Project: TENSOR-BASED INDIVIDUALIZED TREATMENT RULE (TITR)

1. Overview

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We present implementation for estimation of TITR model. For low-dimensional setting, the estimate is obtained by an alternating update algorithm based on Tucker decomposition. For sparse scenario, estimation is performed using Tucker decomposition with potentially distinct ranks for treatment-free and treatment-related effects, combined with L1-regularization to induce sparsity, and an ADMM-based optimization algorithm.

2. Simulation

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• low_d.py: Functions for low-dimensional setting.
• sparse_testing.py: Functions for sparse scenario. (bootstrap code)
• README.txt: Description of the two code files above.

The results of Simulation 4.1 (Low-dimensional TITR) can be reproduced by running

TITR_code/Simu/low_d.py

This code includes the following functions:

• tensor_r_tucker(): tensor regression
• generate_data(): generate data
• pi_opt(), pi_hat(), pi_random(), pi_notensor(): Return average outcome value and corresponding MSEs under different treatment rules.
• select_rank(): select rank for tensor regression by BIC
• replication(): replication of the simulation

To reproduce Table 1 and Table 2 under different parameter settings, modify the arguments in replication():

• n0 sample size, R0 true rank, p tensor dimension, sigma noise level.

The results of Simulation 4.2 (Sparse TITR) and 4.3 (Effectiveness of Inference Procedure) can be reproduced by running

TITR_code/Simu/sparse_testing.py

This code includes the following functions:

• ADMM(): ADMM algorithm to compute factor matrix
• tensor_r_lasso(): tensor regression
• generate_data(): generate true coefficients when dimension = (50,50,2)
• generate_sample(): generate data sample for given coefficients
• pi_opt(), pi_hat(), pi_random(), pi_notensor(): Return average outcome value and corresponding MSEs under different treatment rules.
• rankgrid(): grid search for different ranks, return corresponding estimation errors and estimated coefficients.
• select_r(): select rank for tensor regression by loss in test set.
• replication(): replication of the simulation (change argument n0)

We use the wild bootstrap to test whether two treatment-related coefficients are equal to zero.

• bootstrap_value(): calculate bootstrap statistics
• generate_data2(), generate_sample2(): generate coefficients and data sample under dimension (10,10,10), rank (2,2,2) and sigma level 0.5.
• replication(): calculate p-values under different delta. (change argument delta)

3. ADNI Real Data

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For the ADNI real data analysis, please download the dataset from the following link: Google Drive – ADNI Data

After downloading, place the data in the following directory:

TITR_code/Real Data/ADNI haimati image

Then, update the data path accordingly in the notebooks left_image_modeling.ipynb and right_image_modeling.ipynb.

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