Here we include the Python files that can reproduce all the figures presented in both the main text and the supplementary material.

Description of files and folder:

• lib_fast_solve.py : computes the matrix $\sum_{i\in[n]} \mathsf{V}_i$ appeared in Theorems 2.1 and 2.2.

• scatter.py : generates and saves scatter plots with different values of $(n, p, K, q)$. This file is used to produce Figure 1 and Figure S3.

• qqplot.py : generates and saves Q-Q plots and histograms for different $(n, p, K)$ with $q=1$ using $\Omega_{jj}$ or using $\hat \Omega_{jj}$ under three different covariate distributions (Gaussian, Rademacher, SNP). This file is used to produce Figures 2, 3, S1, and S2.

• real_data_example.ipynb: conduct real data analysis on a heart disease data.

• Folder figs: contains all the generated figures from scatter.py and qqplot.py.

Executing the following steps will generate all the figures used in the main text and the supplement. The figures are saved to the folder figs.

1. Run the script scatter.py: this step will generate all the scatter plots (Figure 1 in the main text and Figure S3 in the supplement).

   We need to specify the setting in the script scatter.py:

   • setting = 1: $(n, p, q) = (2000, 600, 1)$
   • setting = 2: $(n, p, q) = (3500, 1000, 1)$
   • setting = 3: $(n, p, q) = (5000, 1500, 1)$
   • setting = 4: $(n, p, q) = (2000, 600, 2)$
   • setting = 5: $(n, p, q) = (2000, 600, 4)$

   This will generate five PDF files with meaningful file names. For example, the name 'scatter_Gaussian_true_n2000_p600_q1.pdf' means the scatter plot generated from the data with Gaussian distributed covariates and the true $\Omega_{jj}$ under the setting $(n, p, q) = (2000, 600, 1)$.

2. Run the script qqplot.py: this step will generate all the Q-Q plots and histograms of P-values (Figures 2, 3 in the main text and Figures S1 and S2 in the supplement).

   We need to specify the values of (dist, omega, K) in the script qqplot.py.

   • There are three options for dist: dist = 'Gaussian' means covariates are sampled from Gaussian distribution; dist = 'Rademacher' means covariates are sampled from Rademacher distribution; dist = 'SNP' means covariates are sampled from SNP distribution. The detailed description of these distributions are given in Section 3 of the main text.
   • There are two options for omega: omega = 'true' means we use the true $\Omega_{jj}$; omega = 'est' means we use the estimated $\Omega_{jj}$ (namely, $\hat \Omega_{jj}$ in equation (2.5) of the main text).
   • There are three options for K: K = 2 means we consider the setting with $(n, p, K) = (4000, 1000, 2)$; K = 3 means we consider the setting with $(n, p, K) = (5000, 1000, 3)$; K = 4 means we consider the setting with $(n, p, K) = (6000, 1000, 4)$.

   Executing the script qqplot.py with the above combinations of (dist, omega, K) will generate PDF files with meaningful file names. For example, the name 'qqplots_Gaussian_true_K2.pdf' means the Q-Q plot generated from the setting with Gaussian covariates using the true $\Omega_{jj}$ with $K=2$, and the name 'hist_Gaussian_est_K2.pdf' means the histogram of p-values generated from the setting with Gaussian covariates using the estimated $\Omega_{jj}$.

All the simulations were run on a cluster of 50 CPU-cores (each is an Intel Xeon E5-2680 v4 @2.40GHz) equipped with a total of 200 GB of RAM.

• We used Python 3.9.6 to run all the simulations.

• The execution times of scripts scatter.py and qqplot.py vary and can range from 2 minutes to 90 minutes. The specific duration depends on the values of $(n, p, K, q)$. Generally, running scripts with larger values of $(n, p, K, q)$ takes longer time. For example, running scatter.py with $(n, p, q) = (2000, 600, 1)$ takes approximately 2 minutes, while executing qqplot.py with $(n, p, K) = (6000, 1000, 4)$ takes around 90 minutes.

@article{tan2024multinomial,
  title={Multinomial logistic regression: Asymptotic normality on null covariates in high-dimensions},
  author={Tan, Kai and Bellec, Pierre C},
  journal={Advances in Neural Information Processing Systems},
  volume={36},
  year={2024}
}