Find the greatest lower bound to reliability.
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.
glb.algebraic(Cov, LoBounds = NULL, UpBounds = NULL)
Cov |
A p * p covariance matrix. Positive definiteness is not checked. |
LoBounds |
A vector L = (l1 ... lp) 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 C0 + Diag(x) implies 0 ≤ xi. |
UpBounds |
A vector u =(u1, . . . , up) of length p with upper bounds to the diagonal elements xi. The default is u = v. |
If C is a p * p-covariance matrix, v = diag(C) its diagonal (i. e. the vector of variances v_i = c_{ii}), C0 = C - Diag(v) is the covariance matrix with 0s substituted in the diagonal and x = the vector (x1, . . . , xp) the educational testing problem is (see e. g., Al-Homidan 2008)
(Sum i = 1 to p xi) -> min
s.t.
C0 + Diag(x) >= 0
(i.e. positive semidefinite) and xi ≤ vi, i = 1 ..., p. This is the same as minimizing the trace of the symmetric matrix
C0 + Diag(x)
s. t. C0 + Diag(x) is positive semidefinite and xi ≤ vi.
The greatest lower bound to reliability is
(sum cij (i \ne j) + sum xi )/ sum cij
Additionally, function glb.algebraic allows the user to change the upper bounds xi ≤ vi to xi ≤ ui and add lower bounds li ≤ xi.
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 α), 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 xi ≤ vi or to fix xi-values by setting LoBounds[i] < -z; UpBounds[i] <- z.
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 |
Andreas Moltner
Center of Excellence for Assessment in Medicine/Baden-Wurttemberg
University of Heidelberg
William Revelle
Department of Psychology
Northwestern University Evanston, Illiniois
https://personality-project.org/revelle.html
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.
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]
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