This function estimates the parameters of the special restricted Gaussian mixture copula model (GMCM) proposed by Li et. al. (2011). It is used to perform reproducibility (or meta) analysis using GMCMs. It features various optimization routines to identify the maximum likelihood estimate of the special GMCMs.

fit.meta.GMCM(u, init.par, method = c("NM", "SANN", "L-BFGS", "L-BFGS-B",
"PEM"), max.ite = 1000, verbose = TRUE, positive.rho = TRUE,
trace.theta = FALSE, ...)

fit.special.GMCM(u, init.par, method = c("NM", "SANN", "L-BFGS",
"L-BFGS-B", "PEM"), max.ite = 1000, verbose = TRUE,
positive.rho = TRUE, trace.theta = FALSE, ...)

## Arguments

u An n by d matrix of test statistics. Rows correspond to features and columns to experiments. Larger values are assumed to be indicative of stronger evidence and reproducibility. A 4-dimensional vector of the initial parameters where, init.par[1] is the mixture proportion of spurious signals, init.par[2] is the mean, init.par[3] is the standard deviation, init.par[4] is the correlation. A character vector of length $$1$$. The optimization method used. Should be either "NM", "SANN", "L-BFGS", "L-BFGS-B", or "PEM" which are abbreviations of Nelder-Mead, Simulated Annealing, limited-memory quasi-Newton method, limited-memory quasi-Newton method with box constraints, and the pseudo EM algorithm, respectively. Default is "NM". See optim for further details. The maximum number of iterations. If the method is "SANN" this is the number of iterations as there is no other stopping criterion. (See optim) Logical. If TRUE, the log-likelihood values are printed. logical. If TRUE, the correlation parameter is restricted to be positive. logical. Extra convergence information is appended as a list to the output returned if TRUE. The exact behavior is dependent on the value of method. If method equals "PEM", the argument is passed to trace.theta in PseudoEMAlgorithm. Otherwise it is passed to the control argument trace in optim. Arguments passed to the control-list in optim or PseudoEMAlgorithm if method is "PEM".

## Value

A vector par of length 4 of the fitted parameters where par[1] is the probability of being from the first (or null) component, par[2] is the mean, par[3] is the standard deviation, and par[4] is the correlation.

If trace.theta is TRUE, then a list is returned where the first entry is as described above and the second entry is the trace information (dependent of method.).

## Details

The "L-BFGS-B" method does not perform a transformation of the parameters.

fit.special.GMCM is simply an alias of fit.meta.gmcm.

## Note

Simulated annealing is strongly dependent on the initial values and the cooling scheme.

See optim for further details.

## References

Li, Q., Brown, J. B. J. B., Huang, H., & Bickel, P. J. (2011). Measuring reproducibility of high-throughput experiments. The Annals of Applied Statistics, 5(3), 1752-1779. doi:10.1214/11-AOAS466

optim

## Author

Anders Ellern Bilgrau <anders.ellern.bilgrau@gmail.com>

## Examples

set.seed(1)

# True parameters
true.par <- c(0.9, 2, 0.7, 0.6)
# Simulation of data from the GMCM model
data <- SimulateGMCMData(n = 1000, par = true.par)
uhat <- Uhat(data$u) # Ranked observed data init.par <- c(0.5, 1, 0.5, 0.9) # Initial parameters # Optimization with Nelder-Mead nm.par <- fit.meta.GMCM(uhat, init.par = init.par, method = "NM") #> Nelder-Mead direct search function minimizer #> function value for initial parameters = 176.343401 #> Scaled convergence tolerance is 2.62772e-06 #> Stepsize computed as 0.219722 #> BUILD 5 226.656977 105.973352 #> LO-REDUCTION 7 215.745222 105.973352 #> EXTENSION 9 176.343401 4.976728 #> LO-REDUCTION 11 133.560182 4.976728 #> EXTENSION 13 114.710466 -45.512205 #> EXTENSION 15 105.973352 -86.690136 #> EXTENSION 17 11.976424 -97.922305 #> LO-REDUCTION 19 4.976728 -97.922305 #> LO-REDUCTION 21 -45.512205 -97.922305 #> LO-REDUCTION 23 -78.907712 -97.922305 #> REFLECTION 25 -86.690136 -105.072127 #> LO-REDUCTION 27 -95.841590 -105.072127 #> LO-REDUCTION 29 -97.114144 -105.072127 #> LO-REDUCTION 31 -97.922305 -105.072127 #> LO-REDUCTION 33 -99.021762 -105.072127 #> LO-REDUCTION 35 -102.447782 -105.072127 #> EXTENSION 37 -102.566757 -109.005176 #> HI-REDUCTION 39 -103.544631 -109.005176 #> LO-REDUCTION 41 -105.010085 -109.005176 #> EXTENSION 43 -105.072127 -111.555517 #> LO-REDUCTION 45 -105.494019 -111.555517 #> EXTENSION 47 -107.926302 -114.706877 #> LO-REDUCTION 49 -108.163783 -114.706877 #> LO-REDUCTION 51 -109.005176 -114.706877 #> REFLECTION 53 -111.555517 -115.585400 #> HI-REDUCTION 55 -113.529332 -115.585400 #> LO-REDUCTION 57 -113.796380 -115.585400 #> EXTENSION 59 -113.979420 -116.852099 #> HI-REDUCTION 61 -114.706877 -116.852099 #> LO-REDUCTION 63 -115.194526 -116.852099 #> EXTENSION 65 -115.255631 -117.583240 #> HI-REDUCTION 67 -115.585400 -117.583240 #> EXTENSION 69 -116.276863 -118.192081 #> LO-REDUCTION 71 -116.284681 -118.192081 #> REFLECTION 73 -116.852099 -118.328874 #> LO-REDUCTION 75 -117.583240 -118.387443 #> LO-REDUCTION 77 -117.737665 -118.387443 #> HI-REDUCTION 79 -118.192081 -118.387443 #> HI-REDUCTION 81 -118.204023 -118.387443 #> HI-REDUCTION 83 -118.294593 -118.387443 #> LO-REDUCTION 85 -118.325318 -118.387443 #> REFLECTION 87 -118.328874 -118.401466 #> LO-REDUCTION 89 -118.331957 -118.401466 #> LO-REDUCTION 91 -118.368285 -118.401466 #> REFLECTION 93 -118.387443 -118.405678 #> EXTENSION 95 -118.396046 -118.435327 #> HI-REDUCTION 97 -118.398167 -118.435327 #> HI-REDUCTION 99 -118.401466 -118.435327 #> EXTENSION 101 -118.405678 -118.467822 #> LO-REDUCTION 103 -118.417378 -118.467822 #> LO-REDUCTION 105 -118.418910 -118.467822 #> EXTENSION 107 -118.435327 -118.480598 #> LO-REDUCTION 109 -118.440627 -118.480598 #> EXTENSION 111 -118.443666 -118.517242 #> EXTENSION 113 -118.467822 -118.589741 #> LO-REDUCTION 115 -118.478467 -118.589741 #> LO-REDUCTION 117 -118.480598 -118.589741 #> EXTENSION 119 -118.517242 -118.728493 #> LO-REDUCTION 121 -118.528818 -118.728493 #> EXTENSION 123 -118.534609 -118.882758 #> EXTENSION 125 -118.589741 -119.285949 #> LO-REDUCTION 127 -118.661470 -119.285949 #> EXTENSION 129 -118.728493 -119.665201 #> REFLECTION 131 -118.882758 -119.717315 #> LO-REDUCTION 133 -119.273816 -119.717315 #> LO-REDUCTION 135 -119.285949 -119.717315 #> HI-REDUCTION 137 -119.586776 -119.717315 #> HI-REDUCTION 139 -119.642925 -119.717315 #> EXTENSION 141 -119.662587 -119.756290 #> LO-REDUCTION 143 -119.665201 -119.756290 #> EXTENSION 145 -119.700026 -119.882576 #> LO-REDUCTION 147 -119.717315 -119.882576 #> EXTENSION 149 -119.751764 -120.001647 #> LO-REDUCTION 151 -119.756290 -120.001647 #> EXTENSION 153 -119.845613 -120.217221 #> LO-REDUCTION 155 -119.882576 -120.217221 #> LO-REDUCTION 157 -119.924760 -120.217221 #> EXTENSION 159 -120.001647 -120.535915 #> LO-REDUCTION 161 -120.176396 -120.535915 #> EXTENSION 163 -120.209100 -120.720566 #> LO-REDUCTION 165 -120.217221 -120.720566 #> EXTENSION 167 -120.525797 -121.072510 #> LO-REDUCTION 169 -120.535915 -121.072510 #> LO-REDUCTION 171 -120.710482 -121.072510 #> REFLECTION 173 -120.720566 -121.132790 #> HI-REDUCTION 175 -120.974799 -121.132790 #> HI-REDUCTION 177 -120.983393 -121.132790 #> LO-REDUCTION 179 -121.034027 -121.132790 #> LO-REDUCTION 181 -121.059421 -121.132790 #> HI-REDUCTION 183 -121.072510 -121.132790 #> LO-REDUCTION 185 -121.089634 -121.132790 #> EXTENSION 187 -121.103641 -121.188826 #> LO-REDUCTION 189 -121.111154 -121.188826 #> LO-REDUCTION 191 -121.122667 -121.188826 #> EXTENSION 193 -121.132790 -121.204569 #> EXTENSION 195 -121.134706 -121.222492 #> EXTENSION 197 -121.180724 -121.336842 #> LO-REDUCTION 199 -121.188826 -121.336842 #> LO-REDUCTION 201 -121.204569 -121.336842 #> LO-REDUCTION 203 -121.222492 -121.336842 #> REFLECTION 205 -121.282353 -121.360958 #> EXTENSION 207 -121.283241 -121.432059 #> EXTENSION 209 -121.329228 -121.555775 #> LO-REDUCTION 211 -121.336842 -121.555775 #> LO-REDUCTION 213 -121.360958 -121.555775 #> EXTENSION 215 -121.432059 -121.725111 #> LO-REDUCTION 217 -121.444713 -121.725111 #> EXTENSION 219 -121.529250 -121.904929 #> EXTENSION 221 -121.555775 -122.133960 #> EXTENSION 223 -121.692835 -122.326879 #> LO-REDUCTION 225 -121.725111 -122.326879 #> LO-REDUCTION 227 -121.904929 -122.326879 #> REFLECTION 229 -122.133960 -122.383178 #> HI-REDUCTION 231 -122.199466 -122.383178 #> EXTENSION 233 -122.211678 -122.561704 #> LO-REDUCTION 235 -122.282918 -122.561704 #> LO-REDUCTION 237 -122.326879 -122.561704 #> REFLECTION 239 -122.383178 -122.648447 #> REFLECTION 241 -122.541034 -122.732974 #> HI-REDUCTION 243 -122.557909 -122.732974 #> LO-REDUCTION 245 -122.561704 -122.732974 #> REFLECTION 247 -122.626764 -122.761473 #> EXTENSION 249 -122.648447 -122.806684 #> LO-REDUCTION 251 -122.689278 -122.806684 #> REFLECTION 253 -122.732974 -122.824853 #> LO-REDUCTION 255 -122.761473 -122.824853 #> LO-REDUCTION 257 -122.782913 -122.824877 #> HI-REDUCTION 259 -122.806684 -122.824877 #> LO-REDUCTION 261 -122.809614 -122.824877 #> LO-REDUCTION 263 -122.824241 -122.830195 #> HI-REDUCTION 265 -122.824791 -122.830195 #> LO-REDUCTION 267 -122.824853 -122.830891 #> HI-REDUCTION 269 -122.824877 -122.830891 #> LO-REDUCTION 271 -122.829721 -122.830891 #> HI-REDUCTION 273 -122.830143 -122.831148 #> LO-REDUCTION 275 -122.830195 -122.831543 #> HI-REDUCTION 277 -122.830342 -122.831543 #> HI-REDUCTION 279 -122.830891 -122.831543 #> HI-REDUCTION 281 -122.831148 -122.831742 #> LO-REDUCTION 283 -122.831400 -122.831760 #> REFLECTION 285 -122.831449 -122.831854 #> HI-REDUCTION 287 -122.831543 -122.831854 #> REFLECTION 289 -122.831742 -122.831879 #> LO-REDUCTION 291 -122.831760 -122.831916 #> HI-REDUCTION 293 -122.831785 -122.831918 #> REFLECTION 295 -122.831854 -122.832122 #> HI-REDUCTION 297 -122.831879 -122.832122 #> LO-REDUCTION 299 -122.831916 -122.832122 #> HI-REDUCTION 301 -122.831918 -122.832134 #> LO-REDUCTION 303 -122.832057 -122.832134 #> LO-REDUCTION 305 -122.832059 -122.832134 #> EXTENSION 307 -122.832103 -122.832258 #> LO-REDUCTION 309 -122.832117 -122.832258 #> LO-REDUCTION 311 -122.832122 -122.832258 #> HI-REDUCTION 313 -122.832134 -122.832258 #> LO-REDUCTION 315 -122.832189 -122.832258 #> EXTENSION 317 -122.832225 -122.832323 #> LO-REDUCTION 319 -122.832237 -122.832323 #> LO-REDUCTION 321 -122.832258 -122.832323 #> REFLECTION 323 -122.832258 -122.832331 #> LO-REDUCTION 325 -122.832304 -122.832331 #> LO-REDUCTION 327 -122.832308 -122.832331 #> LO-REDUCTION 329 -122.832320 -122.832334 #> REFLECTION 331 -122.832323 -122.832336 #> EXTENSION 333 -122.832325 -122.832341 #> REFLECTION 335 -122.832331 -122.832343 #> REFLECTION 337 -122.832334 -122.832346 #> EXTENSION 339 -122.832336 -122.832357 #> LO-REDUCTION 341 -122.832341 -122.832357 #> REFLECTION 343 -122.832343 -122.832361 #> LO-REDUCTION 345 -122.832346 -122.832361 #> REFLECTION 347 -122.832352 -122.832362 #> LO-REDUCTION 349 -122.832354 -122.832362 #> REFLECTION 351 -122.832357 -122.832364 #> REFLECTION 353 -122.832361 -122.832365 #> HI-REDUCTION 355 -122.832362 -122.832365 #> HI-REDUCTION 357 -122.832362 -122.832365 #> Exiting from Nelder Mead minimizer #> 359 function evaluations used if (FALSE) { # Comparison with other optimization methods # Optimization with simulated annealing sann.par <- fit.meta.GMCM(uhat, init.par = init.par, method = "SANN", max.ite = 3000, temp = 1) # Optimization with the Pseudo EM algorithm pem.par <- fit.meta.GMCM(uhat, init.par = init.par, method = "PEM") # The estimates agree nicely rbind("True" = true.par, "Start" = init.par, "NM" = nm.par, "SANN" = sann.par, "PEM" = pem.par) } # Get estimated cluster Khat <- get.IDR(x = uhat, par = nm.par)$Khat
plot(uhat, col = Khat, main = "Clustering\nIDR < 0.05")