Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: I: Algebraic lower bounds

Abstract

reliability bounds, coefficient alpha, non-homogeneous composites,

The asymptotic bias of minimum trace factor analysis, with applications to the greatest lower bound to reliability

“…An interior points algorithm can be downloaded from the internet, see Alizadeh, Haeberly, Nayakkankuppam, Overton & Schmieta (1997). Jackson and Agunwamba (1977) observed that the greatest lower bound to the reliability is the worst possible value of the ratio of true to observed variance, for any given covariance matrix, which is still compatible with the two standard assumptions of classical test theory in the context of lower bounds to reliability: zero covariance between the error parts of the items of the test, and zero covariance between the errors and the true scores. The glb has a direct link to Constrained Minimum Trace Factor Analysis (CMTFA, see Bentler & Woodward, 1980; Ten Berge, Snijders & Zegers, 1981; Shapiro, 1982).…”

“…Guttman (1945) proposed 6 lower bounds, one of which, λ 3 , has become famous as coefficient alpha (Cronbach, 1951). Jackson and Agunwamba (1977) and Ten Berge and Zegers (1978) introduced additional lower bounds. More importantly, Jackson and Agunwamba also demonstrated that there exists a greatest lower bound (glb) to the reliability of a test.…”

Beyond Alpha: Lower Bounds for the Reliability of Tests

“…From (4) it is clear that -given covariances Γ -the reliability is maximal if the trace of the error covariance matrix Γ e is minimal. As Jackson and Agunwamba (1977) remarked, the only restrictions that the classical model imposes on the elements of Γ ε are (1) 0 ≤ Γ eii ≤ Γ ii , and (2) Γ τ = Γ -Γ e is non-negative definite. under these restrictions can be located, the result would give the smallest possible value for the reliability given the covariance matrix Γ; this value is the greatest possible lower bound to the reliability.…”

“…under these restrictions can be located, the result would give the smallest possible value for the reliability given the covariance matrix Γ; this value is the greatest possible lower bound to the reliability. Jackson and Agunwamba (1977) and ten Berge, Snijders and Zegers (1981) described algorithms to find this largest lower bound; however, several well-known lower bounds are first put forth. Guttman (1945) introduced a series of lower bounds called λ 1 through λ 6 .…”

“…Otherwise λ 6 will tend to be less than or equal to λ 2 . Jackson and Agunwamba (1977) reported that λ 6 should be particularly advantageous in the fairly typical situation where the inter-item correlations are positive, moderate in size and somewhat similar. Jackson and Agunwamba (1977) added a seventh bound, called λ 7 : Jackson and Agunwamba remarked that this bound will be substantially better than λ 2 when there is considerable variation among the squared correlations.…”