The point biserial correlation is a statistical measure that quantifies the relationship between a continuous variable and a dichotomous variable. It is a special case of the Pearson product moment correlation coefficient, which is used to measure the strength and direction of the linear relationship between two continuous variables. Why is this so important? The point biserial correlation (\(r_{pb}\)) is the key to bridging the gap between continuous and dichotomous variables - a gap caused by the shift from one GLM model (MRC with continuous) to another (ANOVA with discrete or categorical predictors).
Here is the idea that makes this simple. A dichotomous variable (passed/failed, treatment/control, yes/no) looks categorical, but if we simply code it 0 and 1, we can drop it straight into a Pearson correlation as if it were a number. That is all the point-biserial correlation is: an ordinary Pearson \(r\) where one of the two variables happens to be a 0/1 indicator.
set.seed(14)
studied <- rbinom(50, 1, 0.5) # 0 = didn't study, 1 = studied
score <- 70 + 8 * studied + rnorm(50, 0, 6) # studiers score higher
r_pb <- cor(score, studied) # point-biserial: just Pearson r!
r_pb_asfac <- cor(score, as.numeric(factor(studied)) - 1)
c(point_biserial = round(r_pb, 3), same_thing = round(r_pb_asfac, 3))point_biserial same_thing
0.537 0.537 * point-biserial correlation: just Pearson r with a 0/1 variable.
CORRELATIONS /VARIABLES=score studied.using Statistics
cor(df.score, df.studied) # point-biserial: just Pearson r!No new formula, no new function; cor() did not even notice one variable was a dichotomy. The point-biserial correlation is just Pearson’s \(r\) computed with a 0/1 variable.
Now the payoff. The relationship between a continuous outcome and a two-group predictor can be expressed as (a) a point-biserial correlation, (b) a two-sample t-test, or (c) a simple regression, and they are the identical analysis. Every one gives the same t statistic and p-value:
# (a) test the point-biserial correlation
ct <- cor.test(score, studied)
# (b) the classic two-sample t-test (pooled variance)
tt <- t.test(score ~ studied, var.equal = TRUE)
# (c) regression with the 0/1 predictor
rg <- summary(lm(score ~ studied))
round(c(cor_test_t = ct$statistic,
t_test_t = abs(tt$statistic),
regression_t = abs(rg$coefficients[2, "t value"])), 4)cor_test_t.t t_test_t.t regression_t
4.4105 4.4105 4.4105 * (a) test the point-biserial correlation.
CORRELATIONS /VARIABLES=score studied.
* (b) the classic two-sample t-test (pooled variance).
T-TEST GROUPS=studied(0 1) /VARIABLES=score.
* (c) regression with the 0/1 predictor.
REGRESSION /STATISTICS COEFF R ANOVA /DEPENDENT score /METHOD=ENTER studied.using HypothesisTests, GLM
# (a) test the point-biserial correlation
CorrelationTest(df.score, df.studied)
# (b) the classic two-sample t-test (pooled variance)
EqualVarianceTTest(df.score[df.studied .== 0], df.score[df.studied .== 1])
# (c) regression with the 0/1 predictor
coeftable(lm(@formula(score ~ studied), df))Three “different” techniques, one t statistic. The correlation, the group comparison, and the regression are the same GLM. And the regression coefficient tells you something the correlation does not show directly: the raw difference between the two group means:
means <- tapply(score, studied, mean)
c(regression_slope = coef(lm(score ~ studied))[["studied"]],
mean_difference = as.numeric(means["1"] - means["0"]))regression_slope mean_difference
6.768483 6.768483 The slope on a 0/1 predictor is the difference in group means. (This is the same dummy-coding logic from the ANOVA chapter, previewed here with the simplest possible category.)
Because they are the same analysis, the correlation and the t statistic are tied by a simple formula, handy for converting between an effect size (\(r_{pb}\)) and a significance test (\(t\)):
\[r_{pb} = \frac{t}{\sqrt{t^2 + df}}\]
t_val <- abs(tt$statistic); df <- tt$parameter
r_from_t <- t_val / sqrt(t_val^2 + df)
c(r_pb = round(as.numeric(r_pb), 3), r_from_t = round(as.numeric(r_from_t), 3)) r_pb r_from_t
0.537 0.537 They match. This little equation is why you can always recover an effect size from a reported t (and its df), and vice versa, which is useful when reading other people’s papers.
MRC (regression with continuous predictors) and ANOVA (with categorical predictors) are often taught as if they were unrelated. The point-biserial correlation connects them: once you code a category as numbers, a group difference is a correlation and a t-test is a regression. Every categorical predictor is just a set of 0/1 indicators you can correlate. That idea is the key to the entire General Linear Model.
cor(), then confirm the t from cor.test(), t.test(var.equal = TRUE), and lm() are identical.We have the bridge and the engine. Now we build the machine in earnest, starting with the simplest continuous-predictor case: bivariate regression.