| Title: | Nonparametric ANCOVA Methods |
|---|---|
| Description: | Nonparametric methods for analysis of covariance (ANCOVA) are distribution-free and provide a flexible statistical framework for situations where the assumptions of parametric ANCOVA are violated or when the response variable is ordinal. This package implements several well-known nonparametric ANCOVA procedures, including Quade, Puri and Sen, McSweeney and Porter, Burnett and Barr, Hettmansperger and McKean, Shirley, and Puri-Sen-Harwell-Serlin. The package provides user-friendly functions to apply these methods in practice. These methods are described in Olejnik et al. (1985) <doi:10.1177/0193841X8500900104> and Harwell et al. (1988) <doi:10.1037/0033-2909.104.2.268>. |
| Authors: | Mina Jahangiri [aut, cre] (ORCID: <https://orcid.org/0000-0001-9119-7571>), Ali Taghavi Rad [aut] (ORCID: <https://orcid.org/0009-0007-5529-8712>), Anoshirvan Kazemnejad [aut] (ORCID: <https://orcid.org/0000-0002-0143-9635>), Keith Goldfeld [ctb] (ORCID: <https://orcid.org/0000-0002-0292-8780>), Shayan Mostafaei [ctb] (ORCID: <https://orcid.org/0000-0002-1966-1306>) |
| Maintainer: | Mina Jahangiri <[email protected]> |
| License: | GPL-3 |
| Version: | 0.1.1 |
| Built: | 2026-05-20 07:21:55 UTC |
| Source: | https://github.com/minajahangiri/npancova |
Implements the Burnett and Barr rank-based method for ANCOVA. This method is suitable for models with one response, one covariate, and one grouping variable.
Burnett_Barr(data, formula)Burnett_Barr(data, formula)
data |
A data frame containing the variables specified in the formula. |
formula |
An object of class "formula": a symbolic description of the model to be fitted. The structure should be 'response ~ covariate + group'. |
A list containing the following components:
The summary of the fitted linear model.
The ANOVA table from the fitted model.
The original data frame with added columns for ranks.
Burnett TD, Barr DRJE, Measurement P. A nonparametric analogy of analysis of covariance. 1977;37(2):341-8.
Olejnik SF, Algina JJER. A review of nonparametric alternatives to analysis of covariance. 1985;9(1):51-83.
# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35) ) # 2. Run the Burnett and Barr method results <- Burnett_Barr( formula = response ~ covariate1 + group, data = data ) # 3. View the results print(results) print(results$anova)# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35) ) # 2. Run the Burnett and Barr method results <- Burnett_Barr( formula = response ~ covariate1 + group, data = data ) # 3. View the results print(results) print(results$anova)
Performs the Harwell and Serlin method using ranked response and covariate variables.
Harwell_Serlin(data, formula)Harwell_Serlin(data, formula)
data |
A data frame containing the variables specified in the formula. |
formula |
An object of class "formula": a symbolic description of the model to be fitted. The structure should be 'response ~ covariate1 + ... + group'. |
A list containing the following components:
The summary of the fitted linear model.
The ANOVA table from the fitted model.
The Harwell-Serlin test statistic.
The degrees of freedom for the test.
The p-value of the test.
The original data frame with added columns for ranks.
Harwell MR, Serlin RCJPB. An empirical study of a proposed test of nonparametric analysis of covariance. 1988;104(2):268.
# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the Harwell and Serlin method results <- Harwell_Serlin( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results$p_value) print(paste("Statistic:", results$statistics,"df:", results$df, "P-value:", results$p_value))# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the Harwell and Serlin method results <- Harwell_Serlin( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results$p_value) print(paste("Statistic:", results$statistics,"df:", results$df, "P-value:", results$p_value))
Applies rank-based residual analysis for ANCOVA. This method involves fitting a model of the response on the covariate, calculating residuals, ranking them, and then performing an ANOVA on the (weighted) ranked residuals.
Hettmansperger_McKean(data, formula)Hettmansperger_McKean(data, formula)
data |
A data frame containing the variables specified in the formula. |
formula |
An object of class "formula": a symbolic description of the model to be fitted. The structure should be 'response ~ covariate1 + ... + group'. |
A list containing the following components:
The summary of the initial model fitting response on covariates.
The summary of the model fitting weighted ranked residuals on the group.
The ANOVA table for the model based on weighted ranked residuals.
A data frame of the mean of weighted ranked residuals for each group.
A data frame of the standard deviation of weighted ranked residuals for each group.
The original data frame augmented with residuals, ranked residuals, and weighted ranked residuals.
Hettmansperger TP, McKean JWJT. A robust alternative based on ranks to least squares in analyzing linear models. 1977;19(3):275-84.
Hettmansperger TP, McKean JWJJotASA. A geometric interpretation of inferences based on ranks in the linear model. 1983;78(384):885-93.
# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the Hettmansperger and McKean method results <- Hettmansperger_McKean( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results) print(results$anova)# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the Hettmansperger and McKean method results <- Hettmansperger_McKean( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results) print(results$anova)
Performs rank-based ANCOVA with and without an interaction term between the covariates and the group.
McSweeny_Porter(data, formula)McSweeny_Porter(data, formula)
data |
A data frame containing the variables specified in the formula. |
formula |
An object of class "formula": a symbolic description of the model to be fitted. The structure should be 'response ~ covariate1 + ... + group'. |
A list containing the following components:
Summary of the model with only covariates.
Summary of the model with covariates and group main effects.
The result of an ANOVA test for group effect.
The result of an ANOVA test for interaction effect between group and covariate variables.
Summary of the model including the interaction term.
The original data frame with added columns for ranks.
McSweeney M, Porter AJOp. Small sample properties of nonparametric index of response and rank analysis of covariance. 1971;16.
Olejnik SF, Algina JJER. A review of nonparametric alternatives to analysis of covariance. 1985;9(1):51-83.
# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the McSweeny and Porter method results <- McSweeny_Porter( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results) print(results$group_effect) print(results$interaction_effect)# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the McSweeny and Porter method results <- McSweeny_Porter( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results) print(results$group_effect) print(results$interaction_effect)
Performs the Puri and Sen method for multiple covariates using a biased variance-covariance matrix.
Puri_Sen_MB(data, formula)Puri_Sen_MB(data, formula)
data |
A data frame containing the variables specified in the formula. |
formula |
An object of class "formula": a symbolic description of the model to be fitted. The structure should be 'response ~ covariate1 + ... + group'. |
A list containing the following components:
A vector of residuals for each group.
The biased variance-covariance matrix.
The inverse of the variance-covariance matrix.
The Puri and Sen L-statistic.
The degrees of freedom for the test.
The corresponding p-value of the L-statistic.
The original data frame with added columns for ranks.
Puri ML, Sen PKJAoMS. Analysis of covariance based on general rank scores. 1969;40(2):610-8.
Olejnik SF, Algina JJER. A review of nonparametric alternatives to analysis of covariance. 1985;9(1):51-83.
# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the Puri and Sen (MB) method results <- Puri_Sen_MB( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results) print(paste("Statistic:", results$L_statistic,"df:", results$df, "P-value:", results$p_value))# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the Puri and Sen (MB) method results <- Puri_Sen_MB( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results) print(paste("Statistic:", results$L_statistic,"df:", results$df, "P-value:", results$p_value))
Performs the Puri and Sen method for multiple covariates using an unbiased variance-covariance matrix.
Puri_Sen_MU(data, formula)Puri_Sen_MU(data, formula)
data |
A data frame containing the variables specified in the formula. |
formula |
An object of class "formula": a symbolic description of the model to be fitted. The structure should be 'response ~ covariate1 + ... + group'. |
A list containing the following components:
A vector of residuals for each group.
The unbiased variance-covariance matrix.
The inverse of the variance-covariance matrix.
The Puri and Sen L-statistic.
The degrees of freedom for the test.
The corresponding p-value of the L-statistic.
The original data frame with added columns for ranks.
Puri ML, Sen PKJAoMS. Analysis of covariance based on general rank scores. 1969;40(2):610-8.
Olejnik SF, Algina JJER. A review of nonparametric alternatives to analysis of covariance. 1985;9(1):51-83.
# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the Puri and Sen (MU) method results <- Puri_Sen_MU( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results) print(paste("Statistic:", results$L_statistic,"df:", results$df, "P-value:", results$p_value))# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the Puri and Sen (MU) method results <- Puri_Sen_MU( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results) print(paste("Statistic:", results$L_statistic,"df:", results$df, "P-value:", results$p_value))
Performs the Puri and Sen method for a single covariate using a biased variance-covariance matrix.
Puri_Sen_OB(data, formula)Puri_Sen_OB(data, formula)
data |
A data frame containing the variables specified in the formula. |
formula |
An object of class "formula": a symbolic description of the model to be fitted. The structure should be 'response ~ covariate + group'. |
A list containing the following components:
A vector of residuals for each group.
The biased variance-covariance matrix.
The inverse of the variance-covariance matrix.
The Puri and Sen L-statistic.
The degrees of freedom for the test.
The corresponding p-value of the L-statistic.
The original data frame with added columns for ranks.
Puri ML, Sen PKJAoMS. Analysis of covariance based on general rank scores. 1969;40(2):610-8.
Olejnik SF, Algina JJER. A review of nonparametric alternatives to analysis of covariance. 1985;9(1):51-83.
# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35) ) # 2. Run the Puri and Sen (OB) method results <- Puri_Sen_OB( formula = response ~ covariate1 + group, data = data ) # 3. View the results print(results) print(paste("Statistic:", results$L_statistic,"df:", results$df, "P-value:", results$p_value))# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35) ) # 2. Run the Puri and Sen (OB) method results <- Puri_Sen_OB( formula = response ~ covariate1 + group, data = data ) # 3. View the results print(results) print(paste("Statistic:", results$L_statistic,"df:", results$df, "P-value:", results$p_value))
Performs the Puri and Sen method for a single covariate using an unbiased variance-covariance matrix.
Puri_Sen_OU(data, formula)Puri_Sen_OU(data, formula)
data |
A data frame containing the variables specified in the formula. |
formula |
An object of class "formula": a symbolic description of the model to be fitted. The structure should be 'response ~ covariate + group'. |
A list containing the following components:
A vector of residuals for each group.
The unbiased variance-covariance matrix.
The inverse of the variance-covariance matrix.
The Puri and Sen L-statistic.
The degrees of freedom for the test.
The corresponding p-value of the L-statistic.
The original data frame with added columns for ranks.
Puri ML, Sen PKJAoMS. Analysis of covariance based on general rank scores. 1969;40(2):610-8.
Olejnik SF, Algina JJER. A review of nonparametric alternatives to analysis of covariance. 1985;9(1):51-83.
# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35) ) # 2. Run the Puri and Sen (OU) method results <- Puri_Sen_OU( formula = response ~ covariate1 + group, data = data ) # 3. View the results print(results) print(paste("Statistic:", results$L_statistic,"df:", results$df, "P-value:", results$p_value))# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35) ) # 2. Run the Puri and Sen (OU) method results <- Puri_Sen_OU( formula = response ~ covariate1 + group, data = data ) # 3. View the results print(results) print(paste("Statistic:", results$L_statistic,"df:", results$df, "P-value:", results$p_value))
Performs Quade's ANCOVA using ranked variables and analysis of residuals. The method fits a linear model of the ranked response on the ranked covariates, and then performs an ANOVA on the residuals of that model.
Quade(data, formula)Quade(data, formula)
data |
A data frame containing the variables specified in the formula. |
formula |
An object of class "formula": a symbolic description of the model to be fitted. The structure should be 'response ~ covariate1 + ... + group'. |
A list containing the following components:
Summary of the linear model regressing the ranked response on the ranked covariates.
The summary of the ANOVA model performed on the residuals.
A data frame of the mean of residuals for each group.
A data frame of the standard deviation of residuals for each group.
The summary of the model fitting residuals on the group.
The original data frame augmented with ranked variables and residuals.
Quade DJJotASA. Rank analysis of covariance. 1967;62(320):1187-200.
Olejnik SF, Algina JJER. A review of nonparametric alternatives to analysis of covariance. 1985;9(1):51-83.
# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the Quade method results <- Quade( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results) print(results$anova_summary)# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the Quade method results <- Quade( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results) print(results$anova_summary)
Calculates group and interaction effects based on ranked response and covariate variables using changes in R-squared values between models.
Shirley(data, formula)Shirley(data, formula)
data |
A data frame containing the variables specified in the formula. |
formula |
An object of class "formula": a symbolic description of the model to be fitted. The structure should be 'response ~ covariate1 + ... + group'. |
A list containing components related to the group and interaction effects, including:
The test statistic for the main group effect.
The p-value for the main group effect.
Degrees of freedom for the group effect.
The test statistic for the interaction effect.
The p-value for the interaction effect.
Degrees of freedom for the interaction effect.
Summary of the model with only covariates.
Summary of the model with covariates and group main effects.
Summary of the model including the interaction term.
The original data frame with added columns for ranks.
Burnett TD, Barr DRJE, Measurement P. A nonparametric analogy of analysis of covariance. 1977;37(2):341-8.
Olejnik SF, Algina JJER. A review of nonparametric alternatives to analysis of covariance. 1985;9(1):51-83.
# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the Shirley method results <- Shirley( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results) print(paste("Statistic:", results$statistics_group, "df_group:", results$df_group, "P-value:", results$p_value_group)) print(paste("Statistic:", results$statistics_interaction, "df_interaction:", results$df_interaction, "P-value:", results$p_value_interaction))# 1. Create a sample data frame data <- data.frame( group = c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3), response = c(16, 60, 82, 126, 137, 44, 67, 87, 100, 142, 17, 28, 105, 149, 160), covariate1 = c(26, 10, 42, 49, 55, 21, 28, 5, 12, 58, 1, 19, 41, 48, 35), covariate2 = c(12, 21, 24, 29, 34, 17, 2, 40, 38, 36, 8, 1, 9, 28, 16) ) # 2. Run the Shirley method results <- Shirley( formula = response ~ covariate1 + covariate2 + group, data = data ) # 3. View the results print(results) print(paste("Statistic:", results$statistics_group, "df_group:", results$df_group, "P-value:", results$p_value_group)) print(paste("Statistic:", results$statistics_interaction, "df_interaction:", results$df_interaction, "P-value:", results$p_value_interaction))