Medical, Pharma, Engineering, Science, Technology and Business

Department of Mathematics and Statistics, University of the West Indies, St. Augustine Campus, Trinidad and Tobago

- *Corresponding Author:
- Dialsingh I

Department of Mathematics and Statistics

University of the West Indies

St. Augustine Campus, Trinidad and Tobago

**Tel:**+1 876-927-1660

**E-mail:**[email protected]

**Received Date**: April 1, 2017; **Accepted Date:** April 24, 2017; **Published Date**: April 28, 2017

**Citation: **Dialsingh I, Cedeno SP (2017) Comparison of Methods for Estimating
the Proportion of Null Hypotheses π_{0} in High Dimensional Data When the Test
Statistics is Continuous. J Biom Biostat 8: 343. doi: 10.4172/2155-6180.1000343

**Copyright:** © 2017 Dialsingh I, et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the
original author and source are credited.

**Visit for more related articles at** Journal of Biometrics & Biostatistics

Advances in Genomics have re-energized interest in multiple hypothesis testing procedures but have simultaneously created new methodological and computational challenges. In Genomics for instance, it is now commonplace for experiments to measure expression levels in thousands of genes creating large multiplicity problems when thousands of hypotheses are to be tested simultaneously. Within this context we seek to identify differentially expressed genes, that is, genes whose expression levels are associated with a particular response or covariate of interest. The False Discovery Rate (FDR) is the preferred measure since the Family Wise Error Rates (FWERs) are usually overly restrictive. In the FDR methods, estimation of the proportion of null hypotheses (π0) is an important parameter that needs to be estimated. In this paper, we compare the effectiveness of 12 methods for estimating π0 when the test statistics are continuous using simulated data with independent, weak dependence, and moderate dependence structures.

Family-wise error rate; False discovery rate; High dimensional data; Hypothesis testing; Multiple hypothesis testing; Proportion of null hypotheses

A hypothesis is a claim or assertion either about a population
parameter. In the case of a single hypothesis, we typically test the null
hypothesis H_{0} versus an alternative hypothesis H_{a} based on some test
statistic. We reject H_{0} in favor of H_{a} whenever the test statistic lies in
the rejection region specified by some rejection rule. The test statistic is
a function of the sample data on which the decision to reject or not to
reject H_{0} will be based. Within the realm of hypothesis testing, there are
two possible types of errors that can be committed. A Type I error, is
committed if we reject H_{0} when H_{0} is true while a Type II error occurs
when we fail to reject H_{0} when H_{0} is false, **Table 1** summarizes the error
possibilities.

Declared True | Declared False | |
---|---|---|

H0 True | Correct Decision (1-α) | Type I Error (α) |

H0 False | Type II Error (β) | Correct Decision (1-β) |

**Table 1:** Possible outcomes for a single hypothesis test.

The problem associated with single hypothesis testing is compounded in the realm of multiple hypothesis testing. Multiple hypothesis testing is concerned with controlling the rate of false positives when we are testing multiple hypotheses simultaneously. When conducting multiple hypothesis tests, the probability of making at least one Type I error is substantially higher than the nominal level used for each test, particularly when the number of total tests, m, is large.

It is not unusual to have thousands of tests being done simultaneously. This implies that the probability of getting at least one significant result approaches 1. These methods of multiple hypothesis testing require an adjustment of α in some way so that the probability of observing at least one significant event, due largely to chance, remains below the chosen level of statistical significance.

**Notation**

Tests are typically assumed to be independent, or that the test
statistics are independent and identically distributed, but there are
methods and procedures that deal with cases of dependence. We assume that we are testing m independent null hypotheses, H_{01}, H_{02},
… ,H_{0m} with corresponding p-values p_{1}, p_{2},.., p_{m}, and we call the ith
hypothesis “significant'' if we reject the null hypothesis H_{0i}. **Table 2** summarizes the possible configurations when testing m hypotheses
simultaneously. In this table, V is the number of false rejections (or
false discoveries), U is the number of true non-rejections (or true
acceptances), S is the number of true rejections, and T is the number of
false non-rejections. Here m_{0}, the total number of true null hypotheses,
is fixed but unknown. Though random variables V, S, U, and T are
not observable, the random variables R=S+V and W=U+T, the number
of significant and insignificant tests, respectively, are observable. The
proportion of false rejections is V/R when R>0 and the proportion of
false acceptances is T/W when W>0.

Significant | Not Significant | Total | |
---|---|---|---|

True Null | V | U | m0 |

False Null | S | T | m1 |

Total | R | W | m |

**Table 2:** Possible outcomes for m hypothesis tests.

Using this notation, the FWER rate which is the Probability of making at least one type I error among the m tests is given by:

FWER=Prob (V ≥ 1) or FWER=1 – Prob (V=0) (1)

While the False Discovery Rate (FDR) which is defined as the expected proportion of false rejections is given by:

FDR=E(Q) (2)

where (3)

Many methods have been created modifying the FWER concepts. A review of these techniques are found in [1,2]. The majority of these methods involves the use of step-up or step-down procedures [3-5]. Controlling FWER results in tests with low power. The FDR is less restrictive and allows a certain amount of false positives [6]. A number of methods have been established for estimation of the proportion of null hypotheses when the test statistics are discrete [7]. In this paper, we compare 12 methods for comparing false discovery rates when the test statistics are continuous under varying dependence structures.

**Methods for controlling false discovery rate**

Modern approaches in multiple hypothesis testing focus on False Discovery Rate (FDR) control [6], which is a simple step-up procedure using the ordered p-values of the tests. False Discovery Rate originated in two papers that dealt with multiple hypothesis testing. Schweder and Spjotvoll [8] first suggested that the ranked p-values be plotted. The assessment of the number of true null hypotheses m0 would then be done via an eye fitted straight line starting from the largest p-values. The p-values that deviate (outliers) from this straight line would then correspond to the false null hypotheses. The density of the p-values can be expressed as:

f(p)=π_{0} f_{0}(p) + (1- π_{0}) f1(p) (4)

Where π_{0} and π_{1} represent the proportion of p-values under the
null density f_{0} and under the alternative density f_{1} respectively. For
continuous tests, p-values are uniformly distributed on the interval
(0,1). The distribution of the p-values under the alternative hypothesis
is unknown. Methods for estimating π_{0} when the test statistics are
continuous have been developed by coupling the mixture model with
the assumption that either the density of marginal p-values, f(p), or the
density of p-values under the alternative, f_{1}(p) is non-increasing.

The first procedure than controls FDR is the BH Algorithm [6]. To control FDR at level q, reject all null hypotheses where:

(5)

It has been shown that when the test statistics are continuous and
independent, this procedure controls the FDR at level π_{0} q where π_{0} is the proportion of true null hypotheses. In this procedure, each of
the m hypotheses are treated as null so in this case, π_{0}=1. There are
many other methods that were created. We briefly describe 12 of these
methods for which we do our comparisons on.

**Storey’s Smoother Method**

The proposed estimator is:

(6)

Where λ ϵ [0,1) and is called a tuning parameter [9]. Beyond 0.5 the histogram of the p-values is flat and this suggests that mostly null p-values are there. Taking λ=0 gives which is overly conservative and as λ→ 1 equation (6) above indicates that the variance of increases and this makes the estimated q-values unreliable.

The general algorithm for estimating q-values from the p-values is:

1. Assume that we have m ordered p-values according to their
evidence against the null hypotheses, i.e. p_{(1)} ≤ p_{(2)} ≤…≤ p_{(m)}.

Use equation (6) and compute for various values of λ.

Let f be a natural cubic spline with 3 degrees of freedom of on λ.

Set the estimate of π_{0} to be

Let t be a threshold where 0 < t ≤ 1 calculate:

(7)

2. For i=m-1, m-2, …, 1 calculate:

(8)

3. Equation (8) above gives the estimated q-value for the i^{th} most significant feature.

This method is referred to as “smoother” throughout this paper.

**Storey’s bootstrap method**

Storey’s bootstrap method [10] resamples the m ordered p-values
p_{(1)} ≤ _{p(2)} ≤ … ≤ p_{(m)} with replacement and creates pseudo-data sets for
some number of bootstrap resamples B. The p-values are then calculated
for each pseudo-data set and a bootstrap estimate is done for the FDR.
The method enables a counter to record whether the minimum p-value
from the pseudo-data set is less than or equal to the actual p-value for
each base test. For m tests there would be m such counters. This process
is repeated a large number of times, say B, and the proportion of
resampled data sets where the minimum pseudo p-values is less than or
equal to an actual p-value is the adjusted p-value. This method has the
ability to incorporate all sources of correlation from both the multiple
contrasts and the multivariate structure. The resulting adjusted p-values
incorporate all correlations and distributional characteristics. The
bootstrap method provides strong control when the joint distribution
of the p-values for any subset of the null hypotheses is identical to that
under the complete null hypothesis, that is, when the subset pivotality
condition holds. It always provides weak control of the FWER. This
method is referred to as “st.boot” throughout this paper.

**Two-step procedure of Jiang and Doerge**

This two-step procedure proposed by Jiang and Doerge [11] increase the power of detecting differentially expressed genes in microarray data. The null hypothesis of equality across the mean expression levels for all treatments is tested in step one. In the second step pairwise comparisons are done on genes for which the treatment means were shown to be statistically significant in step one. This approach estimates the overall FDR used in both steps in such a way that the overall FDR is controlled below a pre-specified FDR significance level. The twostep approach has increased power over a one-step procedure and it controls the FDR at the desired level of significance. The algorithm for this two-step procedure is as follows:

1. Test the null hypothesis that a gene is not differentially
expressed across each treatment condition. This can be done using
the global F-test from an ANOVA model for instance. For m tests
corresponding to m genes an FDR control procedure is applied to
control FDR at the α1 level. Tests that produce p-values ≤ some specified c_{1} are considered to be statistically significant. If we get L significant
tests and M genes are declared to have statistically significant treatment
effects then we go to step 2. However if L=0 then we stop and conclude
that there are no statistically significant pairwise comparisons and
therefore no differentially expressed genes.

2. For genes that form collection M we would perform N
pairwise comparisons for each gene. We would apply an FDR control
procedure at the α_{2} level to these L*N tests. Any comparisons found that
produce p-values ≤ some specified c_{2} are considered to be statistically
significant.

It should be noted that this procedure assumes that genes that show no significant treatment effect (step 1) will also show no statistically significant pairwise comparisons (step 2). However, a gene having a significant treatment effect may or may not have statistical significance in pairwise comparisons. This method is referred to as “jiang” throughout this paper.

**Nettleton’s Histogram method**

Nettleton’s Method [12] estimates π_{0} by estimating the proportion
of the observed p-values that follow a uniform distribution. The steps
in the algorithm are as follows:

• Partition the interval [0,1] in B bins of equal width.

Assume all null hypotheses are true, and set

Calculate the expected number of p-values for each bin given the current estimate of the number of true null hypotheses.

Beginning with the leftmost bin, sum the number of p-values in excess of the expected until a bin with no excess is reached.

Use the excess sum as an updated estimate of m_{1}, and then use that
to update the estimate of m_{0}=m - m_{1}.

Return to Step (iii) and repeat the procedure until convergence is reached.

The number of bins is used as a tuning parameter and the authors suggested using B=20. In their simulation study the histogram based estimator with 20 bins tended to be conservative for independent and autoregressive correlation structures in that it tended to overestimate the number of true null hypotheses for certain correlations that was considered in their simulation study. This method is referred to as “histo” throughout this paper.

**Convex decreasing p-value estimate (Langaas)**

Langass and Landqvist’s estimator is based on the non-parametric
MLE of the p-value density [13]. The authors restricted their attention
to decreasing and convex decreasing densities because of their many
mathematically attractive properties. The derived estimator of π_{0} assumed independence of the test statistics. These estimators were
found to be robust with respect to the independence assumption but
also worked well for test statistics that showed moderate levels of
dependence. The aim is to derive a NPMLE (non-parametric maximum
likelihood estimate) for a convex decreasing density on [0, 1). The
mixture density given in (4) is modified by requiring f(p) be a convex
decreasing function with f(1)=0. The algorithm is as follows:

Specify a convex decreasing initial function .

For j=0, 1, 2, … given the current iterate determine using:

(9)

Where

If then the current iterate is optimal by and we are done.

If this optimality is not found then the next iterate is:

(10)

Where

The procedure employed by this algorithm is similar to the “steepest descent” algorithm used to optimize a function on the Euclidean n-space . The next iterate in each step of the algorithm is the optimal convex combination of the current iterate and the mixing density that corresponds to the most negative directional derivative. This method is referred to as “langaas” throughout this paper.

**Robust estimation (Pounds)**

Pounds and Cheng’s estimator of π_{0} [14] when the test statistics are
continuous is given by:

(11)

Where

(12)

is the average p-value for all m hypotheses,

(13)

and min(a,b) is the minimum of a and b. This estimator is biased
upward but the bias is small when pf_{1}(p) is small or when π_{0} is close to
1. This method is referred to as “pounds” throughout this paper.

**Lowest slope method**

This stepwise procedure was developed in ref. [15] first estimates the value of m0 using the data and this estimate is used in the adaptive procedure itemized in the algorithm that follows:

i. Order the p-values p_{(1)} ≤ p_{(2)} ≤ … ≤ p_{(m)} corresponding to null hypotheses H_{(1)}, H_{(2)}, ….. , H_{(m)}.

For level q and i=1, 2, …., k we compare each p(i) with . If no p(i) is found to be smaller we do not reject any null hypotheses and stop.

Compute the slopes

Starting with i=1 proceed as long as S_{i} ≥ S_{i-1}. When we find S_{j} < S_{j-1} stop.

(14)

Then starting with the largest p-value p(m), compare each p(i) with until we arrive at the first p-value satisfying . Wewould reject all k hypotheses whose p values are smaller than p_{(k)}.

This method is referred to as “abh” throughout this paper.

**Sliding linear method**

A sliding linear model (SLIM) to estimate π_{0} in datasets with
dependence structures [16]. Whenever undetected dependence
structures exist they distort the distribution of the p-values making
any estimate of π_{0} less effective. The SLIM method of FDR estimation
is based on a linear model transformed from the non-linear λ
estimator of Storey. The method functions by partitioning the data
into local dependence blocks. This data partitioning and optimization
allows SLIM to utilize information from a broader range of p-value
distributions for π_{0} estimation. The employed optimization scheme
exploits the non-static relationship between the p-values and the
q-values by minimizing the difference between the fractions of tests
called significant by the p-value and q-value methods. SLIM handles
the hidden dependence in the data without the need to empirically
adjust the null p-value distributions. This method is referred to as
“SLIM” throughout this paper.

**SPLOSH method**

The Spacings LOESS histogram (SPLOSH) estimates the conditional false discovery rate (cFDR) [17]. This cFDR is the expected proportion of false positives conditioned on k "significant" findings. SPLOSH is designed to be more stable than Storey’s approach in that the q-value is based on an unstable estimator of the positive false discovery rate (pFDR). SPLOSH is applicable in a wider variety of settings than BUM. The SPLOSH algorithm is as follows:

Let p_{(1)} ≤ p_{(2)} ≤ … ≤ p_{(k)} be k ordered p-values. Let be their adjusted ranks to control the type I error rate.

Compute the midpoint of the interval using:

(14)

1. Apply the arc-sine transform for i=1,2,…,k to the ranked p-values using:

(15)

Apply LOWESS (LOcally WEighted Scatter-plot Smoother) to for j=1,…,u-1to obtain an estimated curve

For j=1,…,u to obtain the estimated derivative of the CDF up to a unitizing constant c.

Let be an estimate of the PDF at pi for i=1,2,…,k. The trapezoidal rule is used to estimate c using:

(16)

where is the sub-interval width..

Let be the CDF and for k=l + 1,..,u let

(17)

be an estimate of obtained using the trapezoidal rule in approximate integration.

Take as the minimum of the pdf.

For i=1, … , k we obtain by substituting

into

(18)

For p-values that are equal to zero we use L’Hôspital’s Rule to justify as an estimate of the cFDR:

(19)

Define a monotone quantity based on the cFDR
estimates r_{(i)} for i=1, … , k.

This method is referred to as “splosh” throughout this paper.

Mixture modeling of the p-value distribution was first proposed in ref. [18]. This method models the p-value distribution as a mixture of a Uniform (0,1) distribution, corresponding to the true null hypotheses and a Beta(α,β) distribution corresponding to the false null hypotheses. The Beta-Uniform Mixture (BUM) Model was proposed by Pounds and Morris [19]. This is basically a mixture of a Uniform(0,1) distribution and a Beta(α,1) distribution. The Beta distribution is chosen because of its flexibility in modeling any distribution on the interval [0,1].

This mixture model assumes independence of gene expression levels across genes. Thus, under the null hypothesis the p-values are uniformly distributed on the interval [0,1] regardless of the statistical test being used, as long as the test is valid, and regardless of the size of the sample. Under the alternative hypothesis the distribution of the p-values tends to cluster closer to zero than one. By referring to the entire distribution of the p-values obtained in the sample we can then address whether there is statistically significant evidence that any of the genes under study exhibits a difference in expression across the groups. This is usually done by performing an omnibus test of whether the observed distribution of the p-values is significantly different from a uniform distribution. This method is referred to as “BUM” throughout this paper.

**Grenander density estimator**

The Grenander density estimator [20] is a non-parametric
maximum likelihood estimator (NPMLE) similar to an empirical
cumulative density function (ECDF). Unlike the ECDF it has the
added constraint of an underlying density. The Grenander estimator
uses the ECDF as an estimator of the associated distribution function
F(p). This estimator is the decreasing piecewise-constant function
equal to the slopes of the least concave majorant (LCM) of the ECDF.
Monotone regression with weights is used to compute the LCM of the
ECDF in order to obtain this estimator. If x_{i} and γ_{i} are used to denote
coordinates for the ECDF and Δx_{i}=x_{i+1} - x_{i} and Δγ_{i}=γ_{i+1} - γ_{i}, then
the slopes of the LCM are given by antitonic regression and the raw
slopes Δx_{i} and Δγ_{i} with weight Δxi [21]. This method is referred to as
“strimmer” throughout this paper.

**Two stage adaptive procedure**

Adaptive procedures first estimate the number of null hypotheses m_{0} and then use this estimate to revise a multiple test procedure.
Knowledge of m_{0} can be used to improve upon the performance of the
FDR controlling procedure. The two stage adaptive procedure makes
use of a linear step-up procedure making use of m ordered p-values p(1) . If this k exists then we
reject all the k hypotheses associated with p_{(1)} ≤ p_{(2)} ≤ … ≤ p_{(k)}, otherwise
do not reject any of the hypotheses. If m_{0} were known, then the linear
set-up procedure with:

(20)

would control the FDR at precisely the desired level q in the independent and continuous case, and would then be more powerful in rejecting hypotheses for which the alternative holds. The adaptive Benjamini- Hochberg procedure [6] at the level q is as follows:

If reject all hypotheses; otherwise, test the hypotheses using the linear set-up procedure at level

1. Use the linear set-up procedure at level q, and if no hypothesis is rejected stop; otherwise, proceed.

Estimate m_{0}(k) using

Starting with k=2 stop when for the first time m_{0}(k)>m_{0}(k-1).

Estimate rounding up to the next highest integer.

Use the linear step-up procedure with

This method is referred to as “TSBKY” throughout this paper.

The authors conducted a simulation study to compare the twelve
(12) methods for estimating the proportion of null hypotheses π_{0} under
varying levels of dependence. Simulations were drawn from a MVN
(μ, Σ) distribution where μ is the mean vector and Σ is the covariance
matrix using the R library MASS. The study involved simulating the
expression values of 1000 “genes” from 20 samples. The first 10 samples
for each “simulated expression value” came from the case subjects and
the last 10 samples came from the control subjects. The simulations
were done for π_{0} values of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95,
0.99, and 1.00. The data was simulated for three cases of dependency
within the genes:

i. Independent Genes - No dependence structure within or
between the genes. In this case the leading diagonal of Σ contained all
1’s and the off-diagonal entries, denoted as ρ, were all zero. A 1000 x
12 matrix of p-values was generated for each value of π_{0}. Each column
of the matrix would represent 1000 p-values extracted from each of the
“m” matrices. This procedure was replicated 1000 times. The p-value
matrices were then used as the input test statistics to compute the
row means and standard deviations for each π_{0} estimation method
being compared. This data was then bound into one matrix for each
estimation method for plotting.

ii. Weak Dependence - In this case the leading diagonal of Σ contained all 1’s and the off-diagonal entries, denoted as ρ, were all 0.1. The simulation was done in the same manner as for independence.

iii. Moderate Dependence - In this case the leading diagonal of Σ contained all 1’s and the off-diagonal entries, denoted as ρ, were all 0.5.

All of the simulations were done using R version 3.2.3. Source codes for most of the estimation methods were available freely online.

In **Figure 1** the solid black line that runs from bottom left to top
right represents “the truth”, meaning that if the true value of π_{0} is 0.10
then the estimated value should correspond to 0.10. Departures
either above or below this 45° line would allow one to make graphical
inferences about the performance of the proposed estimator. This
graphical approach has the advantage of allowing for easy visualization
of the data and identification of possible trends and patterns that it
contains that might not be readily obvious with tabular approaches.
All of the methods compared tended to overestimate for low values
of π_{0}. The estimators all performed better as π_{0} increased. This is well
documented in literature. “SLIM” performed very well for small values
of π_{0} but tended to underestimate as π_{0} increased. It was found that
“smoother”, “st.boot”, “Langaas” and “histo” all performed very well
for most values of π_{0}. “TSBKY” and “abh” overestimated for every
value of π_{0}. For example, **Table 3** shows that when π_{0} was 0.30 “TSBKY”
gave an estimate of = 0.805 with an estimated standard deviation of 0.022 while “abh” gave an estimate of = 0.607 with an estimated
standard deviation of 0.047.

π_{0} |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | ||||||||||||

a | 0.165 | 0.178 | 0.184 | 0.189 | 0.174 | 0.311 | 0.420 | 0.142 | 0.222 | 0.301 | 0.176 | 0.695 |

b | 0.032 | 0.023 | 0.024 | 0.028 | 0.023 | 0.013 | 0.063 | 0.029 | 0.040 | 0.050 | 0.022 | 0.020 |

0.2 | ||||||||||||

a | 0.259 | 0.247 | 0.275 | 0.277 | 0.261 | 0.387 | 0.515 | 0.218 | 0.319 | 0.326 | 0.263 | 0.754 |

b | 0.047 | 0.032 | 0.030 | 0.033 | 0.030 | 0.015 | 0.058 | 0.033 | 0.046 | 0.074 | 0.028 | 0.025 |

0.3 | ||||||||||||

a | 0.349 | 0.356 | 0.366 | 0.371 | 0.351 | 0.464 | 0.607 | 0.301 | 0.403 | 0.356 | 0.352 | 0.805 |

b | 0.056 | 0.035 | 0.034 | 0.037 | 0.034 | 0.014 | 0.047 | 0.037 | 0.053 | 0.007 | 0.032 | 0.022 |

0.4 | ||||||||||||

a | 0.433 | 0.441 | 0.455 | 0.463 | 0.438 | 0.541 | 0.686 | 0.379 | 0.481 | 0.393 | 0.438 | 0.858 |

b | 0.069 | 0.050 | 0.042 | 0.039 | 0.046 | 0.018 | 0.039 | 0.046 | 0.064 | 0.010 | 0.042 | 0.020 |

0.5 | ||||||||||||

a | 0.536 | 0.532 | 0.553 | 0.561 | 0.534 | 0.617 | 0.769 | 0.454 | 0.548 | 0.440 | 0.532 | 0.900 |

b | 0.076 | 0.042 | 0.037 | 0.031 | 0.034 | 0.016 | 0.039 | 0.047 | 0.071 | 0.014 | 0.034 | 0.018 |

0.6 | ||||||||||||

a | 0.625 | 0.628 | 0.644 | 0.653 | 0.628 | 0.696 | 0.832 | 0.545 | 0.620 | 0.534 | 0.622 | 0.939 |

b | 0.072 | 0.038 | 0.034 | 0.029 | 0.034 | 0.016 | 0.031 | 0.039 | 0.071 | 0.020 | 0.034 | 0.019 |

0.7 | ||||||||||||

a | 0.709 | 0.713 | 0.736 | 0.750 | 0.718 | 0.774 | 0.887 | 0.620 | 0.682 | 0.642 | 0.708 | 0.972 |

b | 0.074 | 0.039 | 0.031 | 0.026 | 0.032 | 0.016 | 0.022 | 0.046 | 0.073 | 0.022 | 0.036 | 0.013 |

0.8 | ||||||||||||

a | 0.816 | 0.806 | 0.829 | 0.835 | 0.811 | 0.851 | 0.942 | 0.695 | 0.771 | 0.775 | 0.801 | 0.989 |

b | 0.080 | 0.043 | 0.034 | 0.026 | 0.032 | 0.016 | 0.015 | 0.053 | 0.079 | 0.021 | 0.040 | 0.007 |

0.9 | ||||||||||||

a | 0.905 | 0.892 | 0.914 | 0.918 | 0.899 | 0.925 | 0.978 | 0.771 | 0.834 | 0.869 | 0.887 | 0.997 |

b | 0.071 | 0.036 | 0.030 | 0.018 | 0.029 | 0.017 | 0.009 | 0.042 | 0.074 | 0.028 | 0.034 | 0.003 |

0.95 | ||||||||||||

a | 0.941 | 0.937 | 0.958 | 0.957 | 0.945 | 0.963 | 0.993 | 0.809 | 0.872 | 0.998 | 0.931 | 0.999 |

b | 0.062 | 0.039 | 0.027 | 0.018 | 0.026 | 0.017 | 0.005 | 0.049 | 0.076 | 0.014 | 0.032 | 0.001 |

0.99 | ||||||||||||

a | 0.960 | 0.968 | 0.984 | 0.991 | 0.976 | 0.989 | 0.999 | 0.840 | 0.877 | 1.000 | 0.961 | 0.999 |

b | 0.059 | 0.039 | 0.020 | 0.013 | 0.024 | 0.012 | 0.001 | 0.046 | 0.085 | 0.000 | 0.034 | 0.001 |

1.00 | ||||||||||||

a | 0.967 | 0.976 | 0.990 | 0.996 | 0.986 | 0.994 | 0.999 | 0.835 | 0.864 | 1.000 | 0.973 | 0.999 |

b | 0.051 | 0.034 | 0.017 | 0.009 | 0.020 | 0.009 | 0.001 | 0.046 | 0.086 | 0.000 | 0.031 | 0.001 |

**Table 3:** Mean (a) and Standard Deviation (b) for compared estimation methods using independent test statistics.

**Figure 2** shows the comparison of all twelve methods for weakly
dependent test statistics. Most of the estimators overestimated for small values of π_{0}, however, “TSBKY” and “abh” significantly
overestimated for most values of π_{0}. For example, **Table 4** shows
that when π_{0} was 0.30 “TSBKY” gave an estimate of = 0.814 with
an estimated standard deviation of 0.125 while “abh” gave an estimate
of = 0.619 with an estimated standard deviation of 0.116. For
weak dependent test statistics “BUM” initially overestimated but performed better for π_{0} ≥ 0.60. With weak dependence in the test
statistics “smoother”, “st.boot”, “Langaas” and “histo” all performed
very well for most values of π_{0}.

π_{0} |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | ||||||||||||

a | 0.148 | 0.164 | 0.173 | 0.322 | 0.163 | 0.301 | 0.395 | 0.133 | 0.203 | 0.302 | 0.166 | 0.685 |

b | 0.055 | 0.053 | 0.050 | 1.411 | 0.050 | 0.069 | 0.133 | 0.049 | 0.058 | 0.043 | 0.051 | 0.140 |

0.2 | ||||||||||||

a | 0.255 | 0.267 | 0.275 | 0.280 | 0.264 | 0.391 | 0.530 | 0.225 | 0.319 | 0.338 | 0.267 | 0.760 |

b | 0.070 | 0.064 | 0.065 | 0.068 | 0.062 | 0.083 | 0.156 | 0.063 | 0.067 | 0.061 | 0.063 | 0.153 |

0.3 | ||||||||||||

a | 0.355 | 0.359 | 0.370 | 0.374 | 0.353 | 0.468 | 0.619 | 0.300 | 0.404 | 0.368 | 0.356 | 0.814 |

b | 0.069 | 0.050 | 0.052 | 0.055 | 0.048 | 0.063 | 0.116 | 0.052 | 0.059 | 0.055 | 0.049 | 0.125 |

0.4 | ||||||||||||

a | 0.436 | 0.443 | 0.457 | 0.467 | 0.441 | 0.540 | 0.689 | 0.381 | 0.479 | 0.339 | 0.442 | 0.852 |

b | 0.074 | 0.057 | 0.052 | 0.050 | 0.053 | 0.059 | 0.104 | 0.058 | 0.074 | 0.058 | 0.055 | 0.106 |

0.5 | ||||||||||||

a | 0.530 | 0.536 | 0.553 | 0.563 | 0.537 | 0.622 | 0.778 | 0.459 | 0.556 | 0.452 | 0.532 | 0.903 |

b | 0.076 | 0.054 | 0.051 | 0.050 | 0.051 | 0.048 | 0.088 | 0.055 | 0.074 | 0.057 | 0.052 | 0.077 |

0.6 | ||||||||||||

a | 0.631 | 0.630 | 0.650 | 0.651 | 0.632 | 0.699 | 0.838 | 0.542 | 0.637 | 0.537 | 0.628 | 0.943 |

b | 0.079 | 0.057 | 0.050 | 0.049 | 0.049 | 0.043 | 0.064 | 0.061 | 0.078 | 0.050 | 0.051 | 0.054 |

0.7 | ||||||||||||

a | 0.719 | 0.711 | 0.732 | 0.741 | 0.714 | 0.766 | 0.882 | 0.611 | 0.694 | 0.630 | 0.705 | 0.959 |

b | 0.087 | 0.051 | 0.045 | 0.043 | 0.045 | 0.039 | 0.056 | 0.062 | 0.079 | 0.048 | 0.047 | 0.041 |

0.8 | ||||||||||||

a | 0.817 | 0.800 | 0.824 | 0.825 | 0.803 | 0.847 | 0.937 | 0.696 | 0.775 | 0.747 | 0.795 | 0.984 |

b | 0.100 | 0.063 | 0.062 | 0.053 | 0.060 | 0.041 | 0.032 | 0.059 | 0.083 | 0.052 | 0.060 | 0.021 |

0.9 | ||||||||||||

a | 0.895 | 0.887 | 0.912 | 0.912 | 0.894 | 0.926 | 0.976 | 0.766 | 0.823 | 0.889 | 0.885 | 0.996 |

b | 0.109 | 0.080 | 0.075 | 0.062 | 0.074 | 0.045 | 0.019 | 0.069 | 0.097 | 0.084 | 0.075 | 0.007 |

0.95 | ||||||||||||

a | 0.922 | 0.925 | 0.943 | 0.944 | 0.929 | 0.957 | 0.991 | 0.791 | 0.847 | 0.967 | 0.920 | 0.999 |

b | 0.094 | 0.074 | 0.072 | 0.063 | 0.074 | 0.046 | 0.010 | 0.071 | 0.094 | 0.080 | 0.073 | 0.001 |

0.99 | ||||||||||||

a | 0.957 | 0.963 | 0.973 | 0.979 | 0.969 | 0.982 | 0.999 | 0.821 | 0.837 | 0.988 | 0.959 | 0.999 |

b | 0.074 | 0.063 | 0.055 | 0.050 | 0.059 | 0.037 | 0.005 | 0.060 | 0.091 | 0.055 | 0.063 | 0.001 |

1.00 | ||||||||||||

a | 0.956 | 0.960 | 0.975 | 0.981 | 0.969 | 0.983 | 0.999 | 0.825 | 0.786 | 0.990 | 0.956 | 0.999 |

b | 0.075 | 0.066 | 0.059 | 0.055 | 0.063 | 0.039 | 0.004 | 0.068 | 0.101 | 0.053 | 0.066 | 0.001 |

**Table 4:** Mean (a) and Standard Deviation (b) for compared estimation methods using weak dependent test statistics.

**Figure 3** shows the comparison of all twelve methods for moderately
dependent test statistics. Most of the estimators overestimated for small values of π_{0}, however, “TSBKY” and “abh” significantly
overestimated for most values of π_{0}. For example, **Table 5** shows
that when π_{0} was 0.30 “TSBKY” gave an estimate of = 0.837 with
an estimated standard deviation of 0.228 and “abh” gave an estimate of = 0.668 with an estimated standard deviation of 0.249. In this case
“smoother”, “st.boot”, and “jiang” performed better than most of the
other estimators even though they all had a high standard deviation.
A small standard deviation is highly desirable because this gives us an
indication that the data points are packed very tightly around the mean
value that we are estimating, so there is little spread in the data. The
table shows that all of the methods appear to perform very badly for
π_{0}=0.40, 0.60, 0.90, 0.99, and 1.00. This might be due to the methods
ignoring the correlation effect among the test statistics.

π_{0} |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | ||||||||||||

a | 0.139 | 0.147 | 0.151 | 0.151 | 0.141 | 0.265 | 0.347 | 0.109 | 0.174 | 0.316 | 0.145 | 0.637 |

b | 0.089 | 0.097 | 0.094 | 0.094 | 0.091 | 0.152 | 0.295 | 0.089 | 0.096 | 0.121 | 0.096 | 0.317 |

0.2 | ||||||||||||

a | 0.114 | 0.116 | 0.120 | 5.382 | 0.114 | 0.174 | 0.238 | 0.331 | 0.130 | 0.170 | 0.116 | 0.328 |

b | 0.150 | 0.150 | 0.155 | 52.60 | 0.146 | 0.225 | 0.331 | 0.130 | 0.170 | 0.222 | 0.150 | 0.416 |

0.3 | ||||||||||||

a | 0.368 | 0.372 | 0.385 | 0.389 | 0.366 | 0.495 | 0.668 | 0.303 | 0.409 | 0.438 | 0.370 | 0.837 |

b | 0.130 | 0.120 | 0.127 | 0.143 | 0.117 | 0.146 | 0.249 | 0.112 | 0.115 | 0.167 | 0.120 | 0.228 |

0.4 | ||||||||||||

a | 0.040 | 0.041 | 0.042 | 0.043 | 0.040 | 0.057 | 0.080 | 0.036 | 0.042 | 0.051 | 0.042 | 0.089 |

b | 0.124 | 0.125 | 0.128 | 0.131 | 0.122 | 0.180 | 0.250 | 0.109 | 0.130 | 0.161 | 0.127 | 0.273 |

0.5 | ||||||||||||

a | 0.553 | 0.529 | 0.553 | 0.548 | 0.525 | 0.619 | 0.756 | 0.432 | 0.545 | 0.479 | 0.526 | 0.879 |

b | 0.177 | 0.137 | 0.146 | 0.141 | 0.135 | 0.121 | 0.174 | 0.112 | 0.132 | 0.140 | 0.136 | 0.164 |

0.6 | ||||||||||||

a | 0.382 | 0.368 | 0.386 | 0.378 | 0.368 | 0.425 | 0.496 | 0.304 | 0.352 | 0.342 | 0.370 | 0.559 |

b | 0.348 | 0.324 | 0.341 | 0.330 | 0.324 | 0.352 | 0.409 | 0.267 | 0.305 | 0.288 | 0.325 | 0.452 |

0.7 | ||||||||||||

a | 0.703 | 0.776 | 0.708 | 0.688 | 0.669 | 0.770 | 0.886 | 0.548 | 0.637 | 0.677 | 0.673 | 0.944 |

b | 0.270 | 0.223 | 0.243 | 0.222 | 0.225 | 0.169 | 0.148 | 0.172 | 0.194 | 0.178 | 0.221 | 0.103 |

0.8 | ||||||||||||

a | 0.738 | 0.724 | 0.757 | 0.738 | 0.717 | 0.826 | 0.922 | 0.567 | 0.653 | 0.763 | 0.723 | 0.971 |

b | 0.294 | 0.261 | 0.279 | 0.258 | 0.262 | 0.193 | 0.113 | 0.194 | 0.202 | 0.195 | 0.259 | 0.071 |

0.9 | ||||||||||||

a | 0.359 | 0.367 | 0.374 | 0.376 | 0.360 | 0.419 | 0.476 | 0.275 | 0.318 | 0.416 | 0.366 | 0.488 |

b | 0.421 | 0.421 | 0.429 | 0.430 | 0.416 | 0.446 | 0.490 | 0.312 | 0.357 | 0.448 | 0.420 | 0.500 |

0.95 | ||||||||||||

a | 0.784 | 0.797 | 0.813 | 0.826 | 0.796 | 0.881 | 0.963 | 0.591 | 0.617 | 0.886 | 0.796 | 0.990 |

b | 0.291 | 0.280 | 0.280 | 0.284 | 0.285 | 0.192 | 0.124 | 0.209 | 0.212 | 0.200 | 0.279 | 0.062 |

0.99 | ||||||||||||

a | 0.529 | 0.531 | 0.535 | 0.539 | 0.531 | 0.555 | 0.579 | 0.393 | 0.283 | 0.560 | 0.530 | 0.580 |

b | 0.475 | 0.474 | 0.477 | 0.480 | 0.475 | 0.480 | 0.495 | 0.362 | 0.284 | 0.485 | 0.474 | 0.496 |

1.00 | ||||||||||||

a | 0.547 | 0.554 | 0.560 | 0.564 | 0.554 | 0.591 | 0.629 | 0.381 | 0.258 | 0.596 | 0.553 | 0.639 |

b | 0.458 | 0.461 | 0.463 | 0.468 | 0.462 | 0.462 | 0.479 | 0.332 | 0.245 | 0.467 | 0.460 | 0.482 |

**Table 5:** Mean (a) and Standard Deviation (b) for compared estimation methods using moderately dependent test statistics.

A simulation study to compare 12 methods for estimating the proportion of null hypotheses under different configurations of dependence. Dependence among genes exists. This paper shows which of the methods would be most appropriate under various dependence configurations. The estimation of the proportion of null hypotheses is important especially when it comes to estimation of the false discovery rate. There is no doubt that this field will continue to evolve as datasets becomes larger, the demands on statistics becomes greater, and computing hardware and software improves.

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