7.4 In-Class Exercises
7.4.1 Measurement Invariance
We’ll now pick up where we left off with the At-Home Exercises by testing measurement invariance in the two-group CFA from 7.3.2.4.
As you saw in the lecture, measurement invariance testing allows us to empirically test for differences in the measurement model between the groups. If we can establish measurement invariance, we can draw the following (equivalent) conclusions:
- Our latent constructs are defined equivalently in all groups.
- The participants in every group are interpreting our items in the same way.
- Participants in different groups who have the same values for the observed indicators will also have the same score on the latent variable.
- Between-group differences in latent parameters are due to true differences in the underlying latent constructs and not caused by differences in measurement.
Anytime we make between-group comparisons (e.g., ANOVA, t-tests, moderation by group, etc.) we assume invariant measurement. That is, we assume that the scores we’re comparing have the same meaning in each group. When doing multiple group SEM, however, we’re apprised of the incredibly powerful capability of actually testing this—very important, and often violated—assumption.
The process of testing measurement invariance can get quite complex, but the basic procedure boils down to using model comparison tests to evaluate the plausibility of increasingly strong between-group constraints. For most problems, these constraints amount to the following three levels:
- Configural: The same pattern of free and fixed effects in all groups
- Weak (aka Metric): Configural + Equal factor loadings in all groups
- Strong (aka Scalar): Weak + Equal item intercepts in all groups
You can read more about measurement invariance here and here, and you can find a brief discussion of how to conduct measurement invariance tests in lavaan here.
7.4.1.1
Load the PGDdata2.txt data as you did for the At-Home Exercises.
- Note: Unless otherwise specified, all analyses in Section 7.4.1 use these data.
Click to show code
library(dplyr)
## Load the data:
pgd <- read.table("PGDdata2.txt",
na.strings = "-999",
header = TRUE,
sep = "\t") %>%
filter(!is.na(Kin2))
## Check the results:
head(pgd)
summary(pdg)
str(pgd)## Kin2 b1pss1 b2pss2 b3pss3
## Min. :0.0000 Min. :0.000 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.0000 1st Qu.:0.000 1st Qu.:0.0000 1st Qu.:0.0000
## Median :1.0000 Median :1.000 Median :0.0000 Median :1.0000
## Mean :0.6661 Mean :1.236 Mean :0.4622 Mean :0.9771
## 3rd Qu.:1.0000 3rd Qu.:2.000 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. :1.0000 Max. :5.000 Max. :3.0000 Max. :5.0000
## NA's :1
## b4pss4 b5pss5
## Min. :0.000 Min. :0.0000
## 1st Qu.:0.000 1st Qu.:0.0000
## Median :1.000 Median :0.0000
## Mean :1.009 Mean :0.6761
## 3rd Qu.:2.000 3rd Qu.:1.0000
## Max. :3.000 Max. :3.0000
## NA's :1## 'data.frame': 569 obs. of 6 variables:
## $ Kin2 : int 0 0 1 1 0 1 1 1 1 1 ...
## $ b1pss1: int 1 1 1 1 1 2 1 3 1 1 ...
## $ b2pss2: int 1 0 1 0 1 2 1 2 0 0 ...
## $ b3pss3: int 1 0 1 1 2 2 1 2 1 1 ...
## $ b4pss4: int 1 1 1 1 0 2 2 3 0 1 ...
## $ b5pss5: int 1 0 0 0 0 1 2 3 0 0 ...7.4.1.2
Test configural, weak, and strong invariance using the multiple-group CFA from 7.3.2.4.
- What are your conclusions?
Click to show code
library(lavaan)
library(semTools) # provides the compareFit() function
## Define the syntax for the CFA model:
cfaMod <- 'grief =~ b1pss1 + b2pss2 + b3pss3 + b4pss4 + b5pss5'
## Estimate the configural model:
configOut <- cfa(cfaMod, data = pgd, group = "Kin2")
## Estimate the weak invariance model:
weakOut <- cfa(cfaMod, data = pgd, group = "Kin2", group.equal = "loadings")
## Estimate the strong invariance model:
strongOut <- cfa(cfaMod,
data = pgd,
group = "Kin2",
group.equal = c("loadings", "intercepts")
)
summary(configOut)## lavaan 0.6-18 ended normally after 27 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 30
##
## Number of observations per group: Used Total
## 0 188 190
## 1 379 379
##
## Model Test User Model:
##
## Test statistic 11.317
## Degrees of freedom 10
## P-value (Chi-square) 0.333
## Test statistic for each group:
## 0 8.976
## 1 2.340
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [0]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## grief =~
## b1pss1 1.000
## b2pss2 0.372 0.076 4.922 0.000
## b3pss3 0.938 0.118 7.986 0.000
## b4pss4 0.909 0.116 7.848 0.000
## b5pss5 0.951 0.122 7.774 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 1.346 0.072 18.727 0.000
## .b2pss2 0.441 0.046 9.499 0.000
## .b3pss3 1.059 0.068 15.618 0.000
## .b4pss4 1.122 0.067 16.671 0.000
## .b5pss5 0.745 0.071 10.442 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 0.478 0.067 7.118 0.000
## .b2pss2 0.338 0.037 9.205 0.000
## .b3pss3 0.430 0.060 7.170 0.000
## .b4pss4 0.445 0.060 7.408 0.000
## .b5pss5 0.511 0.068 7.519 0.000
## grief 0.493 0.098 5.007 0.000
##
##
## Group 2 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## grief =~
## b1pss1 1.000
## b2pss2 0.502 0.052 9.597 0.000
## b3pss3 0.785 0.066 11.945 0.000
## b4pss4 0.708 0.062 11.497 0.000
## b5pss5 0.762 0.062 12.185 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 1.182 0.051 23.277 0.000
## .b2pss2 0.475 0.037 12.973 0.000
## .b3pss3 0.934 0.046 20.460 0.000
## .b4pss4 0.955 0.043 22.270 0.000
## .b5pss5 0.644 0.043 14.879 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 0.385 0.043 8.862 0.000
## .b2pss2 0.359 0.029 12.468 0.000
## .b3pss3 0.425 0.039 11.025 0.000
## .b4pss4 0.401 0.035 11.420 0.000
## .b5pss5 0.366 0.034 10.767 0.000
## grief 0.592 0.073 8.081 0.000## lavaan 0.6-18 ended normally after 22 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 30
## Number of equality constraints 4
##
## Number of observations per group: Used Total
## 0 188 190
## 1 379 379
##
## Model Test User Model:
##
## Test statistic 19.275
## Degrees of freedom 14
## P-value (Chi-square) 0.155
## Test statistic for each group:
## 0 14.525
## 1 4.751
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [0]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## grief =~
## b1pss1 1.000
## b2pss2 (.p2.) 0.457 0.043 10.680 0.000
## b3pss3 (.p3.) 0.824 0.057 14.374 0.000
## b4pss4 (.p4.) 0.756 0.054 13.890 0.000
## b5pss5 (.p5.) 0.805 0.056 14.388 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 1.346 0.073 18.353 0.000
## .b2pss2 0.441 0.049 9.045 0.000
## .b3pss3 1.059 0.067 15.828 0.000
## .b4pss4 1.122 0.066 17.102 0.000
## .b5pss5 0.745 0.070 10.688 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 0.434 0.064 6.779 0.000
## .b2pss2 0.327 0.037 8.898 0.000
## .b3pss3 0.449 0.058 7.786 0.000
## .b4pss4 0.480 0.058 8.209 0.000
## .b5pss5 0.539 0.066 8.207 0.000
## grief 0.577 0.086 6.691 0.000
##
##
## Group 2 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## grief =~
## b1pss1 1.000
## b2pss2 (.p2.) 0.457 0.043 10.680 0.000
## b3pss3 (.p3.) 0.824 0.057 14.374 0.000
## b4pss4 (.p4.) 0.756 0.054 13.890 0.000
## b5pss5 (.p5.) 0.805 0.056 14.388 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 1.182 0.050 23.496 0.000
## .b2pss2 0.475 0.036 13.281 0.000
## .b3pss3 0.934 0.046 20.325 0.000
## .b4pss4 0.955 0.043 21.999 0.000
## .b5pss5 0.644 0.044 14.731 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 0.399 0.042 9.474 0.000
## .b2pss2 0.367 0.029 12.831 0.000
## .b3pss3 0.420 0.038 11.019 0.000
## .b4pss4 0.394 0.035 11.304 0.000
## .b5pss5 0.361 0.034 10.686 0.000
## grief 0.561 0.065 8.564 0.000## lavaan 0.6-18 ended normally after 26 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 31
## Number of equality constraints 9
##
## Number of observations per group: Used Total
## 0 188 190
## 1 379 379
##
## Model Test User Model:
##
## Test statistic 23.968
## Degrees of freedom 18
## P-value (Chi-square) 0.156
## Test statistic for each group:
## 0 17.123
## 1 6.846
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [0]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## grief =~
## b1pss1 1.000
## b2pss2 (.p2.) 0.449 0.042 10.562 0.000
## b3pss3 (.p3.) 0.824 0.057 14.467 0.000
## b4pss4 (.p4.) 0.760 0.054 14.007 0.000
## b5pss5 (.p5.) 0.803 0.056 14.457 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 (.12.) 1.332 0.066 20.221 0.000
## .b2pss2 (.13.) 0.505 0.037 13.718 0.000
## .b3pss3 (.14.) 1.055 0.057 18.601 0.000
## .b4pss4 (.15.) 1.081 0.053 20.237 0.000
## .b5pss5 (.16.) 0.756 0.056 13.525 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 0.435 0.064 6.773 0.000
## .b2pss2 0.332 0.037 8.937 0.000
## .b3pss3 0.448 0.058 7.776 0.000
## .b4pss4 0.480 0.059 8.190 0.000
## .b5pss5 0.539 0.066 8.207 0.000
## grief 0.579 0.086 6.701 0.000
##
##
## Group 2 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## grief =~
## b1pss1 1.000
## b2pss2 (.p2.) 0.449 0.042 10.562 0.000
## b3pss3 (.p3.) 0.824 0.057 14.467 0.000
## b4pss4 (.p4.) 0.760 0.054 14.007 0.000
## b5pss5 (.p5.) 0.803 0.056 14.457 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 (.12.) 1.332 0.066 20.221 0.000
## .b2pss2 (.13.) 0.505 0.037 13.718 0.000
## .b3pss3 (.14.) 1.055 0.057 18.601 0.000
## .b4pss4 (.15.) 1.081 0.053 20.237 0.000
## .b5pss5 (.16.) 0.756 0.056 13.525 0.000
## grief -0.144 0.076 -1.911 0.056
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 0.398 0.042 9.469 0.000
## .b2pss2 0.370 0.029 12.872 0.000
## .b3pss3 0.419 0.038 11.014 0.000
## .b4pss4 0.393 0.035 11.276 0.000
## .b5pss5 0.361 0.034 10.700 0.000
## grief 0.561 0.065 8.586 0.000## Test invariance through model comparison tests:
compareFit(configOut, weakOut, strongOut) %>% summary()## ################### Nested Model Comparison #########################
##
## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
## configOut 10 6472.7 6602.9 11.317
## weakOut 14 6472.7 6585.5 19.275 7.9585 0.059083 4 0.09311 .
## strongOut 18 6469.4 6564.9 23.968 4.6931 0.024722 4 0.32026
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## ####################### Model Fit Indices ###########################
## chisq df pvalue rmsea cfi tli srmr aic bic
## configOut 11.317† 10 .333 .022† 0.998† 0.997† .017† 6472.727 6602.937
## weakOut 19.275 14 .155 .036 .993 .990 .038 6472.685 6585.534
## strongOut 23.968 18 .156 .034 .992 .991 .042 6469.378† 6564.866†
##
## ################## Differences in Fit Indices #######################
## df rmsea cfi tli srmr aic bic
## weakOut - configOut 4 0.015 -0.005 -0.006 0.021 -0.041 -17.403
## strongOut - weakOut 4 -0.002 -0.001 0.001 0.004 -3.307 -20.668Click for explanation
- Configural invariance holds.
- The unrestricted, two-group model fits the data very well (\(\chi^2[10] = 11.32\), \(p = 0.333\), \(\textit{RMSEA} = 0.022\), \(\textit{CFI} = 0.998\), \(\textit{SRMR} = 0.017\)).
- Weak invariance holds.
- The model comparison test shows a non-significant loss of fit between the configural and weak models (\(\Delta \chi^2[4] = 7.96\), \(p = 0.093\)).
- Strong invariance holds.
- The model comparison test shows a non-significant loss of fit between the weak and strong models (\(\Delta \chi^2[4] = 4.69\), \(p = 0.32\)).
- The strongly invariant model still fits the data well (\(\chi^2[18] = 23.97\), \(p = 0.156\), \(\textit{RMSEA} = 0.034\), \(\textit{CFI} = 0.992\), \(\textit{SRMR} = 0.042\)).
7.4.2 Testing Between-Group Differences
Once we establish strong invariance, we have empirical evidence that latent mean levels are comparable across groups. Hence, we can test for differences in those latent means. In this section, we’ll conduct the equivalent of a t-test using our two-group CFA model.
More specifically, we want to know if the latent mean of grief differs
significantly between the Kin2 groups. The null and alternative hypotheses for
this test are as follows:
\[ H_0: \alpha_1 = \alpha_2\\ H_1: \alpha_1 \neq \alpha_2 \]
Where, \(\alpha_1\) and \(\alpha_2\) represent the latent means in Group 1 and Group 2, respectively.
7.4.2.1
Use the strongly invariant model from 7.4.1.2 to test the latent mean difference described above.
- What is your conclusion?
Click to show code
## Estimate the model with the latent means equated:
resOut <- cfa(cfaMod,
data = pgd,
group = "Kin2",
group.equal = c("loadings", "intercepts", "means")
)
## Check the results:
summary(resOut)## lavaan 0.6-18 ended normally after 24 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 30
## Number of equality constraints 9
##
## Number of observations per group: Used Total
## 0 188 190
## 1 379 379
##
## Model Test User Model:
##
## Test statistic 27.615
## Degrees of freedom 19
## P-value (Chi-square) 0.091
## Test statistic for each group:
## 0 19.755
## 1 7.860
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [0]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## grief =~
## b1pss1 1.000
## b2pss2 (.p2.) 0.451 0.043 10.591 0.000
## b3pss3 (.p3.) 0.824 0.057 14.450 0.000
## b4pss4 (.p4.) 0.759 0.054 13.976 0.000
## b5pss5 (.p5.) 0.804 0.056 14.448 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 (.12.) 1.234 0.042 29.668 0.000
## .b2pss2 (.13.) 0.461 0.029 15.988 0.000
## .b3pss3 (.14.) 0.974 0.038 25.664 0.000
## .b4pss4 (.15.) 1.006 0.036 27.699 0.000
## .b5pss5 (.16.) 0.676 0.037 18.267 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 0.436 0.064 6.770 0.000
## .b2pss2 0.331 0.037 8.925 0.000
## .b3pss3 0.448 0.058 7.767 0.000
## .b4pss4 0.482 0.059 8.194 0.000
## .b5pss5 0.538 0.066 8.197 0.000
## grief 0.588 0.088 6.714 0.000
##
##
## Group 2 [1]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## grief =~
## b1pss1 1.000
## b2pss2 (.p2.) 0.451 0.043 10.591 0.000
## b3pss3 (.p3.) 0.824 0.057 14.450 0.000
## b4pss4 (.p4.) 0.759 0.054 13.976 0.000
## b5pss5 (.p5.) 0.804 0.056 14.448 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 (.12.) 1.234 0.042 29.668 0.000
## .b2pss2 (.13.) 0.461 0.029 15.988 0.000
## .b3pss3 (.14.) 0.974 0.038 25.664 0.000
## .b4pss4 (.15.) 1.006 0.036 27.699 0.000
## .b5pss5 (.16.) 0.676 0.037 18.267 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .b1pss1 0.399 0.042 9.469 0.000
## .b2pss2 0.370 0.029 12.863 0.000
## .b3pss3 0.419 0.038 11.012 0.000
## .b4pss4 0.394 0.035 11.283 0.000
## .b5pss5 0.361 0.034 10.691 0.000
## grief 0.563 0.066 8.584 0.000Click for explanation
We can equate the latent means by specifying the group.equal = "means"
argument in cfa(). Then, we simply compare this constrained model to the
strong invariance model to get our test of mean differences.
In this case, the means of the grief factor do not significantly differ between
Kin2 groups (\(\Delta \chi^2[1] = 3.65\), \(p = 0.056\)), assuming we
adopt an alpha-level of 0.05 or lower for the test.
7.4.3 Multiple-Group SEM for Moderation
Now, we’re going to revisit the TORA model from the Week 6 In-Class Exercises, and use a multiple-group model to test the moderating effect of sex.
7.4.3.1
Load the data contained in the toradata.csv file.
Before we get to any moderation tests, however, we first need to establish measurement invariance. The first step in any multiple-group analysis that includes latent variables is measurement invariance testing.
7.4.3.2
Test for measurement invariance across sex groups in the three latent variables of the TORA model from 6.4.2.
- Test configural, weak, and strong invariance.
- Test for invariance in all three latent factors simultaneously.
- Is full measurement invariance (i.e., up to and including strong invariance) supported?
Click to show code
tora_cfa <- '
attitudes =~ attit_1 + attit_2 + attit_3
norms =~ norm_1 + norm_2 + norm_3
control =~ control_1 + control_2 + control_3
'
## Estimate the models:
config <- cfa(tora_cfa, data = condom, group = "sex")
weak <- cfa(tora_cfa, data = condom, group = "sex", group.equal = "loadings")
strong <- cfa(tora_cfa,
data = condom,
group = "sex",
group.equal = c("loadings", "intercepts")
)
## Check that everything went well:
summary(config)## lavaan 0.6-18 ended normally after 54 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 60
##
## Number of observations per group:
## woman 161
## man 89
##
## Model Test User Model:
##
## Test statistic 66.565
## Degrees of freedom 48
## P-value (Chi-square) 0.039
## Test statistic for each group:
## woman 42.623
## man 23.941
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [woman]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## attitudes =~
## attit_1 1.000
## attit_2 1.005 0.075 13.427 0.000
## attit_3 -0.965 0.075 -12.878 0.000
## norms =~
## norm_1 1.000
## norm_2 0.952 0.101 9.470 0.000
## norm_3 0.879 0.101 8.742 0.000
## control =~
## control_1 1.000
## control_2 0.794 0.144 5.526 0.000
## control_3 0.989 0.152 6.523 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## attitudes ~~
## norms 0.450 0.087 5.200 0.000
## control 0.468 0.089 5.249 0.000
## norms ~~
## control 0.387 0.079 4.912 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .attit_1 2.839 0.090 31.702 0.000
## .attit_2 2.907 0.084 34.728 0.000
## .attit_3 3.174 0.084 37.969 0.000
## .norm_1 2.832 0.080 35.342 0.000
## .norm_2 2.832 0.079 35.775 0.000
## .norm_3 2.795 0.081 34.694 0.000
## .control_1 2.851 0.082 34.755 0.000
## .control_2 2.857 0.081 35.104 0.000
## .control_3 2.888 0.081 35.877 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .attit_1 0.398 0.059 6.739 0.000
## .attit_2 0.227 0.045 5.011 0.000
## .attit_3 0.294 0.048 6.092 0.000
## .norm_1 0.346 0.065 5.328 0.000
## .norm_2 0.385 0.065 5.962 0.000
## .norm_3 0.513 0.072 7.108 0.000
## .control_1 0.587 0.090 6.531 0.000
## .control_2 0.754 0.096 7.815 0.000
## .control_3 0.557 0.086 6.453 0.000
## attitudes 0.893 0.142 6.273 0.000
## norms 0.688 0.121 5.706 0.000
## control 0.497 0.119 4.184 0.000
##
##
## Group 2 [man]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## attitudes =~
## attit_1 1.000
## attit_2 1.167 0.149 7.843 0.000
## attit_3 -1.060 0.142 -7.443 0.000
## norms =~
## norm_1 1.000
## norm_2 1.070 0.215 4.965 0.000
## norm_3 0.922 0.189 4.869 0.000
## control =~
## control_1 1.000
## control_2 0.995 0.290 3.435 0.001
## control_3 0.949 0.285 3.332 0.001
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## attitudes ~~
## norms 0.086 0.103 0.837 0.403
## control 0.388 0.113 3.430 0.001
## norms ~~
## control 0.200 0.101 1.976 0.048
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .attit_1 3.270 0.117 28.063 0.000
## .attit_2 3.180 0.128 24.905 0.000
## .attit_3 2.787 0.126 22.187 0.000
## .norm_1 3.236 0.131 24.692 0.000
## .norm_2 3.337 0.132 25.293 0.000
## .norm_3 3.303 0.131 25.136 0.000
## .control_1 3.157 0.111 28.415 0.000
## .control_2 3.135 0.129 24.249 0.000
## .control_3 3.213 0.130 24.805 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .attit_1 0.440 0.095 4.631 0.000
## .attit_2 0.405 0.109 3.699 0.000
## .attit_3 0.541 0.112 4.822 0.000
## .norm_1 0.740 0.175 4.230 0.000
## .norm_2 0.647 0.182 3.555 0.000
## .norm_3 0.868 0.175 4.972 0.000
## .control_1 0.673 0.146 4.602 0.000
## .control_2 1.066 0.197 5.417 0.000
## .control_3 1.110 0.199 5.582 0.000
## attitudes 0.768 0.182 4.220 0.000
## norms 0.788 0.242 3.259 0.001
## control 0.426 0.168 2.537 0.011## lavaan 0.6-18 ended normally after 37 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 60
## Number of equality constraints 6
##
## Number of observations per group:
## woman 161
## man 89
##
## Model Test User Model:
##
## Test statistic 68.557
## Degrees of freedom 54
## P-value (Chi-square) 0.088
## Test statistic for each group:
## woman 43.148
## man 25.409
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [woman]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## attitudes =~
## attit_1 1.000
## attit_2 (.p2.) 1.048 0.068 15.413 0.000
## attit_3 (.p3.) -0.995 0.067 -14.762 0.000
## norms =~
## norm_1 1.000
## norm_2 (.p5.) 0.977 0.091 10.708 0.000
## norm_3 (.p6.) 0.889 0.089 9.996 0.000
## control =~
## cntrl_1 1.000
## cntrl_2 (.p8.) 0.843 0.130 6.506 0.000
## cntrl_3 (.p9.) 0.983 0.135 7.306 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## attitudes ~~
## norms 0.431 0.082 5.256 0.000
## control 0.450 0.083 5.395 0.000
## norms ~~
## control 0.378 0.075 5.031 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .attit_1 2.839 0.088 32.194 0.000
## .attit_2 2.907 0.084 34.445 0.000
## .attit_3 3.174 0.084 37.913 0.000
## .norm_1 2.832 0.080 35.504 0.000
## .norm_2 2.832 0.080 35.571 0.000
## .norm_3 2.795 0.080 34.727 0.000
## .control_1 2.851 0.082 34.882 0.000
## .control_2 2.857 0.082 34.746 0.000
## .control_3 2.888 0.080 36.037 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .attit_1 0.408 0.058 6.987 0.000
## .attit_2 0.221 0.045 4.896 0.000
## .attit_3 0.293 0.048 6.128 0.000
## .norm_1 0.353 0.063 5.580 0.000
## .norm_2 0.380 0.064 5.952 0.000
## .norm_3 0.512 0.071 7.178 0.000
## .control_1 0.590 0.088 6.695 0.000
## .control_2 0.744 0.096 7.731 0.000
## .control_3 0.565 0.085 6.663 0.000
## attitudes 0.844 0.129 6.540 0.000
## norms 0.672 0.113 5.921 0.000
## control 0.485 0.110 4.417 0.000
##
##
## Group 2 [man]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## attitudes =~
## attit_1 1.000
## attit_2 (.p2.) 1.048 0.068 15.413 0.000
## attit_3 (.p3.) -0.995 0.067 -14.762 0.000
## norms =~
## norm_1 1.000
## norm_2 (.p5.) 0.977 0.091 10.708 0.000
## norm_3 (.p6.) 0.889 0.089 9.996 0.000
## control =~
## cntrl_1 1.000
## cntrl_2 (.p8.) 0.843 0.130 6.506 0.000
## cntrl_3 (.p9.) 0.983 0.135 7.306 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## attitudes ~~
## norms 0.092 0.114 0.807 0.420
## control 0.425 0.109 3.912 0.000
## norms ~~
## control 0.217 0.103 2.100 0.036
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .attit_1 3.270 0.120 27.254 0.000
## .attit_2 3.180 0.125 25.501 0.000
## .attit_3 2.787 0.125 22.275 0.000
## .norm_1 3.236 0.132 24.423 0.000
## .norm_2 3.337 0.130 25.610 0.000
## .norm_3 3.303 0.132 25.086 0.000
## .control_1 3.157 0.112 28.208 0.000
## .control_2 3.135 0.127 24.750 0.000
## .control_3 3.213 0.131 24.540 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .attit_1 0.419 0.093 4.528 0.000
## .attit_2 0.438 0.099 4.436 0.000
## .attit_3 0.540 0.107 5.057 0.000
## .norm_1 0.704 0.158 4.456 0.000
## .norm_2 0.692 0.153 4.520 0.000
## .norm_3 0.864 0.164 5.271 0.000
## .control_1 0.668 0.139 4.797 0.000
## .control_2 1.110 0.186 5.960 0.000
## .control_3 1.094 0.193 5.663 0.000
## attitudes 0.862 0.166 5.200 0.000
## norms 0.859 0.193 4.443 0.000
## control 0.447 0.137 3.260 0.001## lavaan 0.6-18 ended normally after 60 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 63
## Number of equality constraints 15
##
## Number of observations per group:
## woman 161
## man 89
##
## Model Test User Model:
##
## Test statistic 72.050
## Degrees of freedom 60
## P-value (Chi-square) 0.137
## Test statistic for each group:
## woman 43.961
## man 28.089
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [woman]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## attitudes =~
## attit_1 1.000
## attit_2 (.p2.) 1.028 0.065 15.693 0.000
## attit_3 (.p3.) -0.990 0.065 -15.114 0.000
## norms =~
## norm_1 1.000
## norm_2 (.p5.) 0.998 0.089 11.182 0.000
## norm_3 (.p6.) 0.918 0.088 10.467 0.000
## control =~
## cntrl_1 1.000
## cntrl_2 (.p8.) 0.848 0.126 6.736 0.000
## cntrl_3 (.p9.) 0.987 0.131 7.558 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## attitudes ~~
## norms 0.428 0.081 5.259 0.000
## control 0.454 0.084 5.438 0.000
## norms ~~
## control 0.372 0.073 5.060 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .attit_1 (.25.) 2.864 0.085 33.535 0.000
## .attit_2 (.26.) 2.887 0.083 34.826 0.000
## .attit_3 (.27.) 3.166 0.082 38.500 0.000
## .norm_1 (.28.) 2.816 0.078 36.330 0.000
## .norm_2 (.29.) 2.838 0.078 36.453 0.000
## .norm_3 (.30.) 2.812 0.078 36.253 0.000
## .cntrl_1 (.31.) 2.847 0.078 36.562 0.000
## .cntrl_2 (.32.) 2.859 0.076 37.381 0.000
## .cntrl_3 (.33.) 2.891 0.077 37.531 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .attit_1 0.405 0.058 6.931 0.000
## .attit_2 0.226 0.045 5.051 0.000
## .attit_3 0.291 0.048 6.084 0.000
## .norm_1 0.362 0.062 5.795 0.000
## .norm_2 0.377 0.064 5.931 0.000
## .norm_3 0.506 0.071 7.109 0.000
## .control_1 0.592 0.088 6.743 0.000
## .control_2 0.743 0.096 7.739 0.000
## .control_3 0.566 0.085 6.684 0.000
## attitudes 0.861 0.130 6.607 0.000
## norms 0.650 0.109 5.950 0.000
## control 0.482 0.108 4.477 0.000
##
##
## Group 2 [man]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## attitudes =~
## attit_1 1.000
## attit_2 (.p2.) 1.028 0.065 15.693 0.000
## attit_3 (.p3.) -0.990 0.065 -15.114 0.000
## norms =~
## norm_1 1.000
## norm_2 (.p5.) 0.998 0.089 11.182 0.000
## norm_3 (.p6.) 0.918 0.088 10.467 0.000
## control =~
## cntrl_1 1.000
## cntrl_2 (.p8.) 0.848 0.126 6.736 0.000
## cntrl_3 (.p9.) 0.987 0.131 7.558 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## attitudes ~~
## norms 0.093 0.113 0.825 0.409
## control 0.428 0.109 3.926 0.000
## norms ~~
## control 0.213 0.101 2.102 0.036
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .attit_1 (.25.) 2.864 0.085 33.535 0.000
## .attit_2 (.26.) 2.887 0.083 34.826 0.000
## .attit_3 (.27.) 3.166 0.082 38.500 0.000
## .norm_1 (.28.) 2.816 0.078 36.330 0.000
## .norm_2 (.29.) 2.838 0.078 36.453 0.000
## .norm_3 (.30.) 2.812 0.078 36.253 0.000
## .cntrl_1 (.31.) 2.847 0.078 36.562 0.000
## .cntrl_2 (.32.) 2.859 0.076 37.381 0.000
## .cntrl_3 (.33.) 2.891 0.077 37.531 0.000
## attitds 0.356 0.133 2.680 0.007
## norms 0.480 0.133 3.602 0.000
## control 0.318 0.116 2.733 0.006
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .attit_1 0.420 0.094 4.484 0.000
## .attit_2 0.456 0.100 4.557 0.000
## .attit_3 0.537 0.107 5.023 0.000
## .norm_1 0.724 0.157 4.599 0.000
## .norm_2 0.686 0.153 4.489 0.000
## .norm_3 0.859 0.165 5.220 0.000
## .control_1 0.669 0.139 4.821 0.000
## .control_2 1.109 0.186 5.958 0.000
## .control_3 1.094 0.193 5.664 0.000
## attitudes 0.872 0.167 5.214 0.000
## norms 0.830 0.186 4.455 0.000
## control 0.445 0.136 3.280 0.001## ################### Nested Model Comparison #########################
##
## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
## config 48 6021.5 6232.8 66.565
## weak 54 6011.5 6201.6 68.557 1.9924 0 6 0.9204
## strong 60 6003.0 6172.0 72.050 3.4934 0 6 0.7448
##
## ####################### Model Fit Indices ###########################
## chisq df pvalue rmsea cfi tli srmr aic bic
## config 66.565† 48 .039 .056 .979 .968 .048† 6021.476 6232.764
## weak 68.557 54 .088 .046 .983 .978 .050 6011.469 6201.628
## strong 72.050 60 .137 .040† .986† .983† .051 6002.962† 6171.992†
##
## ################## Differences in Fit Indices #######################
## df rmsea cfi tli srmr aic bic
## weak - config 6 -0.009 0.005 0.010 0.003 -10.008 -31.136
## strong - weak 6 -0.006 0.003 0.006 0.001 -8.507 -29.635Click for explanation
Yes, we can establish full measurement invariance.
- Configural invariance holds.
- The unrestricted, multiple-group model fits the data well (\(\chi^2[48] = 66.56\), \(p = 0.039\), \(\textit{RMSEA} = 0.056\), \(\textit{CFI} = 0.979\), \(\textit{SRMR} = 0.048\)).
- Weak invariance holds.
- The model comparison test shows a non-significant loss of fit between the configural and weak models (\(\Delta \chi^2[6] = 1.99\), \(p = 0.92\)).
- Strong invariance holds.
- The model comparison test shows a non-significant loss of fit between the weak and strong models (\(\Delta \chi^2[6] = 3.49\), \(p = 0.745\)).
- The strongly invariant model still fits the data well (\(\chi^2[60] = 72.05\), \(p = 0.137\), \(\textit{RMSEA} = 0.04\), \(\textit{CFI} = 0.986\), \(\textit{SRMR} = 0.051\)).
Once we’ve established measurement invariance, we can move on to testing hypotheses about between-group differences secure in the knowledge that our latent factors represent the same hypothetical constructs in all groups.
7.4.3.3
Estimate the full TORA model from 6.4.4 as a multiple-group model.
- Use
sexas the grouping variables. - Keep the strong invariance constraints in place.
Click to show code
## Add the structural paths to the model:
tora_sem <- paste(tora_cfa,
'intent ~ attitudes + norms
behavior ~ intent + control',
sep = '\n')
## Estimate the model:
toraOut <- sem(tora_sem,
data = condom,
group = "sex",
group.equal = c("loadings", "intercepts")
)
## Check the results:
summary(toraOut, fit.measures = TRUE, standardized = TRUE, rsquare = TRUE)## lavaan 0.6-18 ended normally after 62 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 79
## Number of equality constraints 17
##
## Number of observations per group:
## woman 161
## man 89
##
## Model Test User Model:
##
## Test statistic 141.903
## Degrees of freedom 92
## P-value (Chi-square) 0.001
## Test statistic for each group:
## woman 83.870
## man 58.033
##
## Model Test Baseline Model:
##
## Test statistic 1378.913
## Degrees of freedom 110
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.961
## Tucker-Lewis Index (TLI) 0.953
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -3470.878
## Loglikelihood unrestricted model (H1) -3399.927
##
## Akaike (AIC) 7065.756
## Bayesian (BIC) 7284.087
## Sample-size adjusted Bayesian (SABIC) 7087.541
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.066
## 90 Percent confidence interval - lower 0.043
## 90 Percent confidence interval - upper 0.087
## P-value H_0: RMSEA <= 0.050 0.114
## P-value H_0: RMSEA >= 0.080 0.137
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.058
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [woman]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## attitudes =~
## attit_1 1.000 0.919 0.816
## attit_2 (.p2.) 1.023 0.066 15.495 0.000 0.940 0.877
## attit_3 (.p3.) -1.016 0.066 -15.434 0.000 -0.935 -0.884
## norms =~
## norm_1 1.000 0.808 0.798
## norm_2 (.p5.) 0.956 0.083 11.551 0.000 0.772 0.766
## norm_3 (.p6.) 0.942 0.084 11.256 0.000 0.761 0.743
## control =~
## cntrl_1 1.000 0.671 0.646
## cntrl_2 (.p8.) 0.846 0.125 6.768 0.000 0.567 0.545
## cntrl_3 (.p9.) 1.008 0.129 7.814 0.000 0.676 0.665
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## intent ~
## attitudes 0.436 0.082 5.335 0.000 0.401 0.403
## norms 0.598 0.100 6.008 0.000 0.483 0.486
## behavior ~
## intent 0.347 0.064 5.436 0.000 0.347 0.351
## control 0.727 0.138 5.274 0.000 0.488 0.496
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## attitudes ~~
## norms 0.425 0.081 5.262 0.000 0.572 0.572
## control 0.464 0.082 5.634 0.000 0.752 0.752
## norms ~~
## control 0.386 0.073 5.320 0.000 0.712 0.712
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .attit_1 (.31.) 2.863 0.084 34.071 0.000 2.863 2.542
## .attit_2 (.32.) 2.884 0.082 35.225 0.000 2.884 2.690
## .attit_3 (.33.) 3.168 0.081 39.065 0.000 3.168 2.995
## .norm_1 (.34.) 2.802 0.076 36.874 0.000 2.802 2.767
## .norm_2 (.35.) 2.830 0.075 37.555 0.000 2.830 2.807
## .norm_3 (.36.) 2.796 0.076 36.725 0.000 2.796 2.730
## .cntrl_1 (.37.) 2.855 0.077 37.078 0.000 2.855 2.749
## .cntrl_2 (.38.) 2.866 0.076 37.909 0.000 2.866 2.753
## .cntrl_3 (.39.) 2.897 0.076 37.969 0.000 2.897 2.849
## .intent (.40.) 2.712 0.078 34.861 0.000 2.712 2.726
## .behavir (.41.) 1.630 0.175 9.289 0.000 1.630 1.658
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .attit_1 0.423 0.059 7.172 0.000 0.423 0.333
## .attit_2 0.266 0.045 5.879 0.000 0.266 0.231
## .attit_3 0.246 0.043 5.671 0.000 0.246 0.219
## .norm_1 0.372 0.060 6.230 0.000 0.372 0.363
## .norm_2 0.420 0.062 6.760 0.000 0.420 0.413
## .norm_3 0.469 0.066 7.063 0.000 0.469 0.448
## .control_1 0.629 0.085 7.426 0.000 0.629 0.583
## .control_2 0.762 0.094 8.088 0.000 0.762 0.703
## .control_3 0.577 0.080 7.238 0.000 0.577 0.558
## .intent 0.374 0.049 7.615 0.000 0.374 0.378
## .behavior 0.391 0.052 7.486 0.000 0.391 0.404
## attitudes 0.845 0.129 6.561 0.000 1.000 1.000
## norms 0.653 0.108 6.048 0.000 1.000 1.000
## control 0.450 0.101 4.457 0.000 1.000 1.000
##
## R-Square:
## Estimate
## attit_1 0.667
## attit_2 0.769
## attit_3 0.781
## norm_1 0.637
## norm_2 0.587
## norm_3 0.552
## control_1 0.417
## control_2 0.297
## control_3 0.442
## intent 0.622
## behavior 0.596
##
##
## Group 2 [man]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## attitudes =~
## attit_1 1.000 0.922 0.815
## attit_2 (.p2.) 1.023 0.066 15.495 0.000 0.943 0.817
## attit_3 (.p3.) -1.016 0.066 -15.434 0.000 -0.937 -0.782
## norms =~
## norm_1 1.000 0.875 0.723
## norm_2 (.p5.) 0.956 0.083 11.551 0.000 0.837 0.679
## norm_3 (.p6.) 0.942 0.084 11.256 0.000 0.825 0.667
## control =~
## cntrl_1 1.000 0.663 0.631
## cntrl_2 (.p8.) 0.846 0.125 6.768 0.000 0.561 0.467
## cntrl_3 (.p9.) 1.008 0.129 7.814 0.000 0.668 0.540
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## intent ~
## attitudes 0.535 0.097 5.497 0.000 0.494 0.446
## norms 0.858 0.111 7.702 0.000 0.751 0.679
## behavior ~
## intent 0.613 0.060 10.188 0.000 0.613 0.706
## control 0.007 0.159 0.045 0.964 0.005 0.005
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## attitudes ~~
## norms 0.060 0.108 0.552 0.581 0.074 0.074
## control 0.426 0.107 3.978 0.000 0.697 0.697
## norms ~~
## control 0.220 0.096 2.291 0.022 0.380 0.380
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .attit_1 (.31.) 2.863 0.084 34.071 0.000 2.863 2.531
## .attit_2 (.32.) 2.884 0.082 35.225 0.000 2.884 2.499
## .attit_3 (.33.) 3.168 0.081 39.065 0.000 3.168 2.645
## .norm_1 (.34.) 2.802 0.076 36.874 0.000 2.802 2.314
## .norm_2 (.35.) 2.830 0.075 37.555 0.000 2.830 2.296
## .norm_3 (.36.) 2.796 0.076 36.725 0.000 2.796 2.261
## .cntrl_1 (.37.) 2.855 0.077 37.078 0.000 2.855 2.719
## .cntrl_2 (.38.) 2.866 0.076 37.909 0.000 2.866 2.385
## .cntrl_3 (.39.) 2.897 0.076 37.969 0.000 2.897 2.344
## .intent (.40.) 2.712 0.078 34.861 0.000 2.712 2.450
## .behavir (.41.) 1.630 0.175 9.289 0.000 1.630 1.696
## attitds 0.369 0.130 2.834 0.005 0.400 0.400
## norms 0.534 0.126 4.246 0.000 0.610 0.610
## control 0.309 0.115 2.691 0.007 0.466 0.466
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .attit_1 0.430 0.091 4.710 0.000 0.430 0.336
## .attit_2 0.443 0.094 4.691 0.000 0.443 0.333
## .attit_3 0.556 0.108 5.130 0.000 0.556 0.388
## .norm_1 0.699 0.137 5.088 0.000 0.699 0.477
## .norm_2 0.819 0.150 5.457 0.000 0.819 0.539
## .norm_3 0.849 0.153 5.538 0.000 0.849 0.555
## .control_1 0.663 0.135 4.908 0.000 0.663 0.602
## .control_2 1.130 0.187 6.025 0.000 1.130 0.782
## .control_3 1.082 0.191 5.673 0.000 1.082 0.708
## .intent 0.363 0.089 4.091 0.000 0.363 0.296
## .behavior 0.460 0.069 6.671 0.000 0.460 0.498
## attitudes 0.850 0.164 5.190 0.000 1.000 1.000
## norms 0.766 0.171 4.495 0.000 1.000 1.000
## control 0.440 0.133 3.301 0.001 1.000 1.000
##
## R-Square:
## Estimate
## attit_1 0.664
## attit_2 0.667
## attit_3 0.612
## norm_1 0.523
## norm_2 0.461
## norm_3 0.445
## control_1 0.398
## control_2 0.218
## control_3 0.292
## intent 0.704
## behavior 0.5027.4.3.4
Conduct an omnibus test to check if sex moderates any of the latent regression
paths in the model from 7.4.3.3.
Click for explanation
## Estimate a restricted model wherein all latent regression paths are equated
## across groups.
toraOut0 <- sem(tora_sem,
data = condom,
group = "sex",
group.equal = c("loadings", "intercepts", "regressions")
)
## Test the constraints:
anova(toraOut, toraOut0)Click for explanation
We can equate the latent regressions by specifying the group.equal = "regressions"
argument in sem(). Then, we simply compare this constrained model to the
unconstrained model from 7.4.3.3 to get our test of moderation.
Equating all regression paths across groups produces a significant loss of fit (\(\Delta \chi^2[4] = 52.86\), \(p < 0.001\)). Therefore, sex must moderate at least some of these paths.
7.4.3.5
Conduct a two-parameter test to check if sex moderates the effects of intent
and control on behavior.
- Use the
lavTestWald()function to conduct your test. - Keep only the weak invariance constraints when estimating the model.
Click to show code
## Add the structural paths to the model and assign labels:
tora_sem <- paste(tora_cfa,
'intent ~ attitudes + norms
behavior ~ c(b1f, b1m) * intent + c(b2f, b2m) * control',
sep = '\n')
## Estimate the model with weak invariance constraints:
toraOut <- sem(tora_sem, data = condom, group = "sex", group.equal = "loadings")
## Check the results:
summary(toraOut, fit.measures = TRUE, standardized = TRUE, rsquare = TRUE)## lavaan 0.6-18 ended normally after 60 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 76
## Number of equality constraints 6
##
## Number of observations per group:
## woman 161
## man 89
##
## Model Test User Model:
##
## Test statistic 119.722
## Degrees of freedom 84
## P-value (Chi-square) 0.006
## Test statistic for each group:
## woman 76.908
## man 42.814
##
## Model Test Baseline Model:
##
## Test statistic 1378.913
## Degrees of freedom 110
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.972
## Tucker-Lewis Index (TLI) 0.963
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -3459.788
## Loglikelihood unrestricted model (H1) -3399.927
##
## Akaike (AIC) 7059.576
## Bayesian (BIC) 7306.078
## Sample-size adjusted Bayesian (SABIC) 7084.172
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.058
## 90 Percent confidence interval - lower 0.032
## 90 Percent confidence interval - upper 0.081
## P-value H_0: RMSEA <= 0.050 0.272
## P-value H_0: RMSEA >= 0.080 0.058
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.047
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [woman]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## attitudes =~
## attit_1 1.000 0.909 0.813
## attit_2 (.p2.) 1.047 0.069 15.249 0.000 0.951 0.882
## attit_3 (.p3.) -1.025 0.068 -15.075 0.000 -0.931 -0.882
## norms =~
## norm_1 1.000 0.824 0.809
## norm_2 (.p5.) 0.936 0.083 11.256 0.000 0.771 0.768
## norm_3 (.p6.) 0.908 0.084 10.810 0.000 0.748 0.734
## control =~
## cntrl_1 1.000 0.690 0.666
## cntrl_2 (.p8.) 0.832 0.126 6.593 0.000 0.574 0.551
## cntrl_3 (.p9.) 1.006 0.131 7.673 0.000 0.694 0.682
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## intent ~
## attituds 0.441 0.083 5.294 0.000 0.400 0.403
## norms 0.580 0.098 5.918 0.000 0.478 0.480
## behavior ~
## intent (b1f) 0.531 0.074 7.127 0.000 0.531 0.524
## control (b2f) 0.490 0.130 3.767 0.000 0.338 0.336
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## attitudes ~~
## norms 0.428 0.081 5.266 0.000 0.572 0.572
## control 0.454 0.082 5.517 0.000 0.723 0.723
## norms ~~
## control 0.384 0.074 5.169 0.000 0.676 0.676
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .attit_1 2.839 0.088 32.237 0.000 2.839 2.541
## .attit_2 2.907 0.085 34.191 0.000 2.907 2.695
## .attit_3 3.174 0.083 38.126 0.000 3.174 3.005
## .norm_1 2.832 0.080 35.280 0.000 2.832 2.780
## .norm_2 2.832 0.079 35.765 0.000 2.832 2.819
## .norm_3 2.795 0.080 34.787 0.000 2.795 2.742
## .control_1 2.851 0.082 34.875 0.000 2.851 2.749
## .control_2 2.857 0.082 34.813 0.000 2.857 2.744
## .control_3 2.888 0.080 35.985 0.000 2.888 2.836
## .intent 2.677 0.078 34.159 0.000 2.677 2.692
## .behavior 1.107 0.207 5.338 0.000 1.107 1.098
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .attit_1 0.423 0.059 7.208 0.000 0.423 0.339
## .attit_2 0.259 0.045 5.713 0.000 0.259 0.223
## .attit_3 0.248 0.044 5.701 0.000 0.248 0.222
## .norm_1 0.359 0.060 5.989 0.000 0.359 0.346
## .norm_2 0.415 0.062 6.711 0.000 0.415 0.411
## .norm_3 0.479 0.067 7.152 0.000 0.479 0.461
## .control_1 0.599 0.085 7.027 0.000 0.599 0.557
## .control_2 0.755 0.095 7.940 0.000 0.755 0.696
## .control_3 0.555 0.081 6.822 0.000 0.555 0.535
## .intent 0.382 0.050 7.657 0.000 0.382 0.386
## .behavior 0.402 0.049 8.152 0.000 0.402 0.396
## attitudes 0.825 0.127 6.495 0.000 1.000 1.000
## norms 0.679 0.112 6.066 0.000 1.000 1.000
## control 0.477 0.106 4.483 0.000 1.000 1.000
##
## R-Square:
## Estimate
## attit_1 0.661
## attit_2 0.777
## attit_3 0.778
## norm_1 0.654
## norm_2 0.589
## norm_3 0.539
## control_1 0.443
## control_2 0.304
## control_3 0.465
## intent 0.614
## behavior 0.604
##
##
## Group 2 [man]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## attitudes =~
## attit_1 1.000 0.921 0.812
## attit_2 (.p2.) 1.047 0.069 15.249 0.000 0.964 0.831
## attit_3 (.p3.) -1.025 0.068 -15.075 0.000 -0.944 -0.787
## norms =~
## norm_1 1.000 0.928 0.753
## norm_2 (.p5.) 0.936 0.083 11.256 0.000 0.869 0.698
## norm_3 (.p6.) 0.908 0.084 10.810 0.000 0.843 0.676
## control =~
## cntrl_1 1.000 0.669 0.634
## cntrl_2 (.p8.) 0.832 0.126 6.593 0.000 0.556 0.464
## cntrl_3 (.p9.) 1.006 0.131 7.673 0.000 0.673 0.547
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## intent ~
## attituds 0.501 0.098 5.134 0.000 0.462 0.432
## norms 0.749 0.112 6.696 0.000 0.695 0.649
## behavior ~
## intent (b1m) 0.344 0.086 4.005 0.000 0.344 0.453
## control (b2m) 0.307 0.168 1.830 0.067 0.205 0.253
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## attitudes ~~
## norms 0.084 0.113 0.742 0.458 0.098 0.098
## control 0.424 0.107 3.960 0.000 0.688 0.688
## norms ~~
## control 0.240 0.102 2.361 0.018 0.387 0.387
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .attit_1 3.270 0.120 27.179 0.000 3.270 2.881
## .attit_2 3.180 0.123 25.848 0.000 3.180 2.740
## .attit_3 2.787 0.127 21.900 0.000 2.787 2.321
## .norm_1 3.236 0.131 24.790 0.000 3.236 2.628
## .norm_2 3.337 0.132 25.309 0.000 3.337 2.683
## .norm_3 3.303 0.132 24.992 0.000 3.303 2.649
## .control_1 3.157 0.112 28.223 0.000 3.157 2.992
## .control_2 3.135 0.127 24.653 0.000 3.135 2.613
## .control_3 3.213 0.130 24.627 0.000 3.213 2.610
## .intent 3.427 0.113 30.233 0.000 3.427 3.205
## .behavior 2.607 0.303 8.614 0.000 2.607 3.211
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .attit_1 0.439 0.092 4.795 0.000 0.439 0.341
## .attit_2 0.417 0.092 4.518 0.000 0.417 0.310
## .attit_3 0.549 0.107 5.120 0.000 0.549 0.381
## .norm_1 0.656 0.137 4.801 0.000 0.656 0.432
## .norm_2 0.792 0.148 5.342 0.000 0.792 0.512
## .norm_3 0.845 0.153 5.504 0.000 0.845 0.543
## .control_1 0.666 0.134 4.955 0.000 0.666 0.598
## .control_2 1.130 0.187 6.053 0.000 1.130 0.785
## .control_3 1.063 0.187 5.669 0.000 1.063 0.701
## .intent 0.385 0.086 4.471 0.000 0.385 0.337
## .behavior 0.399 0.063 6.328 0.000 0.399 0.605
## attitudes 0.849 0.164 5.174 0.000 1.000 1.000
## norms 0.861 0.190 4.528 0.000 1.000 1.000
## control 0.447 0.134 3.334 0.001 1.000 1.000
##
## R-Square:
## Estimate
## attit_1 0.659
## attit_2 0.690
## attit_3 0.619
## norm_1 0.568
## norm_2 0.488
## norm_3 0.457
## control_1 0.402
## control_2 0.215
## control_3 0.299
## intent 0.663
## behavior 0.395## $stat
## [1] 9.85773
##
## $df
## [1] 2
##
## $p.value
## [1] 0.00723471
##
## $se
## [1] "standard"Click for explanation
The Wald test suggest significant moderation (\(\Delta \chi^2[2] = 9.86\), \(p = 0.007\)). Equating these two regression slopes across groups produces a significant loss of fit. Therefore, sex must moderate one or both of these paths.
End of In-Class Exercises