Base SAS 9.1.3 Procedures Guide - Acsu Buffalo

24 Chapter 1. The CORR Procedure
Assuming further that the two samples are from populations with identical correlation,
a combined correlation estimate can be computed. The weighted average of the
corresponding z values is
¯z = (n 1 − 3)z 1 + (n 2 − 3)z 2
n 1 + n 2 − 6
where the weights are inversely proportional to their variances.
Thus, a combined correlation estimate is ¯r = tanh(¯z) and V (¯z) = 1/(n 1 + n 2 − 6).
See Example 1.4 for further illustrations of these applications.
Note that this approach can be extended to include more than two samples.
Cronbach’s Coefficient Alpha
Analyzing latent constructs such as job satisfaction, motor ability, sensory recognition,
or customer satisfaction requires instruments to accurately measure the constructs.
Interrelated items may be summed to obtain an overall score for each participant.
Cronbach’s coefficient alpha estimates the reliability of this type of scale
by determining the internal consistency of the test or the average correlation of items
within the test (Cronbach 1951).
When a value is recorded, the observed value contains some degree of measurement
error. Two sets of measurements on the same variable for the same individual may
not have identical values. However, repeated measurements for a series of individuals
will show some consistency. Reliability measures internal consistency from one set
of measurements to another. The observed value Y is divided into two components,
a true value T and a measurement error E. The measurement error is assumed to be
independent of the true value, that is,
Y = T + E Cov(T, E) = 0
The reliability coefficient of a measurement test is defined as the squared correlation
between the observed value Y and the true value T , that is,
r 2 (Y, T ) =
Cov(Y, T )2 V (T )2
=
V (Y )V (T ) V (Y )V (T ) = V (T )
V (Y )
which is the proportion of the observed variance due to true differences among individuals
in the sample. If Y is the sum of several observed variables measuring the
same feature, you can estimate V (T ). Cronbach’s coefficient alpha, based on a lower
bound for V (T ), is an estimate of the reliability coefficient.
Suppose p variables are used with Y j = T j + E j for j = 1, 2, . . . , p, where Y j is the
observed value, T j is the true value, and E j is the measurement error. The measurement
errors (E j ) are independent of the true values (T j ) and are also independent of