# SAMM: Some Algoritms for Mixed Models In SAMM: Some Algorithms for Mixed Models

## Description

The LMM model of focus can be expressed as

Y=XB+∑_{j=1}^k Z_j G_j+WE,

where Y is the n \times d response variable, X is the n \times q design matrix of q \times d the fixed effects B, Z_j for j=1,2,…,k (k≥q 1) are the n \times q_j design matrices of the q_j \times d random effects G_j, and W is the n \times t design matrix of t \times d residual effecs E. The random effects and the residual are independently distributed, and have matrix variate distributions (G_j\sim N_{q_j \times d}(0_{q_j \times d}, K_j,Σ_j) for j=1,2,…,k and E\sim N_{t \times d}(0_{t \times d}, R,Σ_E)). The matrices K_j, R, Σ_j, Σ_E might be further parametrized. When the response is univariate (d=1) there is no need to specify a structure for Σ_j's andΣ_E. On the other hand, the K_j and R are concordance matrices (covariance matrices with some standardization over the diagonals) and they need to be provided by the user.

Please refer to the examples below and the other help files for more details about the concordance and covariance structures.

!!IMPORTANT NOTE!!: USE SAMM WITH CAUTION, AND AT YOUR OWN RISK! NO GUARANTIES, NO CLAIMS.

PLEASE REPORT BUGS AND RECOMMENDATIONS TO THE MAINTAINER.

## Usage

 1 2 SAMM(Y,X,Zlist,Klist, lambda, W,R,Siglist, corfunc, corfuncfixed, sigfunc, mmalg, tolparconv=1e-10, tolparinv=1e-10,maxiter=1000, geterrors=FALSE, Hinv = FALSE) 

## Arguments

 Y Y is the n \times d response variable X n \times q design matrix of q \times d the fixed effects B Zlist a list object containing Z_j for j=1,2,…,k (k≥q 1), the n \times q_j design matrices of the q_j \times d random effects G_j, Klist a list to specify the concordance matrices K_j for j=1,2,…,k. For each j, the user needs to provide a constant matrix, or a list specifying the concordance structure lambda a scalar shrinkage parameter for shrinkage of variance components (only works with the choice mmalg=”mmmk_ml”). W W is the n \times t design matrix of t \times d residual effecs E R a list to specify the concordance matrix R,the user needs to provide a constant matrix, or a list specifying the concordance structure Siglist a list to specify the covariance structures Σ_j's andΣ_E corfunc a boolian vector specifying whether K_j for j=1,2,…,k and R are functions or given matrices (TRUE for functions) corfuncfixed a boolian vector specifying whether K_j for j=1,2,…,k and R are fixed at the initial parameter values specified sigfunc a boolian vector specifying whether Σ_j for j=1,2,…,k, E are functions or unstructured (TRUE for functions) mmalg The mixed model solving algorithm tolparconv convergence number tolparinv a small scalar to add to the diagonals of positive semidefinite matrices for inversion or for calculating the Cholesky decompositions maxiter Maximum number of iterations geterrors TRUE or FALSE, if true prediction error variances for the random effects are supplied in the output Hinv TRUE or FALSE, if true inverse of H matrix will be returned

## Details

The algorithms in SAMM were mostly concieved based on the ideas in the referenced papers: 1-emm_reml: reference 4. 2-emm_ml: reference 4. 3-dmm_ml: reference 1. 4-dmm_reml: reference 1. 5-dermm_reml1: reference 5 (Fisher-Scoring). 6-dermm_reml2: reference 5 (Average Information). 7-mm_ml: reference 2. 8-emmmk_reml: reference 4. 9-emmmk_ml: reference 4. 10-mmmk_ml: reference 2. 11-emmmv_ml: reference 4. 12-mmmv_ml: reference 3. 13-mmmkmv_ml: reference 2. The table below shows which cases these algorithms can be used. The prefered algorithms are marked by paranthesis and the user still needs to specify it.

 No Univariate One K corfunc sigfunc shrinkage WRWtIdent ALGS 1- + + - 0 - + 1-12 (1) 2- + + - 0 - - 3-7, 10-12 (3) 3- + + 1^* 0 0 0 3-6, 10-11 (3) 4- + + + 0 0 0 10, 13 (10) 5- + - - 0 - + 5, 6, 8, 9, 10, 13 (5) 6- + - - 0 - - 5, 6, 10, 13 (5) 7- + - + 0 - - 5, 6, 10, 13 (5) 8- + - - 0 + 0 10 (10) 9- + - + 0 + 0 10 (10) 10- - + - - - - 11, 12 (11) 11- - + - - - - 12 (12) 12- - - - - - 0 13 (13) 13- - - 2^* 2^* - 0 13 (13) 14- - + 2^* 2^* - 0 13 (13)

-: No. +: Yes. 0: Doesn't matter. 1^*: Only K is a function not R. 2^*: sigmafunc or corfunc.

## Value

A named list object (the output will differ for different algorithms and model types!).

Deniz Akdemir

## References

1-Bates, Douglas M. "lme4: Mixed-effects modeling with R." URL http://lme4. r-forge. r-project. org/book (2010).

2-Zhou, Hua, et al. "MM Algorithms for Variance Components Models." arXiv preprint arXiv:1509.07426 (2015).

3-Zhou, Xiang, and Matthew Stephens. "Efficient algorithms for multivariate linear mixed models in genome-wide association studies." Nature methods 11.4 (2014): 407.

4-Kang, Hyun Min, et al. "Efficient control of population structure in model organism association mapping." Genetics 178.3 (2008): 1709-1723.

5-Gilmour, Arthur R., Robin Thompson, and Brian R. Cullis. "Average information REML: an efficient algorithm for variance parameter estimation in linear mixed models." Biometrics (1995): 1440-1450.

## Examples

  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 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Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain),R=list(diag(ntrain)),Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="emm_reml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout3<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(list("Const",c(0),K)), lambda=0, W=diag(ntrain),R=list(diag(ntrain)),Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F), mmalg="dmm_ml", tolparconv=1e-10, tolparinv=1e-10, maxiter=1000,geterrors=F) samout4<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(list("Const",c(0),K)), lambda=0, W=diag(ntrain),R=list(diag(ntrain)),Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F), mmalg="dmm_reml", tolparconv=1e-10, tolparinv=1e-10, maxiter=1000,geterrors=F) samout5<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain), R=list(diag(ntrain)),Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="mm_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout6<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain),R=list(diag(ntrain)), Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="dermm_reml1", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout7<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain),R=list(diag(ntrain)), Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="dermm_reml2", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout8<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain),R=list(diag(ntrain)), Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="mmmk_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout9<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain),R=list(diag(ntrain)), Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="emmmk_reml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout10<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain),R=list(diag(ntrain)), Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="emmmk_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout11<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain),R=list(diag(ntrain)), Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F), mmalg="emmmv_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) ###########2 R<-diag(ntrain) diag(R)<-diag(R)+.01*rnorm(nrow(R)) samout12<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain),R=list(R), Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="mmmk_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout13<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain),R=list(R), Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="mm_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout14<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain),R=list(R), Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="mmmv_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout15<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain),R=list(R), Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="mmmkmv_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout16<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain),R=list(R),Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F), mmalg="dermm_reml1", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout17<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(K), lambda=0, W=diag(ntrain),R=list(R), Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="dermm_reml2", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout18<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(list("Const",c(0),K)), lambda=0, W=diag(ntrain), R=list(R),Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="dmm_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout19<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z), Klist=list(list("Const",c(0),K)), lambda=0, W=diag(ntrain), R=list(R),Siglist=list(), corfunc=c(F,F), corfuncfixed=c(F,F), sigfunc=c(F),mmalg="dmm_reml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) samout1$Vu samout2$Vu samout3$Vu samout4$Vu samout5$Vu samout6$Vu samout7$Vu samout8$Vu samout9$Vu samout10$Vu samout11$Vgt samout12$Vu samout13$Vu samout14$sigmahatlist[[1]] samout15$sigmahatlist[[1]] samout16$Vu samout17$Vu samout18$Vu samout19$Vu ############################### library(SAMM) n=100 nsample=80 rhotrans=5 ar1cov_cppforR(c(rhotrans),matrix(5)) rho=(2/pi)*atan(rhotrans) rho tan((pi/2)*(rho)) M1<-matrix(rbinom(n*300, 2, .2)-1, nrow=n) K1<-relmatcov_cppforR(c(.01), M1) M2<-matrix(rbinom(n*300, 2, .2)-1, nrow=n) K2<-relmatcov_cppforR(c(0.03), M2) W=(diag(5)[sample(1:5,n, replace=TRUE),]) covY<-3*K1+5*K2+10*(W%*%ar1cov_cppforR(c(rhotrans),matrix(5))%*%t(W)) K1[1:5,1:5] dim(W) dim(ar1cov_cppforR(c(6),matrix(5))) Y<-10+crossprod(chol(covY),rnorm(n)) #training set Trainset<-sample(1:n,nsample) ytrain=Y[Trainset] Xtrain=matrix(rep(1, n)[Trainset], ncol=1) Ztrain=diag(n)[Trainset,] Wtrain=W[Trainset,] samout27<-SAMM(Y=matrix(ytrain,ncol=1),X=Xtrain, Zlist=list(Ztrain, Ztrain), Klist=list(K1,K2), lambda=0, W=Wtrain,R=list(list("ar1",c(0),matrix(5,1,1))), Siglist=list("","",""), corfunc=c(F,F,T), corfuncfixed=c(F,F,F), sigfunc=c(F,F,F),mmalg="mmmk_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) str(samout27) samout28<-SAMM(Y=matrix(ytrain,ncol=1),X=Xtrain, Zlist=list(Ztrain, Ztrain), Klist=list(K1,K2), lambda=0, W=Wtrain,R=list(list("ar1",c(0),matrix(5,1,1))),Siglist=list("","",""), corfunc=c(F,F,T), corfuncfixed=c(F,F,F), sigfunc=c(F,F,F), mmalg="mmmkmv_ml", tolparconv=1e-10, tolparinv=1e-10, maxiter=1000,geterrors=F) str(samout28) samout29<-SAMM(Y=matrix(ytrain,ncol=1),X=Xtrain, Zlist=list(Ztrain, Ztrain), Klist=list(K1,K2), lambda=0, W=Wtrain,R=list("ar1",c(0),matrix(5,1,1)), Siglist=list("","",""), corfunc=c(F,F,T), corfuncfixed=c(F,F,F), sigfunc=c(F,F,F), mmalg="dermm_reml1", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) str(samout29) samout30<-SAMM(Y=matrix(ytrain,ncol=1),X=Xtrain, Zlist=list(Ztrain, Ztrain), Klist=list(K1,K2), lambda=0, W=Wtrain,R=list("ar1",c(0),matrix(5,1,1)), Siglist=list("","",""), corfunc=c(F,F,T), corfuncfixed=c(F,F,F), sigfunc=c(F,F,F),mmalg="dermm_reml2", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) str(samout30) ########################### ###Data: n=100 nsample=90 M1<-matrix(rbinom(n*300, 2, .2)-1, nrow=n) K1<-relmatcov_cppforR(c(.1), M1) Sigma1=ar1cov_cppforR(c(5),matrix(3)) Sigma2=ar1cov_cppforR(c(-2),matrix(3)) W=(diag(5)[sample(1:5,n, replace=T),]) K2=(W%*%ar1cov_cppforR(c(-3),matrix(5))%*%t(W)) covY<-5*kronecker(Sigma1,K1)+10*kronecker(Sigma2,K2) Y<-10+crossprod(chol(covY),rnorm(n*3)) Y<-matrix(Y, ncol=3) samout31<-SAMM(Y=Y,X=matrix(1,n,1), Zlist=list(diag(n)), Klist=list(K1), lambda=0, W=W,R=list(list("ar1",c(0),matrix(5,1,1))), Siglist=list("",""), corfunc=c(F,T), corfuncfixed=c(F,F), sigfunc=c(F,F), mmalg="mmmkmv_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) str(samout31) samout32<-SAMM(Y=Y,X=matrix(1,n,1), Zlist=list(diag(n)), Klist=list(K1), lambda=0, W=W,R=list(list("ar1",c(0),matrix(5,1,1))), Siglist=list("",list("diag",c(0,0,0),matrix(3,1,1))), corfunc=c(F,T), corfuncfixed=c(F,F), sigfunc=c(F,T),mmalg="mmmkmv_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) str(samout32) samout33<-SAMM(Y=Y,X=matrix(1,n,1), Zlist=list(diag(n)), Klist=list(K1), lambda=0, W=W,R=list(list("ar1",c(0),matrix(5,1,1))), Siglist=list("",list("Ident",c(0),matrix(3,1,1))), corfunc=c(F,T), corfuncfixed=c(F,F), sigfunc=c(F,T),mmalg="mmmkmv_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) str(samout33) ########################### n=100 M1<-matrix(rbinom(n*150, 2, .2)-1, nrow=n) K1<-relmatcov_cppforR(c(0), M1) M2<-matrix(rbinom(n*150, 2, .2)-1, nrow=n) K2<-relmatcov_cppforR(c(0), M2) M3<-matrix(rbinom(n*100, 2, .2)-1, nrow=n) K3<-relmatcov_cppforR(c(0), M3) M4<-matrix(rbinom(n*100, 2, .2)-1, nrow=n) K4<-relmatcov_cppforR(c(0), M4) covY<-2*K1+3*K2+1*K3+2*K4+3*diag(n) Y<-10+crossprod(chol(covY),rnorm(n)) #training set Trainset<-sample(1:n,80) y=Y[Trainset] X=matrix(rep(1, n)[Trainset], ncol=1) Z=diag(n)[Trainset,] X=X y=y samout35<-SAMM(Y=matrix(y,ncol=1),X=X,Zlist=list(Z,Z,Z,Z), lambda=0,Klist=list(K1,K2,K3,K4), W=Z,R=list(diag(n)), Siglist=list("","","","",""), corfunc=c(F,F,F,F,F), corfuncfixed=c(T,T,T,T,T),sigfunc=c(F,F,F,F,F), mmalg="mmmk_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) str(samout35) samout36<-SAMM(Y=matrix(y,ncol=1),X=X,Zlist=list(Z,Z,Z,Z), lambda=0.99999,Klist=list(K1,K2,K3,K4), W=Z, R=list(diag(n)),Siglist=list("","","","",""), corfunc=c(F,F,F,F,F), corfuncfixed=c(T,T,T,T,T), sigfunc=c(F,F,F,F,F),mmalg="mmmk_ml", tolparconv=1e-10, tolparinv=1e-10,maxiter=1000,geterrors=F) str(samout36) outmat<-c() for (lambda in seq(0,.999999, length=30)){ samout37<-SAMM(Y=matrix(y,ncol=1),X=X, Zlist=list(Z,Z,Z,Z),lambda=lambda,Klist=list(K1,K2,K3,K4), W=Z,R=list(diag(n)),Siglist=list("","","","",""), corfunc=c(F,F,F,F,F), corfuncfixed=c(T,T,T,T,T), sigfunc=c(F,F,F,F,F),mmalg="mmmk_ml", tolparconv=1e-10, tolparinv=1e-10, maxiter=1000,geterrors=F) outmat<-cbind(outmat,c(samout37$Vu*samout37$weights, samout37$Ve)) } str(samout37) colnames(outmat)<-seq(0,.999999, length=30) maxmat<-max(c(outmat)) minmat<-min(c(outmat)) plot(seq(0,.999999, length=30),outmat[1,], ylim=c(minmat-1, maxmat+1), col=2, type="b") for (i in 2:5){ par(new=T) plot(seq(0,.999999, length=30), outmat[i,], axes=F, ylim=c(minmat-1, maxmat+1), col=i+1, type="b", xlab="", ylab="") } ## End(Not run) 

SAMM documentation built on May 30, 2017, 1:06 a.m.