# Cram Policy part 2

# Cram Policy

In the article “Introduction & Cram Policy Part 1”, we introduced the Cram method, which enables simultaneous learning and evaluation of a binary policy. We outlined the primary parameters of the `cram_policy()` function and demonstrated its application using an example dataset. In this section, we provide a more detailed discussion on how to configure these parameters for various use cases, depending notably on the nature of the dataset and specific policy learning goals.

We begin by simulating a dataset consisting of covariates `X`, a binary treatment assignment `D`, and a continuous outcome `Y`, which we will use to demonstrate the `cram_policy()` function.

    library(data.table)
    # Function to generate sample data with heterogeneous treatment effects:
    # - Positive effect group
    # - Neutral effect group
    # - Adverse effect group
    generate_data <- function(n) {
      X <- data.table(
        binary = rbinom(n, 1, 0.5),                 # Binary variable
        discrete = sample(1:5, n, replace = TRUE),  # Discrete variable
        continuous = rnorm(n)                       # Continuous variable
      )

      # Binary treatment assignment (50% treated)
      D <- rbinom(n, 1, 0.5)

      # Define heterogeneous treatment effects based on X
      treatment_effect <- ifelse(
        X[, binary] == 1 & X[, discrete] <= 2,        # Group 1: Positive effect
        1,
        ifelse(X[, binary] == 0 & X[, discrete] >= 4, # Group 3: Adverse effect
               -1,
               0.1)                                   # Group 2: Neutral effect
      )

      # Outcome depends on treatment effect + noise
      Y <- D * (treatment_effect + rnorm(n, mean = 0, sd = 1)) +
        (1 - D) * rnorm(n)

      return(list(X = X, D = D, Y = Y))
    }

    # Generate a sample dataset
    set.seed(123)
    n <- 1000
    data <- generate_data(n)
    X <- data$X
    D <- data$D
    Y <- data$Y

## Built-in Model

In this example, we demonstrate how to use the built-in modeling options provided by the `cramR` package. We walk through the key parameters that control how `cram_policy()` behaves, including the choice of model, learner type, baseline policy, and batching strategy.

These parameters allow flexibility in configuring the learning process depending on your use case and nature of your dataset.

    # Options for batch:
    # Either an integer specifying the number of batches or a vector/list of batch assignments for all individuals
    batch <- 20

    # Model type for estimating treatment effects
    # Options for model_type: 'causal_forest', 's_learner', 'm_learner'
    # Note: you can also set model_type to NULL and specify custom_fit and custom_predict to use your custom model
    model_type <- "causal_forest"

    # Options for learner_type:
    # if model_type == 'causal_forest', choose NULL
    # if model_type == 's_learner' or 'm_learner', choose between 'ridge', 'fnn' and 'caret'
    learner_type <- NULL

    # Baseline policy to compare against (list of 0/1 for each individual)
    # Options for baseline_policy:
    # A list representing the baseline policy assignment for each individual.
    # If NULL, a default baseline policy of zeros is created.
    # Examples of baseline policy:
    # - All-control baseline: as.list(rep(0, nrow(X))) or NULL
    # - Randomized baseline: as.list(sample(c(0, 1), nrow(X), replace = TRUE))
    baseline_policy <- as.list(rep(0, nrow(X)))

    # Whether to parallelize batch processing (i.e. the cram method learns T policies, with T the number of batches.
    # They are learned in parallel when parallelize_batch is TRUE
    # vs. learned sequentially using the efficient data.table structure when parallelize_batch is FALSE, recommended for light weight training).
    # Defaults to FALSE.
    parallelize_batch <- FALSE

    # Model-specific parameters (more details in the article "Quick Start")
    # Examples: NULL defaults to the following:
    # - causal_forest: list(num.trees = 100)
    # - ridge: list(alpha = 1)
    # - caret: list(formula = Y ~ ., caret_params = list(method = "lm", trControl = trainControl(method = "none")))
    # - fnn (Feedforward Neural Network): see below
    # input_shape <- if (model_type == "s_learner") ncol(X) + 1 else ncol(X)
    # default_model_params <- list(
    #       input_layer = list(units = 64, activation = 'relu', input_shape = input_shape),
    #       layers = list(
    #         list(units = 32, activation = 'relu')
    #       ),
    #       output_layer = list(units = 1, activation = 'linear'),
    #       compile_args = list(optimizer = 'adam', loss = 'mse'),
    #       fit_params = list(epochs = 5, batch_size = 32, verbose = 0)
    #     )
    model_params <- NULL

### How to set model_params?

The `model_params` argument allows you to customize hyperparameters for the model used in policy learning. If left as `NULL`, `cram_policy()` will fall back to sensible defaults depending on the model and learner type. In the following, we give default examples of `model_params` for each `model_type` and `learner_type` to illustrate how you can specify `model_params`. Generally speaking, `model_params` is a list containing all of the parameters used in the underlying model and its values:

- For `model_type = "causal_forest"`:
  When `model_params <- NULL`, the method defaults to `grf::causal_forest()` with `model_params <- list(num.trees = 100)`.

- For `model_type = "s_learner"` or `"m_learner"`, it depends on the `learner_type`:

  - For `learner_type = "ridge"`, when `model_params <- NULL`, the method defaults to `glmnet::cv.glmnet()` with `model_params <- list(alpha = 1)`, corresponding to Ridge regression.

  - For `learner_type = "fnn"` (Feedforward Neural Network):
    A default Keras model is built with the following architecture:

      # Determine the input shape based on model_type
      # For s_learner, the treatment D is added to the covariates to constitute the training data
      # So the input shape is ncol(X) + 1
      input_shape <- if (model_type == "s_learner") ncol(X) + 1 else ncol(X)

      default_model_params <- list(
          input_layer = list(units = 64, activation = 'relu', input_shape = input_shape),
          layers = list(
            list(units = 32, activation = 'relu')
          ),
          output_layer = list(units = 1, activation = 'linear'),
          compile_args = list(optimizer = 'adam', loss = 'mse'),
          fit_params = list(epochs = 5, batch_size = 32, verbose = 0)
        )

  - For `learner_type = "caret"`, it defaults to a linear regression with no re sampling (see https://topepo.github.io/caret/model-training-and-tuning.html for details about the caret train parameters):

      default_model_params <- list(formula = Y ~ ., caret_params = list(method = "lm", trControl = trainControl(method = "none")))

  Please note that the list should contain an element `formula` where `Y` refers to the vector that you provided as input for `Y`, and an element named `caret_params` containing the parameters of your choice to pass to `caret::train` (see full list of parameters here: https://topepo.github.io/caret/model-training-and-tuning.html).

### Case of categorical target Y

When using `caret`, note that if `Y` is categorical and you are using `model_type = "s_learner"`, you need to choose a classification method outputting probabilities i.e. using the key word `classProbs = TRUE` in `trainControl`, see the following as an example with a Random Forest Classifier:

    model_params <- list(formula = Y ~ ., caret_params = list(method = "rf", trControl = trainControl(method = "none", classProbs = TRUE)))

Also note that all data inputs needs to be of numeric types, hence for `Y` categorical, it should contain numeric values representing the class of each observation. No need to use the type `factor` for `cram_policy()`.

Note on M-learner: If `model_type = "m_learner"`, keep in mind that Y is transformed internally. M-learner requires a propensity model and transformed outcomes:

- propensity is an argument of cram_policy, if `NULL`, it defaults to

    propensity <- function(X) {rep(0.5, nrow(X))}

- outcome_transform is an element of model_params accessed as follows:

    outcome_transform <- model_params$m_learner_outcome_transform

If `NULL`, it defaults to

    outcome_transform <- function(Y, D, prop_score) {Y * D / prop_score - Y * (1 - D) / (1 - prop_score)}`, where `prop_score` is `propensity(X)

Thus, the Y transformed might not be categorical even though Y is categorical i.e. you might want to use a regression model.

See the following reference for more details about the M-learner:

Jia, Z., Imai, K., & Li, M. L. (2024). The Cram Method for Efficient Simultaneous Learning and Evaluation, URL: https://www.hbs.edu/ris/Publication%20Files/2403.07031v1_a83462e0-145b-4675-99d5-9754aa65d786.pdf.

You can override any of these defaults by passing a custom list to model_params including any parameter name defined in the underlying package of the model that you chose, namely `grf::causal_forest()`, `glmnet::cv.glmnet()`, `keras` and `caret::train()`

    # Significance level for confidence intervals (default = 95%)
    alpha <- 0.05

    # Run the CRAM policy method
    result <- cram_policy(
      X, D, Y,
      batch = batch,
      model_type = model_type,
      learner_type = learner_type,
      baseline_policy = baseline_policy,
      parallelize_batch = parallelize_batch,
      model_params = model_params,
      alpha = alpha
    )

    # Display the results
    print(result)
    #> $raw_results
    #>                        Metric   Value
    #> 1              Delta Estimate 0.23208
    #> 2        Delta Standard Error 0.05862
    #> 3              Delta CI Lower 0.11718
    #> 4              Delta CI Upper 0.34697
    #> 5       Policy Value Estimate 0.21751
    #> 6 Policy Value Standard Error 0.05237
    #> 7       Policy Value CI Lower 0.11486
    #> 8       Policy Value CI Upper 0.32016
    #> 9          Proportion Treated 0.60500
    #>
    #> $interactive_table
    #>
    #> $final_policy_model
    #> GRF forest object of type causal_forest
    #> Number of trees: 100
    #> Number of training samples: 1000
    #> Variable importance:
    #>     1     2     3
    #> 0.437 0.350 0.213

## Custom Model

To use your own model for policy learning, the trigger is to set `model_type = NULL` and to supply two user-defined functions:

### 1. `custom_fit(X, Y, D, ...)`

This function takes the training data: covariates `X`, outcomes `Y`, and binary treatment indicators `D`, and returns a fitted model object.
You may also define and use additional parameters (e.g., number of folds, regularization settings, etc.) within the function body.

- `X`: a matrix or data frame of features
- `Y`: a numeric outcome vector
- `D`: a binary vector indicating treatment assignment (0 or 1)

Example: Custom X-learner with Ridge regression and 5-fold cross-validation

    library(glmnet)
    #> Loading required package: Matrix
    #> Loaded glmnet 4.1-8

    custom_fit <- function(X, Y, D, n_folds = 5) {
      treated_indices <- which(D == 1)
      control_indices <- which(D == 0)
      X_treated <- X[treated_indices, ]
      Y_treated <- Y[treated_indices]
      X_control <- X[control_indices, ]
      Y_control <- Y[control_indices]
      model_treated <- cv.glmnet(as.matrix(X_treated), Y_treated, alpha = 0, nfolds = n_folds)
      model_control <- cv.glmnet(as.matrix(X_control), Y_control, alpha = 0, nfolds = n_folds)
      tau_control <- Y_treated - predict(model_control, as.matrix(X_treated), s = "lambda.min")
      tau_treated <- predict(model_treated, as.matrix(X_control), s = "lambda.min") - Y_control
      X_combined <- rbind(X_treated, X_control)
      tau_combined <- c(tau_control, tau_treated)
      weights <- c(rep(1, length(tau_control)), rep(1, length(tau_treated)))
      final_model <- cv.glmnet(as.matrix(X_combined), tau_combined, alpha = 0, weights = weights, nfolds = n_folds)
      return(final_model)
    }

### 2. `custom_predict(model, X_new, D_new)`

This function uses the fitted model to generate a binary treatment decision for each individual in `X_new`.

It should return a vector of 0s and 1s, indicating whether to assign treatment (`1`) or not (`0`).
You may also incorporate a custom threshold or post-processing logic within the function.

Example: Apply the decision rule — treat if the estimated CATE is greater than 0

    custom_predict <- function(model, X_new, D_new) {
      cate <- predict(model, as.matrix(X_new), s = "lambda.min")

      # Apply decision rule: treat if CATE > 0
      as.integer(cate > 0)
    }

### 3. Use `cram_policy()` with `custom_fit()` and `custom_predict()`

Once both `custom_fit()` and `custom_predict()` are defined, you can integrate them into the Cram framework by passing them to `cram_policy()` as shown below (do not forget to set `model_type = NULL`):

    experiment_results <- cram_policy(
      X, D, Y,
      batch = 20,
      model_type = NULL,
      custom_fit = custom_fit,
      custom_predict = custom_predict,
      alpha = 0.05
    )
    print(experiment_results)
    #> $raw_results
    #>                        Metric   Value
    #> 1              Delta Estimate 0.22542
    #> 2        Delta Standard Error 0.06004
    #> 3              Delta CI Lower 0.10774
    #> 4              Delta CI Upper 0.34310
    #> 5       Policy Value Estimate 0.21085
    #> 6 Policy Value Standard Error 0.04280
    #> 7       Policy Value CI Lower 0.12696
    #> 8       Policy Value CI Upper 0.29475
    #> 9          Proportion Treated 0.54500
    #>
    #> $interactive_table
    #>
    #> $final_policy_model
    #>
    #> Call:  cv.glmnet(x = as.matrix(X_combined), y = tau_combined, weights = weights,      nfolds = n_folds, alpha = 0)
    #>
    #> Measure: Mean-Squared Error
    #>
    #>     Lambda Index Measure      SE Nonzero
    #> min 0.0395   100  0.9219 0.02942       3
    #> 1se 0.5868    71  0.9505 0.03367       3

## References

- Jia, Z., Imai, K., & Li, M. L. (2024). The Cram Method for Efficient Simultaneous Learning and Evaluation, URL: https://www.hbs.edu/ris/Publication%20Files/2403.07031v1_a83462e0-145b-4675-99d5-9754aa65d786.pdf.
- Künzel, S. R., Sekhon, J. S., Bickel, P. J., & Yu, B. (2019). Metalearners for estimating heterogeneous treatment effects using machine learning. Proceedings of the National Academy of Sciences of the United States of America, 116(10), 4156–4165. https://doi.org/10.1073/pnas.1804597116
- Wager, S., & Athey, S. (2018). Estimation and inference of heterogeneous treatment effects using random forests. Journal of the American Statistical Association, 113(523), 1228-1242.
- Athey, S., & Imbens, G. (2016). Recursive partitioning for heterogeneous causal effects. Proceedings of the National Academy of Sciences, 113(27), 7353-7360.