Cluster samples and choosing a design
The people you want to ask live in a hundred and fifty buildings across a city, and nobody has a list of them. How can you still give every one of them a known, fair chance without knocking on doors all over town?
Where you are 4 in Random sampling
When you reach for it
A stratified random sample splits the population into groups and draws some members at random from every group. That needs a list of every member, and it sends interviewers into every group. Sometimes neither is practical. A city’s housing office may list its 150 apartment buildings, about 40 households each, without listing the households, and an interviewer who must walk to each sampled door spends more time traveling than asking.
Reach for a cluster sample when the population already comes in ready-made groups (buildings, city blocks, classrooms, shipping crates), you can list the groups even if you cannot list the members, and reaching a group is the expensive part. You choose a few whole groups at random and take everyone in them.
The procedure
The definition. Split the population into groups called clusters, so that every member is in exactly one cluster. Take a simple random sample of the clusters, then measure every member of each chosen cluster. The clusters are numbered and drawn exactly as individuals are drawn for a simple random sample: the chance device picks clusters, not people.
Each member’s chance. A member is in the sample exactly when its cluster is chosen. If k of K clusters are drawn, each cluster’s chance is \displaystyle \frac{k}{K}, so each member’s chance is \displaystyle \frac{k}{K} too, in a large cluster or a small one. When clusters differ in size, the sample size is not fixed in advance: it is the total size of whichever clusters come up.
Good clusters look like the population. The estimate changes from sample to sample only through which clusters are drawn. If every cluster holds a mix much like the whole population, any few clusters give nearly the right answer. If each cluster is alike inside and the clusters differ from each other, the answer rides on the luck of the draw, and k clusters behave more like k answers than like hundreds. This is the reverse of stratifying. Strata should be alike inside and different from each other, because every stratum is sampled; clusters should be mixed inside and like each other, because most clusters are never seen.
Cost against precision. For the same number of members, a cluster sample usually varies more than a simple random sample, since members of one cluster tend to resemble each other at least a little. But each member costs far less, because the interviewer goes to k places instead of hundreds, and the savings can buy more clusters. Example 4 puts dollar figures on the trade.
Cluster or stratified? Both designs begin by splitting the population into groups, so they are easily confused. One question settles it: does every group appear in the sample?
| Stratified random sample | Cluster sample | |
|---|---|---|
| Groups sampled | every one | a few, chosen at random |
| Members taken from a sampled group | some, chosen at random | all |
| Good groups are | alike inside, different from each other | mixed inside, like each other |
| Main reason to use it | a steadier estimate, every group represented | lower cost, or no list of members |
Choosing a design. No design is best everywhere. Each one fits a situation:
| Design | Needs | Fits when |
|---|---|---|
| Simple random | a list of every member | members are easy to reach once listed |
| Stratified | a list giving each member’s group | the groups answer differently |
| Cluster | a list of groups | groups are each mixed, and reaching a group is the cost |
| Systematic | an ordered list or stream of members | the order has no repeating pattern |
The systematic design, every so many members from a random start, is the subject of systematic samples. Whatever the design, members missing from the frame get no chance at all, the problem of undercoverage.
Worked examples
Drawing eight buildings
Iratxe works for a made-up city whose housing office lists its 150 apartment buildings, labeled 001 to 150, holding 6{,}000 households in all. She will choose 8 buildings and survey every household in them. Use this row of random digits, from its first digit, to choose the buildings. Then find each household’s chance of being surveyed and the number of households she should expect.
The labels have three digits, so read blocks of three across the gaps: 047, 392, 118, 150, 863, 009, 047, 131, 076, 000, 102, 144. Blocks 392 and 863 are above 150 and 000 is no building’s label, so they are skipped; the second 047 repeats a building already chosen and is skipped too. The buildings are 047, 118, 150, 009, 131, 076, 102 and 144.
A household is surveyed exactly when its building is chosen, so its chance is
about 5.3\%, the same in a large building as in a small one. With about 40 households per building she should expect about 8 \times 40 = 320 households; the exact count depends on which buildings come up.
The estimate from whole buildings
Iratxe’s 8 buildings hold 38, 44, 40, 36, 42, 41, 39 and 45 households. She asks every household whether it would use a proposed bus route, and the numbers saying yes are 21, 25, 19, 22, 23, 20, 24 and 26. Find the sample size, estimate the proportion of all households that would use the route, and estimate how many of the city’s 6{,}000 households that is.
SolutionThe sample is every household in the chosen buildings:
Of these, 21 + 25 + 19 + 22 + 23 + 20 + 24 + 26 = 180 said yes, so the sample proportion is
About 55.4\% of the city’s households are estimated to use the route, about 0.554 \times 6000 \approx 3{,}323 households. The buildings’ own proportions run only from \displaystyle \frac{19}{40} = 0.475 to \displaystyle \frac{24}{39} \approx 0.615: each building looks roughly like the others, as good clusters should, so a different eight buildings would likely give a similar estimate.
Mixed buildings against alike buildings
Lupe pictures a made-up complex of 4 buildings with 2 households each, of sizes 1, 1, 2, 2, 4, 5, 5 and 6 people. She will choose 2 buildings at random and count everyone in them. Compare two ways the same households could be housed, by listing every possible sample mean. Mixed: building A holds sizes 1 and 5, B holds 2 and 6, C holds 1 and 4, D holds 2 and 5. Alike: A holds 1 and 1, B 2 and 2, C 4 and 5, D 5 and 6.
SolutionThe population mean is \displaystyle \frac{26}{8} = 3.25 people. There are 6 ways to choose 2 of the 4 buildings, and since the buildings are the same size, each sample mean is the average of the two building means.
Mixed: the building means are 3, 4, 2.5 and 3.5, so the six sample means are 3.5 (A and B), 2.75, 3.25, 3.25, 3.75 and 3 (C and D). They run from 2.75 to 3.75.
Alike: the building means are 1, 2, 4.5 and 5.5, so the sample means are 1.5 (A and B), 2.75, 3.25, 3.25, 3.75 and 5 (C and D). They run from 1.5 to 5.
Both sets of six means average exactly 3.25, so the design is fair either way. The difference is spread. With mixed buildings every sample lands within 0.5 of the truth; with alike buildings, drawing A and B misses by 1.75 people. The households and the design are the same; only the grouping changed. Alike buildings would make fine strata and poor clusters.
What the savings buy
Ofa prices the survey. Visiting a building costs 200 dollars (travel and an interviewer’s setup) and each interview costs 10 dollars. (a) Find the cost of Iratxe’s cluster sample of 325 households in 8 buildings. (b) In a made-up computer trial, a simple random sample of 325 households, drawn from a list the office would first have to build, lands in 134 different buildings. Find its cost. (c) For that same cost, how many whole buildings of 40 households could a cluster sample cover?
Solution(a) 8(200) + 325(10) = 1600 + 3250 = 4850 dollars.
(b) The households are the same in number but scattered, so nearly every building needs a visit: 134(200) + 325(10) = 26800 + 3250 = 30050 dollars, more than six times as much.
(c) One whole building costs 200 + 40(10) = 600 dollars, and \displaystyle \frac{30050}{600} \approx 50.1, so 50 buildings, about 2{,}000 households.
For equal money the choice is 325 scattered households or 50 whole buildings. If the buildings are each mixed, the 50 buildings give much the steadier estimate. If households in a building answer alike, the 2{,}000 households count for less than their number suggests, but 50 clusters is still many separate draws. A cluster sample’s precision rests mainly on how many clusters it has, so that is where the savings belong.
Cluster, stratified or neither
Name each design and say what the groups are. (a) Lupe picks 2 households at random from every one of the 150 buildings. (b) Ofa labels all 1{,}200 floors of the 150 buildings, picks 30 floors at random and surveys every household on them. (c) The buildings lie in 5 neighborhoods, and Iratxe picks 2 buildings at random in each neighborhood and surveys every household in them. (d) Ofa surveys every household in the 8 buildings nearest the housing office.
Solution(a) Every building appears, with some of its households chosen at random: a stratified random sample with the buildings as strata.
(b) Only some floors appear, each in full: a cluster sample with floors as the clusters. Each household’s chance is \displaystyle \frac{30}{1200} = \frac{1}{40}.
(c) Both at once. Every neighborhood appears, so the neighborhoods are strata; inside each one, a few whole buildings are drawn, so the buildings are clusters. This is a stratified cluster sample.
(d) Whole buildings again, but chosen by distance rather than by chance, so it is not a cluster sample at all: buildings near the office are a convenience sample, and a block of neighboring buildings may share its streets, rents and bus service.
Choosing a design
Choose a design for each survey and give the reasons. (a) The housing office wants the proportion of households whose heat failed at least once last winter. Each building has its own boiler, and the office’s records list every household by building. (b) Lupe wants residents’ views on the bus route, asked at the door, and has only the list of buildings. (c) The water utility will email a two-question survey about water taste; its billing list holds every household’s email address, in account-number order.
Solution(a) A failed boiler fails for the whole building, so households in one building answer alike and buildings differ: good strata, poor clusters. Eight whole buildings would amount to about eight answers. Since the records give each household’s building, stratify by building and draw a few households at random from each.
(b) No list of households exists and interviewers must travel, so a cluster sample of buildings fits. Whether a household would ride the bus depends on who lives there, and each building holds a mix of households, so buildings are reasonable clusters. This is Iratxe’s plan.
(c) Email costs nothing per address and the list is complete, so draw a simple random sample of households from it. Taking every so many addresses down the list from a random start would also serve, unless account numbers follow a pattern tied to the answer.
Boundaries
Clusters are chosen by chance. Whole groups picked for convenience, as in Example 5(d), make a convenience sample. Every cluster needs the same known chance.
Every member is in exactly one cluster. A household listed under two buildings has two chances; a building missing from the list gives its residents none.
Few clusters mean few draws. Precision comes mainly from the number of clusters, not the number of members. Clusters alike inside make a large sample behave like a small one.
It is not a simple random sample. Every member’s chance is \displaystyle \frac{k}{K}, but a sample with a few members from every cluster is impossible. Methods built for simple random samples need adjusting for cluster data, and the sample size varies when clusters differ in size.
Taking everyone is the plain version. When clusters are large, surveyors often draw members at random inside each chosen cluster, sampling in two stages. That design follows the same reasoning but is not developed here.
Where this goes next
Every design here assumes a frame that reaches everyone; what happens when it does not is undercoverage. Random selection decides who a study speaks for, while random assignment in an experiment decides what it can say about cause, the distinction drawn in random assignment and random selection. The stratifying idea of grouping alike units returns in experiments as blocking.
Exercises
Drawing clusters and finding chances
- From the first digit, read the row 12860 40931 28017 99915 00621 45387 in blocks of three to choose 6 of the buildings labeled 001 to 150.
- Iratxe chooses 12 of the 150 buildings at random. Find each household’s chance of being surveyed and the number of households she should expect, at about 40 per building.
- A made-up complex has 24 buildings, and 3 are chosen at random. They hold 30, 18 and 52 households. Find the sample size and the chance of a household in each chosen building.
- Five randomly chosen buildings hold 36, 48, 40, 33 and 43 households, of which 18, 30, 22, 15 and 25 own a car. Estimate the proportion of the city’s households that own a car, and the number among all 6{,}000.
Mixed or alike clusters
- A made-up complex has 4 buildings of 2 households each. The numbers of children per household are 0 and 3 in A, 1 and 2 in B, 0 and 2 in C, 1 and 3 in D. List every possible sample mean for a cluster sample of 2 buildings, and give the range.
- The same eight households are rehoused: 0 and 0 in A, 1 and 1 in B, 2 and 2 in C, 3 and 3 in D. Repeat exercise 5, and explain what changed.
- Would whole buildings make good clusters for each question? (a) Did your building’s elevator break down this month? (b) How many hours a week does the person answering work? (c) What rent does your household pay, when rents differ widely from building to building?
Cost and precision
- Visiting a building costs 250 dollars and each interview 8 dollars. Find the cost of a cluster sample of 10 buildings holding 400 households, and of a simple random sample of 400 households that lands in 140 buildings.
- With the costs of exercise 8 and buildings of 40 households, how many whole buildings can a budget of 12{,}000 dollars cover, and about how many households is that?
Naming and choosing a design
- Ofa labels all 6{,}000 households on the water utility’s list and draws 300 of them at random. Name the design.
- Lupe chooses 15 of the 150 buildings at random and surveys every household in them. Name the design and give each household’s chance.
- The 150 buildings are split into 60 with an elevator and 90 without. Iratxe chooses 4 buildings at random from each group and surveys every household in them. Name the design and give each household’s chance in each group.
- From each of the 150 buildings Ofa chooses 1 floor at random and surveys every household on it. Name the design and say what the strata and the clusters are.
- Choose a design and give reasons. (a) Inspectors must enter homes to check whether each household’s smoke alarm works, and the office lists only buildings. (b) The office wants satisfaction with rent, which differs widely by building, and lists every household by building.
Reasoning
- In Example 1 every household’s chance is \displaystyle \frac{8}{150}. Explain why Iratxe’s sample is still not a simple random sample.
- Lupe says: “Cluster and stratified samples are the same thing; both split the population into groups.” Explain the difference.
- Explain why good strata are alike inside but good clusters are mixed inside.
- Ofa says: “My cluster sample has 325 households, so it is exactly as precise as a simple random sample of 325.” Explain what is wrong.
Answers
- Buildings 128, 093, 017, 150, 062 and 145. Blocks 604 and 999 are above 150 and are skipped, and the second 128 is a repeat; 387 is not needed.
- Chance \displaystyle \frac{12}{150} = \frac{2}{25} = 8\%; about 12 \times 40 = 480 households.
- Sample size 30 + 18 + 52 = 100. Every household’s chance is \displaystyle \frac{3}{24} = \frac{1}{8}, in the small building and the large one alike.
- 18 + 30 + 22 + 15 + 25 = 110 of 36 + 48 + 40 + 33 + 43 = 200, so \displaystyle \frac{110}{200} = 0.55; about 0.55 \times 6000 = 3300 households.
- Building means 1.5, 1.5, 1 and 2; sample means 1.5, 1.25, 1.75, 1.25, 1.75 and 1.5, a range of 1.25 to 1.75. They average 1.5, the population mean.
- Building means 0, 1, 2 and 3; sample means 0.5, 1, 1.5, 1.5, 2 and 2.5, a range of 0.5 to 2.5. The average is still 1.5, but each building is now alike inside and the buildings differ, so the luck of which buildings are drawn moves the mean much more.
- (a) No: everyone in a building shares the elevator and answers alike. (b) Yes, reasonably: hours of work vary from household to household inside any building. (c) No: rent is tied to the building, so buildings differ and are alike inside.
- Cluster: 10(250) + 400(8) = 2500 + 3200 = 5700 dollars. Simple random: 140(250) + 400(8) = 35000 + 3200 = 38200 dollars.
- One building costs 250 + 40(8) = 570 dollars, and \displaystyle \frac{12000}{570} \approx 21.05, so 21 buildings, about 840 households.
- A simple random sample of households: every household is listed and drawn individually.
- A cluster sample with buildings as clusters; each household’s chance is \displaystyle \frac{15}{150} = \frac{1}{10}.
- Stratified (elevator or not), with a cluster sample of buildings inside each stratum. Chances \displaystyle \frac{4}{60} = \frac{1}{15} with an elevator and \displaystyle \frac{4}{90} = \frac{2}{45} without, so the results need weighting, as in a stratified sample with unequal allocation.
- Every building appears, so the buildings are strata; inside each, one whole floor is taken, so floors are the clusters.
- (a) A cluster sample of buildings: there is no household list, inspectors must travel, and alarm upkeep varies from home to home. (b) Stratify by building: rent satisfaction is alike inside a building and differs between buildings, and the list gives each household’s building.
- Every sample is made of whole buildings; a group of households from many buildings, all equally possible in a simple random sample, has no chance at all.
- A stratified sample takes some members from every group; a cluster sample takes every member of a few groups. Good strata are alike inside; good clusters are mixed inside.
- Every stratum is sampled, so strata that are alike inside are pinned down by a few members each. Most clusters are never seen, so each chosen cluster has to stand in for the whole population.
- Households in one building tend to resemble each other, so 325 households from 8 buildings carry less information than 325 scattered ones; the estimate rests on only 8 draws of buildings.