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}}

Also Read

• https://matistics.com/statistics-data-variables/
• https://matistics.com/descriptive-statistics/
• https://matistics.com/1-1-measurement-scale/
• https://matistics.com/point-biserial-correlation-and-biserial-correlation/
• https://matistics.com/2-0-statistics-distributions/
• https://matistics.com/1-2-statistics-population-and-sample/
• https://matistics.com/7-hypothesis-testing/
• https://matistics.com/8-errors-in-hypothesis-testing/
• https://matistics.com/9-one-tailed-hypothesis-test/
• https://matistics.com/10-statistical-power/
• https://matistics.com/11-t-statistics/
• https://matistics.com/12-hypothesis-t-test-one-sample/
• https://matistics.com/13-hypothesis-t-test-2-sample/
• https://matistics.com/14-t-test-for-two-related-samples/
• https://matistics.com/15-analysis-of-variance-anova-independent-measures/
• https://matistics.com/16-anova-repeated-measures/
• https://matistics.com/17-two-factor-anova-independent-measures/
• https://matistics.com/18-correlation/
• https://matistics.com/19-regression/
• https://matistics.com/20-chi-square-statistic/
• https://matistics.com/21-binomial-test/