The phi coefficient is that system of correlation which is computed between two variables, where neither of them is available in a continuous measures and both of them are expressed in the form of natural or genuine dichotomies. This presentation slides describes the concept and procedures to do the computation of phi coefficient of correlation.

1. Phi Coefficient of Correlation

K.THIYAGU,
Assistant Professor,
Department of Education,
Central University of Kerala, Kasaragod

2. Phi coefficient correlation is
suitable for situations in
which neither of the two
variables can be measured
in terms of scores but both
the variables can be
separated in terms of two
categories. When both the
variables are dichotomous
in the same attributes, we
can use the phi coefficient
correlation
Phi Coefficient

Natural Dichotomy

3. The Phi Coefficient
is a measure of association between
two binary variables
(i.e. living/dead, black/white, success/failure).
It is also called
the Yule phi
or
Mean Square Contingency Coefficient
and is used for contingency tables when:
At least one variable is a nominal variable
&
Both variables are dichotomous variables.

4. The phi is a nonparametric statistic
used in cross-tabulated table data
where both variables are
dichotomous.
Dichotomous means that there are
only two possible values for a
variable.
As an example, the variable addressing life has only two levels, “alive” and “not alive” (or dead).
So if a public health department was researching the proportion of newborns born alive versus
born dead, each baby could be born alive or born dead; there are no other possibilities.
Natural Dichotomy

5. Phi Coefficient
Compute correlation / relationship
between two variables which are
genuinely dichotomousGenuine
Dichotomous
Same
attributes
Genuine
Dichotomous

Male Female
Married (A) (B)
Single (C) (D)
Natural Dichotomy

6. Yes No
Yes (A) (B)
No (C) (D)
No Yes
Yes (B) (A)
No (D) (C)
Favourable Unfavourable
Favourable (A) (B)
Unfavourable (C) (D)
If AD is greater than BC, then the correlation is Positive
If BC is greater than AD, then the correlation is negative.
Formula for Phi Coefficient ()
correlation is
Natural Dichotomy
Male Female
Alive (A) (B)
Not alive (C) (D)
Yes No
Male (A) (B)
Female (C) (D)

7. Two binary variables
are considered
positively associated
if most of the data falls
along the diagonal cells
i.e.,
a and d
are larger than
b and c.
In contrast,
two binary variables
are considered
negatively associated
if most of the data
falls off the diagonal.
i.e.,
a and d
are lesser than
b and c.
Yes No
Yes (A) (B)
No (C) (D) Natural Dichotomy

9. General rule of thumb for correlation
coefficients and you can use the same rule for
the Phi coefficient.
• -1.0 to -0.7 strong negative association.
• -0.7 to -0.3 weak negative association.
• -0.3 to +0.3 little or no association.
• +0.3 to +0.7 weak positive association.
• +0.7 to +1.0 strong positive association.
Interpretation of the
Phi coefficient.

10. Types of Correlation Coefficients
Correlation Coefficient Types of Scales
Pearson product-moment Both Scales - Interval (or) Ratio
Spearman rank-order Both Scales - Ordinal
Phi Both scales are Naturally Dichotomous (nominal)
Tetrachoric Both scales are Artificially Dichotomous (nominal)
Point-biserial
One scale Naturally Dichotomous (nominal),
one scale interval (or ratio)
Biserial
One scale Artificially Dichotomous (nominal),
one scale interval (or ratio)
Gamma One scale nominal, one scale ordinal

11. K.THIYAGU, Assistant
Professor, Department of Education,
Central University of Kerala, Kasaragod
Thank You