psych (version 2.1.9)

glb.algebraic: Find the greatest lower bound to reliability.

Description

The greatest lower bound solves the ``educational testing problem". That is, what is the reliability of a test? (See guttman for a discussion of the problem). Although there are many estimates of a test reliability (Guttman, 1945) most underestimate the true reliability of a test.

For a given covariance matrix of items, C, the function finds the greatest lower bound to reliability of the total score using the csdp function from the Rcsdp package.

Usage

glb.algebraic(Cov, LoBounds = NULL, UpBounds = NULL)

Arguments

Cov

A p * p covariance matrix. Positive definiteness is not checked.

LoBounds

A vector \(l =(l_1, \dots, l_p)\) of length p with lower bounds to the diagonal elements \(x_i\). The default l=(0, . . . , 0) does not imply any constraint, because positive semidefiniteness of the matrix \(\tilde{ C} + Diag(x)\) implies \(0 \leq x_i\)

UpBounds

A vector u =(u1, . . . , up) of length p with upper bounds to the diagonal elements xi. The default is u = v.

Value

glb

The algebraic greatest lower bound

solution

The vector x of the solution of the semidefinite program. These are the elements on the diagonal of C.

status

Status of the solution. See documentation of csdp in package Rcsdp. If status is 2 or greater or equal than 4, no glb and solution is returned. If status is not 0, a warning message is generated.

Call

The calling string

Details

If C is a p * p-covariance matrix, v = diag(C) its diagonal (i. e. the vector of variances \(v_i = c_{ii}\)), \(\tilde { C} = C - Diag(v)\) is the covariance matrix with 0s substituted in the diagonal and x = the vector \(x_1, \dots ,x_n\) the educational testing problem is (see e. g., Al-Homidan 2008)

$$\sum_{i=1}^p x_i \rightarrow \min$$

s.t. $$\tilde{ C} + Diag(x) \geq 0$$(i.e. positive semidefinite) and \(x_i \leq v_i, i=1,\dots,p\). This is the same as minimizing the trace of the symmetric matrix $$\tilde{ C}+diag(x)=\left(\begin{array}{llll} x_1 & c_{12} & \ldots & c_{1p} \\ c_{12} & x_2 & \ldots & c_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ c_{1p} & c_{2p} & \ldots & x_p\\ \end{array}\right)$$

s. t. \(\tilde{ C} + Diag(x)\) is positive semidefinite and \(x_i \leq v_i\).

The greatest lower bound to reliability is $$\frac{\sum_{ij} \bar{c_{ij}} + \sum_i x_i}{\sum_{ij}c_{ij}}$$

Additionally, function glb.algebraic allows the user to change the upper bounds \(x_i \leq v_i\) to \(x_i \leq u_i\) and add lower bounds \(l_i \leq x_i\).

The greatest lower bound to reliability is applicable for tests with non-homogeneous items. It gives a sharp lower bound to the reliability of the total test score.

Caution: Though glb.algebraic gives exact lower bounds for exact covariance matrices, the estimates from empirical matrices may be strongly biased upwards for small and medium sample sizes.

glb.algebraic is wrapper for a call to function csdp of package Rcsdp (see its documentation).

If Cov is the covariance matrix of subtests/items with known lower bounds, rel, to their reliabilities (e. g. Cronbachs \(\alpha\)), LoBounds can be used to improve the lower bound to reliability by setting LoBounds <- rel*diag(Cov).

Changing UpBounds can be used to relax constraints \(x_i \leq v_i\) or to fix \(x_i\)-values by setting LoBounds[i] < -z; UpBounds[i] <- z.

References

Al-Homidan S (2008). Semidefinite programming for the educational testing problem. Central European Journal of Operations Research, 16:239-249.

Bentler PM (1972) A lower-bound method for the dimension-free measurement of internal consistency. Soc Sci Res 1:343-357.

Fletcher R (1981) A nonlinear programming problem in statistics (educational testing). SIAM J Sci Stat Comput 2:257-267.

Shapiro A, ten Berge JMF (2000). The asymptotic bias of minimum trace factor analysis, with applications to the greatest lower bound to reliability. Psychometrika, 65:413-425.

ten Berge, Socan G (2004). The greatest bound to reliability of a test and the hypothesis of unidimensionality. Psychometrika, 69:613-625.

See Also

For an alternative estimate of the greatest lower bound, see glb.fa. For multiple estimates of reliablity, see guttman

Examples

# NOT RUN {
Cv<-matrix(c(215, 64, 33, 22,
              64, 97, 57, 25,
              33, 57,103, 36,
              22, 25, 36, 77),ncol=4)

Cv                    # covariance matrix of a test with 4 subtests
Cr<-cov2cor(Cv)       # Correlation matrix of tests
if(!require(Rcsdp)) {print("Rcsdp must be installed to find the glb.algebraic")} else {
 glb.algebraic(Cv)     # glb of total score
glb.algebraic(Cr)      # glb of sum of standardized scores

 w<-c(1,2,2,1)         # glb of weighted total score
 glb.algebraic(diag(w) %*% Cv %*% diag(w))  
alphas <- c(0.8,0,0,0) # Internal consistency of first test is known

glb.algebraic(Cv,LoBounds=alphas*diag(Cv))

                      # Fix all diagonal elements to 1 but the first:

lb <- glb.algebraic(Cr,LoBounds=c(0,1,1,1),UpBounds=c(1,1,1,1))
lb$solution[1]        # should be the same as the squared mult. corr.
smc(Cr)[1] 
}                        

# }