Unit 5: Sampling Design - Subjective Questions

Sampling design is a definite plan or framework used to select a sample from a given population. It specifies the method by which sampling units will be chosen before data collection begins.

Significance of sampling design:

• It ensures that the selected sample adequately represents the target population.
• It reduces the time, cost, and effort required for a study.
• It helps control sampling errors and selection bias.
• It determines the reliability and validity of statistical generalizations.
• It provides a systematic basis for selecting respondents or other units of analysis.

A carefully developed sampling design enables researchers to draw defensible conclusions about a population from observations made on a smaller group.

The sampling design process generally includes the following steps:

1. Define the target population: Clearly identify the complete group about which conclusions are required.
2. Identify the sampling unit: Decide whether the unit is an individual, household, institution, event, or geographical area.
3. Prepare or select the sampling frame: Obtain a list or operational representation of all population units.
4. Determine the sample size: Decide how many units must be included, considering precision, variability, confidence, cost, and time.
5. Specify the sampling method: Select an appropriate probability or non-probability technique.
6. Develop the sample-selection procedure: Establish precise operational rules for choosing units.
7. Execute the design: Select the sample and collect data according to the prescribed procedure.
8. Evaluate the sample: Examine coverage, non-response, bias, and sampling error before drawing conclusions.

A researcher should consider the following factors:

• Research objectives: The population must correspond directly to the questions being investigated.
• Element: The basic entity from which information is required, such as a consumer, employee, or organization.
• Sampling unit: The unit actually selected at a particular stage of sampling.
• Geographical boundaries: The physical area covered by the investigation.
• Time period: The period to which the study and its findings apply.
• Eligibility criteria: Inclusion and exclusion conditions must be clearly stated.
• Accessibility: The population should be operationally reachable through a suitable frame.
• Level of analysis: The researcher must decide whether conclusions concern individuals, groups, institutions, or regions.

An imprecisely defined population can cause coverage errors and make the study's conclusions ambiguous.

A sampling frame is a list, database, map, register, or other operational representation containing the units from which a sample is selected.

Importance:

• It connects the theoretical target population with the actual selection process.
• It enables the use of random selection procedures.
• It helps identify, contact, and organize sampling units.
• It influences the coverage and representativeness of the sample.

Limitations and frame errors:

• Undercoverage: Some eligible population units are missing.
• Overcoverage: Ineligible units are included.
• Duplication: A unit appears more than once and receives a higher probability of selection.
• Outdated information: Entries may no longer be accurate.
• Incorrect classification: Units may be assigned to the wrong categories.

Researchers should evaluate and update the frame, remove duplicate entries, and disclose remaining coverage limitations.

A good sample possesses the following characteristics:

• Representativeness: It reflects the important characteristics of the population.
• Adequate size: It contains enough observations to produce the required level of precision.
• Freedom from systematic bias: No relevant group is consistently favored or excluded.
• Independence: Selection of one unit should not improperly influence another, unless dependence is deliberately built into the design.
• Accuracy and precision: Estimates should be close to population values and show low variability across repeated samples.
• Appropriateness: The selection method should match the research objectives and population structure.
• Practical feasibility: The sample should be obtainable within available time, budget, and administrative resources.
• Measurability of error: In probability sampling, the design should permit estimation of sampling error.

A large sample alone is not necessarily good; representativeness and sound selection procedures are equally important.

• Accuracy refers to how close a sample estimate is to the true population parameter. It is affected by both sampling and non-sampling errors.
• Precision refers to the consistency of estimates across repeated samples. A precise estimator has a small standard error or variance.
• Representativeness indicates how well the composition and relevant characteristics of the sample reflect those of the target population.
• Reliability refers to the stability and consistency of the results when the sampling and measurement procedures are repeated under similar conditions.

For an estimator of a parameter , its mean squared error is:

Thus, a precise estimate may still be inaccurate if it has substantial bias. A good sampling design seeks both low variance and low bias.

Sampling designs are broadly classified into two categories:

1. Probability or random sampling:

• Every population unit has a known and non-zero probability of selection.
• Selection is governed by a random mechanism.
• Sampling error can generally be estimated.
• Statistical inference from the sample to the population is defensible.
• Examples include simple random, systematic, stratified, cluster, and multistage sampling.

2. Non-probability or non-random sampling:

• Selection probabilities are unknown or may be zero for some units.
• Units are selected through convenience, judgment, referrals, or fixed quotas.
• Sampling error cannot be estimated using standard probability theory.
• Generalization to the entire population is limited.
• Examples include convenience, purposive, quota, snowball, and volunteer sampling.

Probability designs emphasize statistical representativeness, whereas non-probability designs emphasize accessibility, expert selection, or exploration.

In simple random sampling, every possible sample of a specified size has an equal chance of being selected from the population.

• With replacement: After a unit is selected, it is returned to the population and may be selected again. Each draw is independent.
• Without replacement: A selected unit is not returned, so it cannot appear again in the same sample. This is more common in surveys.

For a population of size and a sample of size , the inclusion probability of each unit under simple random sampling without replacement is:

Advantages:

• Easy to understand and implement when a complete frame exists.
• Reduces conscious selection bias.
• Supports standard estimates of sampling error.
• Gives every unit an equal selection opportunity.

Limitations:

• Requires a complete and accurate sampling frame.
• May be costly for geographically dispersed populations.
• Small subgroups may be inadequately represented by chance.
• It does not use prior information about population structure.

In systematic random sampling, units are selected at a fixed interval from an ordered sampling frame after choosing a random starting point.

For a population of size and desired sample size , the sampling interval is approximately:

Procedure:

1. Arrange and number all units in the frame.
2. Calculate the interval .
3. Select a random start between and .
4. Choose units numbered until the required sample is obtained.

Advantages: It is simple, quick, economical, and spreads the sample across the frame.

Limitation: If the ordering contains a periodic pattern related to the study variable and that period coincides with , the sample may be biased. Random ordering or careful frame examination can reduce this risk.

In stratified random sampling, the population is divided into internally homogeneous and mutually exclusive groups called strata. A random sample is then selected independently from every stratum.

Proportionate allocation: The sample from each stratum is proportional to its population size. For stratum :

where is the stratum size, is the population size, and is the total sample size. This preserves the population's group proportions.

Disproportionate allocation: The sampling fraction differs across strata. Small but important strata may be oversampled, or allocation may reflect variability and cost.

Benefits of stratification:

• Ensures representation of all specified subgroups.
• Permits separate estimates for each stratum.
• Can improve precision when units within strata are similar.

When disproportionate allocation is used, sampling weights may be required to obtain unbiased population-level estimates.

In cluster sampling, the population is divided into natural groups called clusters, such as schools, villages, city blocks, or branches. A random sample of clusters is selected. Researchers may study all units in the selected clusters or sample units within them.

Difference from stratified sampling:

• In stratified sampling, units are selected from every stratum; in cluster sampling, only selected clusters are studied.
• Strata should be internally homogeneous and different from one another; ideal clusters are internally heterogeneous and resemble the population.
• Stratification primarily improves precision and subgroup representation; clustering primarily reduces travel, listing, and administrative costs.
• The sampling unit in stratified sampling is usually an individual element, while the initial sampling unit in cluster sampling is a group.

Cluster sampling is economical for dispersed populations, but similarity among units within clusters can increase sampling variance.

Multistage sampling selects a sample through two or more successive stages, using progressively smaller sampling units at each stage.

For example, in a national survey of university students:

1. Randomly select states as primary sampling units.
2. Select districts within the chosen states.
3. Select universities within the chosen districts.
4. Select departments within those universities.
5. Randomly select students within the chosen departments.

Advantages:

• Suitable for large, geographically dispersed populations.
• Does not require a complete list of all ultimate population elements at the beginning.
• Reduces travel, listing, and administrative costs.
• Allows different sampling techniques at different stages.

Limitations:

• Design and analysis are more complex.
• Selection probabilities must be tracked across stages.
• Clustering may reduce precision.
• Appropriate weights and design-based variance estimators may be necessary.

Simple random sampling:

• Selects individual units entirely at random.
• Requires a complete population frame.
• Is appropriate for relatively homogeneous populations.

Systematic sampling:

• Selects every th unit after a random start.
• Is operationally simpler than repeated random draws.
• Can be affected by periodicity in the frame.

Stratified sampling:

• Divides the population into strata and samples from every stratum.
• Ensures subgroup representation and may increase precision.
• Requires information for classifying all units into strata.

Cluster sampling:

• Selects natural groups and studies units within selected groups.
• Reduces field costs for geographically dispersed populations.
• May produce larger sampling errors due to within-cluster similarity.

The appropriate technique depends on population structure, frame availability, desired precision, subgroup analysis, cost, and geographical distribution.

Convenience sampling is a non-random technique in which units are selected because they are readily available, accessible, or willing to participate. Examples include surveying nearby students or collecting responses from visitors to a website.

Appropriate uses:

• Exploratory or pilot studies.
• Preliminary questionnaire testing.
• Studies with severe time or budget constraints.
• Situations where broad statistical generalization is not the objective.

Limitations:

• Selection probabilities are unknown.
• Accessible participants may differ systematically from the target population.
• Sampling error cannot be validly estimated through standard probability formulas.
• Findings are vulnerable to selection and volunteer bias.
• Generalization to the population is weak.

Researchers should clearly label such a sample as non-probability based and avoid presenting it as statistically representative.

Purposive sampling, also called judgmental sampling, is a non-random method in which the researcher deliberately selects information-rich cases that meet criteria relevant to the study.

Major forms include:

• Typical-case sampling: Selects cases considered ordinary or representative of a common situation.
• Extreme or deviant-case sampling: Examines unusual successes, failures, or rare cases.
• Critical-case sampling: Selects cases that can provide especially important logical or practical evidence.
• Maximum-variation sampling: Includes diverse cases to identify shared patterns across differences.
• Homogeneous sampling: Focuses on participants with similar characteristics.
• Expert sampling: Selects individuals with specialized knowledge or experience.
• Criterion sampling: Includes all cases satisfying predetermined conditions.

Purposive sampling is valuable in qualitative and specialized research, but its dependence on researcher judgment limits statistical generalization.

In quota sampling, the population is divided into categories based on characteristics such as age, gender, occupation, or location. A fixed number or proportion of respondents is assigned to each category, but interviewers select respondents non-randomly until each quota is filled.

Differences from stratified random sampling:

• Both methods divide the population into relevant subgroups.
• Stratified sampling uses random selection within every stratum; quota sampling uses convenience or interviewer judgment within each category.
• Inclusion probabilities are known in stratified sampling but unknown in quota sampling.
• Sampling error can be estimated for a properly designed stratified sample but not for a quota sample using standard probability methods.
• Quota sampling is usually faster and less expensive, but it is more vulnerable to interviewer and selection bias.

Meeting demographic quotas does not guarantee representativeness on characteristics not used to construct those quotas.

Snowball sampling, or chain-referral sampling, begins with a small number of eligible participants called seeds. These participants refer the researcher to other eligible people, who may provide further referrals. The sample therefore grows through social networks.

Usefulness:

• Helps locate members of populations for which no sampling frame exists.
• Builds access through trusted relationships.
• Is useful for exploratory research involving sensitive identities, rare conditions, or informal networks.
• Can reduce recruitment time and cost.

Limitations:

• Participants with larger personal networks are more likely to be recruited.
• The final sample may overrepresent closely connected groups.
• Initial seed selection can strongly influence the results.
• People outside the sampled networks may be excluded.
• Selection probabilities and sampling error are usually unknown.

Ethical safeguards are essential because referrals must not disclose another person's sensitive status without consent.

Sampling error is the difference between a sample estimate and the true population parameter caused by observing only a sample rather than the entire population. For example, a sample mean may differ from the population mean .

For simple random sampling, the estimated standard error of the sample mean may be written as:

Non-sampling error includes all other errors arising during the research process, such as:

• Incomplete or inaccurate sampling frames.
• Non-response by selected participants.
• Misunderstood questions or dishonest responses.
• Interviewer influence.
• Data-entry, coding, and processing mistakes.
• Faulty instruments or operational definitions.

Increasing sample size generally reduces sampling error, but it does not automatically eliminate non-sampling error. Strong field procedures, instrument testing, follow-up, and quality control are required.

The appropriate sample size depends on:

• Population variability: Greater heterogeneity generally requires a larger sample.
• Desired precision: A smaller acceptable margin of error requires more observations.
• Confidence level: Higher confidence requires a larger sample.
• Population size: It matters particularly when the sample forms a substantial fraction of a finite population.
• Sampling design: Cluster designs often require larger samples, while effective stratification may improve precision.
• Expected non-response: The initial sample should be inflated to compensate for anticipated losses.
• Planned analysis: Subgroup comparisons and complex statistical models require adequate cases in each relevant category.
• Resources: Time, budget, staff, and access impose practical limits.

For estimating a population proportion in a large population, an initial sample-size formula is:

where is the critical value, is the expected proportion, and is the desired margin of error.

A stratified multistage probability design would be suitable because the workforce contains important hierarchical groups and is distributed across branches.

Recommended procedure:

1. Define the target population as all eligible employees in the specified branches and time period.
2. Obtain an updated employee list containing branch, employment level, and contact details.
3. Treat branches as first-stage units. Study all branches if feasible, or randomly select branches with an appropriate probability method.
4. Within each selected branch, divide employees into strata: senior management, middle management, and operational staff.
5. Determine the total sample size based on precision, confidence, variability, analysis needs, and expected non-response.
6. Allocate the sample proportionately across strata, or oversample the smaller management strata if separate comparisons are required.
7. Select employees randomly within each stratum.
8. Track non-response and use follow-up procedures rather than informally replacing respondents.
9. Apply sampling weights if selection probabilities differ across branches or employee categories.

This design guarantees representation of each employment level while retaining the inferential advantages of random sampling.