Choose your reading route
eduKateSG · Why Mathematics? · Horvitz–Thompson Estimation: Inclusion Probabilities and Unbiased Survey Totals
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 8–14
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 15–21
Check assumptions
See where design and calibration matter.Route 4 · Chapters 22–28
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 29–34
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when people, schools, households or businesses have different chances of being sampled? The Horvitz–Thompson estimator divides each observed value by its inclusion probability. In expectation, that inverse-probability contribution restores every population unit exactly once.
Did You Know? A sampled unit with inclusion probability 0.20 has design weight 5. It represents itself plus four comparable population units under the sampling design; it is not five duplicate respondents.
Full chapter index
Chapters 1–7 · Understand the question
Chapters 8–14 · Reproduce the example
Chapters 15–21 · Check assumptions
Chapters 22–28 · Explore applications
Chapters 29–34 · Study and verify
Probability samples often select large establishments, rare groups or geographic units at different rates. A raw sample sum then over-represents units with high selection probability.
For population total T=sum y_i, the estimator is the sum over sampled units of y_i/pi_i, where pi_i is the first-order inclusion probability and every pi_i must be positive.
Statistics Canada describes the design-based unbiased estimator and defines the design weight d_i=1/pi_i. Read the official survey-methodology article.
Suppose sampled units have values 8, 12 and 20 with inclusion probabilities 0.5, 0.25 and 0.4. Their design weights are 2, 4 and 2.5.
Weighted contributions are 16, 48 and 50. The Horvitz–Thompson total estimate is 16+48+50=114.
| Value | pi | Weight | Contribution |
|---|---|---|---|
| 8 | 0.5 | 2 | 16 |
| 12 | 0.25 | 4 | 48 |
| 20 | 0.4 | 2.5 | 50 |
Let I_i be the sample-inclusion indicator. Since E(I_i)=pi_i, E[I_i y_i/pi_i]=y_i. Summing over the population yields the true finite-population total.
| Contribution sum |
|---|
| 114 |
Set every y_i=1. The estimator becomes the sum of design weights, an estimate of population size when the design and frame support it.
A common Hájek mean divides the HT outcome total by the HT population-size total. This ratio is not exactly design-unbiased but can be more stable and scale-correct.
Variance depends on pi_ij, the probability that units i and j are both sampled. First-order weights alone do not generally determine an exact design variance.
Statistics Canada gives the HT variance in terms of first- and second-order inclusion probabilities and pairwise weighted-value differences. Read the official section.
If some pi_ij are zero, an unbiased variance estimator may be unavailable under the usual formula even when the point estimator remains defined. Conservative alternatives require theory, not ad hoc replacement.
Choosing pi_i related to a size measure can reduce variance when that measure predicts y_i. Extremely small pi_i still create large weights and influential contributions.
A unit with pi_i=1 has weight 1 and represents itself. Its certainty status must be handled correctly in variance strata and PSU coding.
The original design weight addresses sampling. Later nonresponse factors use estimated response propensities and add assumptions; they are not known inclusion probabilities.
Calibration changes weights so auxiliary totals match known controls. Final weights may improve precision but no longer equal simple inverse inclusion probabilities.
The unweighted sample sum here is 40, which estimates neither the population total nor the weighted total 114 when selection probabilities differ.
Both divide by a probability-like quantity to retarget an expectation. Survey HT inference concerns a finite population and a sampling design; importance sampling usually concerns a proposal and target distribution.
CHAPTER 18 OF 34 · Check assumptions
18. Comparison with inverse-probability treatment weighting
Causal weights use treatment or observation propensities under causal assumptions. HT design weights arise from known probability sampling and do not by themselves identify a treatment effect.
Check pi_i in (0,1], confirm weight=1/pi_i, inspect weight range and weighted contributions, verify the sample universe and identify certainty and influential units.
Reproduce total 114, enumerate all samples in a tiny design to verify unbiasedness, compare empirical variance with the pi_ij formula and test weight coding.
HT estimation cannot recover units with zero inclusion probability. Large weights, frame errors, nonresponse, measurement error and unavailable joint probabilities can limit precision or validity.
List values, inclusion probabilities, weights and weighted contributions. Add a row for their sums and a separate sheet for pairwise pi_ij.
Unit-test weights 2, 4 and 2.5 and total 114. Enumerate tiny-design samples and average estimates using their design probabilities.
Survey statisticians, auditors, official-data analysts and experimental designers use inclusion-probability estimators.
Create a five-unit population, assign unequal probabilities, enumerate possible samples and verify the long-run average of HT totals.
A weight is not a clone count, known selection probability is not estimated response propensity, and unbiased does not mean low variance.
CHAPTER 27 OF 34 · Explore applications
27. Connection to recent mathematics articles
The survey-domain guide uses zero-valued domain contributions. The design-effect guide explains the precision cost of unequal weights.
Students should derive E(I_i/pi_i)=1. Parents can ask whether a published weight is a base weight or a final adjusted weight.
- Calculate three weights.
- Reproduce total 114.
- Prove unbiasedness.
- Explain pi_ij.
- Diagnose a tiny pi_i.
### What must be positive? Every first-order inclusion probability.
### Is HT always precise? No.
### Are final weights always 1/pi? No.
### Does HT prove causality? No.
Report target population, sampling design, pi_i, weight stages, certainty units, estimator, pi_ij or variance method, influential contributions and nonsampling-error limits.
Multiplying inclusion indicator I_i by y_i/pi_i gives expected contribution y_i. The entire estimator follows by linearity of expectation.
Confirm positive probabilities, correct inverse weights, finite-population target, joint-inclusion variance information, weight-stage labels and no guarantee of small variance.
Continue through the Mathematics Learning Hub. Recalculate one weighted total or calibration constraint and explain what population quantity it represents.
Extended practice laboratory
Practice case 1: Define population
Setup: Start from the article’s Horvitz-Thompson mechanism for case 1. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 2: List values
Setup: Start from the article’s Horvitz-Thompson mechanism for case 2. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 3: List inclusion probabilities
Setup: Start from the article’s Horvitz-Thompson mechanism for case 3. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 4: Invert probabilities
Setup: Start from the article’s Horvitz-Thompson mechanism for case 4. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 5: Calculate contribution 1
Setup: Start from the article’s Horvitz-Thompson mechanism for case 5. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 6: Calculate all contributions
Setup: Start from the article’s Horvitz-Thompson mechanism for case 6. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 7: Sum total
Setup: Start from the article’s Horvitz-Thompson mechanism for case 7. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 8: Set y to one
Setup: Start from the article’s Horvitz-Thompson mechanism for case 8. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 9: Calculate weighted mean
Setup: Start from the article’s Horvitz-Thompson mechanism for case 9. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 10: Add joint probabilities
Setup: Start from the article’s Horvitz-Thompson mechanism for case 10. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 11: Calculate variance
Setup: Start from the article’s Horvitz-Thompson mechanism for case 11. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 12: Add certainty unit
Setup: Start from the article’s Horvitz-Thompson mechanism for case 12. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 13: Use PPS
Setup: Start from the article’s Horvitz-Thompson mechanism for case 13. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 14: Add tiny probability
Setup: Start from the article’s Horvitz-Thompson mechanism for case 14. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 15: Add nonresponse factor
Setup: Start from the article’s Horvitz-Thompson mechanism for case 15. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 16: Add calibration factor
Setup: Start from the article’s Horvitz-Thompson mechanism for case 16. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 17: Compare unweighted
Setup: Start from the article’s Horvitz-Thompson mechanism for case 17. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 18: Compare importance sampling
Setup: Start from the article’s Horvitz-Thompson mechanism for case 18. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 19: Compare causal weights
Setup: Start from the article’s Horvitz-Thompson mechanism for case 19. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 20: Enumerate samples
Setup: Start from the article’s Horvitz-Thompson mechanism for case 20. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 21: Audit universe
Setup: Start from the article’s Horvitz-Thompson mechanism for case 21. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 22: Audit weights
Setup: Start from the article’s Horvitz-Thompson mechanism for case 22. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 23: Audit variance
Setup: Start from the article’s Horvitz-Thompson mechanism for case 23. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 24: Name limits
Setup: Start from the article’s Horvitz-Thompson mechanism for case 24. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 25: Write conclusion
Setup: Start from the article’s Horvitz-Thompson mechanism for case 25. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.