cia5 + c(L6, L6)*cia6 + c(L7, L7)*cia7
#============================================================================================================
#Item Thresholds all constrained to be equal across groups except non-invariant item ****
cia1|c(I1T1, I1T1)*t1; cia1|c(I1T2, I1T2)*t2; cia1|c(I1T3, I1T3)*t3;
cia2|c(I2T1, I2T1)*t1; cia2|c(I2T2, I2T2)*t2; cia2|c(I2T3, I2T3)*t3;
cia3|c(I3T1, I3T1)*t1; cia3|c(I3T2, I3T2)*t2; cia3|c(I3T3M, I3T3F)*t3;
cia4|c(I4T1, I4T1)*t1; cia4|c(I4T2, I4T2)*t2; cia4|c(I4T3, I4T3)*t3;
cia5|c(I5T1, I5T1)*t1; cia5|c(I5T2, I5T2)*t2; cia5|c(I5T3, I5T3)*t3;
cia6|c(I6T1, I6T1)*t1; cia6|c(I6T2M, I6T2F)*t2; cia6|c(I6T3M, I6T3F)*t3;
cia7|c(I7T1, I7T1)*t1; cia7|c(I7T2, I7T2)*t2; cia7|c(I7T3, I7T3)*t3;
#============================================================================================================
#Redidual variances all fixed to one for each group
cia1 ~~ c(1, 1)*cia1; cia2 ~~ c(1, 1)*cia2; cia3 ~~ c(1, 1)*cia3;
cia4 ~~ c(1, 1)*cia4; cia5 ~~ c(1, 1)*cia5; cia6 ~~ c(1, 1)*cia6; cia7 ~~ c(1, 1)*cia7
#============================================================================================================
#Factor variance fixed to 1 for identification in males; estimated for females
IADL ~~ c(1,NA)*IADL
#============================================================================================================
#Factor mean fixed to zero for identification in males; estimated for females ****
IADL ~ c(0,NA)*1
#============================================================================================================
"
thresholdEstimates2 = lavaan(model = thresholdSyntax2, data = iadlData,
ordered = c("cia1", "cia2", "cia3", "cia4", "cia5", "cia6", "cia7"),
std.lv = TRUE, parameterization="theta", group = "sex")lavaan WARNING: 40 bivariate tables have empty cells; to see them, use:
lavInspect(fit, "zero.cell.tables")summary(thresholdEstimates2, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 167 iterations
Used Total
Number of observations per group
male 163 214
female 383 420
Estimator DWLS Robust
Minimum Function Test Statistic 65.498 105.179
Degrees of freedom 51 51
P-value (Chi-square) 0.083 0.000
Scaling correction factor 0.729
Shift parameter for each group:
male 4.564
female 10.723
for simple second-order correction (Mplus variant)
Chi-square for each group:
male 39.524 58.809
female 25.974 46.371
Model test baseline model:
Minimum Function Test Statistic 69203.021 35439.603
Degrees of freedom 42 42
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 1.000 0.998
Tucker-Lewis Index (TLI) 1.000 0.999
Robust Comparative Fit Index (CFI) NA
Robust Tucker-Lewis Index (TLI) NA
Root Mean Square Error of Approximation:
RMSEA 0.032 0.062
90 Percent Confidence Interval 0.000 0.053 0.045 0.079
P-value RMSEA <= 0.05 0.912 0.109
Robust RMSEA NA
90 Percent Confidence Interval NA NA
Standardized Root Mean Square Residual:
SRMR 0.031 0.031
Weighted Root Mean Square Residual:
WRMR 1.249 1.249
Parameter Estimates:
Information Expected
Standard Errors Robust.sem
Group 1 [male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL =~
cia1 (L1) 4.130 0.479 8.620 0.000 4.130 0.972
cia2 (L2) 4.778 0.777 6.151 0.000 4.778 0.979
cia3 (L3) 5.856 0.977 5.993 0.000 5.856 0.986
cia4 (L4) 4.512 0.567 7.958 0.000 4.512 0.976
cia5 (L5) 3.077 0.380 8.099 0.000 3.077 0.951
cia6 (L6) 2.605 0.321 8.112 0.000 2.605 0.934
cia7 (L7) 1.296 0.196 6.600 0.000 1.296 0.792
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL 0.000 0.000 0.000
.cia1 0.000 0.000 0.000
.cia2 0.000 0.000 0.000
.cia3 0.000 0.000 0.000
.cia4 0.000 0.000 0.000
.cia5 0.000 0.000 0.000
.cia6 0.000 0.000 0.000
.cia7 0.000 0.000 0.000
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1|t1 (I1T1) -5.495 0.530 -10.360 0.000 -5.495 -1.293
cia1|t2 (I1T2) -4.247 0.504 -8.419 0.000 -4.247 -0.999
cia1|t3 (I1T3) -1.664 0.455 -3.660 0.000 -1.664 -0.392
cia2|t1 (I2T1) -6.663 0.903 -7.383 0.000 -6.663 -1.365
cia2|t2 (I2T2) -5.895 0.850 -6.934 0.000 -5.895 -1.208
cia2|t3 (I2T3) -4.008 0.743 -5.394 0.000 -4.008 -0.821
cia3|t1 (I3T1) -7.721 1.058 -7.299 0.000 -7.721 -1.300
cia3|t2 (I3T2) -6.662 1.003 -6.642 0.000 -6.662 -1.121
cia3|t3 (I3T3) -1.524 0.674 -2.260 0.024 -1.524 -0.257
cia4|t1 (I4T1) -5.672 0.606 -9.356 0.000 -5.672 -1.227
cia4|t2 (I4T2) -4.122 0.553 -7.457 0.000 -4.122 -0.892
cia4|t3 (I4T3) -1.978 0.513 -3.853 0.000 -1.978 -0.428
cia5|t1 (I5T1) -4.763 0.435 -10.954 0.000 -4.763 -1.472
cia5|t2 (I5T2) -2.787 0.372 -7.485 0.000 -2.787 -0.861
cia5|t3 (I5T3) -1.166 0.341 -3.419 0.001 -1.166 -0.360
cia6|t1 (I6T1) -4.251 0.393 -10.809 0.000 -4.251 -1.524
cia6|t2 (I6T2) -3.707 0.411 -9.025 0.000 -3.707 -1.328
cia6|t3 (I6T3) -2.511 0.391 -6.427 0.000 -2.511 -0.900
cia7|t1 (I7T1) -3.556 0.313 -11.369 0.000 -3.556 -2.172
cia7|t2 (I7T2) -2.931 0.249 -11.770 0.000 -2.931 -1.790
cia7|t3 (I7T3) -2.014 0.202 -9.973 0.000 -2.014 -1.230
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.cia1 1.000 1.000 0.055
.cia2 1.000 1.000 0.042
.cia3 1.000 1.000 0.028
.cia4 1.000 1.000 0.047
.cia5 1.000 1.000 0.096
.cia6 1.000 1.000 0.128
.cia7 1.000 1.000 0.373
IADL 1.000 1.000 1.000
Scales y*:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1 0.235 0.235 1.000
cia2 0.205 0.205 1.000
cia3 0.168 0.168 1.000
cia4 0.216 0.216 1.000
cia5 0.309 0.309 1.000
cia6 0.358 0.358 1.000
cia7 0.611 0.611 1.000
R-Square:
Estimate
cia1 0.945
cia2 0.958
cia3 0.972
cia4 0.953
cia5 0.904
cia6 0.872
cia7 0.627
Group 2 [female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL =~
cia1 (L1) 4.130 0.479 8.620 0.000 3.002 0.949
cia2 (L2) 4.778 0.777 6.151 0.000 3.474 0.961
cia3 (L3) 5.856 0.977 5.993 0.000 4.257 0.974
cia4 (L4) 4.512 0.567 7.958 0.000 3.280 0.957
cia5 (L5) 3.077 0.380 8.099 0.000 2.237 0.913
cia6 (L6) 2.605 0.321 8.112 0.000 1.894 0.884
cia7 (L7) 1.296 0.196 6.600 0.000 0.942 0.686
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL -0.242 0.108 -2.248 0.025 -0.333 -0.333
.cia1 0.000 0.000 0.000
.cia2 0.000 0.000 0.000
.cia3 0.000 0.000 0.000
.cia4 0.000 0.000 0.000
.cia5 0.000 0.000 0.000
.cia6 0.000 0.000 0.000
.cia7 0.000 0.000 0.000
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1|t1 (I1T1) -5.495 0.530 -10.360 0.000 -5.495 -1.737
cia1|t2 (I1T2) -4.247 0.504 -8.419 0.000 -4.247 -1.342
cia1|t3 (I1T3) -1.664 0.455 -3.660 0.000 -1.664 -0.526
cia2|t1 (I2T1) -6.663 0.903 -7.383 0.000 -6.663 -1.843
cia2|t2 (I2T2) -5.895 0.850 -6.934 0.000 -5.895 -1.631
cia2|t3 (I2T3) -4.008 0.743 -5.394 0.000 -4.008 -1.109
cia3|t1 (I3T1) -7.721 1.058 -7.299 0.000 -7.721 -1.765
cia3|t2 (I3T2) -6.662 1.003 -6.642 0.000 -6.662 -1.523
cia3|t3 (I3T3) -5.081 0.916 -5.546 0.000 -5.081 -1.162
cia4|t1 (I4T1) -5.672 0.606 -9.356 0.000 -5.672 -1.654
cia4|t2 (I4T2) -4.122 0.553 -7.457 0.000 -4.122 -1.202
cia4|t3 (I4T3) -1.978 0.513 -3.853 0.000 -1.978 -0.577
cia5|t1 (I5T1) -4.763 0.435 -10.954 0.000 -4.763 -1.944
cia5|t2 (I5T2) -2.787 0.372 -7.485 0.000 -2.787 -1.137
cia5|t3 (I5T3) -1.166 0.341 -3.419 0.001 -1.166 -0.476
cia6|t1 (I6T1) -4.251 0.393 -10.809 0.000 -4.251 -1.985
cia6|t2 (I6T2) -3.011 0.357 -8.443 0.000 -3.011 -1.406
cia6|t3 (I6T3) -1.967 0.333 -5.905 0.000 -1.967 -0.918
cia7|t1 (I7T1) -3.556 0.313 -11.369 0.000 -3.556 -2.588
cia7|t2 (I7T2) -2.931 0.249 -11.770 0.000 -2.931 -2.133
cia7|t3 (I7T3) -2.014 0.202 -9.973 0.000 -2.014 -1.466
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.cia1 1.000 1.000 0.100
.cia2 1.000 1.000 0.077
.cia3 1.000 1.000 0.052
.cia4 1.000 1.000 0.085
.cia5 1.000 1.000 0.167
.cia6 1.000 1.000 0.218
.cia7 1.000 1.000 0.530
IADL 0.529 0.129 4.087 0.000 1.000 1.000
Scales y*:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1 0.316 0.316 1.000
cia2 0.277 0.277 1.000
cia3 0.229 0.229 1.000
cia4 0.292 0.292 1.000
cia5 0.408 0.408 1.000
cia6 0.467 0.467 1.000
cia7 0.728 0.728 1.000
R-Square:
Estimate
cia1 0.900
cia2 0.923
cia3 0.948
cia4 0.915
cia5 0.833
cia6 0.782
cia7 0.470anova(metricEstimates1, thresholdEstimates2)the condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedScaled Chi Square Difference Test (method = "satorra.2000")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
metricEstimates1 34 42.967
thresholdEstimates2 51 65.498 3.951 2.0848 0.1484anova(configuralEstimates, thresholdEstimates2)the condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedScaled Chi Square Difference Test (method = "satorra.2000")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
configuralEstimates 28 26.447
thresholdEstimates2 51 65.498 2.9814 2.5016 0.3087The difference tests showed the scalar model did not fit significantly worse than both the configural model and the metric invariance model. We will now move on to test residual invariance.
Because lavaan does not function like Mplus, we cannot execute this step. We will move onto the test for the structural model.
To test the equality of factor variance across groups, we use our last model and constrain the factor variance.
factorVarSyntax = "
#============================================================================================================
#Factor loadings (AKA slopes/discriminations) from final metric model
IADL =~ c(L1, L1)*cia1 + c(L2, L2)*cia2 + c(L3, L3)*cia3 +
c(L4, L4)*cia4 + c(L5, L5)*cia5 + c(L6, L6)*cia6 + c(L7, L7)*cia7
#============================================================================================================
#Item Thresholds all constrained to be equal across groups except non-invariant item
cia1|c(I1T1, I1T1)*t1; cia1|c(I1T2, I1T2)*t2; cia1|c(I1T3, I1T3)*t3;
cia2|c(I2T1, I2T1)*t1; cia2|c(I2T2, I2T2)*t2; cia2|c(I2T3, I2T3)*t3;
cia3|c(I3T1, I3T1)*t1; cia3|c(I3T2, I3T2)*t2; cia3|c(I3T3M, I3T3F)*t3;
cia4|c(I4T1, I4T1)*t1; cia4|c(I4T2, I4T2)*t2; cia4|c(I4T3, I4T3)*t3;
cia5|c(I5T1, I5T1)*t1; cia5|c(I5T2, I5T2)*t2; cia5|c(I5T3, I5T3)*t3;
cia6|c(I6T1, I6T1)*t1; cia6|c(I6T2M, I6T2F)*t2; cia6|c(I6T3M, I6T3F)*t3;
cia7|c(I7T1, I7T1)*t1; cia7|c(I7T2, I7T2)*t2; cia7|c(I7T3, I7T3)*t3;
#============================================================================================================
#Redidual variances all fixed to one for each group
cia1 ~~ c(1, 1)*cia1; cia2 ~~ c(1, 1)*cia2; cia3 ~~ c(1, 1)*cia3;
cia4 ~~ c(1, 1)*cia4; cia5 ~~ c(1, 1)*cia5; cia6 ~~ c(1, 1)*cia6; cia7 ~~ c(1, 1)*cia7
#============================================================================================================
#Factor variance fixed to 1 for identification in males; estimated for females ****
IADL ~~ c(1,1)*IADL
#============================================================================================================
#Factor mean fixed to zero for identification in males; estimated for females
IADL ~ c(0,NA)*0
#============================================================================================================
"
factorVarEstimates = lavaan(model = factorVarSyntax, data = iadlData,
ordered = c("cia1", "cia2", "cia3", "cia4", "cia5", "cia6", "cia7"),
std.lv = TRUE, parameterization="theta", group = "sex")lavaan WARNING: 40 bivariate tables have empty cells; to see them, use:
lavInspect(fit, "zero.cell.tables")summary(factorVarEstimates, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 144 iterations
Used Total
Number of observations per group
male 163 214
female 383 420
Estimator DWLS Robust
Minimum Function Test Statistic 125.625 133.981
Degrees of freedom 52 52
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.117
Shift parameter for each group:
male 6.428
female 15.105
for simple second-order correction (Mplus variant)
Chi-square for each group:
male 65.439 65.003
female 60.186 68.978
Model test baseline model:
Minimum Function Test Statistic 69203.021 35439.603
Degrees of freedom 42 42
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.999 0.998
Tucker-Lewis Index (TLI) 0.999 0.998
Robust Comparative Fit Index (CFI) NA
Robust Tucker-Lewis Index (TLI) NA
Root Mean Square Error of Approximation:
RMSEA 0.072 0.076
90 Percent Confidence Interval 0.056 0.088 0.060 0.092
P-value RMSEA <= 0.05 0.013 0.004
Robust RMSEA NA
90 Percent Confidence Interval NA NA
Standardized Root Mean Square Residual:
SRMR 0.039 0.039
Weighted Root Mean Square Residual:
WRMR 1.729 1.729
Parameter Estimates:
Information Expected
Standard Errors Robust.sem
Group 1 [male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL =~
cia1 (L1) 3.604 0.301 11.975 0.000 3.604 0.964
cia2 (L2) 3.762 0.490 7.680 0.000 3.762 0.966
cia3 (L3) 4.184 0.487 8.589 0.000 4.184 0.973
cia4 (L4) 3.721 0.349 10.671 0.000 3.721 0.966
cia5 (L5) 2.479 0.207 11.965 0.000 2.479 0.927
cia6 (L6) 2.335 0.242 9.642 0.000 2.335 0.919
cia7 (L7) 1.042 0.130 8.020 0.000 1.042 0.721
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL 0.000 0.000 0.000
.cia1 0.000 0.000 0.000
.cia2 0.000 0.000 0.000
.cia3 0.000 0.000 0.000
.cia4 0.000 0.000 0.000
.cia5 0.000 0.000 0.000
.cia6 0.000 0.000 0.000
.cia7 0.000 0.000 0.000
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1|t1 (I1T1) -5.139 0.469 -10.969 0.000 -5.139 -1.374
cia1|t2 (I1T2) -3.781 0.428 -8.836 0.000 -3.781 -1.011
cia1|t3 (I1T3) -0.935 0.374 -2.501 0.012 -0.935 -0.250
cia2|t1 (I2T1) -5.678 0.705 -8.053 0.000 -5.678 -1.458
cia2|t2 (I2T2) -4.966 0.660 -7.522 0.000 -4.966 -1.276
cia2|t3 (I2T3) -3.078 0.560 -5.497 0.000 -3.078 -0.791
cia3|t1 (I3T1) -5.952 0.659 -9.027 0.000 -5.952 -1.383
cia3|t2 (I3T2) -4.966 0.621 -7.992 0.000 -4.966 -1.154
cia3|t3 (I3T3) -1.104 0.460 -2.399 0.016 -1.104 -0.257
cia4|t1 (I4T1) -4.994 0.512 -9.762 0.000 -4.994 -1.296
cia4|t2 (I4T2) -3.446 0.461 -7.468 0.000 -3.446 -0.894
cia4|t3 (I4T3) -1.210 0.414 -2.923 0.003 -1.210 -0.314
cia5|t1 (I5T1) -4.272 0.365 -11.719 0.000 -4.272 -1.598
cia5|t2 (I5T2) -2.277 0.302 -7.531 0.000 -2.277 -0.852
cia5|t3 (I5T3) -0.632 0.274 -2.306 0.021 -0.632 -0.236
cia6|t1 (I6T1) -4.191 0.403 -10.401 0.000 -4.191 -1.650
cia6|t2 (I6T2) -3.375 0.396 -8.521 0.000 -3.375 -1.328
cia6|t3 (I6T3) -2.286 0.355 -6.440 0.000 -2.286 -0.900
cia7|t1 (I7T1) -3.278 0.277 -11.815 0.000 -3.278 -2.270
cia7|t2 (I7T2) -2.682 0.216 -12.434 0.000 -2.682 -1.857
cia7|t3 (I7T3) -1.785 0.169 -10.561 0.000 -1.785 -1.236
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.cia1 1.000 1.000 0.071
.cia2 1.000 1.000 0.066
.cia3 1.000 1.000 0.054
.cia4 1.000 1.000 0.067
.cia5 1.000 1.000 0.140
.cia6 1.000 1.000 0.155
.cia7 1.000 1.000 0.480
IADL 1.000 1.000 1.000
Scales y*:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1 0.267 0.267 1.000
cia2 0.257 0.257 1.000
cia3 0.232 0.232 1.000
cia4 0.260 0.260 1.000
cia5 0.374 0.374 1.000
cia6 0.394 0.394 1.000
cia7 0.692 0.692 1.000
R-Square:
Estimate
cia1 0.929
cia2 0.934
cia3 0.946
cia4 0.933
cia5 0.860
cia6 0.845
cia7 0.520
Group 2 [female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL =~
cia1 (L1) 3.604 0.301 11.975 0.000 3.604 0.964
cia2 (L2) 3.762 0.490 7.680 0.000 3.762 0.966
cia3 (L3) 4.184 0.487 8.589 0.000 4.184 0.973
cia4 (L4) 3.721 0.349 10.671 0.000 3.721 0.966
cia5 (L5) 2.479 0.207 11.965 0.000 2.479 0.927
cia6 (L6) 2.335 0.242 9.642 0.000 2.335 0.919
cia7 (L7) 1.042 0.130 8.020 0.000 1.042 0.721
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL 0.001 0.120 0.008 0.993 0.001 0.001
.cia1 0.000 0.000 0.000
.cia2 0.000 0.000 0.000
.cia3 0.000 0.000 0.000
.cia4 0.000 0.000 0.000
.cia5 0.000 0.000 0.000
.cia6 0.000 0.000 0.000
.cia7 0.000 0.000 0.000
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1|t1 (I1T1) -5.139 0.469 -10.969 0.000 -5.139 -1.374
cia1|t2 (I1T2) -3.781 0.428 -8.836 0.000 -3.781 -1.011
cia1|t3 (I1T3) -0.935 0.374 -2.501 0.012 -0.935 -0.250
cia2|t1 (I2T1) -5.678 0.705 -8.053 0.000 -5.678 -1.458
cia2|t2 (I2T2) -4.966 0.660 -7.522 0.000 -4.966 -1.276
cia2|t3 (I2T3) -3.078 0.560 -5.497 0.000 -3.078 -0.791
cia3|t1 (I3T1) -5.952 0.659 -9.027 0.000 -5.952 -1.383
cia3|t2 (I3T2) -4.966 0.621 -7.992 0.000 -4.966 -1.154
cia3|t3 (I3T3) -3.601 0.603 -5.976 0.000 -3.601 -0.837
cia4|t1 (I4T1) -4.994 0.512 -9.762 0.000 -4.994 -1.296
cia4|t2 (I4T2) -3.446 0.461 -7.468 0.000 -3.446 -0.894
cia4|t3 (I4T3) -1.210 0.414 -2.923 0.003 -1.210 -0.314
cia5|t1 (I5T1) -4.272 0.365 -11.719 0.000 -4.272 -1.598
cia5|t2 (I5T2) -2.277 0.302 -7.531 0.000 -2.277 -0.852
cia5|t3 (I5T3) -0.632 0.274 -2.306 0.021 -0.632 -0.236
cia6|t1 (I6T1) -4.191 0.403 -10.401 0.000 -4.191 -1.650
cia6|t2 (I6T2) -2.822 0.348 -8.099 0.000 -2.822 -1.111
cia6|t3 (I6T3) -1.583 0.312 -5.066 0.000 -1.583 -0.623
cia7|t1 (I7T1) -3.278 0.277 -11.815 0.000 -3.278 -2.270
cia7|t2 (I7T2) -2.682 0.216 -12.434 0.000 -2.682 -1.857
cia7|t3 (I7T3) -1.785 0.169 -10.561 0.000 -1.785 -1.236
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.cia1 1.000 1.000 0.071
.cia2 1.000 1.000 0.066
.cia3 1.000 1.000 0.054
.cia4 1.000 1.000 0.067
.cia5 1.000 1.000 0.140
.cia6 1.000 1.000 0.155
.cia7 1.000 1.000 0.480
IADL 1.000 1.000 1.000
Scales y*:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1 0.267 0.267 1.000
cia2 0.257 0.257 1.000
cia3 0.232 0.232 1.000
cia4 0.260 0.260 1.000
cia5 0.374 0.374 1.000
cia6 0.394 0.394 1.000
cia7 0.692 0.692 1.000
R-Square:
Estimate
cia1 0.929
cia2 0.934
cia3 0.946
cia4 0.933
cia5 0.860
cia6 0.845
cia7 0.520anova(thresholdEstimates2, factorVarEstimates)the condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedScaled Chi Square Difference Test (method = "satorra.2000")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
thresholdEstimates2 51 65.498
factorVarEstimates 52 125.625 4.2538 0.55946 0.01708 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1anova(configuralEstimates, factorVarEstimates)the condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedthe condition has length > 1 and only the first element will be usedScaled Chi Square Difference Test (method = "satorra.2000")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
configuralEstimates 28 26.447
factorVarEstimates 52 125.625 7.5719 2.5016 0.03695 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1As the factor variance model fit significantly worse than the configural and last threshold model, we can conclude the factor variances are different.
To test the factor mean, we only need to look at the Wald test for the factor means from the last threshold model:
summary(thresholdEstimates2, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 167 iterations
Used Total
Number of observations per group
male 163 214
female 383 420
Estimator DWLS Robust
Minimum Function Test Statistic 65.498 105.179
Degrees of freedom 51 51
P-value (Chi-square) 0.083 0.000
Scaling correction factor 0.729
Shift parameter for each group:
male 4.564
female 10.723
for simple second-order correction (Mplus variant)
Chi-square for each group:
male 39.524 58.809
female 25.974 46.371
Model test baseline model:
Minimum Function Test Statistic 69203.021 35439.603
Degrees of freedom 42 42
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 1.000 0.998
Tucker-Lewis Index (TLI) 1.000 0.999
Robust Comparative Fit Index (CFI) NA
Robust Tucker-Lewis Index (TLI) NA
Root Mean Square Error of Approximation:
RMSEA 0.032 0.062
90 Percent Confidence Interval 0.000 0.053 0.045 0.079
P-value RMSEA <= 0.05 0.912 0.109
Robust RMSEA NA
90 Percent Confidence Interval NA NA
Standardized Root Mean Square Residual:
SRMR 0.031 0.031
Weighted Root Mean Square Residual:
WRMR 1.249 1.249
Parameter Estimates:
Information Expected
Standard Errors Robust.sem
Group 1 [male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL =~
cia1 (L1) 4.130 0.479 8.620 0.000 4.130 0.972
cia2 (L2) 4.778 0.777 6.151 0.000 4.778 0.979
cia3 (L3) 5.856 0.977 5.993 0.000 5.856 0.986
cia4 (L4) 4.512 0.567 7.958 0.000 4.512 0.976
cia5 (L5) 3.077 0.380 8.099 0.000 3.077 0.951
cia6 (L6) 2.605 0.321 8.112 0.000 2.605 0.934
cia7 (L7) 1.296 0.196 6.600 0.000 1.296 0.792
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL 0.000 0.000 0.000
.cia1 0.000 0.000 0.000
.cia2 0.000 0.000 0.000
.cia3 0.000 0.000 0.000
.cia4 0.000 0.000 0.000
.cia5 0.000 0.000 0.000
.cia6 0.000 0.000 0.000
.cia7 0.000 0.000 0.000
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1|t1 (I1T1) -5.495 0.530 -10.360 0.000 -5.495 -1.293
cia1|t2 (I1T2) -4.247 0.504 -8.419 0.000 -4.247 -0.999
cia1|t3 (I1T3) -1.664 0.455 -3.660 0.000 -1.664 -0.392
cia2|t1 (I2T1) -6.663 0.903 -7.383 0.000 -6.663 -1.365
cia2|t2 (I2T2) -5.895 0.850 -6.934 0.000 -5.895 -1.208
cia2|t3 (I2T3) -4.008 0.743 -5.394 0.000 -4.008 -0.821
cia3|t1 (I3T1) -7.721 1.058 -7.299 0.000 -7.721 -1.300
cia3|t2 (I3T2) -6.662 1.003 -6.642 0.000 -6.662 -1.121
cia3|t3 (I3T3) -1.524 0.674 -2.260 0.024 -1.524 -0.257
cia4|t1 (I4T1) -5.672 0.606 -9.356 0.000 -5.672 -1.227
cia4|t2 (I4T2) -4.122 0.553 -7.457 0.000 -4.122 -0.892
cia4|t3 (I4T3) -1.978 0.513 -3.853 0.000 -1.978 -0.428
cia5|t1 (I5T1) -4.763 0.435 -10.954 0.000 -4.763 -1.472
cia5|t2 (I5T2) -2.787 0.372 -7.485 0.000 -2.787 -0.861
cia5|t3 (I5T3) -1.166 0.341 -3.419 0.001 -1.166 -0.360
cia6|t1 (I6T1) -4.251 0.393 -10.809 0.000 -4.251 -1.524
cia6|t2 (I6T2) -3.707 0.411 -9.025 0.000 -3.707 -1.328
cia6|t3 (I6T3) -2.511 0.391 -6.427 0.000 -2.511 -0.900
cia7|t1 (I7T1) -3.556 0.313 -11.369 0.000 -3.556 -2.172
cia7|t2 (I7T2) -2.931 0.249 -11.770 0.000 -2.931 -1.790
cia7|t3 (I7T3) -2.014 0.202 -9.973 0.000 -2.014 -1.230
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.cia1 1.000 1.000 0.055
.cia2 1.000 1.000 0.042
.cia3 1.000 1.000 0.028
.cia4 1.000 1.000 0.047
.cia5 1.000 1.000 0.096
.cia6 1.000 1.000 0.128
.cia7 1.000 1.000 0.373
IADL 1.000 1.000 1.000
Scales y*:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1 0.235 0.235 1.000
cia2 0.205 0.205 1.000
cia3 0.168 0.168 1.000
cia4 0.216 0.216 1.000
cia5 0.309 0.309 1.000
cia6 0.358 0.358 1.000
cia7 0.611 0.611 1.000
R-Square:
Estimate
cia1 0.945
cia2 0.958
cia3 0.972
cia4 0.953
cia5 0.904
cia6 0.872
cia7 0.627
Group 2 [female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL =~
cia1 (L1) 4.130 0.479 8.620 0.000 3.002 0.949
cia2 (L2) 4.778 0.777 6.151 0.000 3.474 0.961
cia3 (L3) 5.856 0.977 5.993 0.000 4.257 0.974
cia4 (L4) 4.512 0.567 7.958 0.000 3.280 0.957
cia5 (L5) 3.077 0.380 8.099 0.000 2.237 0.913
cia6 (L6) 2.605 0.321 8.112 0.000 1.894 0.884
cia7 (L7) 1.296 0.196 6.600 0.000 0.942 0.686
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
IADL -0.242 0.108 -2.248 0.025 -0.333 -0.333
.cia1 0.000 0.000 0.000
.cia2 0.000 0.000 0.000
.cia3 0.000 0.000 0.000
.cia4 0.000 0.000 0.000
.cia5 0.000 0.000 0.000
.cia6 0.000 0.000 0.000
.cia7 0.000 0.000 0.000
Thresholds:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1|t1 (I1T1) -5.495 0.530 -10.360 0.000 -5.495 -1.737
cia1|t2 (I1T2) -4.247 0.504 -8.419 0.000 -4.247 -1.342
cia1|t3 (I1T3) -1.664 0.455 -3.660 0.000 -1.664 -0.526
cia2|t1 (I2T1) -6.663 0.903 -7.383 0.000 -6.663 -1.843
cia2|t2 (I2T2) -5.895 0.850 -6.934 0.000 -5.895 -1.631
cia2|t3 (I2T3) -4.008 0.743 -5.394 0.000 -4.008 -1.109
cia3|t1 (I3T1) -7.721 1.058 -7.299 0.000 -7.721 -1.765
cia3|t2 (I3T2) -6.662 1.003 -6.642 0.000 -6.662 -1.523
cia3|t3 (I3T3) -5.081 0.916 -5.546 0.000 -5.081 -1.162
cia4|t1 (I4T1) -5.672 0.606 -9.356 0.000 -5.672 -1.654
cia4|t2 (I4T2) -4.122 0.553 -7.457 0.000 -4.122 -1.202
cia4|t3 (I4T3) -1.978 0.513 -3.853 0.000 -1.978 -0.577
cia5|t1 (I5T1) -4.763 0.435 -10.954 0.000 -4.763 -1.944
cia5|t2 (I5T2) -2.787 0.372 -7.485 0.000 -2.787 -1.137
cia5|t3 (I5T3) -1.166 0.341 -3.419 0.001 -1.166 -0.476
cia6|t1 (I6T1) -4.251 0.393 -10.809 0.000 -4.251 -1.985
cia6|t2 (I6T2) -3.011 0.357 -8.443 0.000 -3.011 -1.406
cia6|t3 (I6T3) -1.967 0.333 -5.905 0.000 -1.967 -0.918
cia7|t1 (I7T1) -3.556 0.313 -11.369 0.000 -3.556 -2.588
cia7|t2 (I7T2) -2.931 0.249 -11.770 0.000 -2.931 -2.133
cia7|t3 (I7T3) -2.014 0.202 -9.973 0.000 -2.014 -1.466
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.cia1 1.000 1.000 0.100
.cia2 1.000 1.000 0.077
.cia3 1.000 1.000 0.052
.cia4 1.000 1.000 0.085
.cia5 1.000 1.000 0.167
.cia6 1.000 1.000 0.218
.cia7 1.000 1.000 0.530
IADL 0.529 0.129 4.087 0.000 1.000 1.000
Scales y*:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
cia1 0.316 0.316 1.000
cia2 0.277 0.277 1.000
cia3 0.229 0.229 1.000
cia4 0.292 0.292 1.000
cia5 0.408 0.408 1.000
cia6 0.467 0.467 1.000
cia7 0.728 0.728 1.000
R-Square:
Estimate
cia1 0.900
cia2 0.923
cia3 0.948
cia4 0.915
cia5 0.833
cia6 0.782
cia7 0.470Here, we see the estimate of the factor mean for the female group was .611, which was significant (p < .01). Therefore we determine that the female group has a significantly higher mean than the male group, indicating that on average, females are more able to perform the activities noted on the scale.
The extent to which am item factor model measuring independent daily living (with seven observed items) exhibited measurement invariance and structural invariance between men and women was examined using the lavaan package (Rosseel, 2012) in R (R Core Team, 2017). WLSMV estimation including a probit link and the THETA parameterization was used to estimate all models. Thus, model fit statistics describe the fit of the item factor model to the polychoric correlation matrix among the items for each group. Nested model comparisons were conducted using the DIFFTEST procedure included in the package. A configural invariance model was initially specified in which a single factor was estimated simultaneously in each group. The factor variance was fixed to 1 and the factor mean was fixed to 0 in each group for identification, such that all item factor loadings (one per item) and thresholds (three per item given four response options) were then estimated. The residual variances are not uniquely identified in the configural invariance model and as such were all constrained to 1 in both groups. As shown in Table 1, the configural invariance model had good fit. The analysis proceeded by applying parameter constraints in successive models to examine potential decreases in fit resulting from measurement or structural non-invariance between men and women, with men as the reference group.
Equality of the unstandardized item factor loadings between groups was then examined in a metric invariance model. The factor variance was fixed to 1 in men for identification but was freely estimated in women; the factor mean was fixed to 0 in both groups for identification. All factor loadings were constrained equal across groups, all item thresholds were estimated, and all residual variances were constrained to 1 across groups. The metric invariance model did not fit significantly worse than the configural invariance model, DIFFTEST (df = 1.86) = 2.10, p = .32. Therefore, we concluded there was full metric invariance between groups.
Equality of the unstandardized item thresholds across groups was then examined in a scalar invariance model. The factor mean and variance were fixed to 0 and 1, respectively, in men for identification, but the factor mean and variance were then estimated for women. All factor loadings and all item thresholds were constrained equal across groups; all residual variances were still constrained equal to 1 in both groups. The full scalar invariance model A fit significantly worse than the partial metric invariance model, DIFFTEST (2.21) = 8.81 p = .015. Modification indices indicted that the third threshold for item three was a source of misfit. After freeing item 3’s third threshold, the partial scalar invariance model B fit as well as the partial metric invariance model, DIFFTEST (2.08) = 3.95, p = .15.
After achieving partial measurement invariance as was just described, structural invariance was then tested with one additional model. The factor variance in the women (which had been estimated freely) was constrained to 1 (i.e., to be equal to the factor variance in men), resulting in a significant decrease in fit relative to the last partial residual invariance model B, DIFFTEST (.56) = 4.25, p = .02. Thus, women showed significantly less variability in ability to live independently (factor variance of 0.552) than did men (factor variance fixed = 1). The factor mean for women in the partial measurement invariance model was significantly different from 0 (difference = −0.242, SE = 0.11, p < .01), indicating that women were marginally less able to live independently than men (factor mean fixed = 0).
In conclusion, these analyses showed that partial measurement invariance was obtained across men and women-that is, the relationships of the items to the latent factor of independent living were equivalent between men women. Structural invariance was not obtained, such that women were less variable and less able on average than men. Model parameters from the final model are given in Table 2.
(Table 1 would have model comparison tests; Table 2 would have unstandardized and standardized estimates and their SEs)