The norm Package. November 15, Title Analysis of multivariate normal datasets with missing values

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1 The norm Package November 15, 2003 Verion Date 2002/05/06 Title Analyi of multivariate normal dataet with miing value Author Ported to R by Alvaro A. Novo <alvaro@novo-online.net>. Original by Joeph L. Schafer <jl@tat.pu.edu>. Maintainer Alvaro A. Novo <alvaro@novo-online.net> Analyi of multivariate normal dataet with miing value Licene The oftware may be ditributed free of charge and ued by anyone if credit i given. It ha been teted URL jl/mioftwa.html#aut R topic documented:.code.to.na da.norm em.norm getparam.norm imp.norm loglik.norm logpot.norm makeparam.norm mda.norm mdata mi.inference na.to.nglcode ninvwih prelim.norm rngeed Index 16 1

2 2 da.norm.code.to.na Change miing value code to NA Change miing value code to NA. It called from prelim.norm..code.to.na(x, mvcode) x mvcode data object. internal input of prelim.norm. Initial data object with miing value code changed to NA. prelim.norm da.norm Data augmentation for incomplete multivariate normal data Data augmentation under a normal-inverted Wihart prior. If no prior i pecified by the uer, the uual noninformative prior for the multivariate normal ditribution i ued. Thi function imulate one or more iteration of a ingle Markov chain. Each iteration conit of a random imputation of the miing data given the oberved data and the current parameter value (I-tep), followed by a draw from the poterior ditribution of the parameter given the oberved data and the imputed data (P-tep). da.norm(, tart, prior, tep=1, howit=false, return.ymi=false) tart ummary lit of an incomplete normal data matrix produced by the function tarting value of the parameter. Thi i a parameter vector in packed torage, uch a one created by the function makeparam.norm. One obviou choice for a tarting value i an ML etimate or poterior mode produced by em.norm.

3 da.norm 3 prior tep howit return.ymi optional prior ditribution. Thi i a lit containing the hyperparameter of a normal-inverted Wihart ditribution. In order, the element of the lit are: tau (a calar), m (a calar), mu0 (a vector of length ncol(x), where x i the original matrix of incomplete data), and lambdainv (a matrix of dimenion c(ncol(x),ncol(x))). The element of mu0 and lambdainv apply to the data after tranformation, i.e. after the column have been centered and caled to have mean zero and variance one. If no prior i upplied, the default i the uual noninformative prior for a multivariate normal model: tau=0, m=-1, mu0=arbitrary, and lambdainv = matrix of zero. number of data augmentation iteration to be imulated. if TRUE, report the iteration o the uer can monitor the progre of the algorithm. if TRUE, return the output of the lat I-tep (imputed value of miing data) in addition to the output of the lat P-tep. Thee imputed value are ueful for forming Rao-Blackwellized etimate of poterior ummarie. if return.ymi=false, return a parameter vector, the reult of the lat P-tep. If the value of tep wa large enough to guarantee approximate tationarity, then thi parameter can be regarded a a proper draw from the oberved-data poterior, independent of tart. If return.ymi=true, then thi function return a lit of the following two component: parameter a parameter vector, the reult of the lat P-tep ymi a vector of miing value, the reult of the lat I-tep. The length of thi vector i um(i.na(x)), where x i the original data matrix. The torage order i the ame a that of x[i.na(x)]. WARNING Before thi function may be ued, the random number generator eed mut be initialized with rngeed at leat once in the current S eion. See Chapter 5 of Schafer (1996). rngeed, em.norm, prelim.norm, and getparam.norm. <- prelim.norm(mdata) thetahat <- em.norm() #find the MLE for a tarting value rngeed( ) #et random number generator eed theta <- da.norm(,thetahat,tep=20,howit=true) # take 20 tep getparam.norm(,theta) # look at reult

4 4 em.norm em.norm EM algorithm for incomplete normal data Perform maximum-likelihood etimation on the matrix of incomplete data uing the EM algorithm. Can alo be ued to find a poterior mode under a normal-inverted Wihart prior upplied by the uer. em.norm(, tart, howit=true, maxit=1000, criterion=0.0001, prior) tart howit maxit criterion prior ummary lit of an incomplete normal data matrix produced by the function optional tarting value of the parameter. Thi i a parameter vector in packed torage, uch a one created by the function makeparam.norm. If no tarting value i upplied, em.norm chooe it own tarting value. if TRUE, report the iteration of EM o the uer can monitor the progre of the algorithm. maximum number of iteration performed. The algorithm will top if the parameter till ha not converged after thi many iteration. convergence criterion. The algorithm top when the maximum relative difference in all of the etimated mean, variance, or covariance from one iteration to the next i le than or equal to thi value. optional prior ditribution. Thi i a lit containing the hyperparameter of a normal-inverted Wihart ditribution. In order, the element of the lit are: tau (a calar), m (a calar), mu0 (a vector of length ncol(x)), and lambdainv (a matrix of dimenion c(ncol(x),ncol(x))). The element of mu0 an lambdainv apply to the data after tranformation, i.e. after the column have been centered and caled to have mean zero and variance one. If no prior i upplied, the default i a uniform prior, which reult in maximum-likelihood etimation. Detail The default tarting value take all mean on the tranformed cale to be equal to zero, and covariance matrix on the tranformed cale equal to the identity. All important computation are carried out in double preciion, uing the weep operator. a vector repreenting the maximum-likelihood etimate of the normal parameter. Thi vector contain mean, variance, and covariance on the tranformed cale in packed torage. The parameter can be tranformed back to the original cale and put into a more undertandable format by the function getparam.norm.

5 getparam.norm 5 See Section 5.3 of Schafer (1994). prelim.norm, getparam.norm, and makeparam.norm. <- prelim.norm(mdata) #do preliminary manipulation thetahat <- em.norm() #compute mle getparam.norm(,thetahat,corr=true)$r #look at etimated correlation getparam.norm Extract normal parameter from packed torage Take a parameter vector, uch a one produced by em.norm or da.norm, and return a lit of parameter on the original cale. getparam.norm(, theta, corr=false) theta corr ummary lit of an incomplete normal data matrix created by the function vector of normal parameter expreed on tranformed cale in packed torage, uch a one produced by the function em.norm. if TRUE, compute mean, tandard deviation, and a correlation matrix. If FALSE, compute mean and a covariance matrix. if corr=false, a lit containing the component mu and igma; if corr=true, a lit containing the component mu, dv, and r. The component are: mu igma dv r vector of mean. Element are in the ame order and on the ame cale a the column of the original data matrix, and with name correponding to the column name of the original data matrix. matrix of variance and covariance. vector of tandard deviation. matrix of correlation. prelim.norm and makeparam.norm.

6 6 imp.norm <- prelim.norm(mdata) #do preliminary manipulation thetahat <- em.norm() #compute MLE getparam.norm(,thetahat,corr=true)$r #look at etimated correlation imp.norm Impute miing multivariate normal data Draw miing element of a data matrix under the multivariate normal model and a uerupplied parameter imp.norm(, theta, x) theta x ummary lit of an incomplete normal data matrix x created by the function value of the normal parameter under which the miing data are to be randomly imputed. Thi i a parameter vector in packed torage, uch a one created by em.norm or da.norm. the original data matrix ued to create the ummary lit. If thi argument i not upplied, then the data matrix returned by thi function may diagree lightly with the oberved value in x due to rounding error. Detail Thi function imply perform one I-tep of data augmentation. a matrix of the ame form a x, but with all miing value filled in with imulated value drawn from their predictive ditribution given the oberved data and the pecified parameter. WARNING Before thi function may be ued, the random number generator eed mut be initialized with rngeed at leat once in the current S eion. See Section of Schafer (1996). prelim.norm, makeparam.norm, and rngeed.

7 loglik.norm 7 <- prelim.norm(mdata) #do preliminary manipulation thetahat <- em.norm() #find the mle rngeed( ) #et random number generator eed ximp <- imp.norm(,thetahat,mdata) #impute miing data under the MLE loglik.norm Oberved-data loglikelihood for normal data Evaluate the oberved-data loglikelihood function at a uer-upplied value of the parameter. Thi function i ueful for monitoring the progre of EM and data augmentation. loglik.norm(, theta) theta ummary lit of an incomplete normal data matrix created by the function vector of normal parameter expreed on tranformed cale in packed torage, uch a one produced by the function em.norm. value of the oberved-data loglikelihood See Section of Schafer (1996) prelim.norm and logpot.norm <- prelim.norm(mdata) thetahat <- em.norm() loglik.norm(,thetahat) #do preliminary manipulation #compute MLE #loglikelihood at the MLE

8 8 logpot.norm logpot.norm Oberved-data log-poterior for normal data Evaluate the log of the oberved-data poterior denity at a uer-upplied value of the parameter. Aume a normal-inverted Wihart prior. Thi function i ueful for monitoring the progre of EM and data augmentation. logpot.norm(, theta, prior) theta prior ummary lit of an incomplete normal data matrix created by the function vector of normal parameter expreed on tranformed cale in packed torage, uch a one produced by the function em.norm. optional prior ditribution. Thi i a lit containing the hyperparameter of a normal-inverted Wihart ditribution. In order, the element of the lit are: tau (a calar), m (a calar), mu0 (a vector of length ncol(x), where x i the original matrix of incomplete data), and lambdainv (a matrix of dimenion c(ncol(x),ncol(x))). The element of mu0 and lambdainv apply to the data after tranformation, i.e. after the column have been centered and caled to have mean zero and variance one. If no prior i upplied, the default i the uual noninformative prior for a multivariate normal model: tau=0, m=-1, mu0=arbitrary, and lambdainv = matrix of zero. value of the oberved-data log-poterior denity See Section of Schafer (1996) prelim.norm and loglik.norm <- prelim.norm(mdata) #do preliminary manipulation prior <- lit(0,.5,rep(0,ncol(mdata)),.5*diag(rep(1,ncol(mdata)))) #ridge prior with.5 df thetahat <- em.norm(,prior=prior) #compute poterior mode logpot.norm(,thetahat,prior) #log-poterior at mode

9 makeparam.norm 9 makeparam.norm Convert normal parameter to packed torage Doe the oppoite of getparam.norm. Convert a lit of uer-pecified parameter to a parameter vector uitable for input to function uch a da.norm and em.norm. makeparam.norm(, thetalit) thetalit ummary lit of an incomplete normal data matrix created by the function lit of normal parameter of the ame form a one produced by getparam.norm. If the lit ha two component, the firt mut be the vector of mean and the econd mut be the covariance matrix, where mean and covariance are expreed on the cale of the original data. If the lit ha three component, the firt mut be the vector of mean, the econd mut be the vector of tandard deviation, and the third mut be the correlation matrix. normal parameter in packed torage, uitable for ue a a tarting value for em.norm, mda.norm, or mdamet.norm. prelim.norm and getparam.norm. <- prelim.norm(mdata) #do preliminary manipulation thetahat <- em.norm() #compute mle thetahat <- getparam.norm(,thetahat,corr=true) #extract parameter thetahat$r #look at mle correlation thetahat$r[1,2] <-.5 #tweak a parameter thetahat <- makeparam.norm(,thetahat) #convert to packed torage thetahat <- em.norm(,thetahat) #run EM again from new tarting value

10 10 mda.norm mda.norm Monotone data augmentation for incomplete multivariate normal data Monotone data augmentation under the uual noninformative prior, a decribed in Chapter 6 of Schafer (1996). Thi function imulate one or more iteration of a ingle Markov chain. One iteration conit of a random imputation of the miing data given the oberved data and the current parameter value (I-tep), followed by a draw from the poterior ditribution of the parameter given the oberved data and the imputed data (P-tep). The I-tep impute only enough data to complete a monotone pattern, which typically make thi algorithm converge more quickly than da.norm, particularly when the oberved data are nearly monotone. The order of the variable in the original data matrix determine the monotone pattern to be completed. For fat convergence, it help to order the variable according to their rate of miingne, with the mot oberved (leat miing) variable on the left and the leat oberved variable on the right. mda.norm(, theta, tep=1, howit=false) theta tep howit ummary lit of an incomplete normal data matrix produced by the function tarting value of the parameter. Thi i a parameter vector in packed torage, uch a one created by the function makeparam.norm. One obviou choice for a tarting value i an ML etimate or poterior mode produced by em.norm. number of monotone data augmentation iteration to be imulated. if TRUE, report the iteration o the uer can monitor the progre of the algorithm. Return a parameter vector, the reult of the lat P-tep. If the value of tep wa large enough to guarantee approximate tationarity, then thi parameter can be regarded a a proper draw from the oberved-data poterior, independent of tart. WARNING Before thi function may be ued, the random number generator eed mut be initialized with rngeed at leat once in the current S eion. Chapter 6 of Schafer (1996). rngeed, em.norm, prelim.norm, and getparam.norm.

11 mdata 11 <- prelim.norm(mdata) thetahat <- em.norm() #find the MLE for a tarting value rngeed( ) #et random number generator eed theta <- mda.norm(,thetahat,tep=20,howit=true) # take 20 tep getparam.norm(,theta) # look at reult mdata Dataet with miing value to illutrate ue of package norm Houehold urvey with miing value. See Schafer (1997). Schafer, J.L.(1997) Analyi of Incomplete Multivariate Data, Chapman & Hall, London. ISBN: mi.inference Multiple imputation inference Combine etimate and tandard error from m complete-data analye performed on m imputed dataet to produce a ingle inference. Ue the technique decribed by Rubin (1987) for multiple imputation inference for a calar etimand. mi.inference(et, td.err, confidence=0.95) et td.err confidence a lit of m (at leat 2) vector repreenting etimate (e.g., vector of etimated regreion coefficient) from complete-data analye performed on m imputed dataet. a lit of m vector containing tandard error from the complete-data analye correponding to the etimate in et. deired coverage of interval etimate.

12 12.na.to.nglcode a lit with the following component, each of which i a vector of the ame length a the component of et and td.err: et td.err df ignif lower upper r fminf the average of the complete-data etimate. tandard error incorporating both the between and the within-imputation uncertainty (the quare root of the total variance ). degree of freedom aociated with the t reference ditribution ued for interval etimate. P-value for the two-tailed hypothei tet that the etimated quantitie are equal to zero. lower limit of the (100*confidence)% interval etimate. upper limit of the (100*confidence)% interval etimate. etimated relative increae in variance due to nonrepone. etimated fraction of miing information. METHOD Ue the method decribed on pp of Rubin (1987) for combining the complete-data etimate from m imputed dataet for a calar etimand. Significance level and interval etimate are approximately valid for each one-dimenional etimand, not for all of them jointly. See Rubin (1987) or Schafer (1996), Chapter 4..na.to.nglcode Change NA to ingle preciion miing value code Change NA to ingle preciion miing value code It called internally by other function in the package, e.g., prelim.norm..code.to.na(x, mvcode) x mvcode data object. internal input of prelim.norm. Initial data object with miing value code preciion changed to inlge. prelim.norm

13 ninvwih 13 ninvwih Random normal-inverted Wihart variate Simulate a value from a normal-inverted Wihart ditribution. Thi function may be ueful for obtaining tarting value of the parameter of a multivariate normal ditribution for multiple chain of data augmentation. ninvwih(, param) param ummary lit of an incomplete normal data matrix produced by the function lit of parameter of a normal-inverted Wihart ditribution. In order, the element of the lit are: tau (a calar), m (a calar), mu0 (a vector of length ncol(x)), and lambdainv (a matrix of dimenion c(ncol(x),ncol(x))). When uing thi function to create tarting value for data augmentation, mu0 and lambdainv hould be choen in relation to the data matrix after the column have been centered and caled to have mean zero and variance one. a vector in packed torage repreenting the imulated normal-inverted Wihart variate. Thi vector ha the ame form a parameter vector produced by function uch a em.norm and da.norm, and may be ued directly a a tarting value for thee function. Thi vector can alo be put into a more undertandable format by getparam.norm. WARNING Before thi function may be ued, the random number generator eed mut be initialized with rngeed at leat once in the current S eion. See Section of Schafer (1996). rngeed, getparam.norm, em.norm and da.norm. <- prelim.norm(mdata) #do preliminary manipulation param <- lit(1,.5,rep(0,ncol(mdata)),.5*diag(rep(1,ncol(mdata)))) # give widely dipered value rngeed( ) tart <- ninvwih(,param) # draw a variate thetahat <- em.norm(,tart=tart) # run EM from thi tarting value

14 14 prelim.norm prelim.norm Preliminary manipulation for a matrix of incomplete continuou data. Sort row of x by miingne pattern, and center/cale column of x. Calculate variou bookkeeping quantitie needed for input to other function, uch a em.norm and da.norm. prelim.norm(x) x data matrix containing miing value. The row of x correpond to obervational unit, and the column to variable. Miing value are denoted by NA. a lit of thirteen component that ummarize variou feature of x after the data have been centered, caled, and orted by miingne pattern. Component that might be of interet to the uer include: nmi a vector of length ncol(x) containing the number of miing value for each variable in x. Thi vector ha name that correpond to the column name of x, if any. r matrix of repone indicator howing the miing data pattern in x. Dimenion i (S,p) where S i the number of ditinct miingne pattern in the row of x, and p i the number of column in x. Oberved value are indicated by 1 and miing value by 0. The row name give the number of obervation in each pattern, and the column name correpond to the column name of x. See Section of Schafer (1996). <- prelim.norm(mdata) #do preliminary manipulation $nmi[$co] #look at nmi $r #look at miing data pattern

15 rngeed 15 rngeed Initialize random number generator eed Initialize the eed value for the internal random-number generator ued in miing-data program rngeed(eed) eed a poitive number > = 1, preferably a large integer. NULL. NOTE The random number generator eed mut be et at leat once by thi function before the imulation or imputation function in thi package (da.norm, etc.) can be ued.

16 Index Topic NA.code.to.na, 1.na.to.nglcode, 11 Topic dataet mdata, 10 Topic ditribution da.norm, 2 rngeed, 14 Topic htet mi.inference, 10 Topic multivariate loglik.norm, 6 logpot.norm, 7 mda.norm, 9 ninvwih, 12 prelim.norm, 13 Topic regreion em.norm, 3 getparam.norm, 4 imp.norm, 5 makeparam.norm, 8.code.to.na, 1.na.to.nglcode, 11 da.norm, 2, 12 em.norm, 3, 3, 9, 12 getparam.norm, 3, 4, 4, 8, 9, 12 imp.norm, 5 loglik.norm, 6, 8 logpot.norm, 7, 7 makeparam.norm, 4 6, 8 mda.norm, 9 mdata, 10 mi.inference, 10 ninvwih, 12 prelim.norm, 1, 3 9, 11, 13 rngeed, 3, 6, 9, 12, 14 16