Optimal Stratification of Univariate Populations

  Optimal Stratification of Univariate Populations

Name: Nayana B Menon

Reg.no: 2048128

 

Stratification reduces the variance of sample estimates for population parameters by creating homogeneous strata. Often, surveyors stratify the population using the most convenient variables such as age, sex, region, etc. Such convenient methods often do not produce internally homogeneous strata, hence, the precision of the estimates of the variables of interest could be further improved. Stratification in univariate populations has been explored by numerous researchers, many of whom have proposed competing algorithms that help surveyors determine efficient stratum boundaries.

 

Optimum Stratum boundaries:  The method of choosing the best boundaries that make strata internally homogenous as far as possible is known as optimum stratification. To achieve this, the strata should be constructed in such a way that the strata variances for the characteristic under study be as small as possible. If the frequency distribution of the study variable x is known, the optimum strata boundaries (OSB) could be obtained by cutting the range of the distribution at suitable points. If the frequency distribution of x is unknown, it may be approximated from the past experience or some prior knowledge obtained at a recent study. Many skewed populations have log-normal frequency distribution or may be assumed to follow approximately log-normal frequency distribution.

 

Optimum Sample size: When you perform a survey, the intention is to get a representative image about a number of variables or statements within a certain target group or population. Due to practical reasons (too large, too expensive or too time-consuming etc) it is often difficult to interrogate the total population. In that case a sample is used. This is a selection of respondents chosen in such a way that they represent the total population as well as possible.  It is very important to use a correct sample size. When your sample is too big, this will lead to unnecessary waste of money and time. On the other hand, when it’s too small, your results will not be statistically significant and you will not come to reliable conclusions.

 

Determination of optimum stratum boundaries and optimum sample sizes  to be selected from each stratum are two inherent optimization problems in optimal stratification. Once the optimum stratum boundaries have been determined, optimum sample size can easily be computed using a particular sample allocation method. When stratification is based on a single study variable (y), its distribution can be utilized as the best characteristic to determine the optimum stratum boundaries, i.e., by cutting the range of the distribution at suitable points. The basic consideration involved in determining optimum stratum boundaries is that the strata should be as internally homogenous as possible. Thus, in order to achieve maximum precision, the stratum variances should be as small as possible Cochran (1977).

 

Formulation of the Univariate Stratification Problem

Let the target population of the variable under study be stratified into L strata where the estimation of the mean of this study variable (y) is of interest. If a simple random sample of size nh is to be drawn from h th stratum with sample mean ¯yh, then the stratified sample mean, ¯yst, is given by: 

                                                         

where Wh (stratum weight) is the proportion of the population contained in the hth stratum. When the finite population correction factors are ignored, under the Neyman (1934) allocation, the variance of 

y^¯st is given by: 

 

where S_h² is the stratum variance for the study variable in the h^th where h=1, 2...  L_h=1,2..., L) stratum and n is the preassigned total sample size. For a fixed sample size n, minimizing the expression of the right hand side of equation is equivalent to minimizing

Thus, the objective function could be expressed as a function of boundary points y_h and y_h−1 only. Further defining l_h = y_h − y_h−1; h = 1, 2, ..., L where l_h ≥ 0 denotes the range or width of the h^th stratum and the range of the distribution, d = b − a, is expressed as a function of stratum width as:

                                                       

The h^th stratification point y_h; h = 1, 2, ..., L is then expressed as y_h = yh−1 + l_h and the problem can be treated as an equivalent problem of determining optimum strata widths (OSW), l1, l2, ..., lL. Due to the special nature of functions, the problem may be treated as a function of l_h alone and can be expressed as:

                                                       

 

Optimal Stratification of Univariate Populations – R Programming:

With the simulated data sets, the number of strata (h), fixed sample size (n) and population size (N) were used as the input arguments to the strata.dp() function in the package. When executed, the package outputs the OSB and OSS, amongst other quantities such as stratum weight (W_h), stratum variance (Sh2), etc.

The stratifyR Package Under the proposed method, in order to construct optimum stratum boundaries and optimum sample sizes for a given population, its best-fit frequency distribution needs to be estimated. The problem of OSB is then formulated as a mathematical programming problem, where the objective function is minimised on the range of the data set subject to the constraints Both the estimation of the distribution and the MPP formulation (for the indicated distributions) are implemented in the proposed stratify package.

We use an example to illustrate this in r programming:

SYNTAX:

strata.data(data, h, n, cost=FALSE, ch=NULL)

    data - A vector: data containing every unit of the survey population

    h - A numeric: number of strata to be sampled. The default is 2

    n - A numeric: fixed total sample size

    cost - A logical: stratum cost. Default cost=FALSE.

    ch - A numeric: denotes a vector of stratum costs. Default ch=NULL.

To show the application of the strata.data() function, an example of the command used and its output from the package is given below. The problem uses the ‘mag’ variable from the ‘quakes’ data (with a population of N=1000) available from the datasets package in R. To construct a 2-strata solution with a fixed sample size of n=300, we use the following codes:

data(quakes)
head(quakes)

##      lat   long depth mag stations
## 1 -20.42 181.62   562 4.8       41
## 2 -20.62 181.03   650 4.2       15
## 3 -26.00 184.10    42 5.4       43
## 4 -17.97 181.66   626 4.1       19
## 5 -20.42 181.96   649 4.0       11
## 6 -19.68 184.31   195 4.0       12

mag <- quakes$mag
length(mag)

## [1] 1000

hist(mag) #to see the distribution
# install.packages("stratifyR")
library(stratifyR)

## Warning: package 'stratifyR' was built under R version 3.6.3

## Loading required package: fitdistrplus

## Warning: package 'fitdistrplus' was built under R version 3.6.3

## Loading required package: MASS

## Loading required package: survival

## Loading required package: zipfR

## Warning: package 'zipfR' was built under R version 3.6.3

## Loading required package: actuar

## Warning: package 'actuar' was built under R version 3.6.3

##
## Attaching package: 'actuar'

## The following object is masked from 'package:grDevices':
##
##     cm

## Loading required package: triangle

## Warning: package 'triangle' was built under R version 3.6.3

## Loading required package: mc2d

## Warning: package 'mc2d' was built under R version 3.6.3

## Loading required package: mvtnorm

## Warning: package 'mvtnorm' was built under R version 3.6.3

##
## Attaching package: 'mc2d'

## The following objects are masked from 'package:base':
##
##     pmax, pmin

res <- strata.data(mag, h = 2, n=300) # a 2-strata solution

## The program is running, it'll take some time!

summary(res)

## _____________________________________________
## Optimum Strata Boundaries for h = 2
## Data Range: [4, 6.4] with d = 2.4
## Best-fit Frequency Distribution:  lnorm
## Parameter estimate(s): 
##    meanlog      sdlog
## 1.52681032 0.08503554
## ____________________________________________________
##  Strata  OSB   Wh   Vh  WhSh  nh   Nh   fh
##       1 4.68 0.58 0.03 0.109 140  585 0.24
##       2  6.4 0.42 0.09 0.124 160  415 0.38
##   Total      1.00 0.12 0.233 300 1000 0.30
## ____________________________________________________

# Optimum Strata Boundaries for h = 2
# Data Range: [4, 6.4] with d = 2.4
# Best-fit Frequency Distribution:  lnorm
# Parameter estimate(s): 
# sdlog 0.08503554
# meanlog 1.52681032