Cumulative Distribution Function  Estimation
Description
This function calculates an estimate of the cumulative distribution function (CDF) for the proportion (expressed as percent) and the total of a response variable, where the response variable may be defined for either a finite or an extensive resource. Optionally, for a finite resource, the sizeweighted CDF can be calculated. In addition the standard error of the estimated CDF and confidence bounds are calculated.
Usage
1 2 3 4 5  cdf.est(z, wgt, x=NULL, y=NULL, stratum=NULL, cluster=NULL, wgt1=NULL, x1=NULL,
y1=NULL, popsize=NULL, popcorrect=FALSE, pcfsize=NULL, N.cluster=NULL,
stage1size=NULL, support=NULL, sizeweight=FALSE, swgt=NULL, swgt1=NULL,
vartype="Local", conf=95, cdfval=NULL, pctval=c(5,10,25,50,75,90,95),
check.ind=TRUE, warn.ind=NULL, warn.df=NULL, warn.vec=NULL)

Arguments
z 
the response value for each site. 
wgt 
the final adjusted weight (inverse of the sample inclusion probability) for each site, which is either the weight for a singlestage sample or the stage two weight for a twostage sample. 
x 
xcoordinate for location for each site, which is either the xcoordinate for a singlestage sample or the stage two xcoordinate for a twostage sample. The default is NULL. 
y 
ycoordinate for location for each site, which is either the ycoordinate for a singlestage sample or the stage two ycoordinate for a twostage sample. The default is NULL. 
stratum 
the stratum for each site. The default is NULL. 
cluster 
the stage one sampling unit (primary sampling unit or cluster) code for each site. The default is NULL. 
wgt1 
the final adjusted stage one weight for each site. The default is NULL. 
x1 
the stage one xcoordinate for location for each site. The default is NULL. 
y1 
the stage one ycoordinate for location for each site. The default is NULL. 
popsize 
known size of the resource, which is used to perform ratio adjustment to estimators expressed using measurement units for the resource and to calculate strata proportions for calculating estimates for a stratified sample. For a finite resource, this argument is either the total number of sampling units or the known sum of sizeweights. For an extensive resource, this argument is the measure of the resource, i.e., either known total length for a linear resource or known total area for an areal resource. For a stratified sample this variable must be a vector containing a value for each stratum and must have the names attribute set to identify the stratum codes. The default is NULL. 
popcorrect 
a logical value that indicates whether finite or continuous population correction factors should be employed during variance estimation, where TRUE = use the correction factors and FALSE = do not use the correction factors. The default is FALSE. To employ the correction factor for a singlestage sample, values must be supplied for arguments pcfsize and support. To employ the correction factor for a twostage sample, values must be supplied for arguments N.cluster, stage1size, and support. 
pcfsize 
size of the resource, which is required for calculation of finite and continuous population correction factors for a singlestage sample. For a stratified sample this argument must be a vector containing a value for each stratum and must have the names attribute set to identify the stratum codes. The default is NULLL. 
N.cluster 
the number of stage one sampling units in the resource, which is required for calculation of finite and continuous population correction factors for a twostage sample. For a stratified sample this variable must be a vector containing a value for each stratum and must have the names attribute set to identify the stratum codes. The default is NULL. 
stage1size 
size of the stage one sampling units of a twostage sample, which is required for calculation of finite and continuous population correction factors for a twostage sample and must have the names attribute set to identify the stage one sampling unit codes. For a stratified sample, the names attribute must be set to identify both stratum codes and stage one sampling unit codes using a convention where the two codes are separated by the & symbol, e.g., "Stratum 1&Cluster 1". The default is NULL. 
support 
the support value for each site  the value one (1) for a site from a finite resource or the measure of the sampling unit associated with a site from an extensive resource, which is required for calculation of finite and continuous population correction factors. The default is NULL. 
sizeweight 
a logical value that indicates whether sizeweights should be used in the analysis, where TRUE = use the sizeweights and FALSE = do not use the sizeweights. The default is FALSE. 
swgt 
the sizeweight for each site, which is the stage two sizeweight for a twostage sample. The default is NULL. 
swgt1 
the stage one sizeweight for each site. The default is NULL. 
vartype 
the choice of variance estimator, where "Local" = local mean estimator and "SRS" = SRS estimator. The default is "Local". 
conf 
the confidence level. The default is 95%. 
cdfval 
the set of values at which the CDF is estimated. If a set of values is not provided, then the sorted set of unique values of the response variable is used. The default is NULL. 
pctval 
the set of values at which percentiles are estimated. The default set is: {5, 10, 25, 50, 75, 90, 95}. 
check.ind 
a logical value that indicates whether compatability checking of the input values is conducted, where TRUE = conduct compatibility checking and FALSE = do not conduct compatibility checking. The default is TRUE. 
warn.ind 
a logical value that indicates whether warning messages were generated, where TRUE = warning messages were generated and FALSE = warning messages were not generated. The default is NULL. 
warn.df 
a data frame for storing warning messages. The default is NULL. 
warn.vec 
a vector that contains names of the population type, the subpopulation, and an indicator. The default is NULL. 
Details
This function calculates an estimate of the cumulative distribution function (CDF) for the proportion (expressed as percent) and the total of a response variable, where the response variable may be defined for either a finite or an extensive resource. Optionally, for a finite resource, the sizeweighted CDF can be calculated. In addition the standard error of the estimated CDF and confidence bounds are calculated. The user can supply the set of values at which the CDF is estimated. For the CDF of a proportion, the HorvitzThompson ratio estimator, i.e., the ratio of two HorvitzThompson estimators, is used to calculate the CDF estimate. For the CDF of a total, the user can supply the known size of the resource or the known sum of the sizeweights of the resource, as appropriate. For the CDF of a total when either the size of the resource or the sum of the sizeweights of the resource is provided, the classic ratio estimator is used to calculate the CDF estimate, where that estimator is the product of the known value and the HorvitzThompson ratio estimator. For the CDF of a total when neither the size of the resource nor the sum of the sizeweights of the resource is provided, the HorvitzThompson estimator is used to calculate the CDF estimate. Variance estimates for the estimated CDF are calculated using either the local mean variance estimator or the simple random sampling (SRS) variance estimator. The choice of variance estimator is subject to user control. The local mean variance estimator requires the xcoordinate and the ycoordinate of each site. The SRS variance estimator uses the independent random sample approximation to calculate joint inclusion probabilities. Confidence bounds are calculated using a Normal distribution multiplier. In addition the function uses the estimated CDF to calculate percentile estimates. Estimated confidence bounds for the percentile estimates are calculated. The user can supply the set of values for which percentiles estimates are desired. Optionally, the user can use the default set of percentiles. The function can accommodate a stratified sample. For a stratified sample, separate estimates and standard errors are calculated for each stratum, which are used to produce estimates and standard errors for all strata combined. Strata that contain a single value are removed. For a stratified sample, when either the size of the resource or the sum of the size weights of the resource is provided for each stratum, those values are used as stratum weights for calculating the estimates and standard errors for all strata combined. For a stratified sample when neither the size of the resource nor the sum of the sizeweights of the resource is provided for each stratum, estimated values are used as stratum weights for calculating the estimates and standard errors for all strata combined. The function can accommodate singlestage and twostage samples for both stratified and unstratified sampling designs. Finite population and continuous population correction factors can be utilized in variance estimation. The function checks for compatibility of input values and removes missing values.
Value
If the function was called by the cont.analysis function, then value is a list containing the following components:

Results
 a list composed of two objects:
CDF
 a data frame that contains CDF estimates 
Pct
 a data frame that contains percentile estimates


warn.ind
 a logical value indicating whether warning messages were generated 
warn.df
 a data frame containing warning messages
If the function was called directly, then value is a list containing the following components:

CDF
 a data frame that contains CDF estimates 
Pct
 a data frame that contains percentile estimates
Author(s)
Tom Kincaid Kincaid.Tom@epa.gov
References
DiazRamos, S., D.L. Stevens, Jr., and A.R. Olsen. (1996). EMAP Statistical Methods Manual. EPA/620/R96/XXX. Corvallis, OR: U.S. Environmental Protection Agency, Office of Research and Development, National Health Effects and Environmental Research Laboratory, Western Ecology Division.
Examples
1 2 3 4 5 6 7 8 