csvy {csurvey}

Estimation of Domain Means with Monotonicity or Convexity Constraints

Description

The csvy function performs design-based domain mean estimation with monotonicity and block-monotone shape constraints.

For example, in a one dimensional situation, we assume that \bar{y}_{U_t} are non-decreasing over T domains. If this monotonicity is not used in estimation, the population domain means can be estimated by the Horvitz-Thompson estimator or the Hajek estimator. To use the monotonicity information, this csvy function starts from the Hajek estimates \bar{y}_{S_t} = (\sum_{k\in S_t}y_k/\pi_k)/N_t and the isotonic estimator (\hat{\theta}_1,\ldots,\hat{\theta}_T)^T minimizes the weighted sum of squared deviations from the sample domain means over the set of ordered vectors; that is, \bold{\hat{\theta}} is the minimizer of (\tilde{\bold{y}}_{S} - \bold{\theta})^T \bold{W}_s (\tilde{\bold{y}}_{S} - \bold{\theta}) subject to \bold{A\theta} \geq \bold{0}, where \bold{W}_S is the diagonal matrix with elements \hat{N}_1/\hat{N},\ldots,\hat{N}_D/\hat{N}, and \hat{N} = \sum_{t=1}^T \hat{N}_t and \bold{A} is a m\times T constraint matrix imposing the monotonicity constraint.

Domains can also be formed from multiple covariates. In that case, a grid will be used to represent the domains. For example, if there are two predictors x_1 and x_2, and x_1 has values on D_1 domains: 1,\ldots,D_1, x_2 has values on D_2 domains: 1,\ldots,D_2, then the domains formed by x_1 and x_2 will be a D_1\times D_2 by 2 grid.

To get 100(1-\alpha)\% approximate confidence intervals or surfaces for the domain means, we apply the method in Meyer, M. C. (2018). \hat{p}_J is the estimated probability that the projection of y_s onto \cal C lands on \cal F_J, and the \hat{p}_J values are obtained by simulating many normal random vectors with estimated domain means and covariance matrix I, where I is a M \times M matrix, and recording the resulting sets J.

The user needs to provide a survey design, which is specified by the svydesign function in the survey package, and also a data frame containing the response, predictor(s), domain variable, sampling weights, etc. So far, only stratified sampling design with simple random sampling without replacement (STSI) is considered in the examples in this package.

Note that when there is any empty domain, the user must specify the total number of domains in the nD argument.

Usage

csvy(formula, design, subset=NULL, nD=NULL, family=stats::gaussian(),
    amat=NULL, level=0.95, n.mix=100L, test=TRUE,...)
## S3 method for class 'csvy'
summary(object,...)
## S3 method for class 'csvy'
vcov(object,...)
## S3 method for class 'csvy'
coef(object,...)
## S3 method for class 'csvy'
confint(object, parm=NULL, level = 0.95, type = c("link", "response"),...)
## S3 method for class 'csvy'
predict(object, newdata = NULL, type = c("link", "response"), 
    se.fit = TRUE, level = 0.95, n.mix = 100,...)

Arguments

formula

A formula object which gives a symbolic description of the model to be fitted. It has the form "response ~ predictor". The response is a vector of length n. A predictor can be a non-parametrically modelled variable with a monotonicity or convexity restriction, or a combination of both. In terms of a non-parametrically modelled predictor, the user is supposed to indicate the relationship between the domain mean and a predictor x in the following way:

Assume that \mu is the vector of domain means and x is a predictor:

• incr(x): \mu is increasing in x.

• decr(x): \mu is decreasing in x.

• block.Ord(x): \mu is has a block ordering in x.

design

A survey design, which must be specified by the svydesign routine in the survey package.

subset

Expression to select a subpopulation.

nD

Total number of domains.

family

A parameter indicating the error distribution and link function to be used in the model. It can be a character string naming a family function or the result of a call to a family function. This is borrowed from the glm routine in the stats package. There are four families used in csvy: Gaussian, binomial, poisson, and Gamma.

amat

A k \times M matrix imposing shape constraints in each dimension, where M is the total number of domains. If the user doesn't provide the constraint matrix, a subroutine in the csurvey package will create a constraint matrix according to shape constraints specified in the formula. The default is amat = NULL.

level

Confidence level of the approximate confidence surfaces. The default is 0.95.

n.mix

The number of simulations used to get the approximate confidence intervals or surfaces. If n.mix = 0, no simulation will be done and the face of the final projection will be used to compute the covariance matrix of the constrained estimate. The default is n.mix = 100L.

test

A logical scalar. If test == TRUE, then the p-value for the test H_0:\theta is in V versus H_1:\theta is in C is returned. C is the constraint cone of the form \{\beta: A\beta \ge 0\}, and V is the null space of A. The default is test = TRUE.

...

Other arguments

The coef function returns estimated systematic component of a csvy object.

The confint function returns the confidence interval of a csvy object. If type = "response", then the interval is for the mean; if type = "link", then the interval is for the systematic component.

parm

An argument in the generic confint function in the stats package. For now, this argument is not in use.

The following arguments are used in the predict function.

object

A csvy object.

newdata

A data frame in which to look for variables with which to predict. If omitted, the fitted values are used.

type

If the response is Gaussian, type = "response" and type = "link" give the predicted mean; if the response is binomial, poisson or Gamma, type = "response" gives the predicted mean, and type = "link" gives the predicted systematic component.

se.fit

Logical switch indicating if confidence intervals are required.

Details

For binomial and Poisson families use family=quasibinomial() and family=quasipoisson() to avoid a warning about non-integer numbers of successes. The ‘quasi’ versions of the family objects give the same point estimates and standard errors and do not give the warning.

predict gives fitted values and sampling variability for specific new values of covariates. When newdata are the population mean it gives the regression estimator of the mean, and when newdata are the population totals and total is specified it gives the regression estimator of the population total. Regression estimators of mean and total can also be obtained with calibrate.

Value

The output is a list of values used for estimation, inference and visualization. Main output include:

survey.design

The survey design used in the model.

etahat

Estimated shape-constrained domain systematic component.

etahatu

Estimated unconstrained domain systematic component.

muhat

Estimated shape-constrained domain means.

muhatu

Estimated unconstrained domain means.

lwr

Approximate lower confidence band or surface for the shape-constrained domain mean estimate.

upp

Approximate upper confidence band or surface for the shape-constrained domain mean estimate.

lwru

Approximate lower confidence band or surface for the unconstrained domain mean estimate.

uppu

Approximate upper confidence band or surface for the unconstrained domain mean estimate.

amat

The k \times M constraint matrix imposing shape constraints in each dimension, where M is the total number of domains.

grid

A M \times p grid, where p is the total number of predictors or dimensions.

nd

A vector of sample sizes in all domains.

Ds

A vector of the number of domains in each dimension.

acov

Constrained mixture covariance estimate of domain means.

cov.un

Unconstrained covariance estimate of domain means.

CIC

The cone information criterion proposed in Meyer(2013a). It uses the "null expected degrees of freedom" as a measure of the complexity of the model. See Meyer(2013a) for further details of cic.

CIC.un

The cone information criterion for the unconstrained estimator.

zeros_ps

Index of empty domain(s).

nd

Sample size of each domain.

pval

p-value of the one-sided test.

family

The family parameter defined in a csvy formula.

df.residual

The observed degree of freedom for the residuals of a csvy fit.

df.null

The degree of freedom for the null model of a csvy fit.

domain

Index of each domain in the data set contained in the survey.design object.

null.deviance

The deviance for the null model of a csvy fit.

deviance

The residual deviance of a csvy fit.

Author(s)

Xiyue Liao

References

Xu, X. and Meyer, M. C. (2021) One-sided testing of population domain means in surveys.

Oliva, C., Meyer, M. C., and Opsomer, J.D. (2020) Estimation and inference of domain means subject to qualitative constraints. Survey Methodology

Meyer, M. C. (2018) A Framework for Estimation and Inference in Generalized Additive Models with Shape and Order Restrictions. Statistical Science 33(4) 595–614.

Wu, J., Opsomer, J.D., and Meyer, M. C. (2016) Survey estimation of domain means that respect natural orderings. Canadian Journal of Statistics 44(4) 431–444.

Meyer, M. C. (2013a) Semi-parametric additive constrained regression. Journal of Nonparametric Statistics 25(3), 715.

Lumley, T. (2004) Analysis of complex survey samples. Journal of Statistical Software 9(1) 1–19.

See Also

plotpersp, to create a 3D Plot for a csvy Object

incr, to specify an increasing shape-restriction in a csvy Formula

decr, to specify an decreasing shape-restriction in a csvy Formula

Examples

data(api)

mcat = apipop$meals
for(i in 1:10){mcat[trunc(apipop$meals/10)+1==i] = i}
mcat[mcat==100]=10
D1 = 10

gcat = apipop$col.grad
for(i in 1:10){gcat[trunc(apipop$col.grad/10)+1==i] = i}
gcat[gcat >= 5] = 4
D2 = 4

nsp = c(200,200,200)  ## sample sizes per stratum

es = sample(apipop$snum[apipop$stype=='E'&!is.na(apipop$avg.ed)&!is.na(apipop$api00)],nsp[1])
ms = sample(apipop$snum[apipop$stype=='M'&!is.na(apipop$avg.ed)&!is.na(apipop$api00)],nsp[2])
hs = sample(apipop$snum[apipop$stype=='H'&!is.na(apipop$avg.ed)&!is.na(apipop$api00)],nsp[3])
sid = c(es,ms,hs)

pw = 1:6194*0+4421/nsp[1]
pw[apipop$stype=='M'] = 1018/nsp[2]
pw[apipop$stype=='H'] = 755/nsp[3]

fpc = 1:6194*0+4421
fpc[apipop$stype=='M'] = 1018
fpc[apipop$stype=='H'] = 755

strsamp = cbind(apipop,mcat,gcat,pw,fpc)[sid,]

dstrat = svydesign(ids=~snum, strata=~stype, fpc=~fpc, data=strsamp, weight=~pw)
rds = as.svrepdesign(dstrat, type="JKn")

# Example 1: monotonic in one dimension
ansc1 = csvy(api00~decr(mcat), design=rds, nD=D1)
# checked estimated domain means
# ansc1$muhat

# Example 2: monotonic in three dimensions
D1 = 5
D2 = 5
D3 = 6
Ds = c(D1, D2, D3)
M = cumprod(Ds)[3]

x1vec = 1:D1
x2vec = 1:D2
x3vec = 1:D3
grid = expand.grid(x1vec, x2vec, x3vec)
N = M*100*4
Ns = rep(N/M, M)

mu.f = function(x) {
  mus = x[1]^(0.25)+4*exp(0.5+2*x[2])/(1+exp(0.5+2*x[2]))+sqrt(1/4+x[3])
  mus = as.numeric(mus$Var1)
  return (mus)
}

mus = mu.f(grid)

H = 4
nh = c(180,360,360,540)
n = sum(nh)
Nh = rep(N/H, H)

#generate population
y = NULL
z = NULL

set.seed(1)
for(i in 1:M){
  Ni = Ns[i]
  mui = mus[i]
  ei = rnorm(Ni, 0, sd=1)
  yi = mui + ei
  y = c(y, yi)
  zi = i/M + rnorm(Ni, mean=0, sd=1)
  z = c(z, zi)
}

x1 = rep(grid[,1], times=Ns)
x2 = rep(grid[,2], times=Ns)
x3 = rep(grid[,3], times=Ns)
domain = rep(1:M, times=Ns)

cts = quantile(z, probs=seq(0,1,length=5))
strata = 1:N*0
strata[z >= cts[1] & z < cts[2]] = 1
strata[z >= cts[2] & z < cts[3]] = 2
strata[z >= cts[3] & z < cts[4]] = 3
strata[z >= cts[4] & z <= cts[5]] = 4
freq = rep(N/(length(cts)-1), n)

w0 = Nh/nh
w = 1:N*0
w[strata == 1] = w0[1]
w[strata == 2] = w0[2]
w[strata == 3] = w0[3]
w[strata == 4] = w0[4]
pop = data.frame(y = y, x1 = x1, x2 = x2, x3 = x3, domain = domain, strata = strata, w=w)
ssid = stratsample(pop$strata, c("1"=nh[1], "2"=nh[2], "3"=nh[3], "4"=nh[4]))
sample.stsi = pop[ssid, ,drop=FALSE]
ds = svydesign(id=~1, strata =~strata, fpc=~freq, weights=~w, data=sample.stsi)


#domain means are increasing w.r.t x1, x2 and block monotonic in x3
ord = c(1,1,2,2,3,3)
ans = csvy(y~incr(x1)*incr(x2)*block.Ord(x3,order=ord), design=ds, nD=M, test=FALSE, n.mix=0)

#3D plot of estimated domain means: x1 and x2 with confidence intervals
plotpersp(ans, ci = "both")


#3D plot of estimated domain means: x3 and x2
plotpersp(ans, x3, x2)

#3D plot of estimated domain means: x3 and x2 for each domain of x1
plotpersp(ans, x3, x2, categ="x1")

#3D plot of estimated domain means: x3 and x2 for each domain of x1
plotpersp(ans, x3, x2, categ="x1", NCOL = 3)

# Example 3: unconstrained in one dimension

#no constraint on x1
ans = csvy(y~x1*incr(x2)*incr(x3), design=ds, test=FALSE, n.mix=0) 

#3D plot of estimated domain means: x1 and x2
plotpersp(ans)