Fig 1.
Quadratic dependence of the point-biserial correlation coefficient, rpb.
For the fixed value rpb = 0.2, there is a range for Cohen’s d and the sample size proportion, pA. This ambiguity complicates the interpretation of rpb as an effect size measure.
Fig 2.
Nonoverlap proportion and point-biserial correlation.
Theoretical curves and estimated values for point-biserial correlation, rpb, nonoverlap proportion, ρpb, and sample size adjusted correlation, rpbd, for simulated data with unequal sample sizes (NA : NB = 15000 : 500) and the difference between mean values, . Compared to rpbd, rpb is attenuated due to the confounding effect of the binomial sampling factor. A: Uniform unit width
distributions. B: Standard normal (σ = 1) distributions.
Fig 3.
Projective spaces for the representation of point-biserial correlation.
The point-biserial correlation coefficient, rpb, corresponds to the point on the positive half-circle,
, and the point
on the projective line,
. The homogeneous coordinates
for
correspond to points on the line through the origin. {pA, pB}: sample size proportions, d: Cohen’s d.
Table 1.
Homogeneous coordinates for Pearson correlation.
Fig 4.
Skewed distributions for NHC quality measures.
A. Histogram of ‘Average number of residents per day’ for 15341 nursing homes. B. Two-dimensional Gaussian kernel density estimate of the distribution of ‘Number of outpatient emergency department visits per 1000 long-stay resident days’ (‘Emergency visits’) versus ‘Number of hospitalizations per 1000 long-stay resident days’ (‘Hospitalizations’), with correlation r = 0.37.
Fig 5.
The relation between rpbd and ρpb in rCART.
These graphs display data obtained from association graphs for 380 pairs of quality measures, {(Qi, Qj)|i ≠ j}. A. rpbd effect size for rCART split versus correlation r(Qi, Qj). On average, the largest information gain is obtained when the response and partition variables are highly correlated. B. Correlation r(rpbd, ρpb) between effect size and r(Qi, Qj) for association graphs. There is good correlation between rpbd and ρpb in many cases, but there are exceptions.
Fig 6.
rCART association graphs for effect size.
A,B: ‘Hospitalizations’ response versus ‘Emergency visits’ partition variables, with correlation r(rpbd, ρpb) = 0.93. C,D: ‘Emergency visits’ response versus ‘Hospitalizations’ partition variables, with correlation r(rpbd, ρpb) = 0.49. Bar plot histograms are shown for ‘Emergency visits’ (B inset) and ‘Hospitalizations’ (D inset). rpb: point-biserial correlation coefficient, {pA, pB}: sample size proportions, rpbd: sample size corrected correlation coefficient, ρpb: nonoverlap proportion, (δ, μ): center of mass parameters .
Table 2.
rCART subnode parameters.
Fig 7.
Monte Carlo simulation of the distribution of stochastic effects for point-biserial variation.
2D histograms of MC distributions for (rpbd, μ) (A) and (rpbd, ρpb) (B) for ‘Emergency visits’ response with ‘Hospitalizations’ rCART split value, 3.3 (Table 2). The 1σ error bars for the rpbd histogram (A inset) serve as an indication of convergence for the simulation; the mean for the normal curve corresponds to the observed rpbd value, 0.398. rpbd: sample size corrected correlation, ρpb: nonoverlap proportion, μ: center of mass parameter , number of MC runs: 25, samples per MC run: 4000.