Probability Proportional
to Size Sampling

Outline

Readings

• Probability Proportional to Size (PPS) Sampling, Encyclopedia of Survey Research Methods, 2008 (editor: P Lavrakas) (PDF on Carmen)

Assignments

• Problem Set 4 due Thursday 10/23/2025 11:59pm via Carmen
  (this is after fall break)

So Far: SRS at Each Stage

• One-stage designs

  • Sample \(n\) of \(N\) PSUs (sample = \(\mathcal{S}\))

  • \(\pi_i = \frac{n}{N}\) for PSU \(i\)     →     \(w_i = \frac{N}{n}\)

  • Unbiased estimate of total \(t\):   \(\hat{t} = \sum_{i \in \mathcal{S}} w_i t_i\)

• Two-stage designs

  • Stage 1: Sample \(n\) of \(N\) PSUs (sample = \(\mathcal{S}\))

  • Stage 2: Sample \(m_i\) of \(M_i\) SSUs for sampled PSU \(i\) (sample = \(\mathcal{S}_i\))

  • \(\pi_{ij} = \frac{n}{N}\frac{m_i}{M_i}\) for SSU \(j\) in PSU \(i\)     →     \(w_{ij} = \frac{N}{n}\frac{M_i}{m_i}\)

  • Unbiased estimate of total \(t\):   \(\hat{t}= \sum_{i \in \mathcal{S}} \sum_{j \in \mathcal{S}_i} w_{ij} y_{ij}\)

These designs do not take into account that totals \((t_i)\) are often correlated with the size of PSUs \((M_i)\)

We saw that ratio estimation can improve estimation of \(\bar{y}_U\) – but can we improve on estimation of \(t\)?

Time Machine: Back to Lecture 3 (Sampling Clinics)

• There are \(N=5\) clinics in a town and you want to estimate the average number of patients seen in a day (in the town)

• The population of clinics is:

• True mean number of patients: \(\bar{y}_U = 58\)

• The number of patients seen is correlated with the number of doctors (makes sense!).

• We used this auxiliary data to create a more efficient sampling scheme than an SRS

• That is the idea behind PPS sampling!

Sampling Clinics (con’t)

Example: Sampling Schools

• Consider Activity 13.1, where you sampled students within schools using a 2-stage cluster sampling scheme

• Population: \(N=30\) schools with a total of \(M_0 = \sum_{i = 1}^N M_i = 14{,}989\) students

• Assume it is inexpensive to measure all students in a school, so use 1-stage cluster sampling

• Let’s consider different ways we can sample \(n=2\) schools from this population to estimate \(t\), the total number of females in the population

Tangent: “Pseudo-PSUs”

• Many large-scale surveys have complex designs, with stratification and multi-stage clustering

  • Use lots of strata to increase precision

  • Often, so highly stratified that each stratum contains a small number of PSUs

Sampling Design A: SRS

Design A: SRS (without replacement) of \(n=2\) from the \(N=30\) schools

• I took an SRS and obtained this sample:

• To estimate the total number of females:

  • \(\pi_i = n/N = 2/30 = 1/15\)

  • \(w_i = 1/\pi_i = N/n = 15\)

  • \(\hat{t}= \sum_{i \in \mathcal{S}} w_i t_i = (15 \times 237) + (15 \times 204) = 6615\)

• Note: Could also calculate as:   \(\hat{t}= N \bar{t}= 30 \times (237 + 204)/2 = 30 \times 220.5 = 6615\)

SRS (con’t)

• By properties of SRS estimators (think way back!):

  • \(\displaystyle \widehat{V}(\hat{t})=N^2 \left(1-\frac{n}{N}\right)\frac{s^2}{n}\)

• In my sample:

  • \(s^2 = \frac{1}{n-1} \sum_{i \in \mathcal{S}} (t_i - \bar{t})^2 = \frac{1}{2-1} \left[(237 - 220.5)^2 + (204 - 220.5)^2 \right] = 544.5\)

  • \(\displaystyle \widehat{V}(\hat{t})=N^2 \left(1-\frac{n}{N}\right)\frac{s^2}{n} = 30^2 \left(1 - \frac{2}{30} \right) \frac{544.5}{22} = 228690\)

  • \(\displaystyle \widehat{SE}(\hat{t}) = \sqrt{228690} = 478.2\)

des.srs <- svydesign(id = ~1, weight = ~wt, fpc = ~N, data=samp_srs)
svytotal(~totfemale, design = des.srs)

          total     SE
totfemale  6615 478.22

Probability Proportional to Size (PPS) Sampling

• Think about the efficiency gain in the clinics example – always sampling the much larger clinic resulted in a much smaller variance – generalizing this idea produces PPS sampling

Probability Proportional to Size (PPS) sampling = sample selection methods in which the probability of selection for a sampling unit is directly proportional to a size measure, \(X_i\), which is known for all units before sampling (with “larger” units more likely to be selected)

• We will use throughout this lecture \(X_i = M_i =\) number of SSUs in the PSU

  • I.e., \(X_i\) = the “size” of the PSU = \(M_i\)

• But, other \(X_i\) are possible

  • Example: Sampling businesses using \(X_i\) = last year’s revenue as reported to IRS

• Note that we want to choose an \(X_i\) that is:

  • Proportional to the \(y_i\) that we are trying to estimate

  • Strictly positive \((>0)\)

Sampling Design B: PPS With Replacement

PPSWR = Simplest PPS scheme, where inclusion probabilities are proportional to \(X_i\) and a PSU can be sampled more than once (with replacement)

• Conduct \(n\) independent draws of a single PSU from the population

• The probability of selecting PSU \(i\) on each draw is proportional to \(X_i\)

• A PSU can be sampled more than once (with replacement)

• Probability PSU \(i\) selected on any one draw = (“psi”) = \(\displaystyle \psi_i = \frac{X_i}{\sum_{i=1}^N X_i}\)

Note the difference:

• \(\psi_i\) = probability PSU \(i\) is selected on a single draw

• \(\pi_i\) = probability PSU \(i\) is in the sample (inclusion probability) = \(2 \psi_i - \psi_i^2\)   (via addition rule for probabilities)

• \(\psi_i \ne \pi_i\)

PPSWR (con’t)

• For the population of schools:

• The probability of selection on a single draw, \(\psi_i\), is larger for the larger schools

PPSWR Algorithms

Example of Cumulative Size Method

PPSWR: Estimating the Population Total

Steps

1. Estimate the total based on each draw – i.e., from each sample of size 1

   • Estimated total based on one PSU = \(\frac{\text{total for that PSU}}{\text{probability of selection on 1 draw}} = \frac{t_i}{\psi_i}\)

2. Average these \(n\) estimated totals to produce unbiased estimator of the total = \(\hat{t}_{wr} = \frac{1}{n}\sum_{i \in \mathcal{S}} \frac{t_i}{\psi_i}\)

Example

• My PPSWR sample of \(n=2\) schools:

• Estimated total = \(\hat{t}_{wr} = \frac{1}{n}\sum_{i \in \mathcal{S}} \frac{t_i}{\psi_i} = (7524.5 + 7451.1)/2 = 7487.8\)

Note: If a PSU is selected into the sample more than once, you’ll include its estimate multiple times

PPSWR: Estimating the Variance of the Population Total

PPSWOR: Estimating the Population Total

Horvitz-Thompson Estimator for the total under PPSWOR: \[\hat{t}_{HT} = \sum_{i \in \mathcal{S}} \frac{t_i}{\pi_i} = \sum_{i \in \mathcal{S}} \frac{t_i}{n \psi_i} = \sum_{i \in \mathcal{S}} w_i t_i\]

• For the population of schools, I used Stata to take a PPSWOR sample of \(n=2\):

• My estimate of the total number of females: \[\hat{t}_{HT} = \sum_{i \in \mathcal{S}} w_i t_i = 14.11 \times 267 + 22.37 \times 166 = 7482\]

PPSWOR: Estimating the Variance (con’t)

Problems With These Variance Estimators (con’t)

• For my PPSWOR sample that contains schools 14 and 24:

  • Using with-replacement variance estimation
  • Notice there is no FPC for PPS sampling

des.ppswor <- svydesign(id = ~1, weight = ~wt, data=samp_pps)
svytotal(~totfemale, des.ppswor)

           total     SE
totfemale 7482.1 54.728

• \(\hat{t}_{HT} = 7482\) (same as the hand-calculation)

• \(\widehat{SE}_{WR}(\hat{t}_{HT}) = 54.7\)

• Compare to without-replacement variance estimation:

  • \(\widehat{SE}_{HT}(\hat{t}_{HT}) = \text{N/A}\) (variance estimate is negative!)

  • \(\widehat{SE}_{SYG}(\hat{t}_{HT}) = 53.1\) (slightly smaller than with-replacement estimate)

Activity 14.1 (Parts 1 & 2)

PPS Sampling (Parts 1 & 2)

Comparing the Designs

• Two designs compared for sampling schools: (A) SRS, (B) PPSWOR

• Both provide unbiased estimates of the true total

• In terms of theoretical variances, if size variable is related to \(t\), \(V_{SRS} > V_{PPSWR} > V_{PPSWOR}\)

  • In a single set of samples you might not see the gain (remember: variance is over repeated samples)

• For the schools data, relationship between size and totals:

• Let’s look at our estimates for total number of females and total number of smokers. What do we see?