Package 'dda'

Conditional Direction Dependence Analysis: Independence Properties

Description

cdda.indep evaluates asymmetries of predictor-error independence of competing conditional models (y ~ x * m vs. x ~ y * m with m being a continuous or categorical moderator).

print returns the output of standard linear model coefficients for causally competing target and alternative models.

plot returns graphs of cdda.indep results.

summary returns test statistics from the cdda.indep class object.

Usage

cdda.indep(
  formula = NULL,
  pred = NULL,
  mod = NULL,
  modval = NULL,
  data = list(),
  hetero = FALSE,
  diff = FALSE,
  nlfun = NULL,
  hsic.method = "gamma",
  B = 200,
  boot.type = "perc",
  conf.level = 0.95,
  parallelize = FALSE,
  cores = 1,
  robust = FALSE,
  ...
)

## S3 method for class 'cdda.indep'
print(x, ...)

## S3 method for class 'cdda.indep'
plot(x = NULL, stat = NULL, ylim = NULL, ...)

## S3 method for class 'cdda.indep'
summary(
  object,
  nlfun = FALSE,
  hetero = FALSE,
  hsic = TRUE,
  hsic.diff = FALSE,
  dcor = TRUE,
  dcor.diff = FALSE,
  mi.diff = FALSE,
  ...
)

Arguments

formula	Symbolic formula of the model to be tested or an lm object.
pred	A character indicating the variable name of the predictor which serves as the outcome in the alternative model.
mod	A character indicating the variable name of the moderator.
modval	Characters or a numeric sequence specifying the moderator values used in post-hoc probing. Possible characters include c("mean", "median", "JN"). modval = "mean" tests the interaction effect at the moderator values M – 1SD, M, and M + 1SD; modval = "median" uses Q1, Md, and Q3. The Johnson-Neyman approach is applied when modval = "JN" with conditional effects being evaluated at the boundary values of the significance regions. When a numeric sequence is specified,the pick-a-point approach is used for the selected numeric values.
data	A required data frame containing the variables in the model.
hetero	A logical value indicating whether separate homoscedasticity tests should be returned when using summary, default is FALSE.
diff	A logical value indicating whether differences in HSIC, dCor, and MI values should be computed. Bootstrap confidence intervals are computed using B bootstrap samples.
nlfun	Determines handling of non-linear correlation tests depending on the function used: For cdda.indep: Either a numeric value or a function of .Primitive type used for non-linear correlation tests. When numeric, the value is used in a power transformation. For summary: A logical value indicating whether non-linear correlation tests should be returned in the output. Default is FALSE.
hsic.method	A character indicating the inference method for the Hilbert-Schmidt Independence Criterion. Must be one of the four specifications c("gamma", "eigenvalue", "bootstrap", "permutation"). hsic.method = "gamma" is the default.
B	Number of permutations for separate dCor tests and number of resamples when hsic.method = c("bootstrap", "permutation") or diff = TRUE.
boot.type	A character indicating the type of bootstrap confidence intervals. Must be one of the two specifications c("perc", "bca"). boot.type = "perc" is the default.
conf.level	Confidence level for bootstrap confidence intervals.
parallelize	A logical value indicating whether bootstrapping is performed on multiple cores. Only used if diff = TRUE.
cores	A numeric value indicating the number of cores. Only used if parallelize = TRUE.
robust	A logical value indicating whether Siegel's (1982) repeated median estimator should be used for model estimation. If robust = TRUE repeated median estimation is applied for the causally competing models, otherwise ordinary least squares (OLS) estimation is used.
...	Additional arguments to be passed to the function.
x	An object of class cdda.indep when using print or plot.
stat	A character indicating the CDDA statistic to be plotted with the options c("hsic.diff", "dcor.diff", "mi.diff").
ylim	A numeric vector of length 2 indicating the y-axis limits. If NULL, the function will set the limits automatically.
object	An object of class cdda.indep when using summary.
hsic	A logical value indicating whether separate HSIC tests should be returned when using summary, default is TRUE.
hsic.diff	A logical value indicating whether HSIC difference statistics should be returned when using summary, default is FALSE.
dcor	A logical value indicating whether separate Distance Correlation (dCor) tests should be returned when using summary, default is TRUE.
dcor.diff	A logical value indicating whether dCor difference statistics should be returned when using summary, default is FALSE.
mi.diff	A logical value indicating whether Mutual Information (MI) difference statistics should be returned when using summary, default is FALSE.

Value

A list of class cdda.indep containing the results of CDDA independence tests for pre-specified moderator values.

An object of class cdda.indep with competing model coefficients.

References

Wiedermann, W., & von Eye, A. (2025). Direction Dependence Analysis: Foundations and Statistical Methods. Cambridge, UK: Cambridge University Press.

See Also

dda.indep for an unconditional version.

Examples

set.seed(321)
n <- 700

## --- generate moderator

z <- sort(rnorm(n))
z1 <- z[z <= 0]
z2 <- z[z > 0]

## --- x -> y when z <= 0

x1 <- rchisq(length(z1), df = 4) - 4
e1 <- rchisq(length(z1), df = 3) - 3
y1 <- 0.5 * x1 + e1

## --- y -> x when m z > 0

y2 <- rchisq(length(z2), df = 4) - 4
e2 <- rchisq(length(z2), df = 3) - 3
x2 <- 0.25 * y2 + e2

y <- c(y1, y2); x <- c(x1, x2)

d <- data.frame(x, y, z)

m <- lm(y ~ x * z, data = d)


result <- cdda.indep(m,
                     pred = "x",
                     mod = "z",
                     modval = c(-1, 1),
                     data = d,
                     hetero = TRUE,
                     diff = TRUE,
                     parallelize = TRUE,
                     cores = 2,
                     nlfun = 2,
                     B = 2)
# Note: Only 2 bootstrap samples are created here to lower computation time

## Not run: 
## --- Realistic settings; run time is substantial

result <- cdda.indep(m,
                     pred = "x",
                     mod = "z",
                     modval = c(-1, 1),
                     data = d,
                     hetero = TRUE,
                     diff = TRUE,
                     parallelize = TRUE,
                     cores = 2,
                     nlfun = 2,
                     B = 500)

print(result)
summary(result, hsic.diff = TRUE, dcor.diff = TRUE, mi.diff = TRUE)
plot(result, stat = "dcor.diff")

## End(Not run)


print(result)

plot(result, stat = "dcor.diff")

summary(result, hetero = FALSE)

---

Conditional Directional Dependence Analysis: Variable Distributions

Description

cdda.vardist evaluates variable distributions of competing conditional models (y ~ x * m vs.x ~ y * m with m being a continuous or categorical moderator).

print returns the output of standard linear model coefficients for causally competing target and alternative models.

plot returns graphs of cdda.vardist results.

summary returns test statistics from the cdda.vardist class object.

Usage

cdda.vardist(
  formula,
  pred = NULL,
  mod = NULL,
  data = list(),
  modval = NULL,
  B = 200,
  boot.type = "perc",
  conf.level = 0.95
)

## S3 method for class 'cdda.vardist'
print(x, ...)

## S3 method for class 'cdda.vardist'
plot(x, stat = NULL, ylim = NULL, ...)

## S3 method for class 'cdda.vardist'
summary(object, skew = TRUE, coskew = FALSE, kurt = TRUE, cokurt = FALSE, ...)

Arguments

formula	Symbolic formula of the model to be tested or an lm object
pred	A character indicating the variable name of the predictor which serves as the outcome in the alternative model.
mod	A character indicating the variable name of the moderator.
data	A required data frame containing the variables in the model.
modval	Characters or a numeric sequence specifying the moderator values used in post-hoc probing. Possible characters include c("mean", "median", "JN"). modval = "mean" tests the interaction effect at the moderator values M - 1SD, M, and M + 1SD; modval = "median" uses Q1, Md, and Q3. The Johnson-Neyman approach is applied when modval = "JN" with conditional effects being evaluated at the boundary values of the significance regions. When a numeric sequence is specified, the pick-a-point approach is used for the selected numeric values.
B	Number of bootstrap samples.
boot.type	A character indicating the type of bootstrap confidence intervals. Must be one of the two values c("perc", "bca"). boot.type = "perc" is the default.
conf.level	Confidence level for bootstrap confidence intervals.
x	An object of class cdda.vardist when using print or plot.
...	Additional arguments to be passed to the function.
stat	A character indicating the statistic to be plotted, default is "rhs", with options c("coskew", "cokurt", "rhs", "rcc", "rtanh").
ylim	A numeric vector of length 2 indicating the y-axis limits. If NULL, the function will set the limits automatically.
object	An object of class cdda.vardist when using summary.
skew	A logical value indicating whether skewness differences and separate D'Agostino skewness tests should be returned when using summary, default is TRUE.
coskew	A logical value indicating whether co-skewness differences should be returned when using summary, default is FALSE.
kurt	A logical value indicating whether excess kurtosis differences and Anscombe-Glynn kurtosis tests should be returned when using summary, default is TRUE.
cokurt	A logical value indicating whether co-kurtosis differences should be returned when using summary, default is FALSE.

Value

An object of class cdda.vardist containing the results of conditional direction dependence tests of variable distributions.

References

Wiedermann, W., & von Eye, A. (2025). Direction Dependence Analysis: Foundations and Statistical Methods. Cambridge, UK: Cambridge University Press.

See Also

dda.vardist for an unconditional version.

Examples

set.seed(321)
n <- 700

## --- generate moderator

z <- sort(rnorm(n))
z1 <- z[z <= 0]
z2 <- z[z > 0]

## --- x -> y when z <= 0

x1 <- rchisq(length(z1), df = 4) - 4
e1 <- rchisq(length(z1), df = 3) - 3
y1 <- 0.5 * x1 + e1

## --- y -> x when m z > 0

y2 <- rchisq(length(z2), df = 4) - 4
e2 <- rchisq(length(z2), df = 3) - 3
x2 <- 0.25 * y2 + e2

y <- c(y1, y2); x <- c(x1, x2)

d <- data.frame(x, y, z)

m <- lm(y ~ x * z, data = d)

result <- cdda.vardist(m, pred = "x", mod = "z", B = 50,
                      modval = c(-1, 1), data = d)
# Note: Only 50 bootstrap samples are created here to lower computation time

## Not run: 
## --- Realistic settings; run time is substantial

result <- cdda.vardist(m, pred = "x", mod = "z", B = 500,
                       modval = c(-1, 1), data = d)

print(result)
plot(result, stat = "rtanh", ylim = c(-0.05, 0.05))
summary(result, skew = FALSE, kurt = FALSE, coskew = TRUE)

## End(Not run)

print(result)

plot(result, stat = "rtanh", ylim = c(-0.05, 0.05))

summary(result, skew = FALSE, kurt = FALSE, coskew = TRUE)

---

Bootstrap Aggregated Direction Dependence Analysis (DDA)

Description

dda.bagging performs bootstrap aggregation (bagging) on an existing Direction Dependence Analysis (DDA) object to test the stability and robustness of direction dependence decisions.

Usage

dda.bagging(
  dda_result,
  iter = 100,
  progress = TRUE,
  save_file = NULL,
  alpha = 0.05,
  data = NULL,
  agg_stat = c("mean", "median", "trimmed", "winsorized", "midhinge", "tukey"),
  trim_prob = 0.1,
  win_prob = 0.1,
  inner_B = NULL,
  nlcor.adjust = c("none", "bonferroni")
)

Arguments

dda_result	An output object from any base DDA function (e.g., dda.indep, dda.vardist, or dda.resdist).
iter	Number of bootstrap samples (default: 100).
progress	Logical. Whether to display a progress bar during resampling (default: TRUE).
save_file	Character. Optional file path used to save the resulting object as an R serialized data file (e.g., "results.rds").
alpha	Numeric. Significance level used for causal model selection (default: 0.05).
data	A data.frame containing ALL raw variables used in the original DDA model (outcome, predictor, and any covariates).
agg_stat	Character. Specifies the method used for aggregating test statistics and coefficients across bootstrap samples. Must be one of the following specifications c("mean", "median", "trimmed", "winsorized", "midhinge", "tukey").
trim_prob	Numeric. Proportion of observations to be trimmed from each side of the sampling distribution when agg_stat = "trimmed" (default: 0.10).
win_prob	Numeric. Proportion of observations to be winsorized from each side of the sampling distribution when agg_stat = "winsorized" (default: 0.10).
inner_B	Optional positive integer. Caps the number of inner bootstrap resamples (B) passed to each per-iteration DDA call. NULL (default) keeps whatever B was used in the original DDA call. Every base DDA function runs its own resampling on each of the iter outer iterations, so total cost grows as iter $\times$ B; capping inner_B is the main lever for reducing run time without changing how the outer bagging loop aggregates results. The effect is largest for dda.indep called with diff = TRUE, where each outer iteration runs a full difference-statistic bootstrap, and for dda.resdist and dda.vardist, which bootstrap their confidence intervals internally.
nlcor.adjust	Character. Multiplicity adjustment passed to dda.decisions for the non-linear correlation decision. One of "none" (default) or "bonferroni".

Details

This function uses a fitted DDA output object (obtained from dda.indep, dda.vardist, or dda.resdist) and performs bootstrap aggregation of DDA test statistics. The function computes DDA statistics across iter bootstrap samples and aggregates the results to evaluate the stability and robustness of DDA model selection. p-values obtained from significance tests are aggregated using the harmonic mean p-value approach (Wilson, 2019). Model selection decisions within each bootstrap sample are obtained with dda.decisions (p-value based rules for significance tests, interval-excludes-zero rules for bootstrap difference statistics) and reported as decision proportions across samples.

Run time scales with iter multiplied by the resampling budget of the base DDA call, and the underlying independence statistics are quadratic in the number of observations. Use inner_B to cap the inner budget. Note also that if the original DDA call used parallelize = TRUE, that setting is inherited by every outer iteration, so a new cluster is started iter times; for the small inner bootstraps typical of bagging this is usually slower than running the inner calls serially.

Value

An object of class dda_bagging (with subclasses dda_bagging_indep, dda_bagging_vardist, or dda_bagging_resdist), which contains aggregated and raw results from tests matching the initial DDA function (dda.indep, dda.vardist, or dda.resdist).

References

Wiedermann, W., & von Eye, A. (2025). Direction Dependence Analysis: Foundations and Statistical Methods. Cambridge, UK: Cambridge University Press.

Wilson, D. J. (2019). The harmonic mean p-value for combining dependent tests. Proceedings of the National Academy of Sciences, 116(4), 1195–1200.

See Also

dda.indep, dda.vardist, dda.resdist, dda.decisions

Examples

set.seed(123)
n <- 200
x <- rchisq(n, df = 4) - 4
e <- rchisq(n, df = 3) - 3
y <- 0.5 * x + e
d <- data.frame(x, y)

## --- Fit a base DDA independence model

base_model <- dda.indep(y ~ x, pred = "x", data = d, B = 10, hetero = TRUE)

## --- Bootstrap aggregation of the base model

bagged <- dda.bagging(base_model, data = d, iter = 5, agg_stat = "mean",
  inner_B = 10, progress = FALSE)
# Note: n, B and iter are all kept small here to lower computation time.
# inner_B caps the resampling budget of each outer iteration.

print(bagged)
summary(bagged, show = c("hsic", "dcor"))

## Not run: 
## --- Realistic settings; run time is substantial

base_model <- dda.indep(y ~ x, pred = "x", data = d, B = 500,
  hetero = TRUE, nlfun = 2, diff = TRUE)

bagged <- dda.bagging(base_model, data = d, iter = 200,
  agg_stat = "trimmed", trim_prob = 0.05, progress = TRUE)

print(bagged)
summary(bagged, show = c("hsic", "dcor", "bp"))
print_ols_summary(bagged)

## End(Not run)

---

Model Selection Decisions for a Fitted DDA Object

Description

dda.decisions translates the tests stored in a fitted DDA object (dda.indep, dda.resdist, or dda.vardist) into causal model selection decisions. Significance tests are compared against alpha; difference statistics are decided by whether their bootstrap confidence interval excludes zero. The same rules are used inside dda.bagging, so simulation code can call dda.decisions directly instead of re-implementing decision logic.

Usage

dda.decisions(dda_result, alpha = 0.05, nlcor.adjust = c("none", "bonferroni"))

Arguments

dda_result	An output object from dda.indep, dda.resdist, or dda.vardist.
alpha	Numeric. Significance level used for causal model selection (default: 0.05).
nlcor.adjust	Character. Multiplicity adjustment applied to the smallest non-linear correlation p-value across the three transformations (dda.indep only). "none" (default) uses the smallest p-value directly. "bonferroni" multiplies it by the number of transformations.

Details

Throughout, the target model is x -> y and the alternative model is y -> x; p_yx denotes a p-value obtained under the target model and p_xy a p-value obtained under the alternative model.

Separate independence tests (dda.indep: HSIC, dCor, robust Breusch-Pagan, non-linear correlation):

• p_yx > alpha and p_xy <= alpha: "Target"

• p_yx <= alpha and p_xy > alpha: "Alternative"

• both <= alpha: "Confounding"

• both > alpha: "Undecided"

The non-linear correlation decision combines the three transformations by taking the smallest p-value in each direction, matching the reference implementation. That combination is not adjusted for multiplicity by default; nlcor.adjust = "bonferroni" applies a correction, which lowers the rate of decisive outcomes when neither direction is identified.

Difference statistics (HSIC, dCor, MI) use their bootstrap confidence intervals: an interval above zero speaks for the target model, below zero for the alternative model, otherwise the decision is "Undecided".

Separate normality tests on residuals (dda.resdist, D'Agostino and Anscombe-Glynn) depend on the prob.trans setting of the fitted object. Under prob.trans = FALSE:

• p_yx > alpha and p_xy <= alpha: "Target"

• p_yx <= alpha and p_xy > alpha: "Alternative"

• otherwise: "Undecided"

Under prob.trans = TRUE the roles are reversed (p_yx <= alpha and p_xy > alpha speaks for the target model). The skewness and kurtosis differences reverse in the same way, so that under prob.trans = TRUE a difference below zero speaks for the target model. Co-skewness, co-kurtosis and the likelihood-ratio approximations never reverse: a difference above zero speaks for the target model under both settings.

When dda.resdist is called with B = 0 no bootstrap interval is available for the skewness and kurtosis differences. The asymptotic difference test stored alongside them is used instead, combining its p-value with the sign of the difference.

Separate normality tests on observed variables (dda.vardist):

• outcome p > alpha and predictor p <= alpha: "Target"

• outcome p <= alpha and predictor p > alpha: "Alternative"

• otherwise: "Undecided"

All higher-moment difference intervals in dda.vardist use the interval-excludes-zero rule with values above zero speaking for the target model.

Value

A named character vector of decisions, one element per test available in the object. Values are "Target", "Alternative", "Undecided", "Confounding" (separate independence tests only), or NA when a test result is missing.

See Also

dda.bagging

Examples

set.seed(123)
n <- 200
x <- rchisq(n, df = 4) - 4
e <- rchisq(n, df = 3) - 3
y <- 0.5 * x + e
d <- data.frame(x, y)

fit <- dda.vardist(y ~ x, pred = "x", data = d, B = 50)
dda.decisions(fit, alpha = 0.05)

---

Direction Dependence Analysis: Independence Properties

Description

dda.indep evaluates asymmetries of predictor-error independence of causally competing models (y ~ x vs. x ~ y). print returns DDA test statistics associated with dda.indep objects.

Usage

dda.indep(
  formula,
  pred = NULL,
  data = list(),
  nlfun = NULL,
  hetero = FALSE,
  hsic.method = "gamma",
  diff = FALSE,
  B = 200,
  boot.type = "perc",
  conf.level = 0.95,
  parallelize = FALSE,
  cores = 1,
  robust = FALSE
)

## S3 method for class 'dda.indep'
print(x, ...)

Arguments

formula	Symbolic formula of the model to be tested or a lm object.
pred	A character indicating the variable name of the predictor which serves as the outcome in the alternative model.
data	An optional data frame containing the variables in the model (by default variables are taken from the environment which dda.indep is called from).
nlfun	Either a numeric value or a function of .Primitive type used for non-linear correlation tests. When nlfun is numeric the value is used in a power transformation.
hetero	A logical value indicating whether separate homoscedasticity tests (i.e., standard and robust Breusch-Pagan tests) should be computed.
hsic.method	A character indicating the inference method for the Hilbert-Schmidt Independence Criterion (HSIC). Must be one of the four specifications c("gamma", "eigenvalue", "bootstrap", "permutation"). hsic.method = "gamma" is the default.
diff	A logical value indicating whether differences in HSIC, Distance Correlation (dCor), and MI values should be computed. Bootstrap confidence intervals are computed using B bootstrap samples.
B	Number of permutations for separate dCor tests and number of resamples if hsic.method = c("bootstrap", "permutation") or diff = TRUE.
boot.type	A vector of character strings representing the type of bootstrap confidence intervals. Must be one of the two specifications c("perc", "bca"). boot.type = "perc" is the default.
conf.level	Confidence level for bootstrap confidence intervals.
parallelize	A logical value indicating whether bootstrapping is performed on multiple cores. Only used if diff = TRUE.
cores	A numeric value indicating the number of cores. Only used if parallelize = TRUE.
robust	A logical value indicating whether Siegel's (1982) repeated median estimator should be used for model estimation. If robust = TRUE repeated median estimation is applied for the causally competing models, otherwise ordinary least squares (OLS) estimation is used.
x	An object of class dda.indep when using print.
...	Additional arguments to be passed to the function.

Value

An object of class dda.indep containing the results of DDA independence tests.

References

Wiedermann, W., & von Eye, A. (2025). Direction Dependence Analysis: Foundations and Statistical Methods. Cambridge, UK: Cambridge University Press.

See Also

cdda.indep for a conditional version.

Examples

set.seed(123)
n <- 500
x <- rchisq(n, df = 4) - 4
e <- rchisq(n, df = 3) - 3
y <- 0.5 * x + e
d <- data.frame(x, y)

result <- dda.indep(y ~ x, pred = "x", data = d, parallelize = TRUE, cores = 2,
          nlfun = 2, B = 50, hetero = TRUE, diff = TRUE, robust = FALSE)

print(result)

## Not run: 
## --- Realistic settings; run time is substantial

result <- dda.indep(y ~ x, pred = "x", data = d, parallelize = TRUE,
  cores = 2, nlfun = 2, B = 500, hetero = TRUE, diff = TRUE)

print(result)

## --- Permutation-based HSIC inference

dda.indep(y ~ x, pred = "x", data = d, hsic.method = "permutation", B = 500)

## --- Siegel (1982) repeated median estimation

dda.indep(y ~ x, pred = "x", data = d, robust = TRUE, hetero = TRUE, B = 500)

## End(Not run)

print(result)

---

Direction Dependence Analysis: Residual Distributions

Description

dda.resdist evaluates patterns of asymmetry of error distributions of causally competing models (y ~ x vs. x ~ y). print returns DDA test statistics associated with dda.resdist objects.

Usage

dda.resdist(
  formula,
  pred = NULL,
  data = list(),
  B = 200,
  boot.type = "perc",
  prob.trans = FALSE,
  conf.level = 0.95,
  robust = FALSE
)

## S3 method for class 'dda.resdist'
print(x, ...)

Arguments

formula	Symbolic formula of the target model to be tested or an lm object.
pred	Variable name of the predictor which serves as the outcome in the alternative model.
data	An optional data frame containing the variables in the model (by default variables are taken from the environment which dda.resdist is called from).
B	Number of bootstrap samples.
boot.type	A vector of character strings representing the type of bootstrap confidence intervals. Must be one of the two values c("perc", "bca"); boot.type = "perc" is the default.
prob.trans	A logical value indicating whether a probability integral transformation should be performed prior computation of skewness and kurtosis tests.
conf.level	Confidence level for bootstrap confidence intervals.
robust	A logical value indicating whether Siegel's (1982) repeated median estimator should be used for model estimation. If robust = TRUE repeated median estimation is applied for the causally competing models, otherwise ordinary least squares (OLS) estimation is used.
x	An object of class dda.resdist when using print.
...	Additional arguments to be passed to the method.

Value

An object of class dda.resdist containing the results of direction dependence tests of error distributions.

References

Wiedermann, W., & von Eye, A. (2025). Direction Dependence Analysis: Foundations and Statistical Methods. Cambridge, UK: Cambridge University Press.

Examples

set.seed(123)
n <- 500
x <- rchisq(n, df = 4) - 4
e <- rchisq(n, df = 3) - 3
y <- 0.5 * x + e
d <- data.frame(x, y)

result <- dda.resdist(y ~ x, pred = "x", data = d,
            B = 50, conf.level = 0.90, prob.trans = TRUE, robust = FALSE)

print(result)

## Not run: 
## --- Realistic settings; run time is substantial

result <- dda.resdist(y ~ x, pred = "x", data = d, B = 500,
  boot.type = "bca", conf.level = 0.95, prob.trans = TRUE)

print(result)

## --- Siegel (1982) repeated median estimation

dda.resdist(y ~ x, pred = "x", data = d, B = 500, robust = TRUE)

## End(Not run)

## Not run: 
print(result)

## End(Not run)

---

Direction Dependence Analysis: Variable Distributions

Description

dda.vardist evaluates patterns of asymmetry of variable distributions for causally competing models (y ~ x vs. x ~ y). print returns DDA test statistics associated with dda.vardist objects.

Usage

dda.vardist(
  formula,
  pred = NULL,
  data = list(),
  B = 200,
  boot.type = "perc",
  conf.level = 0.95
)

## S3 method for class 'dda.vardist'
print(x, ...)

Arguments

formula	Symbolic formula of the model to be tested or a lm object.
pred	Variable name of the predictor which serves as the outcome in the alternative model.
data	An optional data frame containing the variables in the model (by default variables are taken from the environment which dda.vardist is called from).
B	Number of bootstrap samples.
boot.type	A character indicating the type of bootstrap confidence intervals. Must be one of the two specifications c("perc", "bca"). boot.type = "perc" is the default.
conf.level	Confidence level for bootstrap confidence intervals.
x	An object of class dda.vardist when using print.
...	Additional arguments to be passed to the function.

Value

An object of class dda.vardist containing the results of direction dependence tests of variable distributions.

References

Wiedermann, W., & von Eye, A. (2025). Direction Dependence Analysis: Foundations and Statistical Methods. Cambridge, UK: Cambridge University Press.

See Also

cdda.vardist for a conditional version.

Examples

set.seed(123)
n <- 500
x <- rchisq(n, df = 4) - 4
e <- rchisq(n, df = 3) - 3
y <- 0.5 * x + e
d <- data.frame(x, y)

result <- dda.vardist(y ~ x, pred = "x", data = d, B = 50)
# Note: Only 50 bootstrap samples are used here to lower computation time

## Not run: 
## --- Realistic settings; run time is substantial

result <- dda.vardist(y ~ x, pred = "x", data = d, B = 500,
  boot.type = "bca", conf.level = 0.95)

print(result)

## End(Not run)

print(result)

---

Print Methods for Bootstrap Aggregated DDA Objects

Description

print returns aggregated test statistics and summaries of bootstrap aggregated Direction Dependence Analysis (DDA) objects. The function supports independence properties (obtained from dda.indep), residual distributions (obtained from dda.resdist), variable distributions (obtained from dda.vardist), and OLS summaries (print_ols_summary).

Usage

## S3 method for class 'dda_bagging_indep'
print(
  x,
  agg_stat = NULL,
  trim_prob = 0.1,
  win_prob = 0.1,
  digits = 4,
  alpha = 0.05,
  ...
)

## S3 method for class 'dda_bagging_resdist'
print(x, agg_stat = NULL, trim_prob = 0.1, win_prob = 0.1, digits = 4, ...)

## S3 method for class 'dda_bagging_vardist'
print(x, agg_stat = NULL, trim_prob = 0.1, win_prob = 0.1, digits = 4, ...)

print_ols_summary(
  object,
  agg_stat = NULL,
  trim_prob = 0.1,
  win_prob = 0.1,
  digits = 4,
  ...
)

Arguments

x	An object of class dda_bagging_indep, dda_bagging_resdist, or dda_bagging_vardist.
agg_stat	Character. Specifies the method used for aggregating test statistics and coefficients across bootstrap samples. Must be one of the following specifications c("mean", "median", "trimmed", "winsorized", "midhinge", "tukey"). If NULL, the method established in dda.bagging() is used.
trim_prob	Numeric. Proportion of observations to be trimmed on each side of the sampling distribution when agg.stat = "trimmed" (default: 0.10).
win_prob	Numeric. Proportion of observations to be winsorized on each side of the sampling distribution when agg.stat = "winsorized" (default: 0.10).
digits	Integer. Number of digits used for rounding (default: 4).
alpha	Numeric. Significance level used for causal model selection (default: 0.05).
...	Additional arguments passed to print.
object	Aggregated bagging output, used by print_ols_summary.

Value

Returns a summary of the bootstrap aggregated DDA object.

Examples

set.seed(123)
n <- 200
x <- rchisq(n, df = 4) - 4
e <- rchisq(n, df = 3) - 3
y <- 0.5 * x + e
d <- data.frame(x, y)

base_model <- dda.indep(y ~ x, pred = "x", data = d, B = 20,
                         hetero = TRUE, nlfun = 2, diff = TRUE)
bagged <- dda.bagging(base_model, data = d, iter = 10, inner_B = 20,
                       agg_stat = "mean", progress = FALSE)

# Default print
print(bagged)

# Override aggregation method at print time
print(bagged, agg_stat = "trimmed", trim_prob = 0.05)

# Print aggregated OLS summaries
print_ols_summary(bagged)


set.seed(123)
n <- 200
x <- rchisq(n, df = 4) - 4
e <- rchisq(n, df = 3) - 3
y <- 0.5 * x + e
d <- data.frame(x, y)

base_rd <- dda.resdist(y ~ x, pred = "x", data = d, B = 20)
bagged_rd <- dda.bagging(base_rd, data = d, iter = 10, inner_B = 20,
                          progress = FALSE)
print(bagged_rd)
print(bagged_rd, agg_stat = "median")


set.seed(123)
n <- 200
x <- rchisq(n, df = 4) - 4
e <- rchisq(n, df = 3) - 3
y <- 0.5 * x + e
d <- data.frame(x, y)

base_vd <- dda.vardist(y ~ x, pred = "x", data = d, B = 20)
bagged_vd <- dda.bagging(base_vd, data = d, iter = 10, inner_B = 20,
                          progress = FALSE)
print(bagged_vd)
print(bagged_vd, agg_stat = "winsorized", win_prob = 0.10)

---

Print Summary for OLS Models of Bootstrap Aggregated DDA Objects

Description

summary_ols returns aggregated ordinary least squares (OLS) regression summaries for the causally competing target and alternative models of a bootstrap aggregated DDA object. Regression coefficients and standard errors are aggregated across bootstrap samples using the method specified in dda.bagging() or overridden via agg.stat.

Usage

summary_ols(
  object,
  agg_stat = NULL,
  trim_prob = 0.1,
  win_prob = 0.1,
  digits = 4,
  ...
)

Arguments

object	Output from dda.bagging().
agg_stat	Character. Specifies the method used for aggregating test statistics and coefficients across bootstrap samples. Must be one of the following specifications c("mean", "median", "trimmed", "winsorized", "midhinge", "tukey"). If NULL, the function uses the method applied with dda.bagging().
trim_prob	Numeric. Proportion of observations to be trimmed on each side of the sampling distribution when agg.stat = "trimmed" (default: 0.10).
win_prob	Numeric. Proportion of observations to be winsorized on each side of the sampling distribution when agg.stat = "winsorized" (default: 0.10).
digits	Integer. Number of digits used for rounding.
...	Additional arguments passed to print.

Examples

set.seed(123)
n <- 200
x <- rchisq(n, df = 4) - 4
e <- rchisq(n, df = 3) - 3
y <- 0.5 * x + e
d <- data.frame(x, y)

base_model <- dda.indep(y ~ x, pred = "x", data = d, B = 20)
bagged <- dda.bagging(base_model, data = d, iter = 10, inner_B = 20,
                       progress = FALSE)

# Print aggregated OLS coefficients for target and alternative models
summary_ols(bagged)

# Override aggregation method
summary_ols(bagged, agg_stat = "median")

---

Summary Methods for Bootstrap Aggregated DDA Objects

Description

summary returns the proportion of model selection decisions (i.e., target model, alternative model, or presence of hidden confounding) across the bootstrap samples for bootstrap aggregated Direction Dependence Analysis (DDA) objects.

Usage

## S3 method for class 'dda_bagging_indep'
summary(object, show = NULL, digits = 2, ...)

## S3 method for class 'dda_bagging_vardist'
summary(object, show = NULL, moment = NULL, digits = 2, ...)

## S3 method for class 'dda_bagging_resdist'
summary(object, show = NULL, moment = NULL, digits = 2, ...)

Arguments

object	An object of class dda_bagging_indep, dda_bagging_vardist, or dda_bagging_resdist.
show	Character vector specifying the DDA statistics used to perform model selection. The function accepts the following specifications c("all", "hsic", "dcor", "mi", "bp", "nlcor", "skew", "kurt", "coskew", "cokurt"). If NULL, shows default statistics based on the object type.
digits	Integer. Number of decimal places to print for proportions (default: 2).
...	Additional arguments passed to summary.
moment	Numeric vector indicating which moments to include in the summary (e.g., c(3, 4)). Applicable only to dda_bagging_vardist and dda_bagging_resdist.

Value

Invisibly returns the original object.

---