-
Notifications
You must be signed in to change notification settings - Fork 2
/
Copy pathmake_combined_FDR_TDR_table_only_WS_ind_dep.tex
347 lines (300 loc) · 21 KB
/
make_combined_FDR_TDR_table_only_WS_ind_dep.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
\documentclass{article}\usepackage[]{graphicx}\usepackage[]{color}
%% maxwidth is the original width if it is less than linewidth
%% otherwise use linewidth (to make sure the graphics do not exceed the margin)
\makeatletter
\def\maxwidth{ %
\ifdim\Gin@nat@width>\linewidth
\linewidth
\else
\Gin@nat@width
\fi
}
\makeatother
\definecolor{fgcolor}{rgb}{0.345, 0.345, 0.345}
\newcommand{\hlnum}[1]{\textcolor[rgb]{0.686,0.059,0.569}{#1}}%
\newcommand{\hlstr}[1]{\textcolor[rgb]{0.192,0.494,0.8}{#1}}%
\newcommand{\hlcom}[1]{\textcolor[rgb]{0.678,0.584,0.686}{\textit{#1}}}%
\newcommand{\hlopt}[1]{\textcolor[rgb]{0,0,0}{#1}}%
\newcommand{\hlstd}[1]{\textcolor[rgb]{0.345,0.345,0.345}{#1}}%
\newcommand{\hlkwa}[1]{\textcolor[rgb]{0.161,0.373,0.58}{\textbf{#1}}}%
\newcommand{\hlkwb}[1]{\textcolor[rgb]{0.69,0.353,0.396}{#1}}%
\newcommand{\hlkwc}[1]{\textcolor[rgb]{0.333,0.667,0.333}{#1}}%
\newcommand{\hlkwd}[1]{\textcolor[rgb]{0.737,0.353,0.396}{\textbf{#1}}}%
\let\hlipl\hlkwb
\usepackage{framed}
\makeatletter
\newenvironment{kframe}{%
\def\at@end@of@kframe{}%
\ifinner\ifhmode%
\def\at@end@of@kframe{\end{minipage}}%
\begin{minipage}{\columnwidth}%
\fi\fi%
\def\FrameCommand##1{\hskip\@totalleftmargin \hskip-\fboxsep
\colorbox{shadecolor}{##1}\hskip-\fboxsep
% There is no \\@totalrightmargin, so:
\hskip-\linewidth \hskip-\@totalleftmargin \hskip\columnwidth}%
\MakeFramed {\advance\hsize-\width
\@totalleftmargin\z@ \linewidth\hsize
\@setminipage}}%
{\par\unskip\endMakeFramed%
\at@end@of@kframe}
\makeatother
\definecolor{shadecolor}{rgb}{.97, .97, .97}
\definecolor{messagecolor}{rgb}{0, 0, 0}
\definecolor{warningcolor}{rgb}{1, 0, 1}
\definecolor{errorcolor}{rgb}{1, 0, 0}
\newenvironment{knitrout}{}{} % an empty environment to be redefined in TeX
\usepackage{alltt}
\usepackage[margin=0.5in]{geometry}
\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
\begin{document}
\section*{Simulations: Overview}
The following 4 functions are considered for $\pi_0(x)$:
\begin{knitrout}
\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}
{\centering \includegraphics[width=\maxwidth]{Figures/Simulation_scenarios-1}
}
\end{knitrout}
\noindent We performed 200 simulations in each scenario.
\\
\noindent We estimated false discovery rates (FDR) and true positive rates (TPR) percentages for a nominal FDR of 5\%.
We considered both the theoretical and empirical nulls for the Scott method.
For III and IV, a dummy variable was used for $x_{2}$, along with linear or spline terms (with 3 df) for $x_1$.
\clearpage
\subsection*{Independent test statistics}
We first generated independent test statistics.
\\
\noindent For the beta distribution, we generated the p-values directly from Beta(1,20). For the other distributions, we generated the test statistics and calculated the p-values from them.
For the t-test, we considered 2 groups of 6 (so 2x6 = 10 df) and used the t-statistics instead of the z-statistics for the Scott method. For the chisquared test, 1 df corresponds to a 2x2 table, 4 df to a 3x3 table. We used the z-statistics obtained from back-transforming the p-values for the Scott method for the beta and the chisquared cases.
\\
BL = Boca-Leek, Scott T = Scott theoretical null, Scott E = Scott empirical null
\subsubsection*{1,000 tests}
% latex table generated in R 3.3.1 by xtable 1.8-2 package
% Thu Jun 22 11:11:21 2017
\begin{table}[ht]
\centering
\begin{tabular}{lll|lllll|lllll}
\hline
&&& \multicolumn{5}{c}{FDR} & \multicolumn{5}{c}{TPR}\\
$\pi_0(x)$ & Dist. under $H_1$ & Reg. model & BL & Scott T & Scott E & Storey & BH & BL & Scott T & Scott E & Storey & BH \\
\hline
I & Beta(1,20) & Linear & 5.0 & 90.0 & 84.0 & 5.2 & 3.9 & 0.2 & 100.0 & 95.9 & 0.2 & 0.1 \\
II & Beta(1,20) & Linear & 4.8 & 92.6 & 85.9 & 4.8 & 4.1 & 0.2 & 100.0 & 98.0 & 0.1 & 0.1 \\
II & Beta(1,20) & Spline & 6.5 & 92.6 & 86.6 & 4.8 & 4.1 & 0.2 & 100.0 & 98.3 & 0.1 & 0.1 \\
III & Beta(1,20) & Linear & 5.2 & 94.9 & 88.9 & 5.4 & 5.4 & 0.2 & 100.0 & 97.5 & 0.2 & 0.2 \\
III & Beta(1,20) & Spline & 6.2 & 94.9 & 89.4 & 5.4 & 5.4 & 0.3 & 100.0 & 97.6 & 0.2 & 0.2 \\
IV & Beta(1,20) & Linear & 6.4 & 56.7 & & 5.1 & 3.4 & 12.2 & 100.0 & & 5.4 & 0.3 \\
IV & Beta(1,20) & Spline & 7.9 & 56.7 & & 5.1 & 3.4 & 15.4 & 100.0 & & 5.4 & 0.3 \\
\hline
I & Norm & Linear & 5.0 & 5.2 & 6.6 & 4.9 & 4.4 & 51.0 & 50.9 & 49.7 & 50.8 & 49.7 \\
II & Norm & Linear & 5.4 & 5.7 & 8.1 & 5.3 & 4.9 & 48.5 & 63.5 & 61.3 & 47.6 & 47.0 \\
II & Norm & Spline & 5.6 & 5.9 & 8.3 & 5.3 & 4.9 & 49.3 & 63.5 & 61.5 & 47.6 & 47.0 \\
III & Norm & Linear & 5.8 & 5.9 & 9.9 & 5.4 & 5.1 & 45.1 & 60.3 & 57.9 & 44.0 & 43.4 \\
III & Norm & Spline & 5.9 & 6.0 & 10.1 & 5.4 & 5.1 & 45.6 & 60.9 & 58.2 & 44.0 & 43.4 \\
IV & Norm & Linear & 5.0 & 4.9 & 2.4 & 4.7 & 2.8 & 71.6 & 71.8 & 60.6 & 71.2 & 65.4 \\
IV & Norm & Spline & 5.2 & 5.0 & 2.4 & 4.7 & 2.8 & 72.0 & 71.9 & 60.7 & 71.2 & 65.4 \\
\hline
I & T & Linear & 5.7 & 21.3 & 23.4 & 5.5 & 4.8 & 15.7 & 55.4 & 56.9 & 15.2 & 13.6 \\
II & T & Linear & 4.8 & 20.7 & 23.8 & 5.0 & 4.4 & 13.0 & 64.5 & 65.5 & 11.6 & 10.6 \\
II & T & Spline & 4.7 & 21.1 & 24.5 & 5.0 & 4.4 & 13.8 & 64.8 & 65.6 & 11.6 & 10.6 \\
III & T & Linear & 6.2 & 26.8 & 31.0 & 5.9 & 5.4 & 9.4 & 54.6 & 54.7 & 8.2 & 7.6 \\
III & T & Spline & 6.8 & 27.3 & 31.3 & 5.9 & 5.4 & 10.0 & 55.2 & 55.3 & 8.2 & 7.6 \\
IV & T & Linear & 5.0 & 9.3 & 2.8 & 4.7 & 2.9 & 52.5 & 72.9 & 44.4 & 52.0 & 40.3 \\
IV & T & Spline & 5.4 & 9.3 & 2.8 & 4.7 & 2.9 & 53.0 & 73.0 & 44.6 & 52.0 & 40.3 \\
\hline
I & Chisq 1 df & Linear & 5.0 & 90.0 & 85.5 & 4.8 & 4.4 & 51.2 & 100.0 & 98.7 & 50.9 & 49.7 \\
II & Chisq 1 df & Linear & 4.8 & 92.6 & 89.4 & 4.8 & 4.4 & 48.3 & 100.0 & 99.6 & 47.1 & 46.3 \\
II & Chisq 1 df & Spline & 5.0 & 92.6 & 90.0 & 4.8 & 4.4 & 48.9 & 100.0 & 99.6 & 47.1 & 46.3 \\
III & Chisq 1 df & Linear & 5.0 & 94.9 & 93.8 & 4.9 & 4.8 & 44.3 & 100.0 & 99.7 & 43.1 & 42.5 \\
III & Chisq 1 df & Spline & 5.3 & 94.9 & 93.9 & 4.9 & 4.8 & 44.8 & 100.0 & 99.7 & 43.1 & 42.5 \\
IV & Chisq 1 df & Linear & 5.1 & 56.7 & & 4.7 & 2.8 & 71.6 & 100.0 & & 71.1 & 65.1 \\
IV & Chisq 1 df & Spline & 5.3 & 56.7 & & 4.7 & 2.8 & 71.9 & 100.0 & & 71.1 & 65.1 \\
\hline
I & Chisq 4 df & Linear & 5.3 & 90.0 & 83.5 & 5.4 & 4.8 & 30.8 & 100.0 & 95.3 & 30.6 & 29.6 \\
II & Chisq 4 df & Linear & 5.3 & 92.6 & 89.6 & 5.3 & 5.0 & 28.4 & 100.0 & 98.5 & 27.5 & 26.7 \\
II & Chisq 4 df & Spline & 5.4 & 92.6 & 89.9 & 5.3 & 5.0 & 29.2 & 100.0 & 98.6 & 27.5 & 26.7 \\
III & Chisq 4 df & Linear & 5.9 & 94.9 & 92.4 & 5.4 & 5.3 & 24.8 & 100.0 & 98.3 & 24.0 & 23.4 \\
III & Chisq 4 df & Spline & 5.9 & 94.9 & 93.0 & 5.4 & 5.3 & 25.2 & 100.0 & 98.7 & 24.0 & 23.4 \\
IV & Chisq 4 df & Linear & 5.1 & 56.7 & 55.9 & 4.7 & 2.8 & 52.3 & 100.0 & 98.8 & 51.7 & 44.5 \\
IV & Chisq 4 df & Spline & 5.5 & 56.7 & 55.9 & 4.7 & 2.8 & 52.7 & 100.0 & 98.8 & 51.7 & 44.5 \\
\hline
\end{tabular}
\end{table}
\clearpage
\subsubsection*{10,000 tests}
% latex table generated in R 3.3.1 by xtable 1.8-2 package
% Thu Jun 22 11:11:21 2017
\begin{table}[ht]
\centering
\begin{tabular}{lll|lllll|lllll}
\hline
&&& \multicolumn{5}{c}{FDR} & \multicolumn{5}{c}{TPR}\\
$\pi_0(x)$ & Dist. under $H_1$ & Reg. model & BL & Scott T & Scott E & Storey & BH & BL & Scott T & Scott E & Storey & BH \\
\hline
I & Beta(1,20) & Linear & 3.7 & 90.0 & 90.0 & 3.7 & 3.6 & 0.0 & 100.0 & 100.0 & 0.0 & 0.0 \\
II & Beta(1,20) & Linear & 3.1 & 92.6 & 92.6 & 3.1 & 3.0 & 0.0 & 100.0 & 100.0 & 0.0 & 0.0 \\
II & Beta(1,20) & Spline & 3.1 & 92.6 & 92.6 & 3.1 & 3.0 & 0.0 & 100.0 & 100.0 & 0.0 & 0.0 \\
III & Beta(1,20) & Linear & 4.0 & 94.9 & 94.9 & 3.5 & 3.5 & 0.0 & 100.0 & 100.0 & 0.0 & 0.0 \\
III & Beta(1,20) & Spline & 4.5 & 94.9 & 94.9 & 3.5 & 3.5 & 0.0 & 100.0 & 100.0 & 0.0 & 0.0 \\
IV & Beta(1,20) & Linear & 4.4 & 56.9 & & 4.8 & 2.5 & 1.2 & 100.0 & & 0.5 & 0.0 \\
IV & Beta(1,20) & Spline & 5.0 & 56.9 & & 4.8 & 2.5 & 2.0 & 100.0 & & 0.5 & 0.0 \\
\hline
I & Norm & Linear & 5.0 & 5.0 & 5.9 & 5.0 & 4.5 & 50.6 & 50.6 & 52.1 & 50.7 & 49.6 \\
II & Norm & Linear & 4.9 & 5.2 & 5.3 & 4.9 & 4.6 & 48.4 & 63.9 & 62.9 & 47.3 & 46.6 \\
II & Norm & Spline & 4.9 & 5.2 & 5.3 & 4.9 & 4.6 & 48.8 & 64.0 & 63.0 & 47.3 & 46.6 \\
III & Norm & Linear & 4.9 & 5.2 & 5.5 & 4.9 & 4.7 & 44.2 & 60.2 & 59.3 & 43.5 & 43.0 \\
III & Norm & Spline & 4.9 & 5.2 & 5.4 & 4.9 & 4.7 & 44.4 & 60.6 & 59.7 & 43.5 & 43.0 \\
IV & Norm & Linear & 4.8 & 5.0 & 2.3 & 4.8 & 2.8 & 71.3 & 71.8 & 62.2 & 71.2 & 65.3 \\
IV & Norm & Spline & 4.8 & 5.0 & 2.3 & 4.8 & 2.8 & 71.3 & 71.8 & 62.2 & 71.2 & 65.3 \\
\hline
I & T & Linear & 5.2 & 21.7 & 20.8 & 5.1 & 4.7 & 14.1 & 55.3 & 53.2 & 14.1 & 12.6 \\
II & T & Linear & 4.6 & 20.0 & 19.9 & 4.9 & 4.5 & 11.5 & 65.7 & 65.4 & 10.2 & 9.2 \\
II & T & Spline & 4.5 & 20.2 & 20.1 & 4.9 & 4.5 & 12.0 & 65.7 & 65.4 & 10.2 & 9.2 \\
III & T & Linear & 4.9 & 24.7 & 26.8 & 5.2 & 5.2 & 6.8 & 62.5 & 63.7 & 6.0 & 5.5 \\
III & T & Spline & 4.8 & 24.8 & 26.9 & 5.2 & 5.2 & 7.0 & 62.6 & 63.9 & 6.0 & 5.5 \\
IV & T & Linear & 4.8 & 9.3 & 1.2 & 4.8 & 2.9 & 51.8 & 72.8 & 28.5 & 51.6 & 40.2 \\
IV & T & Spline & 4.8 & 9.3 & 1.2 & 4.8 & 2.9 & 51.9 & 72.9 & 28.6 & 51.6 & 40.2 \\
\hline
I & Chisq 1 df & Linear & 5.0 & 90.0 & 90.0 & 5.0 & 4.5 & 50.7 & 100.0 & 100.0 & 50.6 & 49.6 \\
II & Chisq 1 df & Linear & 4.9 & 92.6 & 92.6 & 5.0 & 4.6 & 48.2 & 100.0 & 100.0 & 47.2 & 46.4 \\
II & Chisq 1 df & Spline & 4.8 & 92.6 & 92.6 & 5.0 & 4.6 & 48.6 & 100.0 & 100.0 & 47.2 & 46.4 \\
III & Chisq 1 df & Linear & 5.0 & 94.9 & 94.9 & 5.0 & 4.8 & 44.0 & 100.0 & 100.0 & 43.2 & 42.7 \\
III & Chisq 1 df & Spline & 5.0 & 94.9 & 94.9 & 5.0 & 4.8 & 44.2 & 100.0 & 100.0 & 43.2 & 42.7 \\
IV & Chisq 1 df & Linear & 4.8 & 56.9 & & 4.8 & 2.8 & 71.1 & 100.0 & & 71.0 & 65.2 \\
IV & Chisq 1 df & Spline & 4.8 & 56.9 & & 4.8 & 2.8 & 71.2 & 100.0 & & 71.0 & 65.2 \\
\hline
I & Chisq 4 df & Linear & 5.0 & 90.0 & 90.0 & 5.0 & 4.5 & 29.7 & 100.0 & 100.0 & 29.7 & 28.7 \\
II & Chisq 4 df & Linear & 4.9 & 92.6 & 92.6 & 5.0 & 4.7 & 28.0 & 100.0 & 100.0 & 27.1 & 26.5 \\
II & Chisq 4 df & Spline & 4.9 & 92.6 & 92.6 & 5.0 & 4.7 & 28.4 & 100.0 & 100.0 & 27.1 & 26.5 \\
III & Chisq 4 df & Linear & 5.2 & 94.9 & 94.9 & 5.2 & 5.0 & 24.3 & 100.0 & 100.0 & 23.6 & 23.2 \\
III & Chisq 4 df & Spline & 5.2 & 94.9 & 94.9 & 5.2 & 5.0 & 24.4 & 100.0 & 100.0 & 23.6 & 23.2 \\
IV & Chisq 4 df & Linear & 4.7 & 56.9 & 57.1 & 4.7 & 2.8 & 51.8 & 100.0 & 100.0 & 51.7 & 44.8 \\
IV & Chisq 4 df & Spline & 4.7 & 56.9 & 57.1 & 4.7 & 2.8 & 51.9 & 100.0 & 100.0 & 51.7 & 44.8 \\
\hline
\end{tabular}
\end{table}
\clearpage
\subsection*{Dependent test statistics - 1,000 tests}
We next generated independent test statistics. We used multivariate normal and t distributions (10 df for the t-distribution).
We considered block-diagnonal matrices with the number of blocks equal to 20 or 10 and the within-block correlation, $\rho$, of 0.2, 0.5, or 0.9. Thus, 20 blocks meant a block size of 50 tests (lesser dependence) and 10 blocks a block size of 100 tests (more dependence).
\\
BL = Boca-Leek, Scott T = Scott theoretical null, Scott E = Scott empirical null
% latex table generated in R 3.3.1 by xtable 1.8-2 package
% Thu Jun 22 11:11:21 2017
\begin{table}[ht]
\centering
\begin{tabular}{lll|lllll|lllll}
\hline
&&& \multicolumn{5}{c}{FDR} & \multicolumn{5}{c}{TPR}\\
$\pi_0(x)$ & Dist. under $H_1$ & Reg. model & BL & Scott T & Scott E & Storey & BH & BL & Scott T & Scott E & Storey & BH \\
\hline
I & N, 20 blocks, $\rho$=0.2 & Linear & 5.3 & 6.2 & 6.8 & 5.0 & 4.4 & 51.5 & 51.4 & 48.4 & 51.3 & 50.1 \\
II & N, 20 blocks, $\rho$=0.2 & Linear & 5.2 & 6.9 & 8.0 & 5.1 & 4.6 & 48.6 & 63.4 & 59.3 & 47.6 & 46.5 \\
II & N, 20 blocks, $\rho$=0.2 & Spline & 5.7 & 8.3 & 9.2 & 5.1 & 4.6 & 49.2 & 63.3 & 59.6 & 47.6 & 46.5 \\
III & N, 20 blocks, $\rho$=0.2 & Linear & 5.5 & 7.6 & 9.3 & 5.2 & 4.8 & 45.1 & 60.0 & 56.0 & 44.0 & 43.2 \\
III & N, 20 blocks, $\rho$=0.2 & Spline & 5.7 & 9.6 & 10.6 & 5.2 & 4.8 & 45.9 & 60.2 & 56.3 & 44.0 & 43.2 \\
IV & N, 20 blocks, $\rho$=0.2 & Linear & 5.3 & 5.3 & 2.5 & 4.9 & 2.9 & 71.8 & 71.9 & 61.0 & 71.4 & 65.6 \\
IV & N, 20 blocks, $\rho$=0.2 & Spline & 5.6 & 5.5 & 2.5 & 4.9 & 2.9 & 72.0 & 71.9 & 61.1 & 71.4 & 65.6 \\
\hline
I & N, 20 blocks, $\rho$=0.5 & Linear & 6.4 & 10.0 & 10.7 & 6.0 & 5.2 & 52.0 & 51.7 & 47.6 & 51.6 & 50.3 \\
II & N, 20 blocks, $\rho$=0.5 & Linear & 6.1 & 12.4 & 13.5 & 5.7 & 5.1 & 48.4 & 62.8 & 57.6 & 47.3 & 46.2 \\
II & N, 20 blocks, $\rho$=0.5 & Spline & 7.1 & 18.7 & 20.4 & 5.7 & 5.1 & 49.5 & 62.6 & 58.0 & 47.3 & 46.2 \\
III & N, 20 blocks, $\rho$=0.5 & Linear & 5.6 & 11.5 & 15.9 & 5.2 & 4.6 & 45.4 & 59.6 & 56.6 & 44.0 & 43.2 \\
III & N, 20 blocks, $\rho$=0.5 & Spline & 6.6 & 19.9 & 23.6 & 5.2 & 4.6 & 46.2 & 59.0 & 56.9 & 44.0 & 43.2 \\
IV & N, 20 blocks, $\rho$=0.5 & Linear & 5.8 & 6.1 & 2.8 & 5.3 & 3.1 & 72.1 & 72.3 & 59.4 & 71.6 & 65.7 \\
IV & N, 20 blocks, $\rho$=0.5 & Spline & 6.5 & 6.4 & 3.0 & 5.3 & 3.1 & 72.4 & 72.2 & 59.6 & 71.6 & 65.7 \\
\hline
I & N, 20 blocks, $\rho$=0.9 & Linear & 9.0 & 17.6 & 36.2 & 6.9 & 5.3 & 53.8 & 53.3 & 57.9 & 52.6 & 50.4 \\
II & N, 20 blocks, $\rho$=0.9 & Linear & 7.8 & 20.0 & 47.5 & 6.4 & 4.9 & 49.6 & 63.8 & 68.0 & 48.0 & 46.2 \\
II & N, 20 blocks, $\rho$=0.9 & Spline & 18.2 & 34.5 & 53.6 & 6.4 & 4.9 & 52.2 & 64.4 & 69.8 & 48.0 & 46.2 \\
III & N, 20 blocks, $\rho$=0.9 & Linear & 6.4 & 23.1 & 48.8 & 5.1 & 4.0 & 47.3 & 60.5 & 67.9 & 46.1 & 44.0 \\
III & N, 20 blocks, $\rho$=0.9 & Spline & 21.5 & 38.4 & 60.5 & 5.1 & 4.0 & 51.0 & 60.9 & 69.7 & 46.1 & 44.0 \\
IV & N, 20 blocks, $\rho$=0.9 & Linear & 7.7 & 8.4 & 6.9 & 6.1 & 3.1 & 73.1 & 73.2 & 57.4 & 72.2 & 65.9 \\
IV & N, 20 blocks, $\rho$=0.9 & Spline & 11.8 & 10.0 & 8.0 & 6.1 & 3.1 & 74.4 & 72.8 & 57.8 & 72.2 & 65.9 \\
\hline
I & N, 10 blocks, $\rho$=0.2 & Linear & 5.4 & 7.8 & 6.1 & 5.1 & 4.4 & 51.6 & 51.6 & 47.3 & 51.2 & 49.9 \\
II & N, 10 blocks, $\rho$=0.2 & Linear & 5.0 & 9.3 & 8.8 & 4.8 & 4.3 & 48.2 & 63.0 & 59.8 & 47.2 & 46.1 \\
II & N, 10 blocks, $\rho$=0.2 & Spline & 5.5 & 13.3 & 11.1 & 4.8 & 4.3 & 49.1 & 62.8 & 59.8 & 47.2 & 46.1 \\
III & N, 10 blocks, $\rho$=0.2 & Linear & 5.2 & 8.6 & 9.8 & 5.0 & 4.5 & 44.6 & 59.5 & 56.4 & 43.4 & 42.7 \\
III & N, 10 blocks, $\rho$=0.2 & Spline & 5.8 & 14.3 & 13.2 & 5.0 & 4.5 & 45.2 & 59.2 & 56.6 & 43.4 & 42.7 \\
IV & N, 10 blocks, $\rho$=0.2 & Linear & 5.3 & 5.7 & 2.4 & 5.0 & 2.9 & 71.8 & 71.8 & 60.4 & 71.4 & 65.5 \\
IV & N, 10 blocks, $\rho$=0.2 & Spline & 5.7 & 5.9 & 2.5 & 5.0 & 2.9 & 72.1 & 71.8 & 60.5 & 71.4 & 65.5 \\
\hline
I & N, 10 blocks, $\rho$=0.5 & Linear & 7.3 & 17.1 & 15.9 & 6.5 & 5.4 & 51.9 & 51.8 & 48.8 & 51.7 & 50.0 \\
II & N, 10 blocks, $\rho$=0.5 & Linear & 5.9 & 20.3 & 19.9 & 5.3 & 4.5 & 48.3 & 62.6 & 61.0 & 46.8 & 45.6 \\
II & N, 10 blocks, $\rho$=0.5 & Spline & 8.6 & 32.5 & 27.7 & 5.3 & 4.5 & 49.2 & 63.3 & 61.4 & 46.8 & 45.6 \\
III & N, 10 blocks, $\rho$=0.5 & Linear & 5.8 & 17.4 & 17.7 & 4.9 & 4.2 & 44.2 & 58.1 & 54.3 & 43.0 & 42.0 \\
III & N, 10 blocks, $\rho$=0.5 & Spline & 8.6 & 32.7 & 30.2 & 4.9 & 4.2 & 45.0 & 58.1 & 55.6 & 43.0 & 42.0 \\
IV & N, 10 blocks, $\rho$=0.5 & Linear & 6.3 & 7.5 & 3.3 & 5.5 & 3.2 & 72.4 & 72.4 & 59.0 & 71.9 & 65.8 \\
IV & N, 10 blocks, $\rho$=0.5 & Spline & 7.6 & 8.3 & 3.8 & 5.5 & 3.2 & 72.7 & 72.1 & 59.3 & 71.9 & 65.8 \\
\hline
I & N, 10 blocks, $\rho$=0.9 & Linear & 14.1 & 30.6 & 45.6 & 6.6 & 4.1 & 55.5 & 54.7 & 65.6 & 53.3 & 50.2 \\
II & N, 10 blocks, $\rho$=0.9 & Linear & 13.3 & 35.5 & 55.9 & 5.9 & 3.3 & 51.1 & 66.5 & 75.8 & 49.0 & 46.1 \\
II & N, 10 blocks, $\rho$=0.9 & Spline & 35.1 & 49.9 & 67.5 & 5.9 & 3.3 & 56.1 & 67.4 & 77.6 & 49.0 & 46.1 \\
III & N, 10 blocks, $\rho$=0.9 & Linear & 13.3 & 33.7 & 66.4 & 5.4 & 3.3 & 45.6 & 58.1 & 75.7 & 43.4 & 40.7 \\
III & N, 10 blocks, $\rho$=0.9 & Spline & 40.7 & 51.5 & 73.0 & 5.4 & 3.3 & 52.0 & 61.6 & 77.4 & 43.4 & 40.7 \\
IV & N, 10 blocks, $\rho$=0.9 & Linear & 11.2 & 12.4 & 12.0 & 7.0 & 3.1 & 74.0 & 73.5 & 63.9 & 72.5 & 65.8 \\
IV & N, 10 blocks, $\rho$=0.9 & Spline & 19.2 & 15.6 & 13.8 & 7.0 & 3.1 & 76.2 & 73.3 & 64.3 & 72.5 & 65.8 \\
\hline
\end{tabular}
\end{table}
% latex table generated in R 3.3.1 by xtable 1.8-2 package
% Thu Jun 22 11:11:22 2017
\begin{table}[ht]
\centering
\begin{tabular}{lll|lllll|lllll}
\hline
&&& \multicolumn{5}{c}{FDR} & \multicolumn{5}{c}{TPR}\\
$\pi_0(x)$ & Dist. under $H_1$ & Reg. model & BL & Scott T & Scott E & Storey & BH & BL & Scott T & Scott E & Storey & BH \\
\hline
I & T, 20 blocks, $\rho$=0.2 & Linear & 1.7 & 9.1 & 7.4 & 1.5 & 0.9 & 8.0 & 51.6 & 57.8 & 7.6 & 5.7 \\
II & T, 20 blocks, $\rho$=0.2 & Linear & 3.2 & 13.9 & 7.3 & 3.2 & 1.8 & 8.0 & 63.8 & 61.0 & 6.8 & 4.5 \\
II & T, 20 blocks, $\rho$=0.2 & Spline & 3.7 & 14.7 & 8.5 & 3.2 & 1.8 & 9.2 & 63.9 & 61.3 & 6.8 & 4.5 \\
III & T, 20 blocks, $\rho$=0.2 & Linear & 2.6 & 13.8 & 9.6 & 2.1 & 1.3 & 4.3 & 59.4 & 60.1 & 3.4 & 2.3 \\
III & T, 20 blocks, $\rho$=0.2 & Spline & 3.6 & 15.1 & 11.0 & 2.1 & 1.3 & 5.2 & 59.7 & 60.3 & 3.4 & 2.3 \\
IV & T, 20 blocks, $\rho$=0.2 & Linear & 2.7 & 5.4 & 2.9 & 2.4 & 1.0 & 55.4 & 71.8 & 65.1 & 54.4 & 44.3 \\
IV & T, 20 blocks, $\rho$=0.2 & Spline & 3.0 & 5.4 & 2.8 & 2.4 & 1.0 & 56.0 & 71.9 & 65.1 & 54.4 & 44.3 \\
\hline
I & T, 20 blocks, $\rho$=0.5 & Linear & 1.7 & 10.3 & 11.0 & 1.5 & 1.0 & 8.6 & 51.6 & 57.4 & 8.2 & 5.9 \\
II & T, 20 blocks, $\rho$=0.5 & Linear & 3.5 & 16.3 & 11.9 & 3.3 & 2.1 & 7.7 & 64.2 & 61.7 & 6.6 & 4.5 \\
II & T, 20 blocks, $\rho$=0.5 & Spline & 4.7 & 19.5 & 16.6 & 3.3 & 2.1 & 9.1 & 63.9 & 62.1 & 6.6 & 4.5 \\
III & T, 20 blocks, $\rho$=0.5 & Linear & 3.2 & 17.6 & 13.0 & 2.3 & 1.5 & 5.0 & 59.3 & 59.0 & 3.6 & 2.6 \\
III & T, 20 blocks, $\rho$=0.5 & Spline & 4.4 & 23.4 & 20.5 & 2.3 & 1.5 & 5.6 & 59.6 & 59.5 & 3.6 & 2.6 \\
IV & T, 20 blocks, $\rho$=0.5 & Linear & 2.7 & 5.5 & 3.0 & 2.3 & 1.0 & 55.3 & 71.9 & 64.7 & 54.3 & 44.4 \\
IV & T, 20 blocks, $\rho$=0.5 & Spline & 3.2 & 5.8 & 3.1 & 2.3 & 1.0 & 55.8 & 71.9 & 64.8 & 54.3 & 44.4 \\
\hline
I & T, 20 blocks, $\rho$=0.9 & Linear & 3.0 & 14.5 & 29.0 & 1.5 & 0.9 & 11.5 & 51.7 & 64.1 & 9.9 & 6.2 \\
II & T, 20 blocks, $\rho$=0.9 & Linear & 3.8 & 20.9 & 45.7 & 2.3 & 1.9 & 10.2 & 64.9 & 70.6 & 7.7 & 5.0 \\
II & T, 20 blocks, $\rho$=0.9 & Spline & 15.8 & 32.1 & 54.6 & 2.3 & 1.9 & 14.2 & 64.7 & 70.5 & 7.7 & 5.0 \\
III & T, 20 blocks, $\rho$=0.9 & Linear & 5.2 & 23.9 & 49.7 & 3.2 & 1.4 & 7.3 & 60.7 & 63.5 & 5.6 & 3.1 \\
III & T, 20 blocks, $\rho$=0.9 & Spline & 19.0 & 35.1 & 60.6 & 3.2 & 1.4 & 10.6 & 61.7 & 65.5 & 5.6 & 3.1 \\
IV & T, 20 blocks, $\rho$=0.9 & Linear & 3.6 & 6.6 & 7.5 & 2.4 & 1.0 & 56.1 & 72.2 & 67.5 & 54.6 & 44.3 \\
IV & T, 20 blocks, $\rho$=0.9 & Spline & 8.6 & 7.5 & 8.0 & 2.4 & 1.0 & 58.4 & 72.0 & 67.2 & 54.6 & 44.3 \\
\hline
I & T, 10 blocks, $\rho$=0.2 & Linear & 1.8 & 9.9 & 7.8 & 1.6 & 0.8 & 8.3 & 51.3 & 57.2 & 8.0 & 5.9 \\
II & T, 10 blocks, $\rho$=0.2 & Linear & 3.4 & 15.0 & 8.1 & 3.4 & 1.5 & 7.3 & 63.1 & 61.3 & 6.4 & 4.3 \\
II & T, 10 blocks, $\rho$=0.2 & Spline & 4.0 & 16.7 & 9.9 & 3.4 & 1.5 & 8.6 & 63.2 & 61.5 & 6.4 & 4.3 \\
III & T, 10 blocks, $\rho$=0.2 & Linear & 2.2 & 15.2 & 9.5 & 1.6 & 1.2 & 3.7 & 58.7 & 59.4 & 3.0 & 1.9 \\
III & T, 10 blocks, $\rho$=0.2 & Spline & 2.7 & 18.0 & 12.7 & 1.6 & 1.2 & 4.2 & 58.5 & 59.7 & 3.0 & 1.9 \\
IV & T, 10 blocks, $\rho$=0.2 & Linear & 2.6 & 5.5 & 2.8 & 2.4 & 1.0 & 54.8 & 71.5 & 64.6 & 53.9 & 43.9 \\
IV & T, 10 blocks, $\rho$=0.2 & Spline & 3.0 & 5.6 & 2.8 & 2.4 & 1.0 & 55.4 & 71.5 & 64.7 & 53.9 & 43.9 \\
\hline
I & T, 10 blocks, $\rho$=0.5 & Linear & 2.2 & 13.5 & 14.2 & 1.6 & 0.9 & 9.3 & 50.8 & 57.4 & 8.5 & 6.1 \\
II & T, 10 blocks, $\rho$=0.5 & Linear & 3.3 & 19.2 & 13.6 & 3.4 & 1.7 & 7.9 & 63.1 & 61.2 & 7.0 & 4.4 \\
II & T, 10 blocks, $\rho$=0.5 & Spline & 6.2 & 27.6 & 21.3 & 3.4 & 1.7 & 9.9 & 63.5 & 61.3 & 7.0 & 4.4 \\
III & T, 10 blocks, $\rho$=0.5 & Linear & 2.3 & 23.4 & 21.5 & 1.3 & 0.7 & 4.4 & 58.0 & 59.5 & 3.0 & 2.1 \\
III & T, 10 blocks, $\rho$=0.5 & Spline & 3.8 & 35.9 & 31.4 & 1.3 & 0.7 & 5.6 & 58.1 & 60.1 & 3.0 & 2.1 \\
IV & T, 10 blocks, $\rho$=0.5 & Linear & 3.1 & 6.1 & 3.4 & 2.5 & 1.0 & 54.4 & 71.4 & 63.5 & 53.4 & 43.2 \\
IV & T, 10 blocks, $\rho$=0.5 & Spline & 4.3 & 6.6 & 3.8 & 2.5 & 1.0 & 55.3 & 71.2 & 64.0 & 53.4 & 43.2 \\
\hline
I & T, 10 blocks, $\rho$=0.9 & Linear & 7.7 & 23.0 & 38.0 & 1.6 & 1.0 & 14.9 & 51.5 & 70.9 & 11.4 & 6.7 \\
II & T, 10 blocks, $\rho$=0.9 & Linear & 10.1 & 31.5 & 50.0 & 4.1 & 1.7 & 12.4 & 65.4 & 76.2 & 11.1 & 6.0 \\
II & T, 10 blocks, $\rho$=0.9 & Spline & 41.7 & 43.6 & 60.7 & 4.1 & 1.7 & 22.4 & 68.2 & 78.9 & 11.1 & 6.0 \\
III & T, 10 blocks, $\rho$=0.9 & Linear & 12.7 & 36.2 & 62.9 & 2.2 & 1.3 & 11.0 & 60.5 & 77.2 & 5.8 & 2.6 \\
III & T, 10 blocks, $\rho$=0.9 & Spline & 43.0 & 48.4 & 71.0 & 2.2 & 1.3 & 19.3 & 62.9 & 78.7 & 5.8 & 2.6 \\
IV & T, 10 blocks, $\rho$=0.9 & Linear & 6.2 & 9.2 & 11.1 & 3.2 & 1.0 & 56.3 & 72.1 & 68.3 & 54.2 & 42.4 \\
IV & T, 10 blocks, $\rho$=0.9 & Spline & 15.1 & 10.8 & 11.8 & 3.2 & 1.0 & 59.3 & 71.8 & 68.3 & 54.2 & 42.4 \\
\hline
\end{tabular}
\end{table}
\end{document}