What type of sampling technique in which the population is subdivided into several strata?

7.3 Advantages of Stratified Sampling

Stratified sampling is used in most large-scale surveys because of its various advantages, some of which are described below:

Estimation of subpopulations: In cases where the estimates of the population characteristics are needed not only for the entire population but also for its different subpopulations, one should treat such subpopulations as strata. For example, in a national unemployment survey, the government may be interested in estimating unemployment figures for the entire country as well as at provincial levels. In this case, each province can be taken as a stratum.

Administrative convenience: The agency conducting the survey may stratify the population such that the survey can be supervised in an efficient manner, e.g., the agency can appoint separate supervisors to conduct survey for each of the strata separately.

Representativeness of sample: In stratified sampling, formation of strata and allocation of samples to different strata may de done in such a way that the sample can represent the population with respect to the characteristics under study. For instance, if we want to select a sample of students from a school, which represents the different races of South Africa, a simple random sampling without replacement (SRSWOR) sample from the entire school may not be representative. In this situation, a stratified sampling using different racial groups as strata is expected to provide a more representative sample than an SRSWOR sample from the entire school.

Efficiency: Stratification may increase efficiency of the estimates by forming strata in such a way that each stratum becomes homogeneous with respect to the characteristic under study. Suitable sampling schemes to the respective strata may increase efficiencies of the estimators.

Improved quality of data: Improved quality of data may be obtained by employing different types of investigators to different strata. For example, investigators knowing local languages may be deployed to the rural areas, whereas in urban areas investigators knowing English may be more advantageous.

5.4 Stratified Sampling

Stratified sampling, also sometimes called quota sampling, is akin to systematic sampling in that a predetermined number of samples are taken from each of the M subregions, but the method of selection Nm is quite different. As with systematic sampling, one seeks

(5.43)〈z〉=∫Vz(x)f(x)dx=∑m=1M∫V mz(x)f(x)dx.

Proceeding exactly as in systematic sampling (see Section 5.3), the stratified sampling estimator of 〈z〉 is

(5.44)z¯strat=∑m=1Mpmz¯m=∑m=1MpmNm∑im=1Nmz(xim).

From Eq. (5.32), the variance of this estimator is

(5.45)σ2(z¯strat)=∑m=1Mp m2Nmσm2(z),

with σm2 (z) given by Eq. (5.34).

So far, this is the same as systematic sampling. But in stratified sampling the Nm are not predetermined but are chosen so as to minimize the variance of the estimator. To minimize σ2(z¯strat) with the restraint that ∑m=1M Nm=N, one seeks the unrestricted minimum of

(5.46)∑m=1Mpm2σm2(z)Nm+λ∑m=1M Nm,

where λ is a Lagrange multiplier. The variation of this quantity with Nj is then set to zero so that

(5.47)0=δδNj {∑m=1M[pm2σm2 Nm+λNm]}=∑m=1M δδNj[pm2σm2Nm+λNm]=−pj2σj2Nj2 +λ.

From this quantity, the optimum number of samples from each region is found to be Nj=pjσj/ λ. Rearranging and summing this result gives

(5.48)λ∑j=1MNj=λ N=∑j=1Mpjσj≡〈σj〉,

from which the optimum number of samples that minimizes σ2(z¯strat) is

(5.49)Nj=pjσj(z)〈 σj〉Npjσj(z)∑j =1NpjσjN.

Thus, in stratified sampling, the number of points sampled in each subregion is proportional to the variance of z in the region. Regions in which z varies rapidly are sampled more often than regions in which z is relatively constant. Substitution of this result (with j replaced by m) into Eq. (5.45) gives the minimum variance of z¯strat, namely,

(5.50) σ2(z¯strat)=1N (∑m=1Mpmσm)2=12 〈σm〉2.

Unfortunately, the parameters σj(z) and 〈σj〉 are usually not known a priori. It is not a good idea to use some of the same histories used to determine z¯strat to also determine σj, because these results are correlated and very strange results can occur. One approach to estimate the optimum values of Nj is to estimate σj from a small preliminary run to obtain rough estimates, with consideration given to the trade-off between the desired precision and the cost of sampling.

A more serious limitation for stratified sampling is the subdivision of V for high-dimensional integrals, say d ≥ 5. Division of each axis into k segments causes the number of subvolumes, Md, to explode with increasing k and greatly confounds accurate estimates of the needed variances for each subregion.

Stratified Sampling

Stratified sampling designs involve partitioning a population into strata based on a certain characteristic that is known for every sampling unit in the population, and then selecting samples independently from each stratum. This design offers flexibility of sampling methods in different strata and gains improved precision of estimates of target parameters when each stratum is composed of units that are relatively homogenous. In practice, strata are often determined by field conditions, such as geographic regions or administrative structures. Stratified sampling design also enables specific segments of the population to be easily targeted. For example, in the combined NAEP national and state samples (Rust, 2004), the design treated each jurisdiction in the United States as a stratum; therefore, state results can be separately reported and be compared to each other.

Stratified sampling is often made with disproportionate sample allocation across strata, meaning that the stratum proportions in the sample do not represent the corresponding proportions in the population. To remedy this problem, stratum weights, defined for each stratum as the proportion of the size of that stratum to the total population, need to be applied in estimation. When this is done, the mean or proportion estimators for each stratum are unbiased and the estimator for the population, which equals the sum across strata of the stratum weights multiplied together with the corresponding stratum estimator, is also unbiased. Existence of an unbiased estimator for each stratum implies that the average of the estimates over all possible samples in the stratum for the same design is equal to the true parameter being estimated. If stratum weights are not properly computed, it introduces bias in estimation. In general, the weighted mean of a proper stratified sample has less variability than the arithmetic mean of an SRS sample of the population. The stratum weights should also be employed in variance estimation for stratified data.

18.5.3.1 Population Ratio

Consider a stratified sampling where two units are selected at random from each of the stratum. The conventional combined stratified ratio estimator of the population ratio R = Y/X is Rˆcom=Yˆst/Xˆs t, where Yˆst=∑h=1LNhy¯h, Xˆst=∑ h=1LNhx¯h. An estimator R based on αth half-sample is Rˆα(com)=Yˆα/ Xˆα, where Yˆα=∑h=1LNh{ϕhαyh2+(1−ϕ hα)yh2} and Xˆα=∑h=1LNh{ϕhαxh2 +(1−ϕhα)xh2}. The estimator for R based on a set of k balanced half-samples is given by

RˆBRR=1k∑α=1k Yˆα/Xˆα

An estimator for the variance of Rˆcom based on k balanced half-samples is

VˆBR R=1k∑α=1k(Rˆ α(com)−Rˆcom)2

Clearly, VˆBRR is quite different from the conventional variance estimator

Vˆ st(Rˆcom)=14Xˆst2∑h=1LNh2 {(yh2−yh2)−Rˆ com(xh2−xh2)}2 .

5.2 Stratification and RRT

Stratification is known to have its own advantages. Researchers in the RR field automatically extended the available results on Randomized Response to stratified sampling and allocation of sample size, etc. However, we note that some of these extensions are of theoretical nature and it is difficult to envisage their adaptability in practical situations.

Hong et al. (1994) envisaged a straight forward stratified extension of RRT under proportional allocation of sample size to strata applying the same random device in each stratum. Kim and Warde (2004) maintain that Hong et al.’s proportionate sampling could lead to higher costs, though an expression for variance of the suggested estimator could be easily derived. They suggest Neyman's optimum allocation subject to ∑ni=n, the total sample size. However, they do not derive the cost-optimum allocation due to Mahalanobis (1944) based on a simple linear cost constraint. It should be noted that all these extensions of allocations, being theoretical in nature, are difficult to use in practical applications. Kim and Warde (2004) cite certain situations but the optimum allocation depends on unknown quantities. One then needs to make model assumptions and derive user-friendly near-optimum allocations. Though this approach leads to meaningful results in classical situations (Rao, 2010), it is difficult to look for a related auxiliary information in the case of RR models to postulate a super population model. Extending the scrambled RR model proposed by Eichhorn and Hayre (1983), Mahajan et al. (1994), and Mahajan and Verma (2004) obtained optimum points of stratification on an auxiliary variable for SRSWR in each stratum using data on scrambled sensitive character. In the recent past, there have been several publications on these and related aspects but one feels that these extensions are more of an academic interest. Another recent line of theoretical research is to use scrambled responses in the presence of auxiliary information initiated by Singh et al. (2015). With a good choice of related auxiliary information, these results will add to the practical value.

16.1 Introduction

The use of auxiliary information in estimating population mean or total is well known in the field of survey sampling. Various survey sampling schemes such as stratified sampling, cluster sampling, and multistage sampling are frequently used, among them two-stage sampling has the benefit of saving time, cost, and effort. As mentioned in Salinas et al. (2018), the two-stage sampling method is an improvement over cluster sampling when it is not possible or easy to enumerate all the units from the selected clusters. A solution to this difficulty is to select clusters, called first-stage units (FSUs), from the given population of interest and select subsamples from the selected clusters called second-stage units (SSUs). Assuming heterogeneous groups, this technique of sampling helps to increase the precision of the resultant estimates. It is easy to collect information from a few units within the selected FSUs, saving the cost of survey. Assume the population of interest Ω={1,2,…,N} consists of N nonoverlapping clusters, called FSUs. The whole population is divided as Ω={Ω1,Ω2,…,ΩN}, where Ωi denotes the ith cluster of size Mi, for i=1,2,…,N such that Ω=∪i=1NΩi and M= ∑i=1NMi. Särndal et al. (1992) consider three situations with the auxiliary information in two-stage sampling. For the first situation, the auxiliary variable is available for all the FSUs, the second situation has the auxiliary variable for all the units in the population. Lastly, the third situation has the auxiliary variable available for all elements in the selected FSUs. For clarity, assume the simplest and most practical design where the FSUs are selected by simple random and without replacement (SRSWOR) and the SSUs are selected by simple random and with replacement (SRSWR) sampling schemes. Also assume that the population means of the auxiliary variable for the selected FSUs are known or available. The auxiliary information at the individual level may or may not be known. For simplicity of results, focus is put on the use of a single auxiliary variable. The application of two-stage sampling can involve various situations to the interest of the investigator. For example, in the agricultural sectors selecting villages as FSUs, and farmers at the SSUs; in education, selecting departments as FSUs, and faculty as SSUs. In politics, selecting blocks as FSUs and dwellings as SSUs. In the health sector, FSUs could be hospitals and SSUs could be doctors. At a city level study, FSUs could be households and SSUs could be family members.

In the next section, we introduce notations and some basic results related to two-stage sampling.

Task: Perform Any Planned Sampling Regime

Types of sampling regimes are as follows:

Random sampling

It is performed to draw a representative group of data rows from a larger data population.

Stratified sampling

It is performed when a population is composed of two more groups that can be grouped together according to the same criteria (e.g., geographic location).

Oversampling and undersampling

It is performed to produce the same number of rows for each target class. See Chapter 4 for more information on this topic.

Assign case weights or prior probabilities to specific target classes, instead of balancing data sets with a rare target class.

Some analytic tools can balance data sets with rare target classes by using differential weights or using probability of occurrence of the target classes (prior probabilities) to govern the effect of the classification operation. See Chapter 4 for more information.

13.8.1 Stratified sampling

Stratified sampling was first introduced in Section 7.3, and we now have the tools to motivate its use. Stratified sampling works by subdividing the integration domain Λ into n nonoverlapping regions Λ1, Λ2, …, Λn. Each region is called a stratum, and they must completely cover the original domain:

∪i=1nΛi=Λ.

To draw samples from Λ, we will draw ni samples from each Λi, according to densities pi inside each stratum. A simple example is supersampling a pixel. With stratified sampling, the area around a pixel is divided into a k × k grid, and a sample is drawn from each grid cell. This is better than taking k2 random samples, since the sample locations are less likely to clump together. Here we will show why this technique reduces variance.

Sampler 421

Sampler::Request1DArray() 423

Sampler::Request2DArray() 423

Within a single stratum Λi, the Monte Carlo estimate is

Fi=1ni ∑j=1nifXi,j piXi,j,

where Xi, j is the jth sample drawn from density pi. The overall estimate is F=∑i=1nviFi, where vi is the fractional volume of stratum i (vi ∈ (0, 1]).

The true value of the integrand in stratum i is

μi=EfXi, j=1vi∫Λifxdx,

and the variance in this stratum is

σi2=1vi∫Λifx−μi2dx.

Thus, with ni samples in the stratum, the variance of the per-stratum estimator is σi2/ni. This shows that the variance of the overall estimator is

VF=V∑viFi=∑VviFi=∑vi2VFi=∑vi2σi2ni.

If we make the reasonable assumption that the number of samples ni is proportional to the volume vi, then we have ni = viN, and the variance of the overall estimator is

VFN=1N∑viσi2.

To compare this result to the variance without stratification, we note that choosing an unstratified sample is equivalent to choosing a random stratum I according to the discrete probability distribution defined by the volumes vi and then choosing a random sample X in ΛI. In this sense, X is chosen conditionally on I, so it can be shown using conditional probability that

VF=ExViF+VxEiF=1N∑viσi2+∑viμi−Q,

where Q is the mean of f over the whole domain Λ. See Veach (1997) for a derivation of this result.

There are two things to notice about this expression. First, we know that the right-hand sum must be nonnegative, since variance is always nonnegative. Second, it demonstrates that stratified sampling can never increase variance. In fact, stratification always reduces variance unless the right-hand sum is exactly 0. It can only be 0 when the function f has the same mean over each stratum Λi. In fact, for stratified sampling to work best, we would like to maximize the right-hand sum, so it is best to make the strata have means that are as unequal as possible. This explains why compact strata are desirable if one does not know anything about the function f. If the strata are wide, they will contain more variation and will have μi closer to the true mean Q.

Figure 13.17 shows the effect of using stratified sampling versus a uniform random distribution for sampling ray directions for glossy reflection. There is a reasonable reduction in variance at essentially no cost in running time.

The main downside of stratified sampling is that it suffers from the same “curse of dimensionality” as standard numerical quadrature. Full stratification in D dimensions with S strata per dimension requires SD samples, which quickly becomes prohibitive. Fortunately, it is often possible to stratify some of the dimensions independently and then randomly associate samples from different dimensions, as was done in Section 7.3. Choosing which dimensions are stratified should be done in a way that stratifies dimensions that tend to be most highly correlated in their effect on the value of the integrand (Owen 1998). For example, for the direct lighting example in Section 13.7.1, it is far more effective to stratify the (x, y) pixel positions and to stratify the (θ, ϕ) ray direction— stratifying (x, θ) and (y, ϕ) would almost certainly be ineffective.

Another solution to the curse of dimensionality that has many of the same advantages of stratification is to use Latin hypercube sampling (also introduced in Section 7.3), which can generate any number of samples independent of the number of dimensions. Unfortunately, Latin hypercube sampling isn’t as effective as stratified sampling at reducing variance, especially as the number of samples taken becomes large. Nevertheless, Latin hypercube sampling is provably no worse than uniform random sampling and is often much better.

3.3 Latin hypercube sampling

The sampling technique using MCS and QMCS discussed in aforementioned sections is powerful and robust in quantifying uncertainty, albeit at the cost of computation. Latin hypercube sampling (LHS) is a stratified sampling scheme used to reduce the number of simulations in quantifying response uncertainty. In this ED method, the input space is partitioned in different “strata,” and a representative value is selected from each stratum. The representative values for the domain are then combined to check for any repetition in the complete simulation. Following steps are used in LHS to generate random samples from an input vector X (Box 15.2).

BOX 15.2

LHS scheme

Step 1. Partition the input sample space of each random variable (RV) into L ranges of equal probability = 1/L. It is not necessary to divide the domain with equal probability

Step 2. Generate one representative random sample from each range. Sometimes, the midvalue is used instead of a random sample from range

Step 3. Randomly select one value from L values of each RV to get the first sample s1

Step 4. Randomly select one value from the remaining L—one value of each RV to get the second sample s2, and so on upto L sample sL

Step 5. Repeat Steps 1 to 4 for all the RVs

Step 6. The rest sampling technique is the same as in MCS

MATLAB built-in functions for LHS design:

X = lhsdesign(L,K) gives L random samples of each X1,X2, …, XK in an L × K matrix

X = lhsdesign(L,K,‘smooth’,‘off’) gives the median value for each stratum

X = lhsdesign(L,K,‘iterations',J) runs the simulation for J iterations

lhsnorm generates multivariate Gaussian distributions

8.2 Stratification

Stratification means the population of N units is first divided into homogeneous and mutually exclusive groups called strata. Then an independent random sample of the required size is selected from each stratum. In short, the hth stratum in stratified sampling design consists of Nh units, where h=1,2,…,L so that

(8.1)∑h=1LNh=N

Let yhi be the value of the study variable for the ith unit in the hth stratum, i=1,2,…,Nh, so the hth stratum population mean is given by

(8.2) Y¯h=Nh−1∑i=1N hyhi,forh=1,2 ,…,L

Obviously, using the concept of weighted average, the true mean of the whole population can be written as

(8.3)Y¯= N1Y¯1+N2Y¯2+⋯+NLY¯LN1+N2+⋯+NL =N1Y¯1+N2Y ¯2+⋯+NLY¯LN=N1NY¯1+N2NY¯2+⋯+NLNY¯L=Ω1Y¯1+Ω2Y¯2+⋯+ΩLY¯L=∑h=1LΩhY¯h

where

(8.4)Ωh=NhN

is the known proportion of population units falling within the hth stratum. Consider drawing a sample sh of size nh using a simple random sampling (SRS) scheme from the hth stratum consisting of Nh units, such that

(8.5)∑h=1Lnh=n

is the required sample size.

Assume the value of the ith unit of the study variable selected from the hth stratum is denoted by yhi and i=1,2,…,nh. An unbiased estimator of population mean Y¯ is given by

(8.6) y¯st=∑h=1LΩhy¯h

where

(8.7)y¯h=nh−1∑i=1nhyhi

denotes the hth stratum sample mean.

Also assume the value of the auxiliary variable for the ith unit selected from the hth stratum is denoted by xhi, where i=1,2,…,nh. An unbiased estimator of the population mean

(8.8) X¯=∑h=1LΩhX¯h

is given by

(8.9)x¯st=∑h=1LΩhx¯h

where

(8.10)x¯ h=nh−1∑i=1nhxhi

denotes the hth stratum sample mean, and

(8.11)X¯h=Nh−1 ∑i=1Nhxhi

denotes the known hth stratum population mean of the auxiliary variable. For more information about stratified random sampling, one could refer to Neyman (1934).