glb.algebraic function

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)

Arguments

• 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 $0 \le xi.$
• UpBounds: A vector u =(u1, . . . , up) of length p with upper bounds to the diagonal elements xi. The default is u = v.

Details

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)

$$\sum_{i=1}^p x_i \rightarrow \min(Sum i = 1 to p xi) -> min$$

s.t.

$$\tilde{ C} + Diag(x) \geq 0C0 + Diag(x) >= 0$$

(i.e. positive semidefinite) and $xi \le vi, i = 1 ..., p$. This is the same as minimizing the trace of the symmetric matrix

$$\tilde{ C}+diag(x)=\left(\begin{array}{llll}x_1 & c_{12} & \ldots & c_{1p} \\c_{12} & x_2 & \ldots & c_{2p} \\\vdots & \vdots & \ddots & \vdots \\c_{1p} & c_{2p} & \ldots & x_p\\\end{array}\right)C0 + Diag(x)$$

s. t. $C0 + Diag(x)$ is positive semidefinite and $xi \le vi$.

The greatest lower bound to reliability is

$$\frac{\sum_{ij} \bar{c_{ij}} + \sum_i x_i}{\sum_{ij}c_{ij}}(sum cij (i \ne j) + sum xi )/ sum cij$$

Additionally, function glb.algebraic allows the user to change the upper bounds $xi \le vi$ to $xi \le ui$ and add lower bounds $li \le xi$.

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 $\alpha$), 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 $xi \le vi$ or to fix $xi$-values by setting LoBounds[i] < -z; UpBounds[i] <- z.

Returns

• 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

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.

Author(s)

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

See Also

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

Examples

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] 
}