Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

518 CHAPTER 15 | Correlation
measures the effect size. Also, as we noted in Chapter 10, the values for t and r 2 are directly
related. In fact, either can be calculated from the other by the equations
t2
r 2 5
t 2 1 df
and t 2 5
r 2
s1 2 r 2 dydf
where df is the degrees of freedom for the t statistic.
However, you should note that r 2 is determined entirely by the size of the correlation,
whereas t is influenced by the size of the correlation and the size of the sample.
For example, a correlation of r = 0.30 produces r 2 = 0.09 (9%) no matter how large the
sample may be. On the other hand, a point-biserial correlation of r = 0.30 for a total
sample of 10 people (n = 5 in each group) produces a nonsignificant value of t = 0.791.
If the sample is increased to 50 people (n = 25 in each group), the same correlation produces
a significant t value of t = 4.75. Although t and r are related, they are measuring
different things.
■ The Phi-Coefficient
When both variables (X and Y) measured for each individual are dichotomous, the correlation
between the two variables is called the phi-coefficient. To compute phi (Φ), you follow
a two-step procedure.
1. Convert each of the dichotomous variables to numerical values by assigning a 0 to
one category and a 1 to the other category for each of the variables.
2. Use the regular Pearson formula with the converted scores.
This process is demonstrated in the following example.
EXAMPLE 15.15
A researcher is interested in examining the relationship between birth-order position and
personality. A random sample of n = 8 individuals is obtained, and each individual is classified
in terms of birth-order position as first-born or only child vs. later-born. Then each
individual’s personality is classified as either introvert or extrovert.
The original measurements are then converted to numerical values by the following
assignments:
Birth Order
Personality
1st or only child = 0 Introvert = 0
Later-born child = 1 Extrovert = 1
The original data and the converted scores are as follows:
Original Data
Converted Scores
Birth Order X Personality Y Birth Order X Personality Y
1st Introvert 0 0
3rd Extrovert 1 1
Only Extrovert 0 1
2nd Extrovert 1 1
4th Extrovert 1 1
2nd Introvert 1 0
Only Introvert 0 0
3rd Extrovert 1 1