Point Biserial Correlation Analysis

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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).

Point Biserial Correlation Analysis

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.

Point Biserial Correlation Analysis

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 –

Point Biserial Correlation Analysis

      • Y0 = mean score for data pairs for x=0,
      • Y1 = mean score for data pairs for x=1,
      • Sx = standard deviation for the entire test,
      • po = 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  rb
  • If there are two sets  of data :

X  (xi =  0     or     1)       – Dichotomous data

Y = {y1, ……………, yn} – Continuous data

Formula: Biserial correlation coefficient  (rb)

Formula

Where

          • m0 = mean of yi 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 yi 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}}

 

 

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