Introduction to p3d

Introduction

The p-value is a continuous measure of statistical evidence, yet in practice it is often reduced to a binary decision: p < 0.05 is “significant”, p ≥ 0.05 is “not significant”. This dichotomous thinking contributes to the reproducibility crisis in science.

The p3d package visualizes the p-value of a single-sample t-test as a three-dimensional surface over two parameters:

• Effect size (d, Cohen’s d)
• Sample size (n)

This vignette demonstrates the core functionality of the package.

The p-value surface

The central function of the package is p_surface(). Given a range of effect sizes and sample sizes, it computes the p-value for each combination:

surface <- p_surface(
  d = seq(-2, 2, 0.1),
  n = seq(5, 200, 5)
)

str(surface, max.level = 1)
#> List of 6
#>  $ d     : num [1:41] -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 ...
#>  $ n     : num [1:40] 5 10 15 20 25 30 35 40 45 50 ...
#>  $ p     : num [1:41, 1:40] 0.0111 0.0132 0.0158 0.0191 0.0232 ...
#>  $ d_crit: num [1:40] 1.242 0.715 0.554 0.468 0.413 ...
#>  $ alpha : num 0.05
#>  $ grid  :'data.frame':  1640 obs. of  5 variables:
#>   ..- attr(*, "out.attrs")=List of 2

The result is a list with several components:

• d: the vector of effect sizes
• n: the vector of sample sizes
• p: a matrix of p-values (dimensions length(d) × length(n))
• d_crit: critical effect sizes at α = 0.05 for each n
• alpha: the significance level
• grid: a tidy data frame with all combinations

surface$p[surface$d == 0.5, surface$n == 30]
#> [1] 0.01043739

surface$d_crit[surface$n == 30]
#> [1] 0.3734061

Static 3D visualization

The simplest way to visualize the surface is plot_p_static(), which uses base R graphics and has no external dependencies:

plot_p_static(surface)

This is ideal for PDF output, journal figures, and printed reports.

Interactive 3D visualization

For interactive exploration, use plot_p_surface() with the plotly engine:

plot_p_surface(surface, engine = "plotly")

Alternatively, the rgl engine provides a native R window:

plot_p_surface(surface, engine = "rgl")

Both engines allow rotation, zooming, and panning.

2D comparison

For comparison with existing 2D packages such as pvaluefunctions, use p_curve_2d():

curve_30 <- p_curve_2d(d = seq(-2, 2, 0.1), n = 30)
head(curve_30)
#>      d            p significant
#> 1 -2.0 8.021139e-12        TRUE
#> 2 -1.9 2.650857e-11        TRUE
#> 3 -1.8 9.099432e-11        TRUE
#> 4 -1.7 3.247269e-10        TRUE
#> 5 -1.6 1.205427e-09        TRUE
#> 6 -1.5 4.654771e-09        TRUE

The 2D curve is mathematically equivalent to a slice of the 3D surface at fixed n.

Real-data example

Let’s apply p3d to a real dataset. The mtcars dataset contains fuel consumption (mpg) for 32 automobiles. Suppose we want to test whether the mean mpg is different from 20:

x <- mtcars$mpg
d_obs <- (mean(x) - 20) / sd(x)
n_obs <- length(x)

cat("Observed effect size:", round(d_obs, 4), "\n")
#> Observed effect size: 0.015
cat("Sample size:", n_obs, "\n")
#> Sample size: 32

The p-value for this single observation is:

2 * (1 - pt(abs(d_obs) * sqrt(n_obs), df = n_obs - 1))
#> [1] 0.9327606

By placing this observation on the 3D surface, we can see how the same effect size would behave at different sample sizes:

surface_real <- p_surface(
  d = seq(-0.5, 0.5, 0.01),
  n = seq(5, 200, 5)
)

plot_p_static(surface_real, theta = 30, phi = 20)

Conclusion

The p3d package provides a three-dimensional perspective on the p-value. By visualizing the p-value simultaneously across effect size and sample size, it reveals:

1. The continuous nature of the p-value (it is not a binary switch)
2. The moderating effect of sample size on statistical significance
3. The interaction between effect size and sample size

This complements traditional 2D visualizations and helps weaken the dichotomous thinking that plagues statistical practice.

References

• Helske, J., Helske, S., Cooper, M., Ynnerman, A., & Besançon, L. (2021). Can Visualization Alleviate Dichotomous Thinking? Effects of Visual Representations on the Cliff Effect. IEEE Transactions on Visualization and Computer Graphics, 27(8), 3496-3509.

• Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum Associates.