Cumulative Distribution Function - Estimation

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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 size-weighted CDF can be calculated. In addition the standard error of the estimated CDF and confidence bounds are calculated.

Usage

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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 single-stage sample or the stage two weight for a two-stage sample.

x

x-coordinate for location for each site, which is either the x-coordinate for a single-stage sample or the stage two x-coordinate for a two-stage sample. The default is NULL.

y

y-coordinate for location for each site, which is either the y-coordinate for a single-stage sample or the stage two y-coordinate for a two-stage 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 x-coordinate for location for each site. The default is NULL.

y1

the stage one y-coordinate 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 size-weights. 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 single-stage sample, values must be supplied for arguments pcfsize and support. To employ the correction factor for a two-stage 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 single-stage 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 two-stage 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 two-stage sample, which is required for calculation of finite and continuous population correction factors for a two-stage 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 size-weights should be used in the analysis, where TRUE = use the size-weights and FALSE = do not use the size-weights. The default is FALSE.

swgt

the size-weight for each site, which is the stage two size-weight for a two-stage sample. The default is NULL.

swgt1

the stage one size-weight 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 size-weighted 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 Horvitz-Thompson ratio estimator, i.e., the ratio of two Horvitz-Thompson 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 size-weights of the resource, as appropriate. For the CDF of a total when either the size of the resource or the sum of the size-weights 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 Horvitz-Thompson ratio estimator. For the CDF of a total when neither the size of the resource nor the sum of the size-weights of the resource is provided, the Horvitz-Thompson 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 x-coordinate and the y-coordinate 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 size-weights 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 single-stage and two-stage 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

Diaz-Ramos, S., D.L. Stevens, Jr., and A.R. Olsen. (1996). EMAP Statistical Methods Manual. EPA/620/R-96/XXX. Corvallis, OR: U.S. Environmental Protection Agency, Office of Research and Development, National Health Effects and Environmental Research Laboratory, Western Ecology Division.

Examples

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z <- rnorm(100, 10, 1)
wgt <- runif(100, 10, 100)
cdfval <- seq(min(z), max(z), length=20)
cdf.est(z, wgt, vartype="SRS", cdfval=cdfval)

x <- runif(100)
y <- runif(100)
cdf.est(z, wgt, x, y, cdfval=cdfval)

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