6.2 Tau-Equivalent Models#
Essentially Tau-Equivalent Model#
The Essentially tau-equivalent measurement model is also quite flexible but it has one more restriction compared to the Tau Congeneric measurement model. It assumes that
items differ in their difficulty
items are equivalent in their discrimination power
items vary in their reliability
We therefore obtain estimates for the intercepts (Intercepts section) and for the errors (Variances section). A Latent Variables section is still present, but all loadings are fixed to 1.
Fit the model#
Usage#
This notebook fits essentially tau-equivalent and tau-equivalent models to the Data_EmotionalClarity.dat items. After loading the prepared item matrix, the models are specified in lavaan and compared via fit indices. FIrst we load our packages and our data.
# General imports
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
# Rpy2 imports
from rpy2 import robjects as ro
from rpy2.robjects import pandas2ri, numpy2ri
from rpy2.robjects.packages import importr
# Automatic conversion of arrays and dataframes
pandas2ri.activate()
numpy2ri.activate()
# Set random seed for reproducibility
ro.r('set.seed(123)')
# Ipython extenrsion for magix plotting
%load_ext rpy2.ipython
# R imports
importr('base')
importr('lavaan')
importr('psych')
importr('stats')
# Load data
file_name = "data/Data_EmotionalClarity.dat"
dat = pd.read_csv(file_name, sep="\t")
dat2 = dat.iloc[:, 1:7]
ro.globalenv['dat2'] = dat2
dat2.describe()
The rpy2.ipython extension is already loaded. To reload it, use:
%reload_ext rpy2.ipython
| item_1 | item_2 | item_3 | item_4 | item_5 | item_6 | |
|---|---|---|---|---|---|---|
| count | 238.000000 | 238.000000 | 238.000000 | 238.000000 | 238.000000 | 238.000000 |
| mean | 1.504005 | 1.422903 | 1.392156 | 1.304696 | 1.346359 | 1.305712 |
| std | 0.359060 | 0.368799 | 0.392299 | 0.407597 | 0.376931 | 0.383313 |
| min | 0.201307 | 0.046884 | 0.047837 | 0.038259 | 0.162969 | 0.061095 |
| 25% | 1.281558 | 1.185936 | 1.155308 | 1.052121 | 1.129379 | 1.075682 |
| 50% | 1.525186 | 1.422746 | 1.369148 | 1.289783 | 1.347554 | 1.286473 |
| 75% | 1.737302 | 1.651186 | 1.643980 | 1.551804 | 1.567729 | 1.557722 |
| max | 2.437029 | 2.403697 | 2.455821 | 2.441999 | 2.408026 | 2.244108 |
# Specify the model
ro.r("mete = 'eta=~ item_1 + 1*item_2 + 1*item_3 + 1*item_4 + 1*item_5 + 1*item_6'")
# Fit the model
ro.r('fitmete = sem(mete, data=dat2, meanstructure=TRUE)')
# Print the output of the model for interpretation
summary_fitmete = ro.r("summary(fitmete, fit.measures=TRUE, standardized=TRUE)")
print(summary_fitmete)
lavaan 0.6-19 ended normally after 12 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 13
Number of observations 238
Model Test User Model:
Test statistic 16.949
Degrees of freedom 14
P-value (Chi-square) 0.259
Model Test Baseline Model:
Test statistic 435.847
Degrees of freedom 15
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.993
Tucker-Lewis Index (TLI) 0.992
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -435.870
Loglikelihood unrestricted model (H1) -427.396
Akaike (AIC) 897.740
Bayesian (BIC) 942.880
Sample-size adjusted Bayesian (SABIC) 901.674
Root Mean Square Error of Approximation:
RMSEA 0.030
90 Percent confidence interval - lower 0.000
90 Percent confidence interval - upper 0.073
P-value H_0: RMSEA <= 0.050 0.737
P-value H_0: RMSEA >= 0.080 0.023
Standardized Root Mean Square Residual:
SRMR 0.053
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
eta =~
item_1 1.000 0.253 0.682
item_2 1.000 0.253 0.689
item_3 1.000 0.253 0.664
item_4 1.000 0.253 0.656
item_5 1.000 0.253 0.657
item_6 1.000 0.253 0.650
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item_1 1.504 0.024 62.432 0.000 1.504 4.047
.item_2 1.423 0.024 59.697 0.000 1.423 3.870
.item_3 1.392 0.025 56.265 0.000 1.392 3.647
.item_4 1.305 0.025 52.077 0.000 1.305 3.376
.item_5 1.346 0.025 53.815 0.000 1.346 3.488
.item_6 1.306 0.025 51.677 0.000 1.306 3.350
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item_1 0.074 0.008 9.254 0.000 0.074 0.535
.item_2 0.071 0.008 9.186 0.000 0.071 0.525
.item_3 0.081 0.009 9.410 0.000 0.081 0.559
.item_4 0.085 0.009 9.475 0.000 0.085 0.570
.item_5 0.085 0.009 9.468 0.000 0.085 0.569
.item_6 0.088 0.009 9.518 0.000 0.088 0.577
eta 0.064 0.007 9.004 0.000 1.000 1.000
You can see that the output looks very similar to that of the Tau Congeneric measurement model. Interpretation of the intercepts (Intercepts section) and the errors (Variances section) remains the same. The difference is that the loadings (Latent Variables section) are fixed to 1, implying equal discriminatory power for all items. Graphically, this results in parallel slopes across items. The fit indices are interpreted as before for the Tau Congeneric model.
Compare model fit#
Next, let’s compare the models we just fitted.
# recreate the tau-congeneric
ro.r("mtc = 'eta =~ item_1 + item_2 + item_3 + item_4 + item_5 + item_6'")
ro.r('fitmtc = sem(mtc, data=dat2, meanstructure=TRUE)')
# Essentially tau-equivalent model (notice the 1* in the model)
ro.r("mete = 'eta=~ item_1 + 1*item_2 + 1*item_3 + 1*item_4 + 1*item_5 + 1*item_6'")
ro.r('fitmete = sem(mete, data=dat2, meanstructure=TRUE)')
# Perform anova and print indexes
anova_mete_mtc = ro.r("anova(fitmete, fitmtc)")
print(anova_mete_mtc)
Chi-Squared Difference Test
Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
fitmtc 9 900.36 962.86 9.5683
fitmete 14 897.74 942.88 16.9488 7.3805 0.044726 5 0.1938
According to the BIC and AIC the more restricted Essentially tau-equivalent model has a better model fit compared to the Tau Congeneric measurement model (as lower values for AIC and BIC indicate better model fit). The \(\chi^2\) Test however suggests that there are no significant differences in model fit as indicated by p > .05. This result is not too surprising as we already saw quite similar loading estimates across items in the Tau Congeneric measurement model (see Tau-congeneric notebook). Therefore, restricting the loadings to equivalence isn’t too much of a deviation from the Tau Congeneric measurement model (which does not restrict the loadings), resulting in a insignificant difference in model fit.
Tau-Equivalent Model#
The Tau-equivalent measurement model has one more restriction compared to the Essentially tau-equivalent model. It assumes that
items are equivalent in their difficulty
items are equivalent in their discrimination power
items vary in their reliability
We therefore obtain only the error estimates (Variances section). A Latent Variables and an Intercepts section still appear, but all loadings and intercepts are fixed.
Fit the model#
Since we need to code multiple lines in R at once, we will code such syntax as follows:
# The model
ro.r("""
mte = 'eta =~ item_1 + 1*item_2 + 1*item_3 + 1*item_4 + 1*item_5 + 1*item_6
item_1 ~ a*1
item_2 ~ a*1
item_3 ~ a*1
item_4 ~ a*1
item_5 ~ a*1
item_6 ~ a*1'
""")
everythin within the “”” r_code_here “”” will be passed as R code.
# Specify the model
ro.r("""
mte = 'eta =~ item_1 + 1*item_2 + 1*item_3 + 1*item_4 + 1*item_5 + 1*item_6
item_1 ~ a*1
item_2 ~ a*1
item_3 ~ a*1
item_4 ~ a*1
item_5 ~ a*1
item_6 ~ a*1'
""")
# Fit the model
ro.r('fitmte <- sem(mte, data=dat2, meanstructure=TRUE, estimator="ML")')
# Print the output of the model for interpretation
summary_fitmte = ro.r("summary(fitmte, fit.measures=TRUE, standardized=TRUE)")
print(summary_fitmte)
lavaan 0.6-19 ended normally after 13 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 13
Number of equality constraints 5
Number of observations 238
Model Test User Model:
Test statistic 100.116
Degrees of freedom 19
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 435.847
Degrees of freedom 15
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.807
Tucker-Lewis Index (TLI) 0.848
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -477.454
Loglikelihood unrestricted model (H1) -427.396
Akaike (AIC) 970.908
Bayesian (BIC) 998.686
Sample-size adjusted Bayesian (SABIC) 973.329
Root Mean Square Error of Approximation:
RMSEA 0.134
90 Percent confidence interval - lower 0.109
90 Percent confidence interval - upper 0.160
P-value H_0: RMSEA <= 0.050 0.000
P-value H_0: RMSEA >= 0.080 1.000
Standardized Root Mean Square Residual:
SRMR 0.111
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
eta =~
item_1 1.000 0.252 0.637
item_2 1.000 0.252 0.683
item_3 1.000 0.252 0.663
item_4 1.000 0.252 0.639
item_5 1.000 0.252 0.655
item_6 1.000 0.252 0.634
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item_1 (a) 1.381 0.018 76.284 0.000 1.381 3.485
.item_2 (a) 1.381 0.018 76.284 0.000 1.381 3.736
.item_3 (a) 1.381 0.018 76.284 0.000 1.381 3.630
.item_4 (a) 1.381 0.018 76.284 0.000 1.381 3.496
.item_5 (a) 1.381 0.018 76.284 0.000 1.381 3.585
.item_6 (a) 1.381 0.018 76.284 0.000 1.381 3.471
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item_1 0.093 0.010 9.529 0.000 0.093 0.594
.item_2 0.073 0.008 9.138 0.000 0.073 0.534
.item_3 0.081 0.009 9.318 0.000 0.081 0.560
.item_4 0.092 0.010 9.513 0.000 0.092 0.592
.item_5 0.085 0.009 9.387 0.000 0.085 0.571
.item_6 0.095 0.010 9.548 0.000 0.095 0.598
eta 0.064 0.007 8.880 0.000 1.000 1.000
Again, the output looks very similar to the previous ones. The interpretation also is equivalent to before. The only difference is that the loadings (Latent Variables section) and the intercept (Intercepts section) are fixed, meaning that we assume that all items have the same discriminatory power and the same difficulty. Graphically speaking, this means that the slopes and the intercepts of the items are equivalent. The interpretation of the fit indices is analogous to the Tau Congeneric measurement model (see above).
Compare model fit#
As before, we can use the anova() function to compare the model fits.
# Perform anova and print indexes
anova_mete_mte = ro.r("anova(fitmete, fitmte)")
print(anova_mete_mte)
Chi-Squared Difference Test
Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
fitmete 14 897.74 942.88 16.949
fitmte 19 970.91 998.69 100.116 83.168 0.25629 5 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
In this comparison, the more restricted Tau-equivalent model has significantly worse fit compared to the Essentially tau-equivalent model as indicated by the significant differences in \(\chi^2\). Also AIC and BIC favor the more flexible model.