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eduKateSG · Why Mathematics? · Multistage Sampling: Conditional Probabilities and Final Survey Weights
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 a population is reached through schools, classes, households or other nested units rather than one flat list? Multistage sampling multiplies conditional selection probabilities across stages. The final weight records the whole path from population frame to observed person.
Did You Know? A student selected with probability 0.2 inside a school is not a 1-in-5 population sample if the school itself had only a 0.1 chance of selection. The final probability is 0.02.
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
A national list of every student may be unavailable or expensive. Designers can sample schools, then classes, then students using frames created at each selected stage.
If a cluster is selected with pi1 and a unit inside it with conditional pi2|1, final inclusion probability is pi=pi1×pi2|1.
The U.S. Census Bureau states that SIPP selects primary sampling units and then addresses within sampled PSUs. Read the official sampling page.
A school has first-stage probability 0.10. Within that selected school, a student has conditional probability 0.20.
The student’s population inclusion probability is 0.10×0.20=0.02.
| School probability | Student conditional probability | Final probability |
|---|---|---|
| 0.1 | 0.2 | 0.02 |
The inverse final probability is 1/0.02=50. The student represents 50 population students under the design before later adjustments.
| Final probability | Base weight |
|---|---|
| 0.02 | 50 |
With additional class probability 0.5 and student probability 0.4, after school probability 0.1, final pi=0.1×0.5×0.4=0.02.
The second-stage list exists only inside selected first-stage units. Its completeness and size must be recorded during fieldwork.
Select large clusters with higher probability, then sample a fixed number of people within each. Proper choices can make final person probabilities closer to equal.
A cluster selected with probability one contributes stage weight one. Later within-cluster sampling still creates a final weight above one.
Census SIPP weighting includes a base selection component and adjustments for subsampling within clusters. Read the official weighting description.
People in the same school or household tend to resemble one another. More observations inside one cluster often add less independent information than observations across clusters.
PSUs may be stratified before selection, and addresses may be stratified inside PSUs. Variance estimation needs every relevant stratum and cluster code.
A high sampling fraction can matter at any without-replacement stage. Corrections are stage-specific and need the correct frame size.
Response probability is not part of known selection probability. Nonresponse factors multiply later and rely on additional assumptions.
Multistage designs move through unit levels. Two-phase designs gather different information in successive samples, often with the same units nested by phase.
Selecting clusters and observing every element inside them is one-stage cluster sampling. Subsampling elements makes the design multistage.
Ordinary software treats records as independent. Survey analysis must specify weights, strata, PSUs and any lower-stage information required by the method.
Reconcile stage frames, probabilities, weights, certainty units, cluster IDs, sample counts, response and final weight distributions.
Reproduce 0.02 and 50, build a three-stage tree, simulate clustered outcomes and compare naive with design-aware standard errors.
Correct probability multiplication cannot fix incomplete cluster frames, unmeasured within-cluster units, nonresponse, measurement error or too few PSUs.
Store every stage ID, frame size, selected count, conditional probability, stage weight, cumulative probability and cumulative weight.
Build probability trees, unit-test products and inverses, validate counts at each node and analyze with design-aware variance estimators.
Household surveys, education research, health studies and audits use multistage designs.
Simulate schools, classes and students with intraclass correlation; change the number of schools and students per school under fixed cost.
A within-school sampling rate is not a population probability, more pupils in one school do not equal more schools, and final weights are not just the last-stage inverse.
CHAPTER 27 OF 34 · Explore applications
27. Connection to recent mathematics articles
The PPS guide explains first-stage cluster selection. The design-effect guide explains dependence.
Students should draw every branch before multiplying. Parents can ask whether a study selected schools, classes and pupils with known probabilities.
- Multiply two stages.
- Invert the result.
- Add a third stage.
- Identify a certainty cluster.
- Explain clustering variance.
### Are stage probabilities added? No, they are multiplied.
### Is the final weight the last-stage inverse? No.
### Can a cluster be certain? Yes.
### Are ordinary standard errors valid? Usually not.
Report frames and unit levels, conditional probabilities, replacement rules, certainty selections, cumulative weights, strata, clusters, response adjustments, FPCs and variance method.
A population unit must pass every selection gate, so its final chance equals the product of conditional gate probabilities.
Confirm complete stage frames, recorded conditional probabilities, product weights, certainty and PPS rules, enough PSUs, cluster-aware variance and clear separation of selection from response.
Continue through the Mathematics Learning Hub. Recalculate one selection probability and explain which population units the resulting weight represents.
Extended practice laboratory
Practice case 1: Define unit levels
Setup: Start from the article’s multistage-sampling 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: Build stage-one frame
Setup: Start from the article’s multistage-sampling 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: Select cluster
Setup: Start from the article’s multistage-sampling 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: Calculate pi1
Setup: Start from the article’s multistage-sampling 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: Build stage-two frame
Setup: Start from the article’s multistage-sampling 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: Select unit
Setup: Start from the article’s multistage-sampling 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: Calculate pi2
Setup: Start from the article’s multistage-sampling 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: Multiply probabilities
Setup: Start from the article’s multistage-sampling 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: Invert weight
Setup: Start from the article’s multistage-sampling 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 third stage
Setup: Start from the article’s multistage-sampling 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: Add PPS first stage
Setup: Start from the article’s multistage-sampling 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 cluster
Setup: Start from the article’s multistage-sampling 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: Add stratification
Setup: Start from the article’s multistage-sampling 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 FPC
Setup: Start from the article’s multistage-sampling 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
Setup: Start from the article’s multistage-sampling 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: Compare two phase
Setup: Start from the article’s multistage-sampling 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 cluster census
Setup: Start from the article’s multistage-sampling 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: Simulate dependence
Setup: Start from the article’s multistage-sampling 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: Increase clusters
Setup: Start from the article’s multistage-sampling 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: Increase within cluster
Setup: Start from the article’s multistage-sampling 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 frames
Setup: Start from the article’s multistage-sampling 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 multistage-sampling 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 multistage-sampling 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 multistage-sampling 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 multistage-sampling 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.