Package 'ipd'

Calculation of the matrix A based on single dataset

Description

A function for the calculation of the matrix A based on single dataset

Usage

A(
  X,
  Y,
  quant = NA,
  theta,
  method = c("ols", "quantile", "mean", "logistic", "poisson")
)

Arguments

X	Array or data.frame containing covariates
Y	Array or data.frame of outcomes
quant	quantile for quantile estimation
theta	parameter theta
method	indicates the method to be used for M-estimation. Options include "mean", "quantile", "ols", "logistic", and "poisson".

Value

matrix A based on single dataset

---

Augment data from an ipd fit

Description

Augment data from an ipd fit

Usage

## S3 method for class 'ipd'
augment(x, data = x@data_u, ...)

Arguments

x	An object of class ipd.
data	A data.frame to augment; defaults to x@data_u.
...	Ignored.

Value

The data.frame with columns .fitted and .resid.

Examples

dat <- simdat()

fit <- ipd(Y - f ~ X1, method = "pspa", model = "ols",

    data = dat, label = "set_label")

augmented_df <- augment(fit)

head(augmented_df)

---

Estimate PPI++ Power Tuning Parameter

Description

Calculates the optimal value of lhat for the prediction-powered confidence interval for GLMs.

Usage

calc_lhat_glm(
  grads,
  grads_hat,
  grads_hat_unlabeled,
  inv_hessian,
  coord = NULL,
  clip = FALSE
)

Arguments

grads	(matrix): n x p matrix gradient of the loss function with respect to the parameter evaluated at the labeled data.
grads_hat	(matrix): n x p matrix gradient of the loss function with respect to the model parameter evaluated using predictions on the labeled data.
grads_hat_unlabeled	(matrix): N x p matrix gradient of the loss function with respect to the parameter evaluated using predictions on the unlabeled data.
inv_hessian	(matrix): p x p matrix inverse of the Hessian of the loss function with respect to the parameter.
coord	(int, optional): Coordinate for which to optimize lhat. If None, it optimizes the total variance over all coordinates. Must be in (1, ..., d) where d is the shape of the estimand.
clip	(boolean, optional): Whether to clip the value of lhat to be non-negative. Defaults to False.

Value

(float): Optimal value of lhat in [0,1].

---

Chen & Chen Logistic

Description

Helper function for Chen & Chen logistic regression estimation.

Usage

chen_logistic(X_l, Y_l, f_l, X_u, f_u, intercept = TRUE)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
intercept	(Logical): Do the design matrices include intercept columns? Default is TRUE.

Details

Another look at statistical inference with machine learning-imputed data (Gronsbell et al., 2026) doi:10.48550/arXiv.2411.19908

Value

(list): A list containing the following:

est

(vector): vector of Chen & Chen logistic regression coefficient estimates.

se

(vector): vector of standard errors of the coefficients.

Examples

dat <- simdat(model = "logistic")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

chen_logistic(X_l, Y_l, f_l, X_u, f_u, intercept = TRUE)

---

Chen & Chen OLS

Description

Helper function for Chen & Chen OLS estimation

Usage

chen_ols(X_l, Y_l, f_l, X_u, f_u, intercept = TRUE)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
intercept	(Logical): Do the design matrices include intercept columns? Default is TRUE.

Details

Another look at statistical inference with machine learning-imputed data (Gronsbell et al., 2026) doi:10.48550/arXiv.2411.19908

Value

(list): A list containing the following:

est

(vector): vector of Chen & Chen OLS regression coefficient estimates.

se

(vector): vector of standard errors of the coefficients.

Examples

dat <- simdat(model = "ols")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

chen_ols(X_l, Y_l, f_l, X_u, f_u, intercept = TRUE)

---

Chen & Chen Poisson

Description

Helper function for Chen & Chen Poisson regression estimation.

Usage

chen_poisson(X_l, Y_l, f_l, X_u, f_u, intercept = TRUE)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
intercept	(Logical): Do the design matrices include intercept columns? Default is TRUE.

Details

Another look at statistical inference with machine learning-imputed data (Gronsbell et al., 2026) doi:10.48550/arXiv.2411.19908

Value

(list): A list containing the following:

est

(vector): vector of Chen & Chen Poisson regression coefficient estimates.

se

(vector): vector of standard errors of the coefficients.

Examples

dat <- simdat(model = "poisson")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

chen_poisson(X_l, Y_l, f_l, X_u, f_u, intercept = TRUE)

---

Empirical CDF of the Data

Description

Computes the empirical CDF of the data.

Usage

compute_cdf(Y, grid, w = NULL)

Arguments

Y	(matrix): n x 1 matrix of observed data.
grid	(matrix): Grid of values to compute the CDF at.
w	(vector, optional): n-vector of sample weights.

Value

(list): Empirical CDF and its standard deviation at the specified grid points.

---

Empirical CDF Difference

Description

Computes the difference between the empirical CDFs of the data and the predictions.

Usage

compute_cdf_diff(Y, f, grid, w = NULL)

Arguments

Y	(matrix): n x 1 matrix of observed data.
f	(matrix): n x 1 matrix of predictions.
grid	(matrix): Grid of values to compute the CDF at.
w	(vector, optional): n-vector of sample weights.

Value

(list): Difference between the empirical CDFs of the data and the predictions and its standard deviation at the specified grid points.

---

Initial estimation

Description

est_ini function for initial estimation

Usage

est_ini(
  X,
  Y,
  quant = NA,
  method = c("ols", "quantile", "mean", "logistic", "poisson")
)

Arguments

X	Array or data.frame containing covariates
Y	Array or data.frame of outcomes
quant	quantile for quantile estimation
method	indicates the method to be used for M-estimation. Options include "mean", "quantile", "ols", "logistic", and "poisson".

Value

initial estimator

---

Glance at an ipd fit

Description

Glance at an ipd fit

Usage

## S3 method for class 'ipd'
glance(x, ...)

Arguments

x	An object of class ipd.
...	Ignored.

Value

A one-row tibble summarizing the fit.

Examples

dat <- simdat()

fit <- ipd(Y - f ~ X1, method = "pspa", model = "ols",

    data = dat, label = "set_label")

glance(fit)

---

Inference on Predicted Data (ipd)

Description

The main wrapper function to conduct ipd using various methods and models, and returns a list of fitted model components.

Usage

ipd(
  formula,
  method,
  model,
  data,
  label = NULL,
  unlabeled_data = NULL,
  intercept = TRUE,
  alpha = 0.05,
  alternative = "two-sided",
  na_action = "na.fail",
  ...
)

Arguments

formula	An object of class formula: a symbolic description of the model to be fitted. Must be of the form Y - f ~ X, where Y is the name of the column corresponding to the observed outcome in the labeled data, f is the name of the column corresponding to the predicted outcome in both labeled and unlabeled data, and X corresponds to the features of interest (i.e., X = X1 + ... + Xp). See 1. Formula in the Details below for more information.
method	The IPD method to be used for fitting the model. Must be one of "chen", "pdc", "postpi_analytic", "postpi_boot", "ppi", "ppi_a", "ppi_plusplus", or "pspa". See 3. Method in the Details below for more information.
model	The type of downstream inferential model to be fitted, or the parameter being estimated. Must be one of "mean", "quantile", "ols", "logistic", or "poisson". See 4. Model in the Details below for more information.
data	A data.frame containing the variables in the model, either a stacked data frame with a specific column identifying the labeled versus unlabeled observations (label), or only the labeled data set. Must contain columns for the observed outcomes (Y), the predicted outcomes (f), and the features (X) needed to specify the formula. See 2. Data in the Details below for more information.
label	A string, int, or logical specifying the column in the data that distinguishes between the labeled and unlabeled observations. See the Details section for more information. If NULL, unlabeled_data must be specified. See 2. Data in the Details below for more information.
unlabeled_data	(optional) A data.frame of unlabeled data. If NULL, label must be specified. Specifying both the label and unlabeled_data arguments will result in an error message. If specified, must contain columns for the predicted outcomes (f), and the features (X) needed to specify the formula. See 2. Data in the Details below for more information.
intercept	Logical. Should an intercept be included in the model? Default is TRUE.
alpha	The significance level for confidence intervals. Default is 0.05.
alternative	A string specifying the alternative hypothesis. Must be one of "two-sided", "less", or "greater".
na_action	(string, optional) How missing covariate data should be handled. Currently "na.fail" and "na.omit" are accommodated. Defaults to "na.fail".
...	Additional arguments to be passed to the fitting function. See the Details section for more information. See 5. Auxiliary Arguments and 6. Other Arguments in the Details below for more information.

Details

1. Formula:

The ipd function uses one formula argument that specifies both the calibrating model (e.g., PostPI "relationship model", PPI "rectifier" model) and the inferential model. These separate models will be created internally based on the specific method called.

2. Data:

The data can be specified in two ways:

1. Single data argument (data) containing a stacked data.frame and a label identifier (label).

2. Two data arguments, one for the labeled data (data) and one for the unlabeled data (unlabeled_data).

For option (1), provide one data argument (data) which contains a stacked data.frame with both the unlabeled and labeled data and a label argument that specifies the column identifying the labeled versus the unlabeled observations in the stacked data.frame (e.g., label = "set_label" if the column "set_label" in the stacked data denotes which set an observation belongs to).

NOTE: Labeled data identifiers can be:

String

"l", "lab", "label", "labeled", "labelled", "tst", "test", "true"

Logical

TRUE

Factor

Non-reference category (i.e., binary 1)

Unlabeled data identifiers can be:

String

"u", "unlab", "unlabeled", "unlabelled", "val", "validation", "false"

Logical

FALSE

Factor

Non-reference category (i.e., binary 0)

For option (2), provide separate data arguments for the labeled data set (data) and the unlabeled data set (unlabeled_data). If the second argument is provided, the function ignores the label identifier and assumes the data provided are not stacked.

NOTE: Not all columns in data or unlabeled_data may be used unless explicitly referenced in the formula argument or in the label argument (if the data are passed as one stacked data frame).

3. Method:

Use the method argument to specify the fitting method:

"chen"

Gronsbell et al. (2026) Chen and Chen Correction

"pdc"

Gan et al. (2024) Prediction Decorrelated Inference

"postpi_analytic"

Wang et al. (2020) Post-Prediction Inference (PostPI) Analytic Correction

"postpi_boot"

Wang et al. (2020) Post-Prediction Inference (PostPI) Bootstrap Correction

"ppi"

Angelopoulos et al. (2023) Prediction-Powered Inference (PPI)

"ppi_a"

Gronsbell et al. (2025) PPI "All" Correction

"ppi_plusplus"

Angelopoulos et al. (2023) PPI++

"pspa"

Miao et al. (2023) Assumption-Lean and Data-Adaptive Post-Prediction Inference (PSPA)

4. Model:

Use the model argument to specify the type of downstream inferential model or parameter to be estimated:

"mean"

Mean value of a continuous outcome

"quantile"

qth quantile of a continuous outcome

"ols"

Linear regression coefficients for a continuous outcome

"logistic"

Logistic regression coefficients for a binary outcome

"poisson"

Poisson regression coefficients for a count outcome

The ipd wrapper function will concatenate the method and model arguments to identify the required helper function, following the naming convention "method_model".

5. Auxiliary Arguments:

The wrapper function will take method-specific auxiliary arguments (e.g., q for the quantile estimation models) and pass them to the helper function through the "..." with specified defaults for simplicity.

6. Other Arguments:

All other arguments that relate to all methods (e.g., alpha, ci.type), or other method-specific arguments, will have defaults.

Value

a summary of model output.

An S4 object of class IPD with the following slots:

coefficients

Named numeric vector of estimated parameters.

se

Named numeric vector of standard errors.

ci

A matrix of confidence intervals, with columns lower and upper.

coefTable

A data.frame summarizing Estimate, Std. Error, z-value, and Pr(>|z|) (glm-style).

fit

The raw output list returned by the method-specific helper function.

formula

The formula used for fitting the IPD model.

data_l

The labeled data.frame used in the analysis.

data_u

The unlabeled data.frame used in the analysis.

method

A character string indicating which IPD method was applied.

model

A character string indicating the downstream inferential model.

intercept

A logical indicating whether an intercept was included.

Examples

#-- Generate Example Data

dat <- simdat(n = c(300, 300, 300), effect = 1, sigma_Y = 1)

head(dat)

formula <- Y - f ~ X1

#-- Chen and Chen Correction (Gronsbell et al., 2026)

ipd(formula,
  method = "chen", model = "ols",
  data = dat, label = "set_label"
)

#-- Prediction Decorrelated Inference (Gan et al., 2024)

ipd(formula,
  method = "chen", model = "ols",
  data = dat, label = "set_label"
)

#-- PostPI Analytic Correction (Wang et al., 2020)

ipd(formula,
  method = "postpi_analytic", model = "ols",
  data = dat, label = "set_label"
)

#-- PostPI Bootstrap Correction (Wang et al., 2020)

nboot <- 200

ipd(formula,
  method = "postpi_boot", model = "ols",
  data = dat, label = "set_label", nboot = nboot
)

#-- PPI (Angelopoulos et al., 2023)

ipd(formula,
  method = "ppi", model = "ols",
  data = dat, label = "set_label"
)

#-- PPI "All" (Gronsbell et al., 2025)

ipd(formula,
  method = "ppi_a", model = "ols",
  data = dat, label = "set_label"
)

#-- PPI++ (Angelopoulos et al., 2023)

ipd(formula,
  method = "ppi_plusplus", model = "ols",
  data = dat, label = "set_label"
)

#-- PSPA (Miao et al., 2023)

ipd(formula,
  method = "pspa", model = "ols",
  data = dat, label = "set_label"
)

---

ipd: S4 class for inference on predicted data results

Description

ipd: S4 class for inference on predicted data results

Slots

coefficients

Numeric vector of parameter estimates.

se

Numeric vector of standard errors.

ci

Numeric matrix of confidence intervals.

coefTable

Data frame summarizing Estimate, Std. Error, z value, and Pr(>|z|).

fit

The raw list returned by the helper function.

formula

The formula used (class "formula").

data_l

The labeled data (data.frame).

data_u

The unlabeled data (data.frame).

method

Character; which IPD method was used.

model

Character; which downstream model was fitted.

intercept

Logical; was an intercept included?

---

Gradient of the link function

Description

link_grad function for gradient of the link function

Usage

link_grad(t, method = c("ols", "logistic", "poisson"))

Arguments

t	t
method	indicates the method to be used for M-estimation. Options include "mean", "quantile", "ols", "logistic", and "poisson".

Value

gradient of the link function

---

Hessians of the link function

Description

link_Hessian function for Hessians of the link function

Usage

link_Hessian(t, method = c("logistic", "poisson"))

Arguments

t	t
method	indicates the method to be used for M-estimation. Options include "mean", "quantile", "ols", "logistic", and "poisson".

Value

Hessians of the link function

---

Log1p Exponential

Description

Computes the natural logarithm of 1 plus the exponential of the input, to handle large inputs.

Usage

log1pexp(x)

Arguments

x	(vector): A numeric vector of inputs.

Value

(vector): A numeric vector where each element is the result of log(1 + exp(x)).

---

Logistic Regression Gradient and Hessian

Description

Computes the statistics needed for the logstic regression-based prediction-powered inference.

Usage

logistic_get_stats(
  est,
  X_l,
  Y_l,
  f_l,
  X_u,
  f_u,
  w_l = NULL,
  w_u = NULL,
  use_u = TRUE
)

Arguments

est	(vector): Point estimates of the coefficients.
X_l	(matrix): Covariates for the labeled data set.
Y_l	(vector): Labels for the labeled data set.
f_l	(vector): Predictions for the labeled data set.
X_u	(matrix): Covariates for the unlabeled data set.
f_u	(vector): Predictions for the unlabeled data set.
w_l	(vector, optional): Sample weights for the labeled data set.
w_u	(vector, optional): Sample weights for the unlabeled data set.
use_u	(bool, optional): Whether to use the unlabeled data set.

Value

(list): A list containing the following:

grads

(matrix): n x p matrix gradient of the loss function with respect to the coefficients.

grads_hat

(matrix): n x p matrix gradient of the loss function with respect to the coefficients, evaluated using the labeled predictions.

grads_hat_unlabeled

(matrix): N x p matrix gradient of the loss function with respect to the coefficients, evaluated using the unlabeled predictions.

inv_hessian

(matrix): p x p matrix inverse Hessian of the loss function with respect to the coefficients.

---

Sample expectation of psi

Description

mean_psi function for sample expectation of psi

Usage

mean_psi(
  X,
  Y,
  theta,
  quant = NA,
  method = c("ols", "quantile", "mean", "logistic", "poisson")
)

Arguments

X	Array or data.frame containing covariates
Y	Array or data.frame of outcomes
theta	parameter theta
quant	quantile for quantile estimation
method	indicates the method to be used for M-estimation. Options include "mean", "quantile", "ols", "logistic", and "poisson".

Value

sample expectation of psi

---

Sample expectation of PSPA psi

Description

mean_psi_pop function for sample expectation of PSPA psi

Usage

mean_psi_pop(
  X_l,
  X_u,
  Y_l,
  f_l,
  f_u,
  w,
  theta,
  quant = NA,
  method = c("ols", "quantile", "mean", "logistic", "poisson")
)

Arguments

X_l	Array or data.frame containing observed covariates in labeled data.
X_u	Array or data.frame containing observed or predicted covariates in unlabeled data.
Y_l	Array or data.frame of observed outcomes in labeled data.
f_l	Array or data.frame of predicted outcomes in labeled data.
f_u	Array or data.frame of predicted outcomes in unlabeled data.
w	weights vector PSPA linear regression (d-dimensional, where d equals the number of covariates).
theta	parameter theta
quant	quantile for quantile estimation
method	indicates the method to be used for M-estimation. Options include "mean", "quantile", "ols", "logistic", and "poisson".

Value

sample expectation of PSPA psi

---

Ordinary Least Squares

Description

Computes the ordinary least squares coefficients.

Usage

ols(X, Y, return_se = FALSE)

Arguments

X	(matrix): n x p matrix of covariates.
Y	(vector): p-vector of outcome values.
return_se	(bool, optional): Whether to return the standard errors of the coefficients.

Value

(list): A list containing the following:

theta

(vector): p-vector of ordinary least squares estimates of the coefficients.

se

(vector): If return_se == TRUE, return the p-vector of standard errors of the coefficients.

---

OLS Gradient and Hessian

Description

Computes the statistics needed for the OLS-based prediction-powered inference.

Usage

ols_get_stats(
  est,
  X_l,
  Y_l,
  f_l,
  X_u,
  f_u,
  w_l = NULL,
  w_u = NULL,
  use_u = TRUE
)

Arguments

est	(vector): Point estimates of the coefficients.
X_l	(matrix): Covariates for the labeled data set.
Y_l	(vector): Labels for the labeled data set.
f_l	(vector): Predictions for the labeled data set.
X_u	(matrix): Covariates for the unlabeled data set.
f_u	(vector): Predictions for the unlabeled data set.
w_l	(vector, optional): Sample weights for the labeled data set.
w_u	(vector, optional): Sample weights for the unlabeled data set.
use_u	(boolean, optional): Whether to use the unlabeled data set.

Value

(list): A list containing the following:

grads

(matrix): n x p matrix gradient of the loss function with respect to the coefficients.

grads_hat

(matrix): n x p matrix gradient of the loss function with respect to the coefficients, evaluated using the labeled predictions.

grads_hat_unlabeled

(matrix): N x p matrix gradient of the loss function with respect to the coefficients, evaluated using the unlabeled predictions.

inv_hessian

(matrix): p x p matrix inverse Hessian of the loss function with respect to the coefficients.

---

One-step update for obtaining estimator

Description

optim_est function for One-step update for obtaining estimator

Usage

optim_est(
  X_l,
  X_u,
  Y_l,
  f_l,
  f_u,
  w,
  theta,
  quant = NA,
  method = c("ols", "quantile", "mean", "logistic", "poisson")
)

Arguments

X_l	Array or data.frame containing observed covariates in labeled data.
X_u	Array or data.frame containing observed or predicted covariates in unlabeled data.
Y_l	Array or data.frame of observed outcomes in labeled data.
f_l	Array or data.frame of predicted outcomes in labeled data.
f_u	Array or data.frame of predicted outcomes in unlabeled data.
w	weights vector PSPA linear regression (d-dimensional, where d equals the number of covariates).
theta	parameter theta
quant	quantile for quantile estimation
method	indicates the method to be used for M-estimation. Options include "mean", "quantile", "ols", "logistic", and "poisson".

Value

estimator

---

One-step update for obtaining the weight vector

Description

optim_weights function for One-step update for obtaining estimator

Usage

optim_weights(
  X_l,
  X_u,
  Y_l,
  f_l,
  f_u,
  w,
  theta,
  quant = NA,
  method = c("ols", "quantile", "mean", "logistic", "poisson")
)

Arguments

X_l	Array or data.frame containing observed covariates in labeled data.
X_u	Array or data.frame containing observed or predicted covariates in unlabeled data.
Y_l	Array or data.frame of observed outcomes in labeled data.
f_l	Array or data.frame of predicted outcomes in labeled data.
f_u	Array or data.frame of predicted outcomes in unlabeled data.
w	weights vector PSPA linear regression (d-dimensional, where d equals the number of covariates).
theta	parameter theta
quant	quantile for quantile estimation
method	indicates the method to be used for M-estimation. Options include "mean", "quantile", "ols", "logistic", and "poisson".

Value

weights

---

PDC Logistic

Description

Helper function for PDC logistic regression estimation.

Usage

pdc_logistic(X_l, Y_l, f_l, X_u, f_u, intercept = TRUE)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
intercept	(Logical): Do the design matrices include intercept columns? Default is TRUE.

Details

Prediction de-correlated inference: A safe approach for post-prediction inference (Gan et al., 2024) doi:10.1111/anzs.12429

Value

(list): A list containing the following:

est

(vector): vector of PDC logistic regression coefficient estimates.

se

(vector): vector of standard errors of the coefficients.

Examples

dat <- simdat(model = "logistic")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

pdc_logistic(X_l, Y_l, f_l, X_u, f_u, intercept = TRUE)

---

PDC OLS

Description

Helper function for PDC OLS estimation.

Usage

pdc_ols(X_l, Y_l, f_l, X_u, f_u, intercept = TRUE)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
intercept	(Logical): Do the design matrices include intercept columns? Default is TRUE.

Details

Prediction de-correlated inference: A safe approach for post-prediction inference (Gan et al., 2024) doi:10.1111/anzs.12429

Value

(list): A list containing the following:

est

(vector): vector of PDC OLS regression coefficient estimates.

se

(vector): vector of standard errors of the coefficients.

Examples

dat <- simdat(model = "ols")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

pdc_ols(X_l, Y_l, f_l, X_u, f_u, intercept = TRUE)

---

PDC Poisson

Description

Helper function for PDC Poisson regression estimation.

Usage

pdc_poisson(X_l, Y_l, f_l, X_u, f_u, intercept = TRUE)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
intercept	(Logical): Do the design matrices include intercept columns? Default is TRUE.

Details

Prediction de-correlated inference: A safe approach for post-prediction inference (Gan et al., 2024) doi:10.1111/anzs.12429

Value

(list): A list containing the following:

est

(vector): vector of PDC Poisson regression coefficient estimates.

se

(vector): vector of standard errors of the coefficients.

Examples

dat <- simdat(model = "poisson")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

pdc_poisson(X_l, Y_l, f_l, X_u, f_u, intercept = TRUE)

---

PostPI OLS (Analytic Correction)

Description

Helper function for PostPI OLS estimation (analytic correction)

Usage

postpi_analytic_ols(X_l, Y_l, f_l, X_u, f_u, original = FALSE)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
original	(boolean): Logical argument to use original method from Wang et al. (2020). Defaults to FALSE; TRUE retained for posterity.

Details

Methods for correcting inference based on outcomes predicted by machine learning (Wang et al., 2020) doi:10.1073/pnas.2001238117

Value

A list of outputs: estimate of the inference model parameters and corresponding standard error estimate.

Examples

dat <- simdat(model = "ols")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

postpi_analytic_ols(X_l, Y_l, f_l, X_u, f_u)

---

PostPI Logistic Regression (Bootstrap Correction)

Description

Helper function for PostPI logistic regression (bootstrap correction)

Usage

postpi_boot_logistic(X_l, Y_l, f_l, X_u, f_u, nboot = 100, se_type = "par")

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
nboot	(integer): Number of bootstrap samples. Defaults to 100.
se_type	(string): Which method to calculate the standard errors. Options include "par" (parametric) or "npar" (nonparametric). Defaults to "par".

Details

Methods for correcting inference based on outcomes predicted by machine learning (Wang et al., 2020) doi:10.1073/pnas.2001238117

Value

A list of outputs: estimate of inference model parameters and corresponding standard error based on both parametric and non-parametric bootstrap methods.

Examples

dat <- simdat(model = "logistic")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

postpi_boot_logistic(X_l, Y_l, f_l, X_u, f_u, nboot = 200)

---

PostPI OLS (Bootstrap Correction)

Description

Helper function for PostPI OLS estimation (bootstrap correction)

Usage

postpi_boot_ols(
  X_l,
  Y_l,
  f_l,
  X_u,
  f_u,
  nboot = 100,
  se_type = "par",
  rel_func = "lm",
  scale_se = TRUE,
  n_t = Inf
)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
nboot	(integer): Number of bootstrap samples. Defaults to 100.
se_type	(string): Which method to calculate the standard errors. Options include "par" (parametric) or "npar" (nonparametric). Defaults to "par".
rel_func	(string): Method for fitting the relationship model. Options include "lm" (linear model), "rf" (random forest), and "gam" (generalized additive model). Defaults to "lm".
scale_se	(boolean): Logical argument to scale relationship model error variance. Defaults to TRUE; FALSE option is retained for posterity.
n_t	(integer, optional) Size of the dataset used to train the prediction function (necessary if n_t < nrow(X_l). Defaults to Inf.

Details

Methods for correcting inference based on outcomes predicted by machine learning (Wang et al., 2020) doi:10.1073/pnas.2001238117

Value

A list of outputs: estimate of inference model parameters and corresponding standard error based on both parametric and non-parametric bootstrap methods.

Examples

dat <- simdat(model = "ols")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>

  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

postpi_boot_ols(X_l, Y_l, f_l, X_u, f_u, nboot = 200)

---

PPI "All" OLS

Description

Helper function for PPI "All" for OLS estimation

Usage

ppi_a_ols(X_l, Y_l, f_l, X_u, f_u, w_l = NULL, w_u = NULL)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
w_l	(ndarray, optional): Sample weights for the labeled data set. Defaults to a vector of ones.
w_u	(ndarray, optional): Sample weights for the unlabeled data set. Defaults to a vector of ones.

Details

Another look at statistical inference with machine learning-imputed data (Gronsbell et al., 2026) doi:10.48550/arXiv.2411.19908

Value

(list): A list containing the following:

est

(vector): vector of PPI OLS regression coefficient estimates.

se

(vector): vector of standard errors of the coefficients.

rectifier_est

(vector): vector of the rectifier OLS regression coefficient estimates.

Examples

dat <- simdat()

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>

  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

ppi_a_ols(X_l, Y_l, f_l, X_u, f_u)

---

PPI Logistic Regression

Description

Helper function for PPI logistic regression

Usage

ppi_logistic(X_l, Y_l, f_l, X_u, f_u, opts = NULL)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
opts	(list, optional): Options to pass to the optimizer. See ?optim for details.

Details

Prediction Powered Inference (Angelopoulos et al., 2023) doi:10.1126/science.adi6000

Value

(list): A list containing the following:

est

(vector): vector of PPI logistic regression coefficient estimates.

se

(vector): vector of standard errors of the coefficients.

rectifier_est

(vector): vector of the rectifier logistic regression coefficient estimates.

var_u

(matrix): covariance matrix for the gradients in the unlabeled data.

var_l

(matrix): covariance matrix for the gradients in the labeled data.

grads

(matrix): matrix of gradients for the labeled data.

grads_hat_unlabeled

(matrix): matrix of predicted gradients for the unlabeled data.

grads_hat

(matrix): matrix of predicted gradients for the labeled data.

inv_hessian

(matrix): inverse Hessian matrix.

Examples

dat <- simdat(model = "logistic")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>

  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

ppi_logistic(X_l, Y_l, f_l, X_u, f_u)

---

PPI Mean Estimation

Description

Helper function for PPI mean estimation

Usage

ppi_mean(Y_l, f_l, f_u, alpha = 0.05, alternative = "two-sided")

Arguments

Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
alpha	(scalar): type I error rate for hypothesis testing - values in (0, 1); defaults to 0.05.
alternative	(string): Alternative hypothesis. Must be one of "two-sided", "less", or "greater".

Details

Prediction Powered Inference (Angelopoulos et al., 2023) doi:10.1126/science.adi6000

Value

tuple: Lower and upper bounds of the prediction-powered confidence interval for the mean.

Examples

dat <- simdat(model = "mean")

form <- Y - f ~ 1

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

ppi_mean(Y_l, f_l, f_u)

---

PPI OLS

Description

Helper function for prediction-powered inference for OLS estimation

Usage

ppi_ols(X_l, Y_l, f_l, X_u, f_u, w_l = NULL, w_u = NULL)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
w_l	(ndarray, optional): Sample weights for the labeled data set. Defaults to a vector of ones.
w_u	(ndarray, optional): Sample weights for the unlabeled data set. Defaults to a vector of ones.

Details

Prediction Powered Inference (Angelopoulos et al., 2023) doi:10.1126/science.adi6000

Value

(list): A list containing the following:

est

(vector): vector of PPI OLS regression coefficient estimates.

se

(vector): vector of standard errors of the coefficients.

rectifier_est

(vector): vector of the rectifier OLS regression coefficient estimates.

Examples

dat <- simdat()

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>

  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

ppi_ols(X_l, Y_l, f_l, X_u, f_u)

---

PPI++ Logistic Regression

Description

Helper function for PPI++ logistic regression

Usage

ppi_plusplus_logistic(
  X_l,
  Y_l,
  f_l,
  X_u,
  f_u,
  lhat = NULL,
  coord = NULL,
  opts = NULL,
  w_l = NULL,
  w_u = NULL
)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
lhat	(float, optional): Power-tuning parameter (see doi:10.48550/arXiv.2311.01453). The default value, NULL, will estimate the optimal value from the data. Setting lhat = 1 recovers PPI with no power tuning, and setting lhat = 0 recovers the classical point estimate.
coord	(int, optional): Coordinate for which to optimize lhat = 1. If NULL, it optimizes the total variance over all coordinates. Must be in (1, ..., d) where d is the dimension of the estimand.
opts	(list, optional): Options to pass to the optimizer. See ?optim for details.
w_l	(ndarray, optional): Sample weights for the labeled data set. Defaults to a vector of ones.
w_u	(ndarray, optional): Sample weights for the unlabeled data set. Defaults to a vector of ones.

Details

PPI++: Efficient Prediction Powered Inference (Angelopoulos et al., 2023) doi:10.48550/arXiv.2311.01453

Value

(list): A list containing the following:

est

(vector): vector of PPI++ logistic regression coefficient estimates.

se

(vector): vector of standard errors of the coefficients.

lambda

(float): estimated power-tuning parameter.

rectifier_est

(vector): vector of the rectifier logistic regression coefficient estimates.

var_u

(matrix): covariance matrix for the gradients in the unlabeled data.

var_l

(matrix): covariance matrix for the gradients in the labeled data.

grads

(matrix): matrix of gradients for the labeled data.

grads_hat_unlabeled

(matrix): matrix of predicted gradients for the unlabeled data.

grads_hat

(matrix): matrix of predicted gradients for the labeled data.

inv_hessian

(matrix): inverse Hessian matrix.

Examples

dat <- simdat(model = "logistic")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>

  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

ppi_plusplus_logistic(X_l, Y_l, f_l, X_u, f_u)

---

PPI++ Logistic Regression (Point Estimate)

Description

Helper function for PPI++ logistic regression (point estimate)

Usage

ppi_plusplus_logistic_est(
  X_l,
  Y_l,
  f_l,
  X_u,
  f_u,
  lhat = NULL,
  coord = NULL,
  opts = NULL,
  w_l = NULL,
  w_u = NULL
)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
lhat	(float, optional): Power-tuning parameter (see doi:10.48550/arXiv.2311.01453). The default value, NULL, will estimate the optimal value from the data. Setting lhat = 1 recovers PPI with no power tuning, and setting lhat = 0 recovers the classical point estimate.
coord	(int, optional): Coordinate for which to optimize lhat = 1. If NULL, it optimizes the total variance over all coordinates. Must be in (1, ..., d) where d is the dimension of the estimand.
opts	(list, optional): Options to pass to the optimizer. See ?optim for details.
w_l	(ndarray, optional): Sample weights for the labeled data set. Defaults to a vector of ones.
w_u	(ndarray, optional): Sample weights for the unlabeled data set. Defaults to a vector of ones.

Details

PPI++: Efficient Prediction Powered Inference (Angelopoulos et al., 2023) doi:10.48550/arXiv.2311.01453

Value

(vector): vector of prediction-powered point estimates of the logistic regression coefficients.

Examples

dat <- simdat(model = "logistic")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>

  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

ppi_plusplus_logistic_est(X_l, Y_l, f_l, X_u, f_u)

---

PPI++ Mean Estimation

Description

Helper function for PPI++ mean estimation

Usage

ppi_plusplus_mean(
  Y_l,
  f_l,
  f_u,
  alpha = 0.05,
  alternative = "two-sided",
  lhat = NULL,
  coord = NULL,
  w_l = NULL,
  w_u = NULL
)

Arguments

Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
alpha	(scalar): type I error rate for hypothesis testing - values in (0, 1); defaults to 0.05.
alternative	(string): Alternative hypothesis. Must be one of "two-sided", "less", or "greater".
lhat	(float, optional): Power-tuning parameter (see doi:10.48550/arXiv.2311.01453). The default value, NULL, will estimate the optimal value from the data. Setting lhat = 1 recovers PPI with no power tuning, and setting lhat = 0 recovers the classical point estimate.
coord	(int, optional): Coordinate for which to optimize lhat = 1. If NULL, it optimizes the total variance over all coordinates. Must be in (1, ..., d) where d is the dimension of the estimand.
w_l	(ndarray, optional): Sample weights for the labeled data set. Defaults to a vector of ones.
w_u	(ndarray, optional): Sample weights for the unlabeled data set. Defaults to a vector of ones.

Details

PPI++: Efficient Prediction Powered Inference (Angelopoulos et al., 2023) doi:10.48550/arXiv.2311.01453

Value

tuple: Lower and upper bounds of the prediction-powered confidence interval for the mean.

Examples

dat <- simdat(model = "mean")

form <- Y - f ~ 1

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>

  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

ppi_plusplus_mean(Y_l, f_l, f_u)

---

PPI++ Mean Estimation (Point Estimate)

Description

Helper function for PPI++ mean estimation (point estimate)

Usage

ppi_plusplus_mean_est(
  Y_l,
  f_l,
  f_u,
  lhat = NULL,
  coord = NULL,
  w_l = NULL,
  w_u = NULL
)

Arguments

Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
lhat	(float, optional): Power-tuning parameter (see doi:10.48550/arXiv.2311.01453). The default value, NULL, will estimate the optimal value from the data. Setting lhat = 1 recovers PPI with no power tuning, and setting lhat = 0 recovers the classical point estimate.
coord	(int, optional): Coordinate for which to optimize lhat = 1. If NULL, it optimizes the total variance over all coordinates. Must be in (1, ..., d) where d is the dimension of the estimand.
w_l	(ndarray, optional): Sample weights for the labeled data set. Defaults to a vector of ones.
w_u	(ndarray, optional): Sample weights for the unlabeled data set. Defaults to a vector of ones.

Details

PPI++: Efficient Prediction Powered Inference (Angelopoulos et al., 2023) doi:10.48550/arXiv.2311.01453

Value

float or ndarray: Prediction-powered point estimate of the mean.

Examples

dat <- simdat(model = "mean")

form <- Y - f ~ 1

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>

  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

ppi_plusplus_mean_est(Y_l, f_l, f_u)

---

PPI++ OLS

Description

Helper function for PPI++ OLS estimation

Usage

ppi_plusplus_ols(
  X_l,
  Y_l,
  f_l,
  X_u,
  f_u,
  lhat = NULL,
  coord = NULL,
  w_l = NULL,
  w_u = NULL
)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
lhat	(float, optional): Power-tuning parameter (see doi:10.48550/arXiv.2311.01453). The default value, NULL, will estimate the optimal value from the data. Setting lhat = 1 recovers PPI with no power tuning, and setting lhat = 0 recovers the classical point estimate.
coord	(int, optional): Coordinate for which to optimize lhat = 1. If NULL, it optimizes the total variance over all coordinates. Must be in (1, ..., d) where d is the dimension of the estimand.
w_l	(ndarray, optional): Sample weights for the labeled data set. Defaults to a vector of ones.
w_u	(ndarray, optional): Sample weights for the unlabeled data set. Defaults to a vector of ones.

Details

PPI++: Efficient Prediction Powered Inference (Angelopoulos et al., 2023) doi:10.48550/arXiv.2311.01453

Value

(list): A list containing the following:

est

(vector): vector of PPI++ OLS regression coefficient estimates.

se

(vector): vector of standard errors of the coefficients.

lambda

(float): estimated power-tuning parameter.

rectifier_est

(vector): vector of the rectifier OLS regression coefficient estimates.

var_u

(matrix): covariance matrix for the gradients in the unlabeled data.

var_l

(matrix): covariance matrix for the gradients in the labeled data.

grads

(matrix): matrix of gradients for the labeled data.

grads_hat_unlabeled

(matrix): matrix of predicted gradients for the unlabeled data.

grads_hat

(matrix): matrix of predicted gradients for the labeled data.

inv_hessian

(matrix): inverse Hessian matrix.

Examples

dat <- simdat(model = "ols")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>

  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

ppi_plusplus_ols(X_l, Y_l, f_l, X_u, f_u)

---

PPI++ OLS (Point Estimate)

Description

Helper function for PPI++ OLS estimation (point estimate)

Usage

ppi_plusplus_ols_est(
  X_l,
  Y_l,
  f_l,
  X_u,
  f_u,
  lhat = NULL,
  coord = NULL,
  w_l = NULL,
  w_u = NULL
)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
lhat	(float, optional): Power-tuning parameter (see doi:10.48550/arXiv.2311.01453). The default value, NULL, will estimate the optimal value from the data. Setting lhat = 1 recovers PPI with no power tuning, and setting lhat = 0 recovers the classical point estimate.
coord	(int, optional): Coordinate for which to optimize lhat = 1. If NULL, it optimizes the total variance over all coordinates. Must be in (1, ..., d) where d is the dimension of the estimand.
w_l	(ndarray, optional): Sample weights for the labeled data set. Defaults to a vector of ones.
w_u	(ndarray, optional): Sample weights for the unlabeled data set. Defaults to a vector of ones.

Details

PPI++: Efficient Prediction Powered Inference (Angelopoulos et al., 2023) doi:10.48550/arXiv.2311.01453

Value

(vector): vector of prediction-powered point estimates of the OLS coefficients.

Examples

dat <- simdat(model = "ols")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>

  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

ppi_plusplus_ols_est(X_l, Y_l, f_l, X_u, f_u)

---

PPI++ Quantile Estimation

Description

Helper function for PPI++ quantile estimation

Usage

ppi_plusplus_quantile(
  Y_l,
  f_l,
  f_u,
  q,
  alpha = 0.05,
  exact_grid = FALSE,
  w_l = NULL,
  w_u = NULL
)

Arguments

Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
q	(float): Quantile to estimate. Must be in the range (0, 1).
alpha	(scalar): type I error rate for hypothesis testing - values in (0, 1); defaults to 0.05.
exact_grid	(bool, optional): Whether to compute the exact solution (TRUE) or an approximate solution based on a linearly spaced grid of 5000 values (FALSE).
w_l	(ndarray, optional): Sample weights for the labeled data set. Defaults to a vector of ones.
w_u	(ndarray, optional): Sample weights for the unlabeled data set. Defaults to a vector of ones.

Details

PPI++: Efficient Prediction Powered Inference (Angelopoulos et al., 2023) doi:10.48550/arXiv.2311.01453

Value

tuple: Lower and upper bounds of the prediction-powered confidence interval for the quantile.

Examples

dat <- simdat(model = "quantile")

form <- Y - f ~ X1

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>

  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

ppi_plusplus_quantile(Y_l, f_l, f_u, q = 0.5)

---

PPI++ Quantile Estimation (Point Estimate)

Description

Helper function for PPI++ quantile estimation (point estimate)

Usage

ppi_plusplus_quantile_est(
  Y_l,
  f_l,
  f_u,
  q,
  exact_grid = FALSE,
  w_l = NULL,
  w_u = NULL
)

Arguments

Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
q	(float): Quantile to estimate. Must be in the range (0, 1).
exact_grid	(bool, optional): Whether to compute the exact solution (TRUE) or an approximate solution based on a linearly spaced grid of 5000 values (FALSE).
w_l	(ndarray, optional): Sample weights for the labeled data set. Defaults to a vector of ones.
w_u	(ndarray, optional): Sample weights for the unlabeled data set. Defaults to a vector of ones.

Details

PPI++: Efficient Prediction Powered Inference (Angelopoulos et al., 2023) doi:10.48550/arXiv.2311.01453

Value

(float): Prediction-powered point estimate of the quantile.

Examples

dat <- simdat(model = "quantile")

form <- Y - f ~ 1

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>

  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

ppi_plusplus_quantile_est(Y_l, f_l, f_u, q = 0.5)

---

PPI Quantile Estimation

Description

Helper function for PPI quantile estimation

Usage

ppi_quantile(Y_l, f_l, f_u, q, alpha = 0.05, exact_grid = FALSE)

Arguments

Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
q	(float): Quantile to estimate. Must be in the range (0, 1).
alpha	(scalar): type I error rate for hypothesis testing - values in (0, 1); defaults to 0.05.
exact_grid	(bool, optional): Whether to compute the exact solution (TRUE) or an approximate solution based on a linearly spaced grid of 5000 values (FALSE).

Details

Prediction Powered Inference (Angelopoulos et al., 2023) doi:10.1126/science.adi6000

Value

tuple: Lower and upper bounds of the prediction-powered confidence interval for the quantile.

Examples

dat <- simdat(model = "quantile")

form <- Y - f ~ X1

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>

  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>

  matrix(ncol = 1)

ppi_quantile(Y_l, f_l, f_u, q = 0.5)

---

Print ipd fit

Description

Print ipd fit

Usage

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

Arguments

x	An object of class ipd.
...	Ignored.

Value

Invisibly returns x.

---

Print summary.ipd

Description

Print summary.ipd

Usage

## S3 method for class 'summary.ipd'
print(x, ...)

Arguments

x	An object of class summary.ipd.
...	Ignored.

Value

Invisibly returns x.

---

Estimating equation

Description

psi function for estimating equation

Usage

psi(
  X,
  Y,
  theta,
  quant = NA,
  method = c("ols", "quantile", "mean", "logistic", "poisson")
)

Arguments

X	Array or data.frame containing covariates
Y	Array or data.frame of outcomes
theta	parameter theta
quant	quantile for quantile estimation
method	indicates the method to be used for M-estimation. Options include "mean", "quantile", "ols", "logistic", and "poisson".

Value

estimating equation

---

PSPA Logistic Regression

Description

Helper function for PSPA logistic regression

Usage

pspa_logistic(X_l, Y_l, f_l, X_u, f_u, weights = NA, alpha = 0.05)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of binary labeled outcomes.
f_l	(vector): n-vector of binary predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of binary predictions in the unlabeled data.
weights	(array): p-dimensional array of weights vector for variance reduction. PSPA will estimate the weights if not specified.
alpha	(scalar): type I error rate for hypothesis testing - values in (0, 1); defaults to 0.05

Details

Post-prediction adaptive inference (Miao et al. 2023) doi:10.48550/arXiv.2311.14220

Value

A list of outputs: estimate of inference model parameters and corresponding standard error.

Examples

dat <- simdat(model = "logistic")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

pspa_logistic(X_l, Y_l, f_l, X_u, f_u)

---

PSPA Mean Estimation

Description

Helper function for PSPA mean estimation

Usage

pspa_mean(Y_l, f_l, f_u, weights = NA, alpha = 0.05)

Arguments

Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
weights	(array): 1-dimensional array of weights vector for variance reduction. PSPA will estimate the weights if not specified.
alpha	(scalar): type I error rate for hypothesis testing - values in (0, 1); defaults to 0.05.

Details

Post-prediction adaptive inference (Miao et al., 2023) doi:10.48550/arXiv.2311.14220

Value

A list of outputs: estimate of inference model parameters and corresponding standard error.

Examples

dat <- simdat(model = "mean")

form <- Y - f ~ 1

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

pspa_mean(Y_l = Y_l, f_l = f_l, f_u = f_u)

---

PSPA OLS Estimation

Description

Helper function for PSPA OLS for linear regression

Usage

pspa_ols(X_l, Y_l, f_l, X_u, f_u, weights = NA, alpha = 0.05)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
weights	(array): p-dimensional array of weights vector for variance reduction. PSPA will estimate the weights if not specified.
alpha	(scalar): type I error rate for hypothesis testing - values in (0, 1); defaults to 0.05.

Details

Post-prediction adaptive inference (Miao et al., 2023) doi:10.48550/arXiv.2311.14220

Value

A list of outputs: estimate of inference model parameters and corresponding standard error.

Examples

dat <- simdat(model = "ols")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

pspa_ols(X_l, Y_l, f_l, X_u, f_u)

---

PSPA Poisson Regression

Description

Helper function for PSPA Poisson regression

Usage

pspa_poisson(X_l, Y_l, f_l, X_u, f_u, weights = NA, alpha = 0.05)

Arguments

X_l	(matrix): n x p matrix of covariates in the labeled data.
Y_l	(vector): n-vector of count labeled outcomes.
f_l	(vector): n-vector of binary predictions in the labeled data.
X_u	(matrix): N x p matrix of covariates in the unlabeled data.
f_u	(vector): N-vector of binary predictions in the unlabeled data.
weights	(array): p-dimensional array of weights vector for variance reduction. PSPA will estimate the weights if not specified.
alpha	(scalar): type I error rate for hypothesis testing - values in (0, 1); defaults to 0.05

Details

Post-prediction adaptive inference (Miao et al., 2023) doi:10.48550/arXiv.2311.14220

Value

A list of outputs: estimate of inference model parameters and corresponding standard error.

Examples

dat <- simdat(model = "poisson")

form <- Y - f ~ X1

X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
  matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

pspa_poisson(X_l, Y_l, f_l, X_u, f_u)

---

PSPA Quantile Estimation

Description

Helper function for PSPA quantile estimation

Usage

pspa_quantile(Y_l, f_l, f_u, q, weights = NA, alpha = 0.05)

Arguments

Y_l	(vector): n-vector of labeled outcomes.
f_l	(vector): n-vector of predictions in the labeled data.
f_u	(vector): N-vector of predictions in the unlabeled data.
q	(float): Quantile to estimate. Must be in the range (0, 1).
weights	(array): 1-dimensional array of weights vector for variance reduction. PSPA will estimate the weights if not specified.
alpha	(scalar): type I error rate for hypothesis testing - values in (0, 1); defaults to 0.05.

Details

Post-prediction adaptive inference (Miao et al., 2023) doi:10.48550/arXiv.2311.14220

Value

A list of outputs: estimate of inference model parameters and corresponding standard error.

Examples

dat <- simdat(model = "quantile")

form <- Y - f ~ 1

Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |> matrix(ncol = 1)

f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |> matrix(ncol = 1)

f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
  matrix(ncol = 1)

pspa_quantile(Y_l = Y_l, f_l = f_l, f_u = f_u, q = 0.5)

---

PSPA M-Estimation for ML-predicted labels

Description

pspa_y function conducts post-prediction M-Estimation.

Usage

pspa_y(
  X_l = NA,
  X_u = NA,
  Y_l,
  f_l,
  f_u,
  alpha = 0.05,
  weights = NA,
  quant = NA,
  intercept = FALSE,
  method
)

Arguments

X_l	Array or data.frame containing observed covariates in labeled data.
X_u	Array or data.frame containing observed or predicted covariates in unlabeled data.
Y_l	Array or data.frame of observed outcomes in labeled data.
f_l	Array or data.frame of predicted outcomes in labeled data.
f_u	Array or data.frame of predicted outcomes in unlabeled data.
alpha	Specifies the confidence level as 1 - alpha for confidence intervals.
weights	weights vector PSPA linear regression (d-dimensional, where d equals the number of covariates).
quant	quantile for quantile estimation
intercept	Boolean indicating if the input covariates' data contains the intercept (TRUE if the input data contains)
method	indicates the method to be used for M-estimation. Options include "mean", "quantile", "ols", "logistic", and "poisson".

Value

A summary table presenting point estimates, standard error, confidence intervals (1 - alpha), P-values, and weights.

---

Rectified CDF

Description

Computes the rectified CDF of the data.

Usage

rectified_cdf(Y_l, f_l, f_u, grid, w_l = NULL, w_u = NULL)

Arguments

Y_l	(vector): Gold-standard labels.
f_l	(vector): Predictions corresponding to the gold-standard labels.
f_u	(vector): Predictions corresponding to the unlabeled data.
grid	(vector): Grid of values to compute the CDF at.
w_l	(vector, optional): Sample weights for the labeled data set.
w_u	(vector, optional): Sample weights for the unlabeled data set.

Value

(vector): Rectified CDF of the data at the specified grid points.

---

Rectified P-Value

Description

Computes a rectified p-value.

Usage

rectified_p_value(
  rectifier,
  rectifier_std,
  imputed_mean,
  imputed_std,
  null = 0,
  alternative = "two-sided"
)

Arguments

rectifier	(float or vector): Rectifier value.
rectifier_std	(float or vector): Rectifier standard deviation.
imputed_mean	(float or vector): Imputed mean.
imputed_std	(float or vector): Imputed standard deviation.
null	(float, optional): Value of the null hypothesis to be tested. Defaults to 0.
alternative	(str, optional): Alternative hypothesis, either 'two-sided', 'larger' or 'smaller'.

Value

(float or vector): The rectified p-value.

---

Show an ipd object

Description

Display a concise summary of an ipd S4 object, including method, model, formula, and a glm-style coefficient table.

Usage

## S4 method for signature 'ipd'
show(object)

Arguments

object

An object of S4 class ipd.

Value

Invisibly returns object after printing.

---

Variance-covariance matrix of the estimation equation

Description

Sigma_cal function for variance-covariance matrix of the estimation equation

Usage

Sigma_cal(
  X_l,
  X_u,
  Y_l,
  f_l,
  f_u,
  w,
  theta,
  quant = NA,
  method = c("ols", "quantile", "mean", "logistic", "poisson")
)

Arguments

X_l	Array or data.frame containing observed covariates in labeled data.
X_u	Array or data.frame containing observed or predicted covariates in unlabeled data.
Y_l	Array or data.frame of observed outcomes in labeled data.
f_l	Array or data.frame of predicted outcomes in labeled data.
f_u	Array or data.frame of predicted outcomes in unlabeled data.
w	weights vector PSPA linear regression (d-dimensional, where d equals the number of covariates).
theta	parameter theta
quant	quantile for quantile estimation
method	indicates the method to be used for M-estimation. Options include "mean", "quantile", "ols", "logistic", and "poisson".

Value

variance-covariance matrix of the estimation equation

---

Simulate the data for testing the functions

Description

sim_data_y for simulation with ML-predicted Y

Usage

sim_data_y(r = 0.9, binary = FALSE)

Arguments

r	imputation correlation
binary	simulate binary outcome or not

Value

simulated data

---

Data generation function for various underlying models

Description

Data generation function for various underlying models

Usage

simdat(
  n = c(300, 300, 300),
  effect = 1,
  sigma_Y = 1,
  model = "ols",
  shift = 0,
  scale = 1
)

Arguments

n	Integer vector of size 3 indicating the sample sizes in the training, labeled, and unlabeled data sets, respectively
effect	Regression coefficient for the first variable of interest for inference. Defaults is 1.
sigma_Y	Residual variance for the generated outcome. Defaults is 1.
model	The type of model to be generated. Must be one of "mean", "quantile", "ols", "logistic", or "poisson". Default is "ols".
shift	Scalar shift of the predictions for continuous outcomes (i.e., "mean", "quantile", and "ols"). Defaults to 0.
scale	Scaling factor for the predictions for continuous outcomes (i.e., "mean", "quantile", and "ols"). Defaults to 1.

Details

The simdat function generates three datasets consisting of independent realizations of $Y$ (for model = "mean" or "quantile"), or $\{Y, \boldsymbol{X}\}$ (for model = "ols", "logistic", or "poisson"): a training dataset of size $n_t$, a labeled dataset of size $n_l$, and an unlabeled dataset of size $n_u$. These sizes are specified by the argument n.

NOTE: In the unlabeled data subset, outcome data are still generated to facilitate a benchmark for comparison with an "oracle" model that uses the true $Y^{\mathcal{U}}$ values for estimation and inference.

Generating Data

For "mean" and "quantile", we simulate a continuous outcome, $Y \in \mathbb{R}$, with mean given by the effect argument and error variance given by the sigma_y argument.

For "ols", "logistic", or "poisson" models, predictor data, $\boldsymbol{X} \in \mathbb{R}^4$ are simulated such that the $i$th observation follows a standard multivariate normal distribution with a zero mean vector and identity covariance matrix:

$\boldsymbol{X_i} = (X_{i1}, X_{i2}, X_{i3}, X_{i4}) \sim \mathcal{N}_4(\boldsymbol{0}, \boldsymbol{I}).$

For "ols", a continuous outcome $Y \in \mathbb{R}$ is simulated to depend on $X_1$ through a linear term with the effect size specified by the effect argument, while the other predictors, $\boldsymbol{X} \setminus X_1$, have nonlinear effects:

$Y_i = effect \times Z_{i1} + \frac{1}{2} Z_{i2}^2 + \frac{1}{3} Z_{i3}^3 + \frac{1}{4} Z_{i4}^2 + \varepsilon_y,$

and $\varepsilon_y \sim \mathcal{N}(0, sigma_y)$, where the sigma_y argument specifies the error variance.

For "logistic", we simulate:

$\Pr(Y_i = 1 \mid \boldsymbol{X}) = logit^{-1}(effect \times Z_{i1} + \frac{1}{2} Z_{i2}^2 + \frac{1}{3} Z_{i3}^3 + \frac{1}{4} Z_{i4}^2 + \varepsilon_y)$

and generate:

$Y_i \sim Bern[1, \Pr(Y_i = 1 \mid \boldsymbol{X})]$

where $\varepsilon_y \sim \mathcal{N}(0, sigma\_y)$.

For "poisson", we simulate:

$\lambda_Y = exp(effect \times Z_{i1} + \frac{1}{2} Z_{i2}^2 + \frac{1}{3} Z_{i3}^3 + \frac{1}{4} Z_{i4}^2 + \varepsilon_y)$

and generate:

$Y_i \sim Poisson(\lambda_Y)$

Generating Predictions

To generate predicted outcomes for "mean" and "quantile", we simulate a continuous variable with mean given by the empirical mean of the training data and error variance given by the sigma_y argument.

For "ols", we fit a generalized additive model (GAM) on the simulated training dataset and calculate predictions for the labeled and unlabeled datasets as deterministic functions of $\boldsymbol{X}$. Specifically, we fit the following GAM:

$Y^{\mathcal{T}} = s_0 + s_1(X_1^{\mathcal{T}}) + s_2(X_2^{\mathcal{T}}) + s_3(X_3^{\mathcal{T}}) + s_4(X_4^{\mathcal{T}}) + \varepsilon_p,$

where $\mathcal{T}$ denotes the training dataset, $s_0$ is an intercept term, and $s_1(\cdot)$, $s_2(\cdot)$, $s_3(\cdot)$, and $s_4(\cdot)$ are smoothing spline functions for $X_1$, $X_2$, $X_3$, and $X_4$, respectively, with three target equivalent degrees of freedom. Residual error is modeled as $\varepsilon_p$.

Predictions for labeled and unlabeled datasets are calculated as:

$f(\boldsymbol{X}^{\mathcal{L}\cup\mathcal{U}}) = \hat{s}_0 + \hat{s}_1(X_1^{\mathcal{L}\cup\mathcal{U}}) + \hat{s}_2(X_2^{\mathcal{L}\cup\mathcal{U}}) + \hat{s}_3(X_3^{\mathcal{L}\cup\mathcal{U}}) + \hat{s}_4(X_4^{\mathcal{L}\cup\mathcal{U}}),$

where $\hat{s}_0, \hat{s}_1, \hat{s}_2, \hat{s}_3$, and $\hat{s}_4$ are estimates of $s_0, s_1, s_2, s_3$, and $s_4$, respectively.

NOTE: For continuous outcomes, we provide optional arguments shift and scale to further apply a location shift and scaling factor, respectively, to the predicted outcomes. These default to shift = 0 and scale = 1, i.e., no location shift or scaling.

For "logistic", we train k-nearest neighbors (k-NN) classifiers on the simulated training dataset for values of $k$ ranging from 1 to 10. The optimal $k$ is chosen via cross-validation, minimizing the misclassification error on the validation folds. Predictions for the labeled and unlabeled datasets are obtained by applying the k-NN classifier with the optimal $k$ to $\boldsymbol{X}$.

Specifically, for each observation in the labeled and unlabeled datasets:

$\hat{Y} = \operatorname{argmax}_c \sum_{i \in \mathcal{N}_k} I(Y_i = c),$

where $\mathcal{N}_k$ represents the set of $k$ nearest neighbors in the training dataset, $c$ indexes the possible classes (0 or 1), and $I(\cdot)$ is an indicator function.

For "poisson", we fit a generalized linear model (GLM) with a log link function to the simulated training dataset. The model is of the form:

$\log(\mu^{\mathcal{T}}) = \gamma_0 + \gamma_1 X_1^{\mathcal{T}} + \gamma_2 X_2^{\mathcal{T}} + \gamma_3 X_3^{\mathcal{T}} + \gamma_4 X_4^{\mathcal{T}},$

where $\mu^{\mathcal{T}}$ is the expected count for the response variable in the training dataset, $\gamma_0$ is the intercept, and $\gamma_1$, $\gamma_2$, $\gamma_3$, and $\gamma_4$ are the regression coefficients for the predictors $X_1$, $X_2$, $X_3$, and $X_4$, respectively.

Predictions for the labeled and unlabeled datasets are calculated as:

$\hat{\mu}^{\mathcal{L} \cup \mathcal{U}} = \exp(\hat{\gamma}_0 + \hat{\gamma}_1 X_1^{\mathcal{L} \cup \mathcal{U}} + \hat{\gamma}_2 X_2^{\mathcal{L} \cup \mathcal{U}} + \hat{\gamma}_3 X_3^{\mathcal{L} \cup \mathcal{U}} + \hat{\gamma}_4 X_4^{\mathcal{L} \cup \mathcal{U}}),$

where $\hat{\gamma}_0$, $\hat{\gamma}_1$, $\hat{\gamma}_2$, $\hat{\gamma}_3$, and $\hat{\gamma}_4$ are the estimated coefficients.

Value

A data.frame containing n rows and columns corresponding to the labeled outcome (Y), the predicted outcome (f), a character variable (set_label) indicating which data set the observation belongs to (training, labeled, or unlabeled), and four independent, normally distributed predictors (X1, X2, X3, and X4), where applicable.

Examples

#-- Mean

dat_mean <- simdat(c(100, 100, 100),
  effect = 1, sigma_Y = 1,
  model = "mean"
)

head(dat_mean)

#-- Linear Regression

dat_ols <- simdat(c(100, 100, 100),
  effect = 1, sigma_Y = 1,
  model = "ols"
)

head(dat_ols)

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