# Scale development

I am in the process of developing a scale that is supposed to measure construct X with 4 factors. I have already found support for a 4-factor model to the data using exploratory factor analysis.

Now, I'm thinking of performing a confirmatory factor analysis that specifies a one-factor hierarchical structure with 4 facets. Is that the right model to be specifying for my purposes? Perhaps I should also specify another model where there are only four factors?

In addition, how will you specify the formula for this model using lavaan in R? Is this the right way to write out the measurement model?

model <- '   Fac1 =~ Q1 + Q2 + Q4   Fac2 =~ Q12 + Q15 + Q16
Fac3 =~ Q25 + Q26 + Q27    Fac4 =~ Q31 + Q32 + Q35
Fac5 =~ 1*Fac1 + 1*Fac2 + 1*Fac3 + 1*Fac4
Fac5 ~~ Fac5 '


Now, I'm thinking of performing a confirmatory factor analysis that specifies a one-factor hierarchical structure with 4 facets. Is that the right model to be specifying for my purposes? Perhaps I should also specify another model where there are only four factors?

The 4-factor CFA with simple structure will be less constrained than a hierarchical CFA because you replace 4*3/2=6 factor covariances with 4 higher-order factor loadings. You can compare the nested models using a likelihood ratio test (with the lavTestLRT() function) of the $$H_0$$ that the higher-order factor model is sufficient to explain the correlations among the 4 lower-order factors.

Because you used the data to decide on the 4-factor structure, you should gather an independent sample to validate your original findings.

In addition, how will you specify the formula for this model using lavaan in R? Is this the right way to write out the measurement model?

Write your equations on separate lines, i.e., each operator can have multiple variables on the lefthand or righthand side, but there can only be one operator (e.g., =~) per line.

model <- '   Fac1 =~ Q1 + Q2 + Q4
Fac2 =~ Q12 + Q15 + Q16
Fac3 =~ Q25 + Q26 + Q27
Fac4 =~ Q31 + Q32 + Q35
Fac5 =~ 1*Fac1 + 1*Fac2 + 1*Fac3 + 1*Fac4
Fac5 ~~ Fac5 '


Note that you are fixing all higher-order loadings to 1, postulating the latent indicators are essentially tau-equivalent. That is also testable against the data, if you compare the model above (with a LRT) to one in which the loadings are free to differ across latent indicators (as they do across manifest indicators of first-order factors).