Package {SCAtools}

SCAtools: Direction-Aware Sufficient Condition Analysis

Description

Tests whether a high or low level of a condition is enough to guarantee a high or low level of an outcome. Such a claim predicts that one corner of the condition-outcome scatter plot is empty of observations. The package estimates the boundary of that empty corner, its size as a share of the scope (the effect size), a permutation test against random pairing, and a threshold table giving the condition level that guarantees each outcome level, in actual units, percentages, percentiles or standard deviations. Each of the four directions (high or low condition, high or low outcome) is stated as a claim and mapped to its empty corner. Because a condition can rule out low outcomes while barely moving the average, results can be set beside ordinary least-squares and median regression slopes, showing whether the data hold an average effect, a sufficiency effect or both. Also provides plots, influential-case diagnostics, scope-sensitivity and resampling checks, random-data generation and power analysis for planning studies. Frontiers are estimated with the 'NCA' package.

The method

A condition is sufficient for an outcome when the condition guarantees the outcome. In a scatter plot of condition against outcome, such a claim rules out one corner: high X sufficient for high Y, for instance, leaves no cases with high X and low Y. Necessary condition analysis (NCA; Dul, 2016) estimates the empty corner of a necessity claim. By contraposition a sufficiency claim is a necessity claim about the opposite levels – high X sufficient for high Y is low X necessary for low Y – so its empty corner can be estimated with the same frontier techniques. SCAtools states the four sufficiency directions as claims, maps each to the corner it leaves empty, and reports the effect size, permutation test and sufficiency thresholds in the vocabulary of condition analysis in degree, following Chan (in press), From Necessity to Sufficiency and Necessity-and-Sufficiency: Extending Condition Analysis in Degree.

What it is for

• Finding the condition level that guarantees each outcome level, reported in actual units, percentages, percentiles or standard deviations (sca_thresholds()).

• Measuring an effect that regression does not show: a condition can rule out low outcomes while barely moving the average. sca_compare() reports the two side by side.

• Comparing several conditions, each with its own direction, in one table (sca_table()).

• Planning a study: sca_power() estimates the sample size a sufficiency effect of a given size needs.

How to use it

1. Fit the claim with sca_analysis(), naming the condition, the outcome and the direction, and the scope if theory gives one. Use test.rep for the permutation test; sca() runs 1000 permutations and draws the plot in one step.

2. Read the results with sca_table(), sca_thresholds() and sca_plot(); sca_output() prints and draws them together.

3. Set the result beside regression with sca_compare() and sca_locationshift().

4. Check it with sca_test_plot(), sca_scope(), sca_scope_sensitivity(), sca_resample() and sca_outliers().

Reading the result

A fitted model describes the empty-space pattern in the sample under the stated scope and frontier technique, and its thresholds are empirical rules for the population and design studied. Claims beyond the sample – that the condition causes the outcome, that future cases will follow the frontier, or that the pattern holds once common causes are controlled – rest on the research design. Each direction predicts a single empty corner, which assumes the relationship is monotone, so plot the data before choosing one.

Regression as a reference

An ordinary least-squares or median regression line can be drawn beside the frontier as a central-tendency reference. It summarises the average effect and is never a component of a sufficiency claim. Chan (in press), Mean Effects Are Not Sufficiency Effects: How Sufficient Condition Analysis Complements Regression Analysis, sets out why the two kinds of effect are reported side by side; sca_compare() and sca_four_configurations follow it.

The engine

Frontier estimation, effect sizes, threshold estimation, permutation tests, random-data generation and power analysis are delegated to the NCA package by Jan Dul and Govert Buijs. SCAtools contributes the mapping from a logical sufficiency direction to the physically empty corner, SCA-specific tables and labels, direction-aware reporting scales, and the shared vocabulary of condition analysis in degree that the NSCA package also uses.

Author(s)

Maintainer: Young Chan youngchanresearcher@gmail.com

Authors:

• Young Chan youngchanresearcher@gmail.com

References

Chan, Y. (in press). From Necessity to Sufficiency and Necessity-and-Sufficiency: Extending Condition Analysis in Degree. University of Leeds.

Chan, Y. (in press). Mean Effects Are Not Sufficiency Effects: How Sufficient Condition Analysis Complements Regression Analysis. University of Leeds.

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

See Also

Useful links:

• https://github.com/youngchanresearcher/SCAtools

• Report bugs at https://github.com/youngchanresearcher/SCAtools/issues

Quick sufficient condition analysis

Description

This convenience wrapper is identical to sca_analysis() but marks the result for automatic plotting when it is printed interactively, and runs a permutation test by default.

Usage

sca(
  data,
  x,
  y,
  direction = "HH",
  ceilings = c("ce_fdh", "cr_fdh"),
  reference = NULL,
  test.rep = 1000,
  test.seed = NULL
)

Arguments

Details

test.rep defaults to zero in sca_analysis(), where every inferential choice is made deliberately, and to 1000 here, where a quick result should still carry statistical evidence. NSCA::nsca() takes the same default for the same reason.

It runs that test whether or not you asked for one, so pass test.seed to pin the p-value. An unseeded permutation test draws from the clock and reports a different number on every run, which is not a number to put in a paper; the Settings block says which of the two you got, and says so again beneath the results when the p-values are unpinned.

Value

An object of class sca_result.

Examples

# sca() runs the default 1000 permutations, which takes longer than
# an example should; sca_analysis() runs none unless asked.
iv <- subset(sca_four_configurations, configuration == "IV")
fit <- sca(iv[seq(1, nrow(iv), by = 5), ], x = "X", y = "Y",
           direction = "HH", test.seed = 1)
fit

Sufficient condition analysis

Description

Fits empty-space frontiers for directional sufficiency statements. The logical statement is converted to the physically empty scatter-plot corner by contraposition and passed to NCA::nca_analysis().

Usage

sca_analysis(
  data,
  x,
  y,
  direction = "HH",
  ceilings = c("ce_fdh", "cr_fdh"),
  reference = NULL,
  custom = NULL,
  scope = NULL,
  threshold.x = "percentage.range",
  threshold.y = "percentage.range",
  convention = c("absolute", "directional"),
  steps = 10,
  step.size = NULL,
  cutoff = 0,
  qr.tau = 0.95,
  reference.tau = 0.5,
  test.rep = 0,
  test.p_confidence = 0.95,
  test.p_threshold = 0.05,
  test.seed = NULL,
  relevance = NULL,
  na.action = c("pairwise", "listwise")
)

Arguments

Details

Threshold values are always requested from the engine in actual units and converted to the reporting scale by SCAtools, so that a single, stated convention governs every axis and every direction. See sca_scales() for the available scales and sca_thresholds() for the resulting table.

Value

An object of class sca_result containing the NCA engine results and SCA-specific direction metadata.

Examples

iv <- subset(sca_four_configurations, configuration == "IV")
fit <- sca_analysis(iv, x = "X", y = "Y", direction = "HH",
                    scope = c(0, 100, 0, 100))
fit

# Report the condition axis as percentiles instead
sca_thresholds(fit, scale = "percentile")

# sca_table() shows the same model with a permutation test added.

Whether the boundary of a corner is a ceiling or a floor

Description

Boundary type follows the empty corner and nothing else. A corner at the top means the frontier bounds the data from above, which is a ceiling; at the bottom it is a floor. For sufficiency this makes the boundary a floor whenever the outcome level is high and a ceiling whenever it is low, which is why the technique names CE-FDH and CR-FDH keep the word "ceiling" even where the fitted boundary is a floor.

Usage

sca_boundary_type(corner)

Arguments

Value

A character vector: "ceiling" or "floor".

See Also

sca_corner_location(), sca_terms()

Examples

sca_boundary_type(1:4)

Validate a set of directions

Description

Accepts one direction for every condition, or a single direction to be applied to all of them, and refuses anything else by name.

Usage

sca_check_direction(direction, n = length(direction))

Arguments

Value

The validated directions, recycled to length n.

See Also

sca_corner(), sca_corner_map()

Examples

sca_check_direction("hh ")
sca_check_direction("HH", n = 3)

Average effect beside sufficiency in degree

Description

Reports the ordinary least-squares coefficient and the sufficiency effect size for the same relationship, and classifies the pair into the four configurations of the condition analysis in degree framework.

Usage

sca_compare(
  model,
  x = NULL,
  ceiling = NULL,
  beta_cut = .sca_beta_cut_default,
  d_cut = NULL
)

Arguments

Value

A data frame with one row per condition.

What is being compared

beta is the standardised regression coefficient: how far the expected outcome moves with the condition. effect_size is the share of the declared scope left empty in the corner the sufficiency claim forbids. Neither implies the other. A relationship can have a strong average effect and no empty corner, or a clean empty corner and almost no slope.

The four configurations

I, both

The average effect and the empty corner both reach their cut. Two separate findings about one relationship.

II, sufficiency only

A flat slope with a largely empty corner. Typically a level shift or a threshold: beyond some level of the condition the low-outcome cases stop occurring, without further gain above it.

III, average effect only

A slope with counterexamples still present. The condition raises the expected outcome without being enough for it – the reading most often mistaken for sufficiency.

IV, neither

No evidence for either reading.

Configurations II and IV are indistinguishable in a design that reports only a regression coefficient, because both have a flat slope. Separating them is what this table is for.

The cuts

beta_cut defaults to 0.30, the lower edge of the medium band in Cohen (1988) for a standardised coefficient. d_cut defaults to the relevance threshold given to sca_analysis() when one was pre-specified – the researcher has already stated their criterion and this function should not override it – and otherwise to 0.10, the lower edge of the medium band in Dul (2016). The two numbers differ because the two conventions do; each is the bottom of "medium" in its own tradition. Whichever cuts were used are returned as columns, so a table always carries the lines that produced its own classification.

These are conventions, not benchmarks calibrated for this design.

effect_size_null and effect_size_excess sit beside the classification for that reason. A condition can clear d_cut on an effect size that is mostly the envelope fitting sample extremes – a noise condition at n = 70 reached 0.104 against the conventional 0.10 with a floor of 0.057 – and the label would read "dominant" either way. The cut is left where the convention puts it; what the table adds is the number that qualifies it.

References

Chan, Y. (in press). Mean Effects Are Not Sufficiency Effects: How Sufficient Condition Analysis Complements Regression Analysis. University of Leeds.

Chan, Y. (in press). From Necessity to Sufficiency and Necessity-and-Sufficiency: Extending Condition Analysis in Degree. University of Leeds.

See Also

sca_locationshift(), sca_reference(), sca_table()

Examples

# Configuration II: a flat slope with a largely empty corner
ii <- subset(sca_four_configurations, configuration == "II")
fit <- sca_analysis(ii, x = "X", y = "Y", direction = "HH",
                    ceilings = "ce_fdh", scope = c(0, 100, 0, 100))
sca_compare(fit)

Convert a sufficiency direction to its physical empty corner

Description

The first letter of a direction refers to X and the second to Y. For example, "HH" means high X is sufficient for high Y. The returned corner is a geometric location in the original, unflipped scatter plot.

Usage

sca_corner(direction = c("HH", "LH", "HL", "LL"))

Arguments

Value

An integer vector of physical empty corners.

Examples

sca_corner("HH")
sca_corner(c("HH", "LH", "HL", "LL"))

Where a physical corner sits on the plot

Description

Where a physical corner sits on the plot

Usage

sca_corner_location(corner)

Arguments

Value

A character vector: "upper-left", "upper-right", "lower-left" or "lower-right".

See Also

sca_boundary_type(), sca_corner_map()

Examples

sca_corner_location(1:4)

Complete mapping between SCA directions and physical empty corners

Description

Reproduces the sufficiency half of Table 1 of the condition analysis in degree framework, and adds the corner numbering the frontier engine uses. relationship states the direction as a claim about the theorised X-Y relationship; empty_space names what the expected empty space would contain; boundary_type says whether the boundary separating it is a ceiling or a floor.

Usage

sca_corner_map()

Details

The direction is a claim about the theorised relationship, not about the shape of the estimated frontier, which may be a step function or a straight line under any of the four types. What the four types do assume is that the relationship is monotone, because that is what makes the expected empty space a single corner of the plot. Theory may instead posit a curvilinear or U-shaped relationship – a condition that is sufficient over part of its range and not over the rest – and then no one corner is the expected empty space, the four directions cannot express the claim, and an effect size computed for any of them is answering a question that was not asked. Those relationships are outside the scope of this package. Plot the data before choosing a direction.

Value

A data frame that distinguishes SCA's logical direction number from the physical empty corner used by the frontier engine.

References

Chan, Y. (in press). From Necessity to Sufficiency and Necessity-and-Sufficiency: Extending Condition Analysis in Degree. University of Leeds.

Examples

sca_corner_map()

Convert a physical empty corner to its sufficiency direction

Description

Convert a physical empty corner to its sufficiency direction

Usage

sca_direction(corner = 1:4)

Arguments

Value

A character vector of sufficiency directions.

Examples

sca_direction(4)
sca_direction(1:4)

What an expected empty space would contain

Description

Fills the variable names into the Table 1 wording for a corner.

Usage

sca_empty_space(corner, x = "X", y = "Y")

Arguments

Value

A character vector, one entry per corner.

See Also

sca_relationship(), sca_corner_map()

Examples

sca_empty_space(4)
sca_empty_space(4, "Training", "Performance")

Extract one value from an SCA result

Description

Extract one value from an SCA result

Usage

sca_extract(model, x = NULL, ceiling = NULL, param = "Effect size")

Arguments

Value

The selected value.

See Also

sca_legacy_names()

Examples

iv <- subset(sca_four_configurations, configuration == "IV")
fit <- sca_analysis(iv, x = "X", y = "Y", direction = "HH",
                    scope = c(0, 100, 0, 100))
sca_extract(fit, param = "Effect size")
sca_extract(fit, ceiling = "cr_fdh", param = "Effect size")
sca_extract(fit, param = "relationship")

Four comparative demonstrations of mean effect against sufficiency effect

Description

The simulated datasets behind the four configurations of Chan (in press), Mean Effects Are Not Sufficiency Effects: How Sufficient Condition Analysis Complements Regression Analysis. Each holds 500 cases on a theoretical scope of 0 to 100 on both axes, and each was built so that one of the four readings of an X-Y relationship holds.

Usage

sca_four_configurations

Format

A data frame with 2000 rows and 3 columns:

configuration

"I", "II", "III" or "IV".

X

The condition, on a theoretical scope of 0 to 100.

Y

The outcome, on a theoretical scope of 0 to 100.

Why these four

A regression coefficient and a sufficiency effect size answer different questions, and neither implies the other. These datasets make the four possible combinations concrete, in the paper's own numbering, which runs from weakest to strongest:

"I"

Neither focal effect. No slope and no empty corner.

"II"

Sufficiency-dominant. A flat slope with a largely empty corner: beyond some level of X the low-Y cases stop occurring. A regression-only design reports this as configuration I – both have a flat slope – which is the point the four demonstrations exist to make.

"III"

Mean-dominant. A slope with counterexamples still present: X raises the expected outcome without being enough for it.

"IV"

Both. A slope and an empty corner. Two findings about one relationship, not one finding stated twice.

The scope these numbers assume

The four configurations were simulated on a 0-100 box on both axes, and every effect size documented for them is an area over that box. Pass it to reproduce them:

fit <- sca_analysis(subset(sca_four_configurations, configuration == "IV"),
                    x = "X", y = "Y", direction = "HH",
                    scope = c(0, 100, 0, 100),
                    ceilings = c("ce_fdh", "cr_fdh"),
                    test.rep = 5000, test.seed = 1)
sca_compare(fit)

Fitting on the observed range instead gives materially smaller effect sizes for the same data. Neither number is a fault in the other, and neither kind of scope is the correct one: they are answers to different questions, and NCA::nca_analysis() itself defaults to the observed range. What the two numbers show is that the denominator is half of the statistic, which is why sca_analysis() records both the scope it was given and the scope it used, and why sca_scope() reports them side by side.

Source

Simulated for the four comparative demonstrations. The same data is shipped as plain text at system.file("extdata", "sca_four_configurations.csv", package = "SCAtools"), so it can be read without R. The original per-configuration CSV files and a base-R reproduction script, which re-implements the CE-FDH frontier, the effect size and the permutation test from their definitions rather than calling this package, accompany the paper.

References

Chan, Y. (in press). Mean Effects Are Not Sufficiency Effects: How Sufficient Condition Analysis Complements Regression Analysis. University of Leeds.

See Also

sca_compare(), sca_locationshift()

Examples

str(sca_four_configurations)
table(sca_four_configurations$configuration)

# The two that a regression cannot tell apart
flat <- subset(sca_four_configurations, configuration %in% c("I", "II"))
vapply(split(flat, flat$configuration),
       function(d) unname(stats::coef(stats::lm(Y ~ X, d))[[2L]]),
       numeric(1L))

Names retired in SCAtools 0.4.0 and 0.5.0

Description

0.4.0 adopts the vocabulary of the condition analysis in degree framework. The threshold a sufficiency analysis reports is a sufficiency threshold, and boundary is reserved for the theoretical line an empty space is separated by, so the argument controlling the strictness of a reported inequality was renamed to inequality. 0.5.0 replaces this package's own shorthand "empty zone" with the framework's "empty space", so empty_zone_area became empty_space_area.

Usage

sca_legacy_names()

Details

Retired column names remain reachable. sca_table() and sca_thresholds() append them as duplicate columns when called with legacy = TRUE, each an exact copy of the current column. sca_extract() no longer accepts them: it stops and names the replacement. The retired arguments boundary, bottleneck.x and bottleneck.y were removed in 0.5.0; use inequality, threshold.x and threshold.y.

Value

A named character vector: names are the retired names, values the replacements.

See Also

sca_terms()

Examples

sca_legacy_names()

Is the frontier a level shift or a threshold?

Description

Places three slopes side by side for the same relationship: the ordinary least-squares slope, the median (quantile regression at tau = 0.5) slope, and the slope of the fitted frontier.

Usage

sca_locationshift(model, x = NULL, ceiling = NULL, tolerance = 0.1)

Arguments

Details

When all three agree, the frontier is a location shift: it is the central tendency moved bodily to the edge of the data, so sufficiency analysis adds a level to what the regression already said and does not add a slope. When the frontier slope departs from the other two, the boundary has a shape of its own and the relationship is not a shift.

This is the natural follow-up to configuration II of sca_compare(), where the average slope is flat but the forbidden corner is empty.

Only frontier techniques that fit a line report a slope. A step frontier – ce_fdh, ce_vrs – has none by construction, so frontier_slope is NA for those and the comparison needs a regression-type frontier such as cr_fdh.

Value

A data frame with one row per condition.

See Also

sca_compare()

Examples

iv <- subset(sca_four_configurations, configuration == "IV")
fit <- sca_analysis(iv, x = "X", y = "Y", direction = "HH",
                    ceilings = "cr_fdh", scope = c(0, 100, 0, 100))
sca_locationshift(fit)

Conventional magnitude bands

Description

The bands Dul (2016) proposes for an NCA effect size and Cohen (1988) for a standardised regression coefficient, and the label a given value falls in. Called with no value it returns the band table.

Usage

sca_magnitude(value = NULL, scale = c("effect_size", "beta"))

Arguments

Value

With value, a character vector of labels, NA where the value is not finite. Without it, a data frame of bands.

What these are, and are not

They are a convention to cite, in the way a reader of a regression coefficient cites Cohen. Reporting "d = 0.35, large by the conventional bands (Dul, 2016)" is a citation, and a normal one.

They are not calibrated for your design. An effect size is the share of the declared scope left empty, so it depends on the scope you declared, the frontier technique, and – because the frontier is fitted to the sample's own extremes – on the sample size. Two studies of the same relationship at n = 30 and n = 800 will not report the same number, and neither is wrong: each describes its own data inside its own scope. What does not follow is that the two can be read off one band table without comment.

The two conventions share their edges and shift their labels

This is worth stating plainly, because the coincidence is a trap.

value    as a coefficient (Cohen)   as an effect size (Dul)
0.15     small                      medium
0.35     medium                     large
0.60     large                      very large

Both use .10, .30 and .50 as edges; the labels sit one band apart. It is why sca_compare() defaults beta_cut to 0.30 and d_cut to 0.10 – each is the lower edge of "medium" in its own tradition, and those are different numbers even though the tradition's edges are not.

Read a band beside the floor

effect_size_null from sca_table() is what random pairing alone produces at that sample size, scope and technique. A band says where a number sits on a conventional scale; the floor says how much of it the geometry would have produced anyway. They answer different questions and belong in the same sentence.

References

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

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

See Also

sca_table(), sca_compare()

Examples

sca_magnitude()
sca_magnitude(scale = "beta")
sca_magnitude(c(0.04, 0.18, 0.35, 0.62))
# The same values under the other convention, one band apart:
sca_magnitude(c(0.04, 0.18, 0.35, 0.62), scale = "beta")

Normalize numeric variables for SCA

Description

Normalize numeric variables for SCA

Usage

sca_normalize(data, scale = NULL, min_max = NULL)

Arguments

Value

Data on the requested scales.

Examples

dat <- data.frame(X = c(0, 2, 4, 10), Y = c(10, 20, 30, 50))
sca_normalize(dat)

Diagnose observations that influence an SCA empty-space effect

Description

Which observations hold up the frontier, and how far the effect size would move without them. A frontier is fixed by the most extreme cases rather than by the central tendency, so a handful of observations can carry a whole claim; this says how few.

Usage

sca_outliers(
  data,
  x = NULL,
  y = NULL,
  direction = "HH",
  ceiling = NULL,
  scope = NULL,
  k = 1,
  min.dif = 0.01,
  max.results = 25,
  condensed = FALSE
)

Arguments

Value

An outlier-sensitivity table whose first column names the condition, or NULL when no observation reaches min.dif.

Two ways to call it

Pass a fitted model and the condition, outcome, direction and scope are taken from it, so the diagnostic cannot quietly describe a different analysis from the one being reported:

fit <- sca_analysis(data, x = c("A", "B"), y = "Y", direction = "HH")
sca_outliers(fit)

Every condition in the model is examined unless x selects some. Passing y, direction or scope alongside a model is an error rather than an override: the model fixes all three already, and a second answer for any of them would produce a table that does not describe the fit it came from.

Passing a data frame instead works as it always has, for the case where there is no fitted model to hand.

Reading it

This is a fragility check, not a cleaning step. An observation that moves the frontier is doing what the method says it should – frontiers are fixed by extreme cases – so finding one is not grounds for removing it. What the table says is how much of the claim rests on how few cases, which belongs in the write-up either way.

See Also

sca_analysis(), sca_compare()

Examples

iv <- subset(sca_four_configurations, configuration == "IV")
fit <- sca_analysis(iv[seq(1, nrow(iv), by = 5), ], x = "X", y = "Y",
                    direction = "HH", ceilings = "ce_fdh",
                    scope = c(0, 100, 0, 100))
sca_outliers(fit)

Display selected SCA output

Description

Display selected SCA output

Usage

sca_output(
  model,
  summaries = TRUE,
  thresholds = FALSE,
  plots = TRUE,
  tests = FALSE,
  selection = NULL,
  ceiling = NULL,
  scale = NULL,
  outcome_scale = NULL,
  convention = NULL,
  inequality = c("strict", "inclusive")
)

Arguments

Value

Invisibly, a list containing generated plots and tables.

Examples

iv <- subset(sca_four_configurations, configuration == "IV")
fit <- sca_analysis(iv, x = "X", y = "Y", direction = "HH",
                    ceilings = "ce_fdh", scope = c(0, 100, 0, 100))
sca_output(fit, thresholds = TRUE)

Plot SCA frontiers

Description

Plot SCA frontiers

Usage

sca_plot(model, x = NULL, ceilings = NULL, shade = TRUE, annotate = TRUE)

Arguments

Value

A ggplot for one condition or a named list of plots for multiple conditions.

Examples

iv <- subset(sca_four_configurations, configuration == "IV")
fit <- sca_analysis(iv, x = "X", y = "Y", direction = "HH",
                    scope = c(0, 100, 0, 100), reference = "ols")
sca_plot(fit)

Estimate power for a directional sufficiency analysis

Description

The frontier slope has to be able to bound the corner the direction implies. A rising line bounds the upper-left and lower-right corners, so it suits HH and LL; a falling line bounds the upper-right and lower-left corners, so it suits LH and HL. A horizontal line is accepted for every direction, as it is by the engine.

Usage

sca_power(
  n = c(20, 50, 100),
  effect = 0.1,
  slope = 1,
  ceiling = "ce_fdh",
  direction = "HH",
  p = 0.05,
  distribution.x = "uniform",
  distribution.y = "uniform",
  rep = 100,
  test.rep = 200,
  test.seed = NULL
)

Arguments

Value

A data frame of estimated power values with SCA direction metadata.

Examples

# Deliberately small: a reporting-grade run uses rep = 100 and
# test.rep = 200 and takes minutes rather than seconds.
power <- sca_power(n = c(20, 40), rep = 5, test.rep = 19, test.seed = 1)
power

Plot SCA power results

Description

Plot SCA power results

Usage

sca_powerplot(
  df_power,
  x.variable = "n",
  x.name = "Sample size",
  x.min = 0,
  x.max = NULL,
  line.variable = "ES",
  line.name = "Effect size"
)

Arguments

Value

A ggplot object.

Examples

power <- sca_power(n = c(20, 40), rep = 5, test.rep = 19, test.seed = 1)
sca_powerplot(power)

Generate data with a directional sufficiency pattern

Description

Generate data with a directional sufficiency pattern

Usage

sca_random(
  n,
  intercepts,
  slopes,
  direction = "HH",
  distribution.x = "uniform",
  distribution.y = "uniform",
  mean.x = 0.5,
  mean.y = 0.5,
  sd.x = 0.2,
  sd.y = 0.2
)

Arguments

Value

A data frame with X condition columns and one Y outcome column.

Examples

set.seed(42)
dat <- sca_random(50, intercepts = 0, slopes = 1, direction = "HH")
head(dat)

Central-tendency reference lines

Description

Fits the central-tendency lines for each condition, on the same observations the frontier was fitted to. "ols" is the ordinary least-squares regression, the line lm(y ~ x) returns. "qr" is a quantile regression at reference.tau, which defaults to 0.5: the median line, which is the central-tendency summary that does not move when one observation does.

Usage

sca_reference(model, x = NULL, line = NULL)

Arguments

Details

Reported separately from sca_table() on purpose. Regression describes how the expected outcome moves with the condition; a sufficiency frontier describes which condition-outcome combinations are absent. Section 6.1 of the condition analysis in degree framework treats these as different targets that are most informative shown side by side rather than merged into one number, and section 4.3 notes that a frontier is fixed by the most extreme observations rather than by the central tendency. A reference line therefore has a slope and an intercept but no empty space, no effect size, no permutation p value and no sufficiency threshold.

"qr" as a reference is not the same thing as "qr" as a frontier, and they are set by different arguments. A frontier qr uses qr.tau, defaults to 0.95 and is meant to sit at the edge of the cloud; a reference qr uses reference.tau, defaults to 0.5 and is meant to sit in the middle of it. Both can be asked for at once, which the engine could not do in a single call: it takes one qr.tau, so the reference is computed here instead.

r_squared and beta are NA for "qr", and pseudo_r1 is NA for "ols". Quantile regression minimises a different loss, so there is no residual variance to take a share of; Koenker and Machado's R1 is the right measure for it, and it is a share of absolute loss at one quantile rather than of variance. Reporting it under the r_squared heading would be one column carrying two meanings, and leaving it out would report a difference in kind as a missing value, so it has a column of its own. A pseudo_r1 and an r_squared are not on a common scale and must not be compared.

Every requested line is computed on request, and the lines are also computed for a model that asked for none, so the table is available either way; drawn records whether the line appears on sca_plot().

Value

A data frame with one row per condition and reference line.

See Also

sca_analysis(), sca_table(), sca_terms()

Examples

iv <- subset(sca_four_configurations, configuration == "IV")
fit <- sca_analysis(iv, x = "X", y = "Y", direction = "HH",
                    ceilings = "ce_fdh", scope = c(0, 100, 0, 100),
                    reference = c("ols", "qr"))
sca_reference(fit)

The direction stated as a claim about the relationship

Description

Gives a direction in the wording of Table 1 of the framework. This is a claim about the theorised relationship, not about the shape of the estimated frontier, which may be a step function or a straight line under any of the four directions.

Usage

sca_relationship(direction, x = "X", y = "Y")

Arguments

Value

A character vector, one entry per direction.

See Also

sca_empty_space(), sca_corner_map()

Examples

sca_relationship("HH")
sca_relationship("HH", "Training", "Performance")

How much the effect size moves when the sample does

Description

Refits one condition on random subsamples of several sizes and reports the distribution of the effect size at each size. Two things become visible: how far d moves under resampling at a given sample size, and how it trends as the sample grows.

Usage

sca_resample(
  model,
  x = NULL,
  ceiling = NULL,
  sizes = c(0.4, 0.6, 0.8),
  reps = 200,
  seed = NULL
)

Arguments

Value

A data frame with one row per subsample size.

Why there is no confidence interval

Deliberately, and for reasons specific to frontier estimators rather than general caution.

The ordinary n-out-of-n bootstrap is inconsistent here. Resampling with replacement duplicates observations, and a duplicated extreme point cannot push the envelope further out than the original did, so the resampled frontier lies inside the fitted one far more often than chance – the resampling distribution does not converge to the sampling distribution. This is the familiar result for DEA and FDH efficiency scores.

Subsampling without replacement, which is what this function does, restores consistency but leaves a bias with a known direction: a subsample of size m < n has a higher expected effect size than the full sample, because the envelope is built from sample extremes and fewer points leave more of the corner empty by accident. A percentile interval read straight off the subsample distribution would therefore sit above the quantity it claims to cover. Correcting for that requires the estimator's convergence rate, which depends on the technique and the dimension and which this package does not assert.

So the table reports the distribution and stops there.

Reading it

effect_size is the full-sample value, repeated on every row as the anchor. subsample_median and the two percentiles describe the subsample distribution at that size.

The spread at one size answers "how much does my d depend on which cases I happened to collect". The trend down the m column answers "how much of my d is the sample being small" – it falls as m rises. Compare it with effect_size_null from sca_table(), which measures the same overstatement a different way, by random pairing at the full sample size.

Cost

One refit per replication per size. The default is 200 by 3, so a call is six hundred fits of one condition; nothing is cached between them.

See Also

sca_scope_sensitivity(), sca_outliers(), sca_table()

Examples

iv <- subset(sca_four_configurations, configuration == "IV")
fit <- sca_analysis(iv[seq(1, nrow(iv), by = 5), ], x = "X", y = "Y",
                    direction = "HH", ceilings = "ce_fdh",
                    scope = c(0, 100, 0, 100))
# A report would use the default reps = 200
sca_resample(fit, reps = 20, seed = 1)

Convert values to an SCA reporting scale

Description

The conversion used internally by sca_thresholds(), exposed so that other packages can express their own quantities on exactly the same scale. This matters whenever two numbers are to be compared: a threshold produced elsewhere is only comparable with an SCA threshold if both were converted by the same rule, including the direction-aware mirroring and the percentile strictness that follows the inequality.

Usage

sca_rescale(
  values,
  reference,
  bounds = range(reference, na.rm = TRUE),
  scale = "percentage.range",
  letter = c("H", "L"),
  convention = c("absolute", "directional"),
  inequality = c("strict", "inclusive")
)

Arguments

Value

A numeric vector on the requested scale. Non-finite inputs return NA.

Examples

x <- c(1, 3, 4, 7, 9)
sca_rescale(4, x, scale = "percentile", letter = "H")
sca_rescale(4, x, scale = "percentile", letter = "L")

Reporting scales available for SCA threshold tables

Description

Lists the scales and conventions understood by sca_analysis() and sca_thresholds(), with a note on how each behaves when the sufficiency direction refers to a low level.

Usage

sca_scales()

Value

A data frame describing each scale.

Examples

sca_scales()

The scope each effect size was divided by

Description

Every effect size in this package is an empty area divided by the area of the X-Y scope, so the scope is half of every number reported. It is not always the scope that was asked for: the NCA engine widens a declared scope until it contains every observation, which changes the denominator without changing what the researcher wrote down.

Usage

sca_scope(model)

Arguments

Details

This reports both. ⁠declared_*⁠ is NA when no scope was declared and the observed range was used. ⁠used_*⁠ is what the engine divided by, and adjusted is TRUE for a condition on which the two differ.

Value

A data frame with one row per condition.

See Also

sca_table(), sca_analysis()

Examples

fit <- sca_analysis(
  subset(sca_four_configurations, configuration == "IV"),
  x = "X", y = "Y", direction = "HH", scope = c(0, 100, 0, 100)
)
sca_scope(fit)

How much of the effect size is the scope

Description

Refits one condition on a series of scopes and reports the effect size each one produces. The scope is the denominator of every effect size and, unlike the frontier, it is chosen rather than estimated, so a reported d is a claim about the data and a claim about the box at the same time. This separates them.

Usage

sca_scope_sensitivity(
  model,
  x = NULL,
  ceiling = NULL,
  widen = c(0, 0.25, 0.5, 1),
  scopes = NULL
)

Arguments

Details

By default the boxes are the observed range widened by a fraction of the observed span on each side, because that is a quantity a reader can judge: "a box half as wide again as the data" means something where a pair of numbers does not. The first row is always the observed range itself: the box the engine uses when no scope is declared, and the smallest box that contains the data, since any smaller one is widened back. When the model was fitted on a declared scope, that box is added as a final row labelled "as fitted (declared)", so the caller can see where their own choice sits among the alternatives instead of working it out by eye.

No column reports a change from a baseline, because which row is the baseline is the caller's question. The observed range answers "how much of this number is box, counting from the smallest box that holds the data"; the scope the model was fitted on answers "how sensitive is my number to the box I chose". Those are different questions and the table cannot silently pick one, so it reports the effect sizes and leaves the subtraction to the reader. The data does not change between rows: every difference in effect_size down the column is the box.

Value

A data frame with one row per scope.

See Also

sca_scope(), sca_table()

Examples

fit <- sca_analysis(
  subset(sca_four_configurations, configuration == "IV"),
  x = "X", y = "Y", direction = "HH", ceilings = "ce_fdh"
)
sca_scope_sensitivity(fit)

Tidy SCA results

Description

One row per condition and frontier technique.

Usage

sca_table(model, legacy = FALSE, p.adjust = "none")

Arguments

Details

Three columns state the geometry in the vocabulary of the condition analysis in degree framework. relationship gives the direction as a claim about the theorised X-Y relationship, as in Table 1 of that framework. empty_space names what the expected empty space would contain. boundary_type says whether the boundary separating it is a ceiling, which limits how high Y can be for a given X, or a floor, which limits how low. For sufficiency the boundary is a floor whenever the outcome level is high and a ceiling whenever it is low, so the technique names CE-FDH and CR-FDH keep the word "ceiling" even where the fitted boundary is a floor.

scope_area is the denominator: effect_size is empty_space_area / scope_area exactly. scope_adjusted is TRUE where the engine had to widen the declared scope to hold every observation, so that the denominator is not the scope as declared. sca_scope() reports both scopes bound by bound.

frontier names the frontier technique the row was fitted with, such as "ce_fdh"; it is the value given in ceilings.

empty_zone_area is an exact copy of empty_space_area under the name it had before 0.5.0. It is kept because NSCA 0.4.5 reads it, and will be removed once a release of NSCA that reads empty_space_area has replaced that version. New code should use empty_space_area.

Value

A data frame with one row for every condition-frontier combination.

Multiple testing

Each condition carries its own permutation test, so a model with k conditions produces k p-values and a model with several frontier techniques produces k of them per technique. None of these is adjusted for multiplicity, and the Settings block printed by print() says so with the count and the arithmetic: were k tests independent, the chance of at least one false positive at alpha is 1 - (1 - \alpha)^k.

Disclosure rather than adjustment, deliberately. Which correction fits depends on what the family is – whether the conditions were pre-specified, whether one significant condition would be reported as the finding or all of them together, whether the analysis is confirmatory at all. Those are properties of the design, not of the data, and a package that guessed at them would be making the methodological claim on the researcher's behalf. Adjust with stats::p.adjust() on the p_value column when the family is known.

The frontier techniques are a different problem and not a family-wise rate: two techniques are two estimates of one claim. Choosing between them after seeing both p-values is a garden of forking paths, which the Settings block also names.

See Also

sca_thresholds(), sca_scope(), sca_corner_map(), sca_legacy_names()

Examples

iv <- subset(sca_four_configurations, configuration == "IV")
fit <- sca_analysis(iv, x = "X", y = "Y", direction = "HH",
                    scope = c(0, 100, 0, 100))
sca_table(fit)

# Column names retired in earlier releases, appended as duplicates
names(sca_table(fit, legacy = TRUE))

# With a permutation test, on every fifth case to keep the example quick.
# A report would use 1000 or more permutations.
small <- iv[seq(1, nrow(iv), by = 5), ]
fit_test <- sca_analysis(small, x = "X", y = "Y", direction = "HH",
                         ceilings = "ce_fdh", scope = c(0, 100, 0, 100),
                         test.rep = 99, test.seed = 1)
sca_table(fit_test)

Terminology of condition analysis in degree

Description

A machine-readable glossary linking the vocabulary of the condition analysis in degree framework to the names this package uses. Terms retired in 0.4.0 are listed with their replacements in sca_legacy_names().

Usage

sca_terms()

Value

A data frame with one row per term.

References

Chan, Y. (in press). From Necessity to Sufficiency and Necessity-and-Sufficiency: Extending Condition Analysis in Degree. University of Leeds.

See Also

sca_legacy_names(), sca_corner_map(), sca_scales()

Examples

sca_terms()

Plot an SCA permutation test

Description

Plot an SCA permutation test

Usage

sca_test_plot(model, x = NULL, ceiling = NULL, bins = 30)

Arguments

Value

A ggplot object.

Examples

iv <- subset(sca_four_configurations, configuration == "IV")
# Every fifth case, to keep the permutation test quick
small <- iv[seq(1, nrow(iv), by = 5), ]
fit <- sca_analysis(small, x = "X", y = "Y", direction = "HH",
                    ceilings = "ce_fdh", scope = c(0, 100, 0, 100),
                    test.rep = 99, test.seed = 1)
sca_test_plot(fit)

Sufficiency threshold table

Description

Turns the fitted frontier into directional sufficient-condition rules. A strict HH row reads ⁠X > threshold => Y > outcome level⁠.

Usage

sca_thresholds(
  model,
  ceiling = NULL,
  x = NULL,
  scale = NULL,
  outcome_scale = NULL,
  convention = NULL,
  inequality = c("strict", "inclusive"),
  digits = 6L,
  legacy = FALSE
)

Arguments

Details

A sufficiency threshold is defined for one outcome target: in a higher X for higher Y relationship it is the lowest X value at which the estimated sufficiency frontier reaches that target. The table is the sufficiency counterpart of an NCA bottleneck table; the format is parallel but the interpretation is not, and the term "bottleneck" is reserved for necessity.

Values are stored in actual units and converted here, so a table can be re-expressed on any scale without refitting the model. Under the "absolute" convention 0 sits at the low end of each axis and the inequality carries the direction; under "directional" the axis is mirrored for a low-level direction, which flips the inequality so that every rule reads as "more of the sufficient thing". See sca_scales().

The status column reports each row in sufficiency terms: "estimable" for a threshold inside the scope, "no_threshold" when no attainable condition value reaches the outcome level, and "always_satisfied" when every observation in scope already lies on the required side.

The inequalities describe an empirical empty-space pattern and should not be read as causal guarantees without an appropriate research design.

Value

A long-form data frame of threshold rules.

See Also

sca_table(), sca_legacy_names(), sca_scales()

Examples

iv <- subset(sca_four_configurations, configuration == "IV")
fit <- sca_analysis(iv, x = "X", y = "Y", direction = "HH",
                    ceilings = "ce_fdh", scope = c(0, 100, 0, 100))
sca_thresholds(fit)
sca_thresholds(fit, scale = "actual", inequality = "inclusive")