Log-ratio DA algorithm

Description

This function implements a simulation-based Data Augmentation (DA) algorithm to impute left-censored values (e.g. values below detection limit, rounded zeros) via coordinates representation of compositional data sets which incorporate the information of the relative covariance structure. Multiple imputation estimates can be also obtained from the output.

Usage

lrDA(X, label = NULL, dl = NULL,
        ini.cov=c("lrEM", "complete.obs", "multRepl"),
        delta = 0.65, n.iters = 1000, m = 1, store.mi = FALSE)

Arguments

Details

After convergence of the Markov chain Monte Carlo (MCMC) iterative process to its steady state, this function imputes left-censored compositional parts by simulated values from their posterior predictive distributions through coordinates representation, given the information from the observed data and the censoring thresholds. It allows for either single (vector form) or multiple (matrix form, same size as X) limits of detection by component. Any threshold value can be set for non-censored elements (e.g. use 0 if no threshold for a particular column or element of the data matrix).

It produces imputed data sets on the same scale as the input data set. If X is not closed to a constant sum, then the results are adjusted to provide a compositionally equivalent data set, expressed in the original scale, which leaves the absolute values of the observed components unaltered.

The common conjugate normal inverted-Wishart distribution with non-informative prior has been assumed for the model parameters in the coordinates space. Under this setting, convergence is expected to be fast (n.iters set to 1000 by default). Besides, considering EM parameter estimates as initial point for the DA algorithm (ini.cov="lrEM") assures faster convergence by starting near the centre of the posterior distribution. Note that the procedure is based on the oblique additive log-ratio (alr) transformation to simplify calculations and alleviates computational burden.

By setting m greater than 1, the procedure also allows for multiple imputations of the censored values drawn at regular intervals after convergence. In this case, in addition to the burn-in period for convergence, n.iters determines the gap, large enough to prevent from correlated values, between successive imputations. The total number of iterations is then n.iters*m. By default, a single imputed data set results from averaging the m imputations in the space of coordinates. If store.mi=TRUE, a list with m imputed data sets is generated instead.

In the case of censoring patterns involving samples containing only one observed component, these are imputed by multiplicative simple replacement (multRepl) and a warning message identifying them is printed.

Value

A data.frame object containing the imputed compositional data set or a list of imputed data sets if multiple imputation is carried out (m>1) and store.mi=TRUE.

References

Palarea-Albaladejo J, Martin-Fernandez JA, Olea, RA. A bootstrap estimation scheme for chemical compositional data with nondetects. Journal of Chemometrics 2014; 28: 585-599.

See Also

zPatterns, lrEM, multRepl, multLN, multKM, cmultRepl

Examples

# Data set closed to 100 (percentages, common dl = 1%)
X <- matrix(c(26.91,8.08,12.59,31.58,6.45,14.39,
              39.73,26.20,0.00,15.22,6.80,12.05,
              10.76,31.36,7.10,12.74,31.34,6.70,
              10.85,46.40,31.89,10.86,0.00,0.00,
              7.57,11.35,30.24,6.39,13.65,30.80,
              38.09,7.62,23.68,9.70,20.91,0.00,
              27.67,7.15,13.05,32.04,6.54,13.55,
              44.41,15.04,7.95,0.00,10.82,21.78,
              11.50,30.33,6.85,13.92,30.82,6.58,
              19.04,42.59,0.00,38.37,0.00,0.00),byrow=TRUE,ncol=6)

# Imputation by single simulated values
X_lrDA <- lrDA(X,label=0,dl=rep(1,6),ini.cov="multRepl",n.iters=150)

# Imputation by multiple imputation (m = 5, one imputation every 150 iterations)
X_milrDA <- lrDA(X,label=0,dl=rep(1,6),ini.cov="multRepl",m=5,n.iters=150)

# Multiple limits of detection by component
mdl <- matrix(0,ncol=6,nrow=10)
mdl[2,] <- rep(1,6)
mdl[4,] <- rep(0.75,6)
mdl[6,] <- rep(0.5,6)
mdl[8,] <- rep(0.5,6)
mdl[10,] <- c(0,0,1,0,0.8,0.7)

X_lrDA2 <- lrDA(X,label=0,dl=mdl,ini.cov="multRepl",n.iters=150)

# Non-closed compositional data set
data(LPdata) # data (ppm/micrograms per gram)
dl <- c(2,1,0,0,2,0,6,1,0.6,1,1,0,0,632,10) # limits of detection (0 for no limit)
LPdata2 <- subset(LPdata,select=-c(Cu,Ni,La))  # select a subset for illustration purposes
dl2 <- dl[-c(5,7,10)]

## Not run:  # May take a little while
LPdata_lrDA <- lrDA(LPdata2,label=0,dl=dl2)
## End(Not run)

Results