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ipd: Inference on Predicted Data

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Overview

ipd is an open-source R software package for the downstream modeling of an outcome and its associated features where a potentially sizable portion of the outcome data has been imputed by an artificial intelligence or machine learning (AI/ML) prediction algorithm. The package implements several recent proposed methods for inference on predicted data (IPD) with a single, user-friendly wrapper function, ipd. The package also provides custom print, summary, tidy, glance, and augment methods to facilitate easy model inspection.

Background

Using predictions from pre-trained algorithms as outcomes in downstream statistical analyses can lead to biased estimates and misleading conclusions. The statistical challenges encountered when drawing inference on predicted data (IPD) include:

  1. Understanding the relationship between predicted outcomes and their true, unobserved counterparts.
  2. Quantifying the robustness of the AI/ML models to resampling or uncertainty about the training data.
  3. Appropriately propagating both bias and uncertainty from predictions into downstream inferential tasks.

Several works have proposed methods for IPD, including post-prediction inference (PostPI) by Wang et al., 2020, prediction-powered inference (PPI) and PPI++ by Angelopoulos et al., 2023a and Angelopoulos et al., 2023b, and post-prediction adaptive inference (PSPA) by Miao et al., 2023. Each method was developed to perform inference on a quantity such as the outcome mean or quantile, or a regression coefficient, when we have:

  1. A dataset consisting of our outcome and features of interst, where the outcome is only observed for a small ‘labeled’ subset and missing for a, typically larger, ‘unlabeled’ subset.
  2. Access to an algorithm to predict the missing outcome in the entire dataset using the fully observed features.
Overview of data and setup for IPD

We can use these methods for IPD to obtain corrected estimates and standard errors by using the predicted outcomes and unlabeled features to augment the labeled subset of the data.

To enable researchers and practitioners interested in these state-of-the-art methods, we have developed the ipd package in R to implement these methods under the umbrella of IPD. This README provides an overview of the package, including installation instructions, basic usage examples, and links to further documentation. The examples show how to generate data, fit models, and use custom methods provided by the package.

Installation

To install the development version of ipd from GitHub, you can use the devtools package:

#-- Install devtools if it is not already installed

install.packages("devtools")   

#-- Install the ipd package from GitHub

devtools::install_github("ipd-tools/ipd")

Usage

We provide a simple example to demonstrate the basic use of the functions included in the ipd package.

Data Generation

You can generate synthetic datasets for different types of regression models using the provided simdat function by specifying the sizes of the datasets, the effect size, residual variance, and the type of model. The function currently supports “mean”, “quantile”, “ols”, “logistic”, and “poisson” models. The simdat function generate a data.frame with three subsets: (1) an independent “training” set with additional observations used to fit a prediction model, and “labeled” and “unlabeled” sets which contain the observed and predicted outcomes and the simulated features of interest.

#-- Load the ipd Library

library(ipd)

#-- Generate Example Data for Linear Regression

set.seed(123)

n <- c(10000, 500, 1000)

dat <- simdat(n = n, effect = 1, sigma_Y = 4, model = "ols")

#-- Print First 6 Rows of Training, Labeled, and Unlabeled Subsets

options(digits=2)

head(dat[dat$set == "training",])
#>       X1    X2    X3     X4     Y  f      set
#> 1 -0.560 -0.56  0.82 -0.356 -0.15 NA training
#> 2 -0.230  0.13 -1.54  0.040 -4.49 NA training
#> 3  1.559  1.82 -0.59  1.152 -1.08 NA training
#> 4  0.071  0.16 -0.18  1.485 -3.67 NA training
#> 5  0.129 -0.72 -0.71  0.634  2.19 NA training
#> 6  1.715  0.58 -0.54 -0.037 -1.42 NA training

head(dat[dat$set == "labeled",])
#>          X1      X2    X3    X4     Y     f     set
#> 10001  2.37 -1.8984  0.20 -0.17  1.40  3.24 labeled
#> 10002 -0.17  1.7428  0.26 -2.05  3.56  1.03 labeled
#> 10003  0.93 -1.0947  0.76  1.25 -3.66  2.37 labeled
#> 10004 -0.57  0.1757  0.32  0.65 -0.56  0.58 labeled
#> 10005  0.23  2.0620 -1.35  1.46 -0.82 -0.15 labeled
#> 10006  1.13 -0.0028  0.23 -0.24  7.30  2.16 labeled

head(dat[dat$set == "unlabeled",])
#>          X1     X2    X3    X4    Y     f       set
#> 10501  0.99 -3.280 -0.39  0.97  8.4  1.25 unlabeled
#> 10502 -0.66  0.142 -1.36 -0.22 -7.2 -1.08 unlabeled
#> 10503  0.58 -1.368 -1.73  0.15  5.6 -0.31 unlabeled
#> 10504 -0.14 -0.728  0.26 -0.23 -4.2  0.91 unlabeled
#> 10505 -0.17 -0.068 -1.10  0.58  2.2 -0.39 unlabeled
#> 10506  0.58  0.514 -0.69  0.97 -1.2  0.76 unlabeled

The simdat function provides observed and unobserved outcomes for both the labeled and unlabeled datasets, though in practice the observed outcomes are not in the unlabeled set. We can visualize the relationships between these variables:

We can see that:

Model Fitting

We compare two non-IPD approaches to analyzing the data to methods included in the ipd package. A summary comparison is provided in the table below, followed by the specific calls for each method:

#>                    Estimate Std.Error
#> Naive                  0.98      0.03
#> Classic                1.10      0.19
#> PostPI (Bootstrap)     1.16      0.18
#> PostPI (Analytic)      1.15      0.18
#> PPI++                  1.12      0.19
#> PSPA                   1.12      0.19

We can see that the IPD methods have similar estimates and standard errors, while the ‘naive’ method has a different estimate and standard errors that are too small.

0.1 ‘Naive’ Regression Using the Predicted Outcomes

#--- Fit the Naive Regression

lm(f ~ X1, data = dat[dat$set == "unlabeled",]) |> 
  
  summary()
#> 
#> Call:
#> lm(formula = f ~ X1, data = dat[dat$set == "unlabeled", ])
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -2.5426 -0.6138 -0.0153  0.6345  2.8907 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   0.8391     0.0297    28.3   <2e-16 ***
#> X1            0.9848     0.0296    33.3   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.94 on 998 degrees of freedom
#> Multiple R-squared:  0.527,  Adjusted R-squared:  0.526 
#> F-statistic: 1.11e+03 on 1 and 998 DF,  p-value: <2e-16

0.2 ‘Classic’ Regression Using only the Labeled Data

#--- Fit the Classic Regression

lm(Y ~ X1, data = dat[dat$set == "labeled",]) |> 
  
  summary()
#> 
#> Call:
#> lm(formula = Y ~ X1, data = dat[dat$set == "labeled", ])
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -15.262  -2.828  -0.094   2.821  11.685 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)    0.908      0.187    4.86  1.6e-06 ***
#> X1             1.097      0.192    5.71  1.9e-08 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 4.2 on 498 degrees of freedom
#> Multiple R-squared:  0.0614, Adjusted R-squared:  0.0596 
#> F-statistic: 32.6 on 1 and 498 DF,  p-value: 1.95e-08

You can fit the various IPD methods to your data and obtain summaries using the provided wrapper function, ipd():

1.1 PostPI Bootstrap Correction (Wang et al., 2020)

#-- Specify the Formula

formula <- Y - f ~ X1

#-- Fit the PostPI Bootstrap Correction

nboot <- 200

ipd::ipd(formula, 
         
  method = "postpi_boot", model = "ols", data = dat, label = "set", 
  
  nboot = nboot) |> 
  
  summary()
#> 
#> Call:
#>  Y - f ~ X1 
#> 
#> Method: postpi_boot 
#> Model: ols 
#> Intercept: Yes 
#> 
#> Coefficients:
#>             Estimate Std.Error Lower.CI Upper.CI
#> (Intercept)    0.866     0.183    0.507     1.22
#> X1             1.164     0.183    0.806     1.52

1.2 PostPI Analytic Correction (Wang et al., 2020)

#-- Fit the PostPI Analytic Correction

ipd::ipd(formula, 
         
  method = "postpi_analytic", model = "ols", data = dat, label = "set") |> 
  
  summary()
#> 
#> Call:
#>  Y - f ~ X1 
#> 
#> Method: postpi_analytic 
#> Model: ols 
#> Intercept: Yes 
#> 
#> Coefficients:
#>             Estimate Std.Error Lower.CI Upper.CI
#> (Intercept)    0.865     0.183    0.505     1.22
#> X1             1.145     0.182    0.788     1.50

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

#-- Fit the PPI Correction

ipd::ipd(formula, 
         
  method = "ppi", model = "ols", data = dat, label = "set") |> 
  
  summary()
#> 
#> Call:
#>  Y - f ~ X1 
#> 
#> Method: ppi 
#> Model: ols 
#> Intercept: Yes 
#> 
#> Coefficients:
#>             Estimate Std.Error Lower.CI Upper.CI
#> (Intercept)    0.871     0.182    0.514     1.23
#> X1             1.122     0.195    0.740     1.50

3. PPI++ (Angelopoulos et al., 2023)

#-- Fit the PPI++ Correction

ipd::ipd(formula, 
         
  method = "ppi_plusplus", model = "ols", data = dat, label = "set") |> 
  
  summary()
#> 
#> Call:
#>  Y - f ~ X1 
#> 
#> Method: ppi_plusplus 
#> Model: ols 
#> Intercept: Yes 
#> 
#> Coefficients:
#>             Estimate Std.Error Lower.CI Upper.CI
#> (Intercept)    0.881     0.182    0.524     1.24
#> X1             1.116     0.187    0.750     1.48

4. Post-Prediction Adaptive Inference (PSPA; Miao et al., 2023)

#-- Fit the PSPA Correction

ipd::ipd(formula, 
         
  method = "pspa", model = "ols", data = dat, label = "set") |> 
  
  summary()
#> 
#> Call:
#>  Y - f ~ X1 
#> 
#> Method: pspa 
#> Model: ols 
#> Intercept: Yes 
#> 
#> Coefficients:
#>             Estimate Std.Error Lower.CI Upper.CI
#> (Intercept)    0.881     0.182    0.524     1.24
#> X1             1.109     0.187    0.743     1.47

Printing and Tidying

The package also provides custom print, summary, tidy, glance, and augment methods to facilitate easy model inspection:

#-- Fit the PostPI Bootstrap Correction

nboot <- 200

fit_postpi <- ipd::ipd(formula, 
         
  method = "postpi_boot", model = "ols", data = dat, label = "set", 
  
  nboot = nboot)
  
#-- Print the Model

print(fit_postpi)
#> 
#> Call:
#>  Y - f ~ X1 
#> 
#> Coefficients:
#> (Intercept)          X1 
#>        0.86        1.15

#-- Summarize the Model

summ_fit_postpi <- summary(fit_postpi)
  
#-- Print the Model Summary

print(summ_fit_postpi)
#> 
#> Call:
#>  Y - f ~ X1 
#> 
#> Method: postpi_boot 
#> Model: ols 
#> Intercept: Yes 
#> 
#> Coefficients:
#>             Estimate Std.Error Lower.CI Upper.CI
#> (Intercept)    0.860     0.183    0.502     1.22
#> X1             1.148     0.182    0.790     1.50

#-- Tidy the Model Output

tidy(fit_postpi)
#>                    term estimate std.error conf.low conf.high
#> (Intercept) (Intercept)     0.86      0.18     0.50       1.2
#> X1                   X1     1.15      0.18     0.79       1.5

#-- Get a One-Row Summary of the Model

glance(fit_postpi)
#>        method model include_intercept nobs_labeled nobs_unlabeled       call
#> 1 postpi_boot   ols              TRUE          500           1000 Y - f ~ X1

#-- Augment the Original Data with Fitted Values and Residuals

augmented_df <- augment(fit_postpi)

head(augmented_df)
#>          X1     X2    X3    X4    Y     f       set .fitted .resid
#> 10501  0.99 -3.280 -0.39  0.97  8.4  1.25 unlabeled   1.992    6.5
#> 10502 -0.66  0.142 -1.36 -0.22 -7.2 -1.08 unlabeled   0.099   -7.3
#> 10503  0.58 -1.368 -1.73  0.15  5.6 -0.31 unlabeled   1.522    4.1
#> 10504 -0.14 -0.728  0.26 -0.23 -4.2  0.91 unlabeled   0.702   -4.9
#> 10505 -0.17 -0.068 -1.10  0.58  2.2 -0.39 unlabeled   0.667    1.5
#> 10506  0.58  0.514 -0.69  0.97 -1.2  0.76 unlabeled   1.521   -2.7

Vignette

For additional details, we provide more use cases and examples in the package vignette:

vignette("ipd")

Feedback

For questions, comments, or any other feedback, please contact the developers (ssalerno@fredhutch.org).

Contributing

Contributions are welcome! Please open an issue or submit a pull request on GitHub. The following method/model combinations are currently implemented:

Method Mean Estimation Quantile Estimation Linear Regression Logistic Regression Poisson Regression Multiclass Regression
PostPI :x: :x: :white_check_mark: :white_check_mark: :x: :x:
PPI :white_check_mark: :white_check_mark: :white_check_mark: :white_check_mark: :x: :x:
PPI++ :white_check_mark: :white_check_mark: :white_check_mark: :white_check_mark: :x: :x:
PSPA :white_check_mark: :white_check_mark: :white_check_mark: :white_check_mark: :white_check_mark: :x:
PSPS :x: :x: :x: :x: :x: :x:
PDC :x: :x: :x: :x: :x: :x:
Cross-PPI :x: :x: :x: :x: :x: :x:
PPBoot :x: :x: :x: :x: :x: :x:
DSL :x: :x: :x: :x: :x: :x:

License

This package is licensed under the MIT License.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
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