Package 'staggered' reference manual

Title:	Efficient Estimation Under Staggered Treatment Timing
Description:	Efficiently estimates treatment effects in settings with randomized staggered rollouts, using tools proposed by Roth and Sant'Anna (2021) <arXiv:2102.01291>.
Authors:	Jonathan Roth [aut], Pedro H.C. Sant'Anna [aut, cre]
Maintainer:	Pedro H.C. Sant'Anna <[email protected]>
License:	GPL-2
Version:	1.1
Built:	2024-09-03 04:35:15 UTC
Source:	https://github.com/cran/staggered

Plug-in efficient Beta hat

Description

compute_Betastar computes the plug-in efficient betahat

Usage

compute_Betastar(
  Ybar_g_list,
  A_theta_list,
  A_0_list,
  S_g_list,
  N_g_list,
  Xvar_list = NULL
)
compute_Betastar(
  Ybar_g_list,
  A_theta_list,
  A_0_list,
  S_g_list,
  N_g_list,
  Xvar_list = NULL
)

Arguments

`Ybar_g_list`	Ybar_g_list
`A_theta_list`	A_theta_list
`A_0_list`	A_0_list
`S_g_list`	S_g_list
`N_g_list`	N_g_list
`Xvar_list`	Xvar_list

Value

betastar Vector of plug-in efficient betahat estimates.

Calculate group level summary statistics

Description

This function computes the mean-vector and covariance matrix of the outcomes for each cohort, where a cohort g is a group of units first treated in period g

Usage

compute_g_level_summaries(df, is_balanced = TRUE)
compute_g_level_summaries(df, is_balanced = TRUE)

Arguments

`df`	A data frame containing panel data with the variables y (an outcome), i (an individual identifier), t (the period in which the outcome is observe), g (the period in which i is first treated, with Inf denoting never treated)
`is_balanced`	If true, the df has previously been balanced so this does not need to be done internally.

Value

Y_bar_list A list of the means of the outcomes for each cohort g

S_g_list A list of covariance matrices for the outcomes for each cohort g

N_g_list A list of the number of observations for each cohort g

g_list A list of when the cohorts were first treated

t_list A list of the the time periods for the outcome. The vector of outcomes corresponds with this order.

Compute Xhat of pre-treatment differences

Description

compute_Xhat computes the vector Xhat of pre-treatment differences given the list of cohort means Ybar_g_list and the list of matrices A_0_list

Usage

compute_Xhat(Ybar_g_list, A_0_list)
compute_Xhat(Ybar_g_list, A_0_list)

Arguments

`Ybar_g_list`	Ybar_g_list
`A_0_list`	A_0_list

Value

Xhat the vector Xhat of pre-treatment differences to be used as regressors

create_A0_list

Description

create_A0_list creates the list of A_0 matrices for Xhat corresponding with all possible comparisons of cohorts before they are treated

Usage

create_A0_list(g_list, t_list)
create_A0_list(g_list, t_list)

Arguments

`g_list`	g_list
`t_list`	t_list

Value

A0_list list of A_0 matrices for Xhat corresponding with all possible comparisons of cohorts before they are treated

Procedural Justice Training Program in the Chicago Police Department

Description

Data from a large-scale procedural justice training program in the Chicago Police Department analyzed by Wood, Tyler, Papachristos, Roth and Sant'Anna (2020) and Roth and Sant'Anna (2021). The data contains a balanced panel of 7,785 police officers in Chicago who were randomly given a procedural justice training on different dates, and who remained in the police force throughout the study period (from January 2011 to December 2016).

Usage

pj_officer_level_balanced
pj_officer_level_balanced

Format

A data frame with 560520 observations (7,785 police officers and 72 months) and 12 variables:

uid: identifier for the police officer
month: month and year of the observation
assigned: month-year of first training assignment
appointed: appointment date
resigned: Date the police officer resigned. NA if he/she did not resigned by the time data was collected
birth_year: Officer's year of birth
assigned_exact: Exact date of first training assignment
complaints: Number of complaints (setlled and sustained)
sustained: Number of sustained complaints
force: Number of times force was used
period: Time period: 1 - 72
first_trained: Time period first exposed to treatment (Treatment cohort/group)

Source

Wood, Tyler, Papachristos, Roth and Sant'Anna (2020) and Roth and Sant'Anna (2021).

References

Roth, Jonatahan, and Sant'Anna, Pedro H. C. (2021), 'Efficient Estimation for Staggered Rollout Designs', arXiv: 2102.01291, https://arxiv.org/abs/2102.01291.

Wood, George, Tyler, Tom R., Papachristos, Andrew P., Roth, Jonathan and Sant'Anna, Pedro H. C. (2020), 'Revised findings for "Procedural justice training reduces police use of force and complaints against officers", doi:10.31235/osf.io/xf32m.

Calculate the efficient adjusted estimator in staggered rollout designs

Description

This functions calculates the efficient estimator for staggered rollout designs proposed by Roth and Sant'Anna.

Usage

staggered(
  df,
  i = "i",
  t = "t",
  g = "g",
  y = "y",
  estimand = NULL,
  A_theta_list = NULL,
  A_0_list = NULL,
  eventTime = 0,
  beta = NULL,
  use_DiD_A0 = ifelse(is.null(A_0_list), TRUE, FALSE),
  return_full_vcv = FALSE,
  return_matrix_list = FALSE,
  use_last_treated_only = FALSE,
  compute_fisher = FALSE,
  num_fisher_permutations = 500,
  skip_data_check = FALSE
)
staggered(
  df,
  i = "i",
  t = "t",
  g = "g",
  y = "y",
  estimand = NULL,
  A_theta_list = NULL,
  A_0_list = NULL,
  eventTime = 0,
  beta = NULL,
  use_DiD_A0 = ifelse(is.null(A_0_list), TRUE, FALSE),
  return_full_vcv = FALSE,
  return_matrix_list = FALSE,
  use_last_treated_only = FALSE,
  compute_fisher = FALSE,
  num_fisher_permutations = 500,
  skip_data_check = FALSE
)

Arguments

`df`	A data frame containing panel data with the variables y (an outcome), i (an individual identifier), t (the period in which the outcome is observe), g (the period in which i is first treated, with Inf denoting never treated)
`i`	The name of column containing the individual (cross-sectional unit) identifier. Default is "i".
`t`	The name of the column containing the time periods. Default is "t".
`g`	The name of the column containing the first period when a particular observation is treated, with Inf denoting never treated. Default is "g".
`y`	The name of the column containing the outcome variable. Default is "y".
`estimand`	The estimand to be calculated: "simple" averages all treated (t,g) combinations with weights proportional to N_g; "cohort" averages the ATEs for each cohort g, and then takes an N_g-weighted average across g; "calendar" averages ATEs for each time period, weighted by N_g for treated units, and then averages across time. "eventstudy" returns the average effect at the ”event-time” given in the parameter EventTime. The parameter can be left blank if a custom parameter is provided in A_theta_list. The argument is not case-sensitive.
`A_theta_list`	This parameter allows for specifying a custom estimand, and should be left as NULL if estimand is specified. It is a list of matrices A_theta_g so that the parameter of interest is sum_g A_theta_g Ybar_g, where Ybar_g = 1/N sum_i Y_i(g)
`A_0_list`	This parameter allow for specifying the matrices used to construct the Xhat vector of pre-treatment differences. If left NULL, the default is to use the scalar set of controls used in Callaway and Sant'Anna. If use_DiD_A0 = FALSE, then it uses the full vector possible comparisons of (g,g') in periods t<g,g'.
`eventTime`	If using estimand = "eventstudy", specify what eventTime you want the event-study parameter for. The default is 0, the period in which treatment occurs. If a vector is provided, estimates are returned for all the event-times in the vector.
`beta`	A coefficient to use for covariate adjustment. If not specified, the plug-in optimal coefficient is used. beta =0 corresponds with the simple difference-in-means. beta = 1 corresponds with the Callaway and Sant'Anna estimator when using the default value of use_DiD_A0 = TRUE.
`use_DiD_A0`	If this parameter is true, then Xhat corresponds with the scalar used by Callaway and Sant'Anna, so the Callaway and Sant'Anna estimator corresponds with beta=1. If it is false, the Xhat is a vector with all possible comparisons of pairs of cohorts before either is treated. The latter option should only be used when the number of possible comparisons is small relative to sample size.
`return_full_vcv`	If this is true and estimand = "eventstudy", then the function returns a list containing the full variance-covariance matrix for the event-plot estimates in addition to the usual dataframe with the estimates
`return_matrix_list`	If true, the function returns a list of the A_0_list and A_theta_list matrices along with betastar. This is used for internal recursive calls to calculate the variance-covariance matrix, and will generally not be needed by the end-user. Default is False.
`use_last_treated_only`	If true, then A_0_list and A_theta_list are created to only make comparisons with the last treated cohorts (as suggested by Sun and Abraham), rather than using not-yet-treated units as comparisons. If set to TRUE (and use_DiD_A0 = TRUE), then beta=1 corresponds with the Sun and Abraham estimator.
`compute_fisher`	If true, computes a Fisher Randomization Test using the studentized estimator.
`num_fisher_permutations`	The number of permutations to use in the Fisher Randomization Test (if compute_fisher = TRUE). Default is 500.
`skip_data_check`	If true, skips checks that the data is balanced and contains the colums i,t,g,y. Used in internal recursive calls to increase speed, but not recommended for end-user.

Value

resultsDF A data.frame containing: estimate (the point estimate), se (the standard error), and se_neyman (the Neyman standard error). If a vector-valued eventTime is provided, the data.frame contains multiple rows for each eventTime and an eventTime column. If return_full_vcv = TRUE and estimand = "eventstudy", the function returns a list containing resultsDF and the full variance covariance for the event-study estimates (vcv) as well as the Neyman version of the covariance matrix (vcv_neyman). (If return_matrix_list = TRUE, it likewise returns a list containing lists of matrices used in the vcv calculation.)

References

Roth, Jonatahan, and Sant'Anna, Pedro H. C. (2021), 'Efficient Estimation for Staggered Rollout Designs', arXiv: 2102.01291, https://arxiv.org/abs/2102.01291.

Examples


# Load some libraries
library(dplyr)
library(purrr)
library(MASS)
set.seed(1234)
# load the officer data and subset it
df <- pj_officer_level_balanced
group_random <- sample(unique(df$assigned), 3)
df <- df[df$assigned %in% group_random,]
# Calculate efficient estimator for the simple weighted average
staggered(df = df,
  i = "uid",
  t = "period",
  g = "first_trained",
  y = "complaints",
  estimand = "simple")
# Calculate efficient estimator for the cohort weighted average
staggered(df = df,
  i = "uid",
  t = "period",
  g = "first_trained",
  y = "complaints",
  estimand = "cohort")
# Calculate efficient estimator for the calendar weighted average
staggered(df = df,
  i = "uid",
  t = "period",
  g = "first_trained",
  y = "complaints",
  estimand = "calendar")
# Calculate event-study coefficients for the first 24 months
# (month 0 is instantaneous effect)
eventPlotResults <- staggered(df = df,
  i = "uid",
  t = "period",
  g = "first_trained",
  y = "complaints",
  estimand = "eventstudy",
  eventTime = 0:23)
eventPlotResults %>% head()

# Load some libraries
library(dplyr)
library(purrr)
library(MASS)
set.seed(1234)
# load the officer data and subset it
df <- pj_officer_level_balanced
group_random <- sample(unique(df$assigned), 3)
df <- df[df$assigned %in% group_random,]
# Calculate efficient estimator for the simple weighted average
staggered(df = df,
  i = "uid",
  t = "period",
  g = "first_trained",
  y = "complaints",
  estimand = "simple")
# Calculate efficient estimator for the cohort weighted average
staggered(df = df,
  i = "uid",
  t = "period",
  g = "first_trained",
  y = "complaints",
  estimand = "cohort")
# Calculate efficient estimator for the calendar weighted average
staggered(df = df,
  i = "uid",
  t = "period",
  g = "first_trained",
  y = "complaints",
  estimand = "calendar")
# Calculate event-study coefficients for the first 24 months
# (month 0 is instantaneous effect)
eventPlotResults <- staggered(df = df,
  i = "uid",
  t = "period",
  g = "first_trained",
  y = "complaints",
  estimand = "eventstudy",
  eventTime = 0:23)
eventPlotResults %>% head()

Calculate the Callaway & Sant'Anna (2020) estimator for staggered rollouts

Description

This functions calculates the Callaway & Sant'Anna (2020) estimator for staggered rollout designs using not-yet-treated units (including never-treated, if available) as controls.

Usage

staggered_cs(
  df,
  i = "i",
  t = "t",
  g = "g",
  y = "y",
  estimand = NULL,
  A_theta_list = NULL,
  A_0_list = NULL,
  eventTime = 0,
  return_full_vcv = FALSE,
  return_matrix_list = FALSE,
  compute_fisher = FALSE,
  num_fisher_permutations = 500,
  skip_data_check = FALSE
)
staggered_cs(
  df,
  i = "i",
  t = "t",
  g = "g",
  y = "y",
  estimand = NULL,
  A_theta_list = NULL,
  A_0_list = NULL,
  eventTime = 0,
  return_full_vcv = FALSE,
  return_matrix_list = FALSE,
  compute_fisher = FALSE,
  num_fisher_permutations = 500,
  skip_data_check = FALSE
)

Arguments

`df`	A data frame containing panel data with the variables y (an outcome), i (an individual identifier), t (the period in which the outcome is observe), g (the period in which i is first treated, with Inf denoting never treated)
`i`	The name of column containing the individual (cross-sectional unit) identifier. Default is "i".
`t`	The name of the column containing the time periods. Default is "t".
`g`	The name of the column containing the first period when a particular observation is treated, with Inf denoting never treated. Default is "g".
`y`	The name of the column containing the outcome variable. Default is "y".
`estimand`	The estimand to be calculated: "simple" averages all treated (t,g) combinations with weights proportional to N_g; "cohort" averages the ATEs for each cohort g, and then takes an N_g-weighted average across g; "calendar" averages ATEs for each time period, weighted by N_g for treated units, and then averages across time. "eventstudy" returns the average effect at the ”event-time” given in the parameter EventTime. The parameter can be left blank if a custom parameter is provided in A_theta_list. The argument is not case-sensitive.
`A_theta_list`	This parameter allows for specifying a custom estimand, and should be left as NULL if estimand is specified. It is a list of matrices A_theta_g so that the parameter of interest is sum_g A_theta_g Ybar_g, where Ybar_g = 1/N sum_i Y_i(g)
`A_0_list`	This parameter allow for specifying the matrices used to construct the Xhat vector of pre-treatment differences. If left NULL, the default is to use the scalar set of controls used in Callaway and Sant'Anna. If use_DiD_A0 = FALSE, then it uses the full vector possible comparisons of (g,g') in periods t<g,g'.
`eventTime`	If using estimand = "eventstudy", specify what eventTime you want the event-study parameter for. The default is 0, the period in which treatment occurs. If a vector is provided, estimates are returned for all the event-times in the vector.
`return_full_vcv`	If this is true and estimand = "eventstudy", then the function returns a list containing the full variance-covariance matrix for the event-plot estimates in addition to the usual dataframe with the estimates
`return_matrix_list`	If true, the function returns a list of the A_0_list and A_theta_list matrices along with betastar. This is used for internal recursive calls to calculate the variance-covariance matrix, and will generally not be needed by the end-user. Default is False.
`compute_fisher`	If true, computes a Fisher Randomization Test using the studentized estimator.
`num_fisher_permutations`	The number of permutations to use in the Fisher Randomization Test (if compute_fisher = TRUE). Default is 500.
`skip_data_check`	If true, skips checks that the data is balanced and contains the colums i,t,g,y. Used in internal recursive calls to increase speed, but not recommended for end-user.

Value

References

Callaway, Brantly, and Sant'Anna, Pedro H. C. (2020), 'Difference-in-Differences with Multiple Time Periods', Forthcoming at the Journal of Econometrics, doi:10.1016/j.jeconom.2020.12.001.

Examples

# Load some libraries
library(dplyr)
library(purrr)
library(MASS)
set.seed(1234)
# load the officer data and subset it
df <- pj_officer_level_balanced
group_random <- sample(unique(df$assigned), 3)
df <- df[df$assigned %in% group_random,]
# We modify the data so that the time dimension is named t,
# the period of treatment is named g,
# the outcome is named y,
# and the individual identifiers are named i
# (this allow us to use default arguments on \code{staggered_cs}).
df <- df %>% rename(t = period, y = complaints, g = first_trained, i = uid)
# Calculate Callaway and Sant'Anna estimator for the simple weighted average
staggered_cs(df = df, estimand = "simple")
# Calculate Callaway and Sant'Anna estimator for the cohort weighted average
staggered_cs(df = df, estimand = "cohort")
# Calculate Callaway and Sant'Anna estimator for the calendar weighted average
staggered_cs(df = df, estimand = "calendar")
# Calculate Callaway and Sant'Anna event-study coefficients for the first 24 months
# (month 0 is instantaneous effect)
eventPlotResults <- staggered_cs(df = df, estimand = "eventstudy", eventTime = 0:23)
eventPlotResults %>% head()

# Load some libraries
library(dplyr)
library(purrr)
library(MASS)
set.seed(1234)
# load the officer data and subset it
df <- pj_officer_level_balanced
group_random <- sample(unique(df$assigned), 3)
df <- df[df$assigned %in% group_random,]
# We modify the data so that the time dimension is named t,
# the period of treatment is named g,
# the outcome is named y,
# and the individual identifiers are named i
# (this allow us to use default arguments on \code{staggered_cs}).
df <- df %>% rename(t = period, y = complaints, g = first_trained, i = uid)
# Calculate Callaway and Sant'Anna estimator for the simple weighted average
staggered_cs(df = df, estimand = "simple")
# Calculate Callaway and Sant'Anna estimator for the cohort weighted average
staggered_cs(df = df, estimand = "cohort")
# Calculate Callaway and Sant'Anna estimator for the calendar weighted average
staggered_cs(df = df, estimand = "calendar")
# Calculate Callaway and Sant'Anna event-study coefficients for the first 24 months
# (month 0 is instantaneous effect)
eventPlotResults <- staggered_cs(df = df, estimand = "eventstudy", eventTime = 0:23)
eventPlotResults %>% head()

Calculate the Sun & Abraham (2020) estimator for staggered rollouts

Description

This functions calculates the Sun & Abraham (2020) estimator for staggered rollout designs using last-treated-treated units (never-treated, if availabe) as controls.

Usage

staggered_sa(
  df,
  i = "i",
  t = "t",
  g = "g",
  y = "y",
  estimand = NULL,
  A_theta_list = NULL,
  A_0_list = NULL,
  eventTime = 0,
  return_full_vcv = FALSE,
  return_matrix_list = FALSE,
  compute_fisher = FALSE,
  num_fisher_permutations = 500,
  skip_data_check = FALSE
)
staggered_sa(
  df,
  i = "i",
  t = "t",
  g = "g",
  y = "y",
  estimand = NULL,
  A_theta_list = NULL,
  A_0_list = NULL,
  eventTime = 0,
  return_full_vcv = FALSE,
  return_matrix_list = FALSE,
  compute_fisher = FALSE,
  num_fisher_permutations = 500,
  skip_data_check = FALSE
)

Arguments

`df`	A data frame containing panel data with the variables y (an outcome), i (an individual identifier), t (the period in which the outcome is observe), g (the period in which i is first treated, with Inf denoting never treated)
`i`	The name of column containing the individual (cross-sectional unit) identifier. Default is "i".
`t`	The name of the column containing the time periods. Default is "t".
`g`	The name of the column containing the first period when a particular observation is treated, with Inf denoting never treated. Default is "g".
`y`	The name of the column containing the outcome variable. Default is "y".
`estimand`	The estimand to be calculated: "simple" averages all treated (t,g) combinations with weights proportional to N_g; "cohort" averages the ATEs for each cohort g, and then takes an N_g-weighted average across g; "calendar" averages ATEs for each time period, weighted by N_g for treated units, and then averages across time. "eventstudy" returns the average effect at the ”event-time” given in the parameter EventTime. The parameter can be left blank if a custom parameter is provided in A_theta_list. The argument is not case-sensitive.
`A_theta_list`	This parameter allows for specifying a custom estimand, and should be left as NULL if estimand is specified. It is a list of matrices A_theta_g so that the parameter of interest is sum_g A_theta_g Ybar_g, where Ybar_g = 1/N sum_i Y_i(g)
`A_0_list`	This parameter allow for specifying the matrices used to construct the Xhat vector of pre-treatment differences. If left NULL, the default is to use the scalar set of controls used in Callaway and Sant'Anna. If use_DiD_A0 = FALSE, then it uses the full vector possible comparisons of (g,g') in periods t<g,g'.
`eventTime`	If using estimand = "eventstudy", specify what eventTime you want the event-study parameter for. The default is 0, the period in which treatment occurs. If a vector is provided, estimates are returned for all the event-times in the vector.
`return_full_vcv`	If this is true and estimand = "eventstudy", then the function returns a list containing the full variance-covariance matrix for the event-plot estimates in addition to the usual dataframe with the estimates
`return_matrix_list`	If true, the function returns a list of the A_0_list and A_theta_list matrices along with betastar. This is used for internal recursive calls to calculate the variance-covariance matrix, and will generally not be needed by the end-user. Default is False.
`compute_fisher`	If true, computes a Fisher Randomization Test using the studentized estimator.
`num_fisher_permutations`	The number of permutations to use in the Fisher Randomization Test (if compute_fisher = TRUE). Default is 500.
`skip_data_check`	If true, skips checks that the data is balanced and contains the colums i,t,g,y. Used in internal recursive calls to increase speed, but not recommended for end-user.

Value

References

Sun, Liyang, and Abraham, Sarah (2020), 'Estimating dynamic treatment effects in event studies with heterogeneous treatment effects', Forthcoming at the Journal of Econometrics, doi:10.1016/j.jeconom.2020.09.006.

Examples

# Load some libraries
library(dplyr)
library(purrr)
library(MASS)
set.seed(1234)
# load the officer data and subset it
df <- pj_officer_level_balanced
group_random <- sample(unique(df$assigned), 3)
df <- df[df$assigned %in% group_random,]
# We modify the data so that the time dimension is named t,
# the period of treatment is named g,
# the outcome is named y,
# and the individual identifiers are named i
#  (this allow us to use default arguments on \code{staggered_cs}).
df <- df %>% rename(t = period, y = complaints, g = first_trained, i = uid)
# Calculate Sun and Abraham estimator for the simple weighted average
staggered_sa(df = df, estimand = "simple")
# Calculate Sun and Abraham estimator for the cohort weighted average
staggered_sa(df = df, estimand = "cohort")
# Calculate Sun and Abraham estimator for the calendar weighted average
staggered_sa(df = df, estimand = "calendar")
# Calculate Sun and Abraham event-study coefficients for the first 24 months
# (month 0 is instantaneous effect)
# eventPlotResults <- staggered_sa(df = df, estimand = "eventstudy", eventTime = 0:23)
# eventPlotResults %>% head()

# Load some libraries
library(dplyr)
library(purrr)
library(MASS)
set.seed(1234)
# load the officer data and subset it
df <- pj_officer_level_balanced
group_random <- sample(unique(df$assigned), 3)
df <- df[df$assigned %in% group_random,]
# We modify the data so that the time dimension is named t,
# the period of treatment is named g,
# the outcome is named y,
# and the individual identifiers are named i
#  (this allow us to use default arguments on \code{staggered_cs}).
df <- df %>% rename(t = period, y = complaints, g = first_trained, i = uid)
# Calculate Sun and Abraham estimator for the simple weighted average
staggered_sa(df = df, estimand = "simple")
# Calculate Sun and Abraham estimator for the cohort weighted average
staggered_sa(df = df, estimand = "cohort")
# Calculate Sun and Abraham estimator for the calendar weighted average
staggered_sa(df = df, estimand = "calendar")
# Calculate Sun and Abraham event-study coefficients for the first 24 months
# (month 0 is instantaneous effect)
# eventPlotResults <- staggered_sa(df = df, estimand = "eventstudy", eventTime = 0:23)
# eventPlotResults %>% head()

Package 'staggered'

Help Index

Plug-in efficient Beta hat

Description

Usage

Arguments

Value

Calculate group level summary statistics

Description

Usage

Arguments

Value

Compute Xhat of pre-treatment differences

Description

Usage

Arguments

Value

create_A0_list

Description

Usage

Arguments

Value

Procedural Justice Training Program in the Chicago Police Department

Description

Usage

Format

Source

References

Calculate the efficient adjusted estimator in staggered rollout designs

Description

Usage

Arguments

Value

References

Examples

Calculate the Callaway & Sant'Anna (2020) estimator for staggered rollouts

Description

Usage

Arguments

Value

References

Examples

Calculate the Sun & Abraham (2020) estimator for staggered rollouts

Description

Usage

Arguments

Value

References

Examples