Point Biserial Correlation Analysis

Asit

3 years ago

What is Point Biserial Correlation?

The point-biserial correlation is a special case of correlation in which one variable is continuous and the other variable is binary (dichotomous).

Dichotomous meaning:

A dichotomous scale is a two-point scale that presents options that are absolutely opposite each other. This type of response scale does not give the respondent an opportunity to be neutral on his answer to a question.

Examples:

Yes – No, True – False, smoker (yes/no), sex (male/female), 0-1 variable.

Examples: Point Biserial Correlation

Are women or men likely to earn more profit as Business entrepreneurs? Is there an association between gender and profit?
Does Vaccine A or Vaccine B improve immunity? Is there an association between the vaccine type and immunity level?
Do dogs react differently to yellow and red lights as food signals? Is there an association between the color and the reaction time?
Are women or men likely to earn more as doctors? Is there an association between gender and earnings as a doctor?

Assumptions: Point Biserial Correlation

The assumptions for Point-Biserial correlation include:
- - Continuous and Binary
  - Normally Distributed
  - No Outliers
  - Equal Variances

Normally Distributed meaning:

In statistics, a normal data distribution (frequency) graph must look like a bell-shaped curve.

Formula: Point Biserial Correlation

Find out the correlation r between –
- - A continuous random variable Y₀ and
  - A binary random variable Y₁ takes the values 0 and 1.

The point-biserial correlation coefficient r is calculated from these data as –

- - Y₀ = mean score for data pairs for x=0,
  - Y₁ = mean score for data pairs for x=1,
  - S_x = standard deviation for the entire test,
  - p_o = proportion of data pairs for x=0,
  - p₁ = proportion of data pairs for x=1,

Properties: Point-Biserial Correlation

The point-Biserial Correlation Coefficient measures the strength of the association of two variables in a single measure ranging from -1 to +1,
Where -1 indicates a perfect negative association, +1 indicates a perfect positive association and 0 indicates no association at all.
In place of Point-Biserial Correlation, Linear Regression Analysis is better suited for randomly independent variables.
A similar problem can also be answered with an independent sample t-test or Mann-Whitney-U or Kruskal-Wallis-H or Chi-Square. These tests fulfill the requirement of normally distributed variables and can analyze the dependency or causal relationship between an independent variable and dependent variables.

Definition: Biserial Correlation

It is the same as the point-biserial correlation coefficient. But this correlation is the true value of the association if samples are really normally distributed.
biserial correlation provides a better estimate.
It is also a correlation coefficient between Dichotomous + Continuous data (same as the Point-biserial correlation coefficient)
It is denoted by r_b
If there are two sets of data :

X (x_i = 0 or 1) – Dichotomous data

Y = {y₁, ……………, y_n} – Continuous data

Formula: Biserial correlation coefficient (r_b)

Where

- - - - m₀ = mean of y_i when X = 0
      - n₀ = number of elements in X which are at X=0
      - p₀ = n₀/n
      - p₁ = n₁/n
      - n₁ = the number of elements in X =1
      - n = n₀+n₁
      - m₁ = mean y_i when X = 1
      - s = population standard deviation of Y
      - y = height of the standard normal distribution at z, where P(z'<z) = q & P(z’>z) = p
      - In Excel y = NORM.S.DIST(NORM.S.INV(p₀),FALSE)

Relation: Point-biserial and biserial correlation

The biserial correlation coefficient can also be computed from the point-biserial correlation coefficient:

Biserial correlation coefficient = $\Large{r_{b} = \frac{r_{pb}\sqrt{p_{o}}p_{1}}{y}}$