Inference on Predicted Data (ipd)

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,
  seed = NULL,
  intercept = TRUE,
  alpha = 0.05,
  alternative = "two-sided",
  n_t = Inf,
  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).
method: The method to be used for fitting the model. Must be one of "postpi_analytic", "postpi_boot", "ppi", "pspa", or "ppi_plusplus".
model: The type of model to be fitted. Must be one of "mean", "quantile", "ols", or "logistic".
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.
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.
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.
seed: (optional) An integer seed for random number generation.
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".
n_t: (integer, optional) Size of the dataset used to train the prediction function (necessary for the "postpi" methods if n_t < nrow(X_l). Defaults to Inf.
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.

Value

a summary of model output.

A list containing the fitted model components:

coefficients: Estimated coefficients of the model
se: Standard errors of the estimated coefficients
ci: Confidence intervals for the estimated coefficients
formula: The formula used to fit the ipd model.
data: The data frame used for model fitting.
method: The method used for model fitting.
model: The type of model fitted.
intercept: Logical. Indicates if an intercept was included in the model.
fit: Fitted model object containing estimated coefficients, standard errors, confidence intervals, and additional method-specific output.
...: Additional output specific to the method used.

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:

Single data argument (data) containing a stacked data.frame and a label identifier (label).
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 specify the column that identifies the labeled versus the unlabeled observations in the stacked data.frame

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 is stacked.

3. Method:

Use the method argument to specify the fitting method:

"postpi": Wang et al. (2020) Post-Prediction Inference (PostPI)
"ppi": Angelopoulos et al. (2023) Prediction-Powered Inference (PPI)
"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 model:

"mean": Mean value of the outcome
"quantile": qth quantile of the outcome
"ols": Linear regression
"logistic": Logistic regression
"poisson": Poisson regression

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.

Examples


#-- Generate Example Data

set.seed(12345)

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

head(dat)
#>           X1          X2         X3          X4         Y  f      set
#> 1  0.5855288 -0.78486098  1.1872102  1.05076285 1.4008570 NA training
#> 2  0.7094660 -2.56005244 -0.3567140 -0.07179733 4.1079201 NA training
#> 3 -0.1093033  0.07280078  1.2122385  0.11673662 1.4501726 NA training
#> 4 -0.4534972  0.75024358 -0.6939527  0.97786651 1.2987926 NA training
#> 5  0.6058875 -0.12824888  1.3560616 -1.03154201 2.5256490 NA training
#> 6 -1.8179560 -0.48786673  0.9057313  2.19912933 0.2889297 NA training

formula <- Y - f ~ X1

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

ipd(formula, method = "postpi_analytic", model = "ols",

    data = dat, label = "set")
#> 
#> Call:
#>  Y - f ~ X1 
#> 
#> Coefficients:
#> (Intercept)          X1 
#>   0.7654975   0.8975856 

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

nboot <- 200

ipd(formula, method = "postpi_boot", model = "ols",

    data = dat, label = "set", nboot = nboot)
#> 
#> Call:
#>  Y - f ~ X1 
#> 
#> Coefficients:
#> (Intercept)          X1 
#>   0.7725950   0.9026754 

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

ipd(formula, method = "ppi", model = "ols",

    data = dat, label = "set")
#> 
#> Call:
#>  Y - f ~ X1 
#> 
#> Coefficients:
#>                  [,1]
#> (Intercept) 0.7899574
#> X1          0.8012371
#> attr(,"names")
#> [1] "(Intercept)" "X1"         

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

ipd(formula, method = "ppi_plusplus", model = "ols",

    data = dat, label = "set")
#> 
#> Call:
#>  Y - f ~ X1 
#> 
#> Coefficients:
#> (Intercept)          X1 
#>   0.7341888   0.7718728 

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

ipd(formula, method = "pspa", model = "ols",

    data = dat, label = "set")
#> 
#> Call:
#>  Y - f ~ X1 
#> 
#> Coefficients:
#> (Intercept)          X1 
#>   0.7306059   0.7744542