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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.
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)a numeric vector (may be a character or category or factor vector
for wtd.table)
vector of numerator frequencies
vector of denominators (numbers of trials)
a numeric vector of weights
specify normwt=TRUE to make weights sum to
length(x) after deletion of NAs. If weights are
frequency weights, then normwt should be FALSE, and if
weights are normalization (aka reliability) weights, then
normwt should be TRUE. In the case of the former, no check
is made that weights are valid frequencies.
set to FALSE to suppress checking for NAs
determines the estimator type; if 'unbiased' (the
default) then the usual unbiased estimate (using Bessel's correction)
is returned, if 'ML' then it is the maximum likelihood estimate
for a Gaussian distribution. In the case of the latter, the
normwt argument has no effect. Uses stats:cov.wt for
both methods.
a vector of quantiles to compute. Default is 0 (min), .25, .5, .75, 1 (max).
For wtd.quantile, type defaults to quantile to use the same
interpolated order statistic method as quantile. Set type to
"(i-1)/(n-1)","i/(n+1)", or "i/n" to use the inverse of the
empirical distribution function, using, respectively, (wt - 1)/T,
wt/(T+1), or wt/T, where wt is the cumulative weight and T is the
total weight (usually total sample size). These three values of
type are the possibilities for wtd.Ecdf. For wtd.table the
default type is "list", meaning that the function is to return a
list containing two vectors: x is the sorted unique values of x
and sum.of.weights is the sum of weights for that x. This is the
default so that you don't have to convert the names attribute of the
result that can be obtained with type="table" to a numeric variable
when x was originally numeric. type="table" for wtd.table
results in an object that is the same structure as those returned from
table. For wtd.loess.noiter the default type is "all",
indicating that the function is to return a list containing all the
original values of x (including duplicates and without sorting) and
the smoothed y values corresponding to them. Set type="ordered
all" to sort by x, and type="evaluate" to evaluate the smooth
only at evaluation equally spaced points between the observed limits
of x.
a numeric vector the same length as x
see loess.smooth. The default is linear (degree=1) and 100 points
to evaluation (if type="evaluate").
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.
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.
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
mean, var, quantile, table, rank, loess.smooth, lowess,
plsmo, Ecdf, somers2, describe
# NOT RUN {
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,])
# }
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