Spearman-Brown’s Predicted Reliability

The Spearman-Brown correction is a specific form of the Spearman-Brown predicted reliability formula. Suppose we have a test with reliability ρ. The reliability ρ′ of the test replicated n times is given by the formula

When n = 2, we have the Spearman-Brown correction for halves of equal length.

Another way to view the Spearman-Brown formula is as follows: suppose that the reliability for a test with m items is ρ, then a test with mn items will have reliability ρ′.

Now suppose that the reliability of a test is ρ = .2 and suppose we have 5 such units (i.e. n = 5), then

If n is increased to 10, then ρ′ = .714 and if n = 50, then ρ′ = .926. We see that the larger the number of units, the larger the reliability.

In fact, we can solve the above equation for n to determine how many units are required to achieve any specified reliability target (provided we know the value of ρ).

Thus, if our goal is to achieve reliability of .75 we would need

Thus, we would need a sample with 12 units to achieve the total reliability target.

Example 1: Suppose that the split-half reliability is .5 based on a test with 12 questions. What will the predicted reliability be if you have 24 (comparable) questions? How many questions to you need to have to get a predicted reliability of .80?

First, we solve the Spearman-Brown predicted reliability formula for ρ as follows:

ρ represents the reliability for one test unit (i.e. one question for this example).

Substituting the values ρ′ = .5 and n = 12, we get

Now we can calculate ρ′ using the Spearman-Brown predicted reliability formula when ρ = .0769 and n = 24.

To achieve a predicted reliability of .80, we use the following approach

Real Statistics Functions: The Real Statistics Resource Pack contains the following functions:

SB_PRED(m, rho, n) = Spearman-Brown predicted reliability based on m items when Spearman-Brown for n items is rho.

SB_SIZE(rho1, rho, n) = the number of items necessary to bring the Spearman-Brown predicted reliability up (or down) to rho1 from n items with Spearman-Brown of rho

We can get the answers for Example 1 by using the following formulas:

SB_PRED(24, .5, 12) = .667

SB_SIZE(.8, .5, 12) = 48

The Spearman-Brown predicted reliability formula takes the following form when the two halves are not equal in length.

Suppose we have n comparable test units, where the correlation between any two of these units is ρ, then the (predicted) reliability for the sum of n1 units with the sum of the other n2 units (where n = n1 + n2) is