Direction-Aware Sufficiency Condition Analysis

Provides a direction-aware interface for analysing bivariate sufficiency statements from empty-space frontier patterns. Logical sufficiency directions (high or low levels of a condition and outcome) are kept separate from the physical location of the empty corner in the scatter plot. Computation is delegated to version 5 of the 'NCA' package based on Dul (2016) , using the contraposition between necessity and sufficiency. Threshold tables are computed in actual units and converted by this package, so percentage, percentile and standard-deviation scales follow one stated reporting convention in every sufficiency direction. Includes tidy summaries, threshold rules, plots, random-data generation, permutation tests, and power analysis. An ordinary least-squares line can be drawn beside the frontier as a central-tendency reference; it is an average-effect summary and never a component of a sufficiency claim. An empty-space pattern alone does not establish causality or deterministic sufficiency.

SCAtools

SCAtools is an experimental R package for direction-aware Sufficiency Condition Analysis (SCA) using empty-space frontiers.

The package keeps two ideas separate:

1. the logical sufficiency statement (HH, LH, HL, or LL); and
2. the physical corner that must be empty in the original X-Y scatter plot.

The mapping follows contraposition:

X sufficient for Y  <=>  not-Y necessary for not-X

For example, High X -> High Y requires the lower-right corner (High X, Low Y) to be empty. It is equivalent to Low X being necessary for Low Y.

Source and issue tracker: https://github.com/youngchanresearcher/SCAtools

Installation

Install the released NCA engine and then the local SCAtools source package:

install.packages(c("NCA", "ggplot2"))
install.packages("SCAtools_0.4.3.tar.gz", repos = NULL, type = "source")

Basic use

library(SCAtools)

set.seed(42)
dat <- sca_random(
  n = 100,
  intercepts = 0,
  slopes = 1,
  direction = "HH"
)

fit <- sca_analysis(
  dat,
  x = "X",
  y = "Y",
  direction = "HH",
  ceilings = c("ce_fdh", "cr_fdh"),
  test.rep = 1000
)

fit
sca_table(fit)
sca_thresholds(fit, ceiling = "ce_fdh", inequality = "strict")
sca_plot(fit)
sca_test_plot(fit, ceiling = "ce_fdh")

# Optional diagnostics
sca_normalize(dat)
sca_outliers(dat, "X", "Y", direction = "HH")

For several X conditions, direction can be a single value or one value per condition. All directions in one model must refer to the same outcome level:

fit <- sca_analysis(
  my_data,
  x = c("resources", "constraints"),
  y = "performance",
  direction = c("HH", "LH")
)

Important interpretation limits

• direction is logical; empty_corner is geometric. They are deliberately not given the same number.
• The effect size is the fitted empty-zone area divided by the stated X-Y scope. It is scope- and frontier-dependent.
• A permutation p-value evaluates whether an empty zone this large is unusual after breaking the X-Y pairing. It does not prove a causal mechanism.
• Observational absence of counterexamples does not by itself establish deterministic sufficiency. Temporal order, design, measurement quality, scope, and out-of-sample validation still matter.
• A fitted threshold is best reported as an empirical frontier rule, not as a causal guarantee, unless the research design justifies that stronger claim.
• For continuous variables, strict inequalities are the exact contrapositive at the frontier. Inclusive inequalities are available only as an explicit reporting convention.
• A row marked no_threshold means no attainable condition value reaches that outcome level, and always_satisfied means every observation in scope already lies on the required side. These are the sufficiency readings of the engine's NN and NA markers, whose meanings invert under contraposition.

Reporting scales

Threshold tables can be reported in actual units, as a percentage of the range or of the maximum, as percentiles, or in standard deviations. Because actual values are retained, a fitted model can be re-expressed without refitting:

fit <- sca_analysis(dat, "X", "Y", direction = "LL", threshold.x = "percentile")

sca_thresholds(fit)                        # percentiles
sca_thresholds(fit, scale = "actual")      # original units
sca_thresholds(fit, scale = "sd")          # standard deviations

Two conventions are available, and they differ only for low-level directions:

Under absolute, 0 sits at the low end of every axis and the inequality carries the direction, so numbers are directly comparable across directions. Under directional, 0 sits at the least sufficient end: a low-level axis is mirrored, the inequality flips with it, and every rule reads as "more of the sufficient thing". sca_scales() lists what is available.

Implementation and attribution

SCAtools delegates frontier estimation, effect-size calculation, threshold estimation, permutation tests, random-data generation, and power analysis to NCA 5.0.2 or later. The wrapper maps sufficiency directions to NCA's physical corners and provides SCA-specific output and graphics. This avoids maintaining a divergent copy of the statistical engine.

Scale conversion is the one place where SCAtools does not delegate. Thresholds are always requested from the engine in actual units and converted here. The engine measures its percentage scales from the low end of the axis but mirrors its percentile scale according to the physical empty corner, and its out-of-range threshold cells use a third convention again. Because the physical corner does not correspond to the logical sufficiency direction, inheriting those conventions produced tables that could not be read consistently. Doing the conversion here means one stated convention governs both axes in all four directions.

NCA is distributed under GPL-3-or-later and is authored by Jan Dul and Govert Buijs. Cite the underlying method and software as appropriate, including:

• Dul, J. (2016). Necessary Condition Analysis (NCA): Logic and methodology of “necessary but not sufficient” causality. Organizational Research Methods, 19(1), 10-52. https://doi.org/10.1177/1094428115584005

The installed package also includes docs/sufficiency-logic.md and examples/four-directions.R for a fuller derivation and a runnable example.

citation("SCAtools") returns both this package and the NCA method article.

The reference line: OLS is not a frontier

reference = "ols" draws an ordinary least-squares regression of the outcome on the condition -- the line lm(y ~ x) returns -- beside the frontier, for comparison only:

fit <- sca_analysis(
  dat, x = "X", y = "Y", direction = "HH",
  ceilings = "ce_fdh",
  reference = "ols"
)

sca_reference(fit)   # intercept, slope, R-squared; no effect size

A regression line describes how the expected outcome moves with the condition. A sufficiency frontier describes which condition-outcome combinations are absent. A frontier is fixed by the most extreme observations, a regression line by the central tendency, so the two are different quantities and are reported separately: the reference line has no empty zone, no effect size, no permutation p value and no sufficiency threshold, and never appears as a row of sca_table().

Before 0.4.1 "ols" was the first entry of the default ceilings. It is not a default any more. Passing it in ceilings still works and moves it here with a warning; passing it alone is an error, because no empty space is left to estimate.

The Traditional Chinese version of this README is installed with the package:

file.show(system.file("docs", "README_zh-TW.md", package = "SCAtools"))