unidim: Several indices of the unidimensionality of a set of variables.

Description

There are a variety of ways of assessing whether a set of items measures one latent trait. unidim is just one more way. If a one factor model holds in the data, then the factor analytic decomposition F implies that FF' should reproduce the correlations with communalities along the diagonal. In this case, the fit FF' should be identical to the correlation matrix minus the uniquenesses. unidim is just the ratio of these two estimates. The higher it is, the more the evidence for unidimensionality. A number of alternative statistics are estimated.

Usage

unidim(x, keys.list = NULL,cor="cor",correct=.5, check.keys = TRUE)

Arguments

An input matrix or data frame. If x is not a correlation matrix, then the correlations are found.

keys.list

If specified, then a number of scales can be tested at once. (See scoreItems for a similar procedure.)

cor

By default, find the Pearson correlation, other options are "spearman","kendall","tet"(for tetrachoric), "poly" (for polychoric), or "mixed"

correct

If using "tetrachoric" or "polychoric" correlations, should we correct empty cells for continuity, and if so, by how much. (See tetrachoric for a discussion of this correction)

check.keys

If TRUE, then items will be keyed based upon their loadings on the first factor. Automatically done if key.list is NULL.

Value

The estimate of unidimensionality which is just the product of

av.r.fit

The fit of the average r to the correlation matrix

fa.fit

The off diagonal fit from fa

alpha

Standardized alpha of the keyed items (after appropriate reversals)

av.r

The average interitem correlation of the keyed items.

median.r

The median value of the iteritem correlations of the keyed items.

Unidim.A

The unidimensional criterion when items are keyed in positive direction.

Unidim

The raw value of the unidimensional criterion

raw.model

The ratio of the FF' model to the sum(R)

adj.model

The ratio of the FF' model to the sum(R) when items are flipped.

itemTotalThe ratio of the sum(R - uniqueness)/sum(R) itemTotal.ASame ratio with flipped items

Details

This is set of exploratory indices that are still under development. A number of test cases suggest that u provides high values when the data are in fact unidimensional, low values when they are not.

The logic is deceptively simple: Unidimensionality implies that a one factor model of the data fits the covariances of the data. If this is the case, then factor model implies R = FF' + U2 will have residuals of 0. Similarly, this also implies that the observed correlations will equal the model. Thus, the sum of the observed correlations (with the diagonal replaced by the communalities) should match the factor model. Compare these two models: R - U2 versus FF'. This is the unidim.A estimate.

Also reported, and probably better, is the fit of the one factor model to the correlations. This is merely the sumsq of the residual correlations/sumsq of the original correlations. When the factor model is perfect, this will be 1.

This works well, but when some of the loadings are very small, even though 1 factor is correct, it is probably not a good idea to think of the items as forming a unidimensional scale.

An alternative model (the av.r.fit statistic) considers the residuals found by subtracting the average correlation from the observed correlations. This will achieve a maximum if the item covariances are all identical (a tau equivalent model).

The product of fa.fit and av.r.fit is the measure of unidimensionality, u.

All of the results are reported in the uni list.

Examples

Run this code

# NOT RUN {
#test the unidimensionality of the five factors of the bfi data set.


unidim(psychTools::bfi,psychTools::bfi.keys) 
unidim(psychTools::ability,psychTools::ability.keys)
#Try a known 3 factor structure
x <- sim.minor(nfact=3,bipolar=FALSE)   #this makes all the items positive
unidim(x$model) 
keys.list <- list(first =c(1:4),second = 5:8,third=9:12,all=1:12)
unidim(x$model,keys.list)

x <- sim.minor(nfact=3)
unidim(x$model,keys.list)   #we flip the negative items 

#what about a hierarchical model?
H <- sim.hierarchical()  # by default, a nice hierarchical model
H.keys <- list(First = paste0("V",1:3),Second=paste0("V",4:6),Third=paste0("V",7:9),
  All = paste0("V",1:9))
unidim(H,H.keys)



# }