Six Lower limits of reliability coefficients are presented.
Usage
gutt(test)
Arguments
test
a matrix or a Dataframe that holds the test response data
Value
The six coefficients Guttman for the test.
Details
Let $S_j^2$ the variances over persons of the n items in the test, and
$S_t^2$ the variance over persons of the sum of the items.
The firt estimate $lambda_1$ can be computed from $L_1 = 1 - (sums_j^2/S_t^2)$
Let $C_2$ the sum of squares of the covariances between items, therefore is
the sum of $n(n-1)/2$ terms. The bound $lambda_2$ is computed by $L_2 = L_1 + (sqrtn/n-1 C_2/S_t^2)$
The third lower bound $lambda_3$ is a modification of $lambda_1$, it is computed
from the $L_3 = n/(n-1) L_1$
Fourth lower bound $lamda_4$ has been interpreted as the greatest split half reliability,
and requires that the test be scored as twohalves. It is calculated from
$L_4 = 2(1 - (s_a^2 + s_b^2)/s_t^2)$ where $S_a^2$ and $S_b^2$ are the respectives variances
of the two parts for the single trial.
For the fifth lower bound $lambda_5$, let $C_2j$ be the sum of the squares of the
covariances of item j with the remaining $n-1$ items, and let $barC_2$ be the largest of
the $C_2j$. Then the coefficient can be computed from $L_5 = L_1 + (2sqrtbarC_2)/S_t^2$
The final bound is based on multiple correlation, let $e_j^2$ be the variance of the errors
of estimate of item j from its linear multiple regression on the remaining n-1 items. Then
$lambda_6$ can be computed from $L_6 = 1 - (sume_j^2)/S_t^2$
References
Guttman, L. (1945). A basis for analyzing test-retest reliability. Psychometrika, 10(4), 255-282.