Day 39

MATH 313: Survey Design and Sampling

Introduction to Survey Sampling Challenges

In this section, we explore various survey sampling designs and the common challenges they encounter, such as measurement inaccuracies, nonresponse, and issues with sampling frames. We also discuss strategies to enhance the reliability of statistical estimates.

Comprehensive Overview of Survey Sampling Designs

Principal Sampling Techniques

  • Simple Random Sampling
  • Stratified Random Sampling
  • Cluster Sampling
  • Systematic Sampling

Predominant Sampling Challenges

  • Biases due to interviewers or measurement instruments
  • Incomplete or inadequate sampling frames
  • Data sensitivity affecting response accuracy
  • Issues arising from nonresponse

Detailed Exploration of Interpenetrating Subsamples

The concept of interpenetrating subsamples involves dividing a sample of \(n\) elements into \(k\) groups, assigning each group to different interviewers. This method aims to neutralize interviewer biases and enhance the accuracy of the sample mean.

Mathematical Formulations

  • Mean of subsample: \(\bar{y}_i = \frac{1}{m} \sum_{j=1}^m y_{ij}\)
  • Composite sample mean: \(\bar{y} = \frac{1}{k} \sum_{i=1}^k \bar{y}_i\)
  • Variance estimation: \(\hat{V}(\bar{y}) = \left(1-\frac{n}{N}\right) \frac{s_k^2}{k}\)

Estimating Parameters in Subpopulations

Surveying subpopulations presents unique challenges, especially when the subpopulation cannot be explicitly defined from the sampling frame.

Techniques for Estimation

  • Mean of subpopulation: \(\bar{y}_1 = \frac{1}{n_1} \sum_{j=1}^{n_1} y_{1j}\)
  • Variance of subpopulation mean: \(\hat{V}(\bar{y}_1) = \left(1-\frac{n_1}{N_1}\right) \frac{s_1^2}{n_1}\)
  • Total for subpopulation: \(N_1 \bar{y}_1 = \frac{N_1}{n_1} \sum_{j=1}^{n_1} y_{1j}\)

Advanced Techniques for Estimation

When the exact number of subpopulation elements (\(N_1\)) is unknown, alternative estimation techniques must be employed.

Formulas for Advanced Estimation

  • Estimation of total without \(N_1\): \(\hat{\tau}_1 = \frac{N}{n} \sum_{j=1}^{n_1} y_{1j}\)
  • Variance of this estimate: \(\hat{V}(\hat{\tau}_1) = N^2 \left(1-\frac{n}{N}\right) \frac{s_n^2}{n}\)

EXERCISE 11.1: A researcher is interested in estimating the average yearly medical expenses per family in a community of 545 families. The researcher has eight assistants available to do the fieldwork. Skill is required to obtain accurate information on medical expenses because some respondents are reluctant to give detailed information on their health. Because the assistants differ in their interviewing abilities, the researcher decides to use eight interpenetrating subsamples of five families each, with one assistant assigned to each subsample. Hence, a simple random sample of 40 families is selected and divided into eight random subsamples. The interviews are conducted and yield the results shown in the accompanying table. Estimate the average medical expenses per family for the past year and place a bound on the error of estimation.

library(readxl)
expenses <- read_excel("data/EXERCISE11.1.XLS")
expenses %>% knitr::kable()
SS1 SS2 SS3 SS4 SS5 SS6 SS7 SS8
101 157 689 322 837 1015 837 327
95 192 432 48 649 864 249 419
310 108 187 93 152 325 1127 291
427 960 512 162 175 470 493 114
680 312 649 495 210 295 218 287 

# Calculate the mean for each subsample
subsample_means <- apply(expenses, 2, mean)
subsample_means
  SS1   SS2   SS3   SS4   SS5   SS6   SS7   SS8 
322.6 345.8 493.8 224.0 404.6 593.8 584.8 287.6 
# Calculate the overall mean
overall_mean <- mean(subsample_means)
overall_mean
[1] 407.125
# Calculate the sample variance of the subsample means
s_k_squared <- var(subsample_means)

# Constants
N <- 545   # Total population
n <- 40    # Sample size
k <- 8     # Number of subsamples

# Calculate the estimated variance of the sample mean
hat_V_y <- (1 - n/N) * s_k_squared / k

# Calculate the bound on the error of estimation
B <- 2 * sqrt(hat_V_y)
B
[1] 93.70338
# 95% CI
overall_mean + c(-1,1)*B
[1] 313.4216 500.8284

Exercise 11.2: An experiment is designed to gauge the emotional reaction to a city’s decision on school desegregation. A simple random sample of 50 people is interviewed, and the emotional reactions are given a score from 1 to 10 . The scale on which scores are assigned runs from extreme anger to extreme joy. Ten interviewers do the questioning and scoring, with each interviewer working on a random subsample (interpenetrating subsample) of five people. Interpenetrating subsamples are used because of the flexible nature of the scoring. The results are given in the accompanying table. Estimate the average score for people in the city and place a bound on the error of estimation.

library(readxl)
expenses <- read_excel("data/EXERCISE11.2.XLS")
expenses %>% knitr::kable()
SS1 SS2 SS3 SS4 SS5 SS6 SS7 SS8 SS9 SS10
5 4 9 8 6 1 6 5 2 9
4 6 8 5 4 5 4 6 4 7
6 5 9 4 5 6 3 7 4 8
1 2 7 6 7 4 5 3 5 6
8 7 5 3 9 7 2 4 3 4