Supplementary file: Comprehensive comparative analysis of local false discovery rate control methods

We consider three different scenarios: basic, mean shift, and scale change. For comparison purpose, we generate data matrix, size of 3000 by 40, each group consisting of 20, respectively. Data structure for one-sided alternative is given in Fig. 2(a) and data structure for two-sided case is given in 2(b). Throughout simulation study, we consider π0 = 0.8 and the standard normal density is used as null density. Furthermore, for each scenario, we consider two different true alternatives: one-sided and two-sided alternative. In case of one-sided alternative, 20 percent of one group belongs to alternative, which is highlighted in red in Fig. 2(a). That is, they are generated from N(μ, σ) where μ > 0. In addition, for two-sided alternative, 10 percent of


Simulation setup
We consider three different scenarios: basic, mean shift, and scale change. For comparison purpose, we generate data matrix, size of 3000 by 40, each group consisting of 20, respectively. Data structure for one-sided alternative is given in Fig. 2(a) and data structure for two-sided case is given in 2(b).
Throughout simulation study, we consider π 0 = 0.8 and the standard normal density is used as null density. Furthermore, for each scenario, we consider two different true alternatives: one-sided and two-sided alternative. In case of one-sided alternative, 20 percent of one group belongs to alternative, which is highlighted in red in Fig. 2(a). That is, they are generated from N (µ, σ 2 ) where µ > 0. In addition, for two-sided alternative, 10 percent of 1 Supplementary Figure 2: Simulation setup: One-sided alternative (left) and two-sided alternative (right).
one group is generated from N (µ, σ 2 ) where µ > 0 and the other 10 percent from N (−µ, σ 2 ). In all scenarios, the size of data is 3000 × 40 and this process is repeated 100 times.

Basic scenario
To generate sample for alternative density, µ = 2.5 and σ 2 = 1.5 2 are used. We generate 3000 × 40 data matrix and then estimate FDR by using all methods. Three different cutoffs (0.05, 0.1, 0.2) are applied to the estimated false discovery rate.

Mean shift scenario
To generate sample for alternative density, three different means, µ = (1, 1.5, 2) are considered. However, variance does not change here. We generate 3000 × 40 data matrix and then estimate local FDR by using all methods. Three cutoff values of (0.05, 0.1, 0.2) are applied to the estimated false discovery rate.

Scale change scenario
Unlike the mean-shift scenario, we change the variance of alternative density. To generate sample for alternative density, three different variances of alternative density, kσ 2 , i.e., k = (2, 3, 4), σ 2 = 1.5 2 . We generate 3000 × 40 data matrix and then estimate FDR by using all methods. Three cutoff values of (0.05, 0.1, 0.2) are applied to the estimated false discovery rate.

Simulation results
For the estimation of fdr1d, three methods are considered: Efron, Ploner1d, and Ploner1dE. For the estimation of fdr2d, we consider two methods such as Ploner2d, and Kim.

Basic scenario
We consider two types of alternative densities, which are well-separated from the true null: one-sided or two-sided. For one-sided alternative, right-tail alternative only is considered. In case of two-sided alternative, symmetric alternative density is considered. We generate 3000 × 40 data matrix and then estimate FDR by using all methods. Three different cutoffs (0.05, 0.1, 0.2) are applied to the estimated false discovery rate.
One-sided alternative We set π 1 = 0.2. To generate sample, N (0, 1) and N (2.5, 1.5 2 ) are used as null and alternative density, respectively. The true density, which generate random samples, is given in Fig. 3 Supplementary Figure 3: Basic scenario: true density. Figure 1 in the main text includes estimated fdr1d. Furthermore, Figure 2 in the main text includes estimated fdr2d: union null(left) and intersection null(right).
For each method, mean and standard error of the estimated FDRs over 100 repetitions are summarized in Table 1. As seen in Table 1, all procedures control local FDR strictly.
Supplementary Table 1 Also, we selected one sample out of 100 and calculated some performance measures for each method when cutoff=0.1 (Table 2). Not surprisingly, all methods in case of basic scenario show very good performance. Two-sided alternative To generate alternative sample, N (2.5, 1.5 2 ) and N (−2.5, 1.5 2 ) are used, i.e., 10 percent from each density. Figure 3 in the main text includes the estimated fdr1d and Figure 4 in main text includes fdr2d for union null (left) and intersection null (right).

Mean shift scenario
As mean value of alternative density, three different means, µ = (1, 1.5, 2) are considered. However, variance does not change here. Again, we consider two types of alternative: onesided or two-sided. We generate 3000 × 40 data matrix and then estimate local FDR by using all methods. Three cutoff values of (0.05, 0.1, 0.2) are applied to the estimated false discovery rate.
One-sided alternative In the mixture model, we set π 1 = 0.2 and random sample for G1 come from N (µ, 1.5 2 ). Three different µ = (1, 1.5, 2) are considered. Fig. 4 includes the true density, which are used to generate random samples for each mean value.
Supplementary Figure 4: Mean shift scenario: True densities for µ 1 = (1, 1.5, 2).  Figures 6 and 7 correspond to µ = 1.5 and µ = 2, respectively. Compared to the results from Basic scenario, we observe relatively small margin between true fdr and estimated fdr1d in the mean shift scenario.