The Phi
Coefficient of Correlation
Correlate Two Binary Variables
In the introductory chapter, we
proposed a classification of data based on the distinction
between binary and continuous variables. When both variables
to be correlated are binary, the phi coefficient of
correlation, described by Udny Yule in his 1912 article in
the Journal of Royal Statistical Society, is the
correct statistics to use.
The numerical value of the phi
coefficient of correlation is identical to that obtained by
the Pearson product-moment coefficient of correlation. The
phi coefficient of correlation is a concept, not, in the
present computer era, a computational device. The derivation
of the phi coefficient from the Pearson product-moment
coefficient of correlation will be used here to gain
additional insight into the properties, assumptions, and
limitations of the product-moment coefficient of
correlation.
Yule's
Conceptualization of the Phi Coefficient
The phi coefficient can be derived from
the formula for the product-moment coefficient of
correlation for obtained scores
by considering several equivalencies
that are true only for binary data. The contingencies
surrounding use of the phi coefficient can be perhaps best
explicated for the example of a data matrix containing only
unique response patterns for binary variables X and Y,
Since all variables in the above table
are binary, their variances can be expressed using the pq
notation and the formula for the coefficient of correlation
can be rewritten as
The r symbol for the product-moment
correlation is subscripted with the Greek character phi to
stress that to use the above formula, the variables
correlated must be binary.
The
Four-Fold Tables
A graphical rendering of the binary
data matrix for the current example is shown as
Since in the case of two binary
variables, the test scores can occupy only four points on
the scatter-plot, it is possible to envelope the Cartesian
coordinates by a four-fold table, shown as
In the four-fold table,
each cell corresponds to a
specific response pattern. The Allen's cell contains the
(0,1)-response pattern, the Beth's cell the
(1,1)-response
pattern, the Cathy's cell the (0,0)-response pattern, and
the Debra's cell the (1,0)-response pattern. Let us borrow
initials of our subject's names and call these response
patterns A, B, C, and D, respectively.
To translate the phi coefficient from
the 'pq' notation to the 'ABCD' notation, consider data
matrix in the above diagram. Using the 'ABCD' notation,
indicating the frequencies
of a four-fold data table, the
phi coefficient of correlation can be rewritten from its 'pq'
notation
into the 'ABCD' notation as
Expressions in both the numerator and
the denominator of the above compound fraction can be put
under the common denominator as
Simplifying the above expression
results in
Since n equals the sum of all matrix
frequencies (i.e., n = A+B+C+D), the phi coefficient can be
written as
and, further simplified, this equation
can be expressed as
The above formula discloses the phi
coefficient of correlation in terms of a simple frequency
count of [0,1]
[1,1]
[0,0] and
[1,0] binary response patterns.
| | Row
Total |
| |
A | B
| A+B |
| C
|
D | C+D |
| Column
Total | A+C | B+D
| |
In terms of proportions instead of frequencies,
and adapting a convention to write frequencies in the upper
case and the proportions in the lower case letters, the
above formula translates to
Since the above proportions are also
the means of their corresponding response patterns, the phi
coefficient of correlation can be also written as
As witnessed by the presented algebraic derivation, the phi coefficient belongs to the family of Pearson's product moment coefficients of correlations.
About
Kittens Killing Rodents: Will They Give Quarter?
A classic heredity vs. environment
experiment involved kittens reared together with a rat or
with other kittens. When the kittens reached adulthood, they
were placed with a rodent and observed whether they killed
the rodent. This experiment (which actually involved 30
kittens) was conducted by Z.Y. Kuo, and described in his
classical article The
genesis of the cat's response to the rat, published in
the Journal of
Comparative Psychology, 1930, 11, 1-30.
Coding
Let's define the
rearing circumstances as the variable X and enter
0 for kittens reared with a rodent and 1 for kittens reared
with other kittens. Next, define the killing behavior as the
variable, Y,
and enter 1 when the cat killed the rodent and 0 when it did
not.
Combinations
There are two rearing
conditions (kitten reared with a rodent and kitten reared
with other kittens) and the variable killing behavior has
two levels (cat killed the rodent and cat did not kill the
rodent). Thus, there are four combinations
(2*2 = 4). The four combinations with respect to
Cartesian coordinates can be positioned as shown below.
| Y | Kill
the Rodent (1) | (0,
1) | (1,
1) |
| Did
not Kill the Rodent (0) | (0,
0) | (1,
0) |
| | | Rear
Together (0) | Rear
Apart (1) |
| | | X |
ABCD Notation
Record the response frequencies corresponding to response patterns (0,1), (1,1), (0,0), and
(1,0).
Pattern
|
Frequency
|
|
01
11
00
10
|
0
2
3
1
|
| |
|
They are usually
presented as a four-fold table
|
|
Reared
Together
(0)
|
Reared
Apart
(1)
|
|
Killed
the Rodent
(1)
|
0
|
2
|
|
Did
Not Kill the Rodent
(())
|
3
|
1
|
Note that no
cat reared with the rodent killed the rodent. For the cats
reared with other cats, two of them killed the rodent and
one did not.
To analyze this type of data, all we
have to do is to correctly orient the for-fold table with
respect to Cartesian coordinates.
and unravel it into the form suitable
to compute a coefficient of correlation.
| Rear
Together
(0) | Rear
Apart
(1) | Row
Total |
| Kill
the Rodent (1) | 0 | 2 | 2 |
| Did
not Kill the Rodent (0) | 3 |
1 | 4 |
| Column
Total | 3 | 3 | 6 |
The phi coefficient of correlation is
computed from the four-fold table of frequencies as [2(3) -
0(1)] / (3*3*2*4)1/2 which equals .71.
PQ Notation
Lets define the
rearing circumstances as the predictor variable X and enter
0 for kittens reared with a rodent and 1 for kittens reared
with other kittens. Next, define the killing behavior as the
criterion variable, Y,
and enter 1 when the cat killed the rodent and 0 when it did
not.
The phi coefficient of correlation is
computed from the
above table as
(.33 - (.50)(.33)) / (.5)(.47)
that is also .71. Note that pq1/2 =
Experiments like Kuo's are
relevant within the context of the nature-nurture
controversy.
Limitations
of the Phi Coefficient of Correlation
The assumptions of normality and
homogeneity can be violated when the categories are
extremely uneven, as in the case of proportions close to
.90, .95 or .10, .05. In these cases, the phi coefficient
can be markedly attenuated. The assumption of linearity
cannot be violated within the context of the phi coefficient
of correlation.
The
Chi Square Test of Statistical Significance
The phi
coefficient of correlation is associated with the chi square
test of significance. Is there any relation between the type
of the
rearing circumstances and their killing behavior?
Hypotheses
The null
hypothesis states that there is no relation between the rear
conditions and the killing behavior. The
alternative hypothesis states that there is a relation
between the rear conditions and the killing behavior.
The
Chi-Square Test
The phi square and chi square coefficients are related. The chi square can be expressed using the square of the phi coefficient as
The
Greek symbol on the left side of the above formula is called
chi and the index is called the chi square. To understand the
nature of the chi square distribution, let us first review the family of the gamma distributions.
Kittens
Killing Rodents Revisited
In
the chapter on the phi coefficient of correlation we
described a classic heredity vs. environment experiment
involved kittens reared together with a rat or with other
kittens. When the kittens reached adulthood, they were
placed with a rodent and observed whether they killed the
rodent.
Coding
The 1s and 0s of the parent vector X indexes the
rearing circumstances. The criterion variable Y indexes
whether the cat killed the rat or it did not.
The
regression analysis for the example is shown in the table
below.
Compute
the Chi-Square Value
The coefficient of determination
is computed as .11
/ .22 which equals .5. Using the chi square test of statistical
significance,
the chi-square equals .5 (6) that is 3.0.
Note
that the t-square ratio is characteristic of the Fisherian
conceptualization of statistical inference with the degrees
of freedom used throughout all computations leading to the
t-square ratio. The chi-square ratio is characteristics of
the Pearsonian conceptualization of statistical inference
where the degrees of freedom are introduced only during the
last phase of the computation of probability associated with
the chi-square.
Degrees
of Freedom
The
degrees of freedom for a contingency table can be computed as
(number of rows - 1)(number of columns - 1). Thus, for a
two-by-two table, (2-1)(2-1) = 1.
Visualize
the Chi-Square Distribution
Locate
the position of the obtained chi square value of 3 in
the chi -square distribution with one degree of freedom.
The
associated probability is .08. The observed (exact)
probability is larger than .05. We can not reject the null
hypothesis. In conclusion, there is no significant
differences between the two rearing circumstances with
respect to their proportion of killing a rodent.
Strength
of the Relationship
The square of the phi coefficient of correlation can be written in terms of the chi square as
as
The phi and chi square coefficients indicate jointly the strength and the significance of a relationship.
For
the example, the strength of the relationship can be
computed as 3 / 6, equal to .50. The phi correlation equals
.71.
Report
the Result
A
two-way contingency table analysis was conducted to evaluate
whether there was any relation between the type
of the
rearing circumstances and the killing behavior. The two variables were
the
rearing circumstances with two levels (kittens reared together with a rat
or with other kittens) and the killing behavior with two levels (the cat killed the rat or
it did not). The rearing circumstances and the killing behavior
(for this arbitrary example) were not significantly related, Pearson
(1,
n = 6) = 3, p > .05,
= .71.
The
Family of Pearson's Coefficients of Correlation
The
correlation between continuous variables is captured by the
product-moment correlation coefficient. The correlation
between continuous and binary variables can be computed by
the point biserial correlation. The correlation between
binary variables is conceptualized by phi correlation
coefficient.
All the coefficients of Pearson's family of
product-moment coefficients are algebraically equivalent and
give identical numerical results. There is no need for a
computer program dedicated to compute the point biserial
coefficient of correlation. Any computer program calculating
the Pearson's product-moment coefficient of correlation will
also correctly calculate the point biserial coefficient of
correlation. Together with the phi coefficient of
correlation, the point biserial belongs to the family of
Pearson's coefficients of correlation.
The importance of the point biserial as
a labor saving solution for obtaining a correlation
coefficient for those special cases where one variable is
binary and the other is continuous, has diminished with the
advent of computerization. However, the point biserial
correlation is of considerable theoretical importance in
that it provides a theoretical basis for the translation
between proportions of variance accounted for by the
coefficients of determination and alienation and the
variances of their constituent variables.
Understanding the
principles behind the computation of the point biserial
coefficient is also essential for understanding the theory
behind the tests of statistical significance. Also, the
different renderings of the correlation coefficient are
indispensable for understanding the relationship between
correlational measures of relationships and tests of
statistical significance. The phi
coefficient of correlation is associated with the chi square
test of significance. The point biserial
coefficient is
associated with the t-test.
Summary
The family of Pearson's coefficients of
correlation can be summarized as
with all coefficients in the above
table being different algebraic renderings of the product
moment coefficient, returning identical numerical values .
