Hmisc (version 5.3-1)

wtd.stats: Weighted Statistical Estimates

Description

These functions compute various weighted versions of standard estimators. In most cases the weights vector is a vector the same length of x, containing frequency counts that in effect expand x by these counts. weights can also be sampling weights, in which setting normwt to TRUE will often be appropriate. This results in making weights sum to the length of the non-missing elements in x. normwt=TRUE thus reflects the fact that the true sample size is the length of the x vector and not the sum of the original values of weights (which would be appropriate had normwt=FALSE). When weights is all ones, the estimates are all identical to unweighted estimates (unless one of the non-default quantile estimation options is specified to wtd.quantile). When missing data have already been deleted for, x, weights, and (in the case of wtd.loess.noiter) y, specifying na.rm=FALSE will save computation time. Omitting the weights argument or specifying NULL or a zero-length vector will result in the usual unweighted estimates.

wtd.mean, wtd.var, and wtd.quantile compute weighted means, variances, and quantiles, respectively. wtd.Ecdf computes a weighted empirical distribution function. wtd.table computes a weighted frequency table (although only one stratification variable is supported at present). wtd.rank computes weighted ranks, using mid--ranks for ties. This can be used to obtain Wilcoxon tests and rank correlation coefficients. wtd.loess.noiter is a weighted version of loess.smooth when no iterations for outlier rejection are desired. This results in especially good smoothing when y is binary. wtd.quantile removes any observations with zero weight at the beginning. Previously, these were changing the quantile estimates.

num.denom.setup is a utility function that allows one to deal with observations containing numbers of events and numbers of trials, by outputting two observations when the number of events and non-events (trials - events) exceed zero. A vector of subscripts is generated that will do the proper duplications of observations, and a new binary variable y is created along with usual cell frequencies (weights) for each of the y=0, y=1 cells per observation.

Usage

wtd.mean(x, weights=NULL, normwt="ignored", na.rm=TRUE)
wtd.var(x, weights=NULL, normwt=FALSE, na.rm=TRUE,
        method=c('unbiased', 'ML'))
wtd.quantile(x, weights=NULL, probs=c(0, .25, .5, .75, 1), 
             type=c('quantile','(i-1)/(n-1)','i/(n+1)','i/n'), 
             normwt=FALSE, na.rm=TRUE)
wtd.Ecdf(x, weights=NULL, 
         type=c('i/n','(i-1)/(n-1)','i/(n+1)'), 
         normwt=FALSE, na.rm=TRUE)
wtd.table(x, weights=NULL, type=c('list','table'), 
          normwt=FALSE, na.rm=TRUE)
wtd.rank(x, weights=NULL, normwt=FALSE, na.rm=TRUE)
wtd.loess.noiter(x, y, weights=rep(1,n),
                 span=2/3, degree=1, cell=.13333, 
                 type=c('all','ordered all','evaluate'), 
                 evaluation=100, na.rm=TRUE)
num.denom.setup(num, denom)

Value

wtd.mean and wtd.var return scalars. wtd.quantile returns a vector the same length as probs. wtd.Ecdf returns a list whose elements x and Ecdf correspond to unique sorted values of x. If the first CDF estimate is greater than zero, a point (min(x),0) is placed at the beginning of the estimates. See above for wtd.table. wtd.rank returns a vector the same length as x (after removal of NAs, depending on na.rm). See above for wtd.loess.noiter.

Details

The functions correctly combine weights of observations having duplicate values of x before computing estimates.

When normwt=FALSE the weighted variance will not equal the unweighted variance even if the weights are identical. That is because of the subtraction of 1 from the sum of the weights in the denominator of the variance formula. If you want the weighted variance to equal the unweighted variance when weights do not vary, use normwt=TRUE. The articles by Gatz and Smith discuss alternative approaches, to arrive at estimators of the standard error of a weighted mean.

wtd.rank does not handle NAs as elegantly as rank if weights is specified.

References

Research Triangle Institute (1995): SUDAAN User's Manual, Release 6.40, pp. 8-16 to 8-17.

Gatz DF, Smith L (1995): The standard error of a weighted mean concentration--I. Bootstrapping vs other methods. Atmospheric Env 11:1185-1193.

Gatz DF, Smith L (1995): The standard error of a weighted mean concentration--II. Estimating confidence intervals. Atmospheric Env 29:1195-1200.

https://en.wikipedia.org/wiki/Weighted_arithmetic_mean

See Also

mean, var, quantile, table, rank, loess.smooth, lowess, plsmo, Ecdf, somers2, describe

Examples

set.seed(1)
x <- runif(500)
wts <- sample(1:6, 500, TRUE)
std.dev <- sqrt(wtd.var(x, wts))
wtd.quantile(x, wts)
death <- sample(0:1, 500, TRUE)
plot(wtd.loess.noiter(x, death, wts, type='evaluate'))
describe(~x, weights=wts)
# describe uses wtd.mean, wtd.quantile, wtd.table
xg <- cut2(x,g=4)
table(xg)
wtd.table(xg, wts, type='table')

# Here is a method for getting stratified weighted means
y <- runif(500)
g <- function(y) wtd.mean(y[,1],y[,2])
summarize(cbind(y, wts), llist(xg), g, stat.name='y')

# Empirically determine how methods used by wtd.quantile match with
# methods used by quantile, when all weights are unity
set.seed(1)
u <-  eval(formals(wtd.quantile)$type)
v <- as.character(1:9)
r <- matrix(0, nrow=length(u), ncol=9, dimnames=list(u,v))

for(n in c(8, 13, 22, 29))
  {
    x <- rnorm(n)
    for(i in 1:5) {
      probs <- sort( runif(9))
      for(wtype in u) {
        wq <- wtd.quantile(x, type=wtype, weights=rep(1,length(x)), probs=probs)
        for(qtype in 1:9) {
          rq <- quantile(x, type=qtype, probs=probs)
          r[wtype, qtype] <- max(r[wtype,qtype], max(abs(wq-rq)))
        }
      }
    }
  }

r

# Restructure data to generate a dichotomous response variable
# from records containing numbers of events and numbers of trials
num   <- c(10,NA,20,0,15)   # data are 10/12 NA/999 20/20 0/25 15/35
denom <- c(12,999,20,25,35)
w     <- num.denom.setup(num, denom)
w
# attach(my.data.frame[w$subs,])