Why Use Two Stages?
A population may be spread across many natural groups, such as schools, clinics, villages, or apartment buildings. Listing every individual in the population and selecting a simple random sample may be difficult or expensive. A cluster design can make data collection more manageable by selecting some groups first. In a two-stage cluster sample, the process then continues: a random sample of individuals is selected from within each chosen group.
The second selection is the defining feature. If a survey includes every individual in each selected group, it is a one-stage cluster sample. If it randomly selects only some individuals within selected groups, it is a two-stage cluster sample. The distinction matters because the designs include different people, require different lists, and can give individuals different chances of selection.
For example, if a city has 40 schools, the first-stage units are the schools. The second-stage units are the individual students on the selected schools’ student lists. The same people are not chosen directly from the entire citywide student list. Instead, the second-stage sample is drawn only from the selected schools.
Designing the Two Stages
A clear design names the target population, the clusters, how clusters are selected, and how individuals are selected within chosen clusters. It also specifies how many clusters and individuals will be sampled. A probability sample requires a defined random selection procedure and a known, nonzero chance of selection for every population individual the design is intended to cover.
State who or what the study is about, and define the groups that partition that population. Each individual should belong to the appropriate cluster for the design.
List the clusters from which the sample will be drawn. Randomly select the planned number of clusters using a specified probability method.
Obtain a list of the eligible individuals in every selected cluster. This list determines who can be sampled at the second stage.
For each selected cluster, randomly choose the planned number of individuals from its list. Record how the selection was made.
Confirm that the cluster list and the within-cluster lists cover the intended population, and record the selection chances at both stages.
Simple random sampling without replacement can be used at either stage. In that case, the specified number of distinct clusters is selected from the cluster list, and distinct individuals are selected from each chosen cluster’s list. Another probability procedure may also be used, but the selection chances must be understood. The method at the second stage cannot be left to convenience, such as choosing whichever individuals are easiest to contact, while still describing the sample as a probability sample.
Worked Example: Sampling Students Through Schools
A hypothetical district has 20 schools and wants to survey students about their daily travel time. The district randomly selects 4 schools, then uses a student roster to randomly select 15 students in each selected school. For this example, suppose every school has 60 eligible students and every selected student is invited to respond.
First stage: The clusters are the 20 schools. A simple random sample without replacement selects 4 of them. Each school has probability \(4/20=0.20\) of being selected.
Second stage: In each selected school, 15 of its 60 students are randomly selected. Conditional on that school being selected, each student there has probability \(15/60=0.25\) of selection.
Overall selection chance: A particular student is included only if their school is selected and then the student is selected from that school. Thus the student’s probability of selection is \(0.20(0.25)=0.05\), or 5%. The intended sample has \(4(15)=60\) students.
This is a two-stage cluster sample because only some students from each selected school are sampled. If the district instead surveyed all 60 students in each of the 4 selected schools, that would be a one-stage cluster sample, with a planned sample of \(4(60)=240\) students.
Combining Selection Chances
The chance that an individual enters a two-stage sample depends on both selections. Let cluster \(j\) have probability \(\pi_j\) of being selected at the first stage. Let individual \(i\), who belongs to cluster \(j\), have conditional probability \(\pi_{i\mid j}\) of selection at the second stage, given that cluster \(j\) was selected. Then the individual’s overall probability of inclusion is the product of these chances.
This result follows from the structure of the design: an individual can be included only when both events occur. The second event is defined within the selected cluster, so its relevant chance is conditional on that cluster having been chosen.
When the first-stage clusters are selected by simple random sampling without replacement, \(m\) of \(M\) clusters are chosen, and each cluster has first-stage probability \(m/M\). If \(n_j\) individuals are selected from the \(N_j\) individuals in cluster \(j\) by simple random sampling without replacement, then each individual in that cluster has conditional probability \(n_j/N_j\). Therefore:
The formula makes clear why a design that samples the same number of individuals from every selected cluster does not always give every individual the same chance. When cluster sizes differ, the within-cluster sampling fractions \(n_j/N_j\) differ. The first-stage chances may be equal across clusters, but the overall chances for individuals need not be.
Worked Example: Unequal Cluster Sizes
A hypothetical organization has 20 branches. It randomly selects 4 branches and randomly samples 10 employees from each chosen branch. One branch has 100 employees and another has 50 employees. Assume all employees are eligible and the employee lists are complete.
A particular employee’s branch has first-stage selection probability \(4/20=0.20\). In the 100-employee branch, the employee’s conditional selection probability is \(10/100=0.10\). Their overall probability is:
Thus, each employee in that branch has a 2% chance of selection. In the 50-employee branch, the within-branch probability is \(10/50=0.20\), so an employee there has overall probability:
That is a 4% chance. The second branch’s employees have twice the individual selection chance of those in the first branch, even though both branches have the same chance of being selected. The design is still a two-stage probability sample, but the individual chances are unequal.
What the Second Stage Changes
The second stage gives the researcher control over how many individuals are contacted within each selected group. That can reduce the burden of collecting information from every member of a chosen cluster. For instance, interviewing every resident of a few large neighborhoods may take more time than interviewing a smaller random sample within each selected neighborhood.
However, selecting a manageable number within a cluster does not by itself guarantee that the sample represents the target population well. The lists and random procedures matter. If a cluster is omitted from the first-stage list, its members cannot be selected by this design. If an individual is missing from a selected cluster’s roster, that individual cannot be selected at the second stage. These are coverage problems in the design, not problems solved simply by adding another random draw.
It is also important to distinguish selection from participation. A randomly selected individual who does not respond was still selected by the design, but their information is unavailable. A random procedure for choosing people does not ensure that everyone responds. The sample design describes who is selected; the achieved responses may differ.
Worked Example: Planning a Two-Stage Clinic Survey
A hypothetical health system wants to survey adult patients who visited its 12 clinics during a defined month. It can make a list of eligible visits for each clinic. The plan is to select 3 clinics at random and then select 25 eligible patients at random from each selected clinic, provided each chosen clinic has at least 25 eligible patients.
Identify the stages. The 12 clinics are the clusters and are the first-stage units. The eligible patients listed within a chosen clinic are the second-stage units.
State the random procedures. Number the 12 clinics and select 3 by simple random sampling without replacement. For each selected clinic, number the eligible patients on its list and select 25 by simple random sampling without replacement. If a clinic has fewer than 25 eligible patients, the plan must be revised in advance; selecting all available patients there would be a different second-stage rule.
Check the sample size and stages. Under the stated condition, the planned sample size is \(3(25)=75\) selected patients. The design is two-stage because the system selects a subset of patients from each selected clinic, rather than surveying every eligible patient in those clinics.
State what is still needed. To calculate a particular patient’s overall selection probability, the system needs the number of eligible patients in that patient’s clinic. The first-stage probability is \(3/12=0.25\), while the conditional second-stage probability is \(25/N_j\) for a clinic with \(N_j\) eligible patients. Thus the patient’s chance is \(0.25(25/N_j)\), which varies if clinic sizes vary.
Common Mistake: Calling Every Cluster Design Two-Stage
A frequent mistake is to say, “We randomly selected several schools, so we used two-stage cluster sampling.” That description names only the first stage. If the survey then includes every student in each selected school, it is one-stage cluster sampling. A two-stage design needs a second random selection of individuals within the selected schools.
A second mistake is to assume that equal numbers from each selected cluster mean equal chances for all individuals. As the unequal-branch example showed, the chance within a cluster is \(n_j/N_j\). If cluster sizes differ while the same \(n_j\) is sampled from each, the within-cluster fractions differ. Check the product of the first- and second-stage probabilities for individuals in each type of cluster before claiming their chances are equal.
A useful description of a proposed design should therefore answer these questions: What are the clusters? How are clusters selected? What list is used inside each selected cluster? How are individuals selected from that list? How many are selected at each stage? These details allow another person to distinguish the design from a one-stage cluster sample and determine the individual selection chances.
Check Your Understanding
Use the definitions and selection-probability result to answer these questions.
- A survey randomly selects 5 apartment buildings and interviews every resident of each chosen building. Is this one-stage or two-stage cluster sampling? Explain.
- A district has 30 schools and randomly selects 6. In each selected school, it randomly samples 12 students from a roster of 48. What is a student’s overall probability of selection, assuming the same roster size in every school?
- Two clusters have 80 and 40 individuals. Four clusters are selected from 20, and 8 individuals are sampled from each selected cluster. Calculate an individual’s overall selection probability in each cluster.
- Why is randomly selecting individuals only from the selected clusters a second stage, rather than a second sample from the whole population?
- A plan says, “Choose some clinics at random, then contact patients who are easiest to reach.” What information or change is needed before this can be described as a two-stage probability sample?