0 C D C + D
Totals: A + C B + D N
With this coding:
phi = (BC - AD)/sqrt((A+B)(C+D)(A+C)(B+D)).
For this example we obtain:
phi = (25-100)/sqrt(15•15•15•15) = -75/225 = -0.33,
indicating a slight correlation. Please note that this is
the Pearson correlation coefficient, just calculated in a
simplified manner. However, the extreme values of |r| = 1
can only be realized when the two row totals are equal and
the two column totals are equal. There are thus ways
of computing the maximal values, if desired.
As product moment correlation coefficients,
the point biserial, phi, and Spearman rho are
all special cases of the Pearson.
However, there are correlation coefficients which are not.
Many of these are more properly called
measures of association, although
they are usually termed coefficients as well.
Three of these are similar to Phi
in that they are for nominal against nominal data,
but these do not require the data to be dichotomous.
One is called Pearson's contingency coefficient
and is termed C whereas the second is
called Cramer's V coefficient.
Both utilize the chi-square statistic
so will be deferred into the next lesson.
However, the Goodman and Kruskal lambda coefficient
does not, but is another commonly used association measure.
There are two flavors, one called symmetric when the researcher
does not specify which variable is the dependent variable and
one called asymmetric which is used when such a designation
is made. We leave the details to any good statistics book.
Another measure of association, the
biserial correlation coefficient,
termed
rb, is similar to the point biserial, but
pits quantitative data against ordinal data, but ordinal data
with an underlying continuity but measured discretely as two values
(dichotomous). An example might be test performance
vs
anxiety, where anxiety is designated as either high or low.
Presumably, anxiety can take on any value inbetween, perhaps beyond,
but it may be difficult to measure. We further assume that
anxiety is normally distributed.
The formula is very similar to the point-biserial but yet different:
rb = (Y1 - Y0)
• (pq/Y) / [sigma]Y,
where
Y0 and
Y1
are the
Y score means for data pairs with
an
x score of 0 and 1, respectively,
q = 1 -
p and
p are the proportions
of data pairs with
x scores of 0 and 1, respectively,
and [sigma]
Y
is the population standard deviation for the
y data,
and
Y is the height of the standardized normal distribution
at the point
z, where P(
z'<
z)=
q and
P(
z'>
z)=
p.
Since the factor involving
p,
q, and the height
is always greater than 1, the biserial is always greater than
the point-biserial.
The
tetrachoric correlation coefficient,
rtet,
is used when both variables are dichotomous, like the phi, but
we need also to be able to assume both variables really are
continuous and normally distributed. Thus it is applied to
ordinal
vs. ordinal data which has this characteristic.
Ranks are discrete so in this manner it differs from the Spearman.
The formula involves a trigonometric function called cosine.
The cosine function, in its simpliest form, is the ratio of
two side lengths in a right triangle, specifically, the
side adjacent to the reference angle divided by the
length of the hypotenuse. The formula is:
rtet =
cos (180/(1 + sqrt(BC/AD)).
The
rank-biserial correlation coefficient,
rrb,
is used for dichotomous nominal data
vs rankings (ordinal).
The formula is usually expressed as
rrb = 2 •(
Y1 -
Y0)/
n,
where
n is the number of data pairs, and
Y0 and
Y1,
again, are the
Y score means for data pairs with
an
x score of 0 and 1, respectively.
These
Y scores are ranks. This formula assumes
no tied ranks are present.
This may be the same as a Somer's D statistic for
which an online calculator is available.
It is often useful to measure a relationship irrespective of
if it is linear or not. The
eta correlation ratio
or
eta coefficient gives us that ability. This statistic
is interpretted similar to the Pearson, but can never be negative.
It utilizes equal width intervals and always exceeds |
r|.
However, even though
r is the same whether we
regress
y on
x or
x on
y,
two possible values for eta can be obtained.
Again, the calculation goes beyond what can be presented
here at the moment.