Sample Surveys

A sample survey is a method of collecting information from a subset of a population, rather than the entire population. This subset, known as a sample, is carefully selected to represent the characteristics of the larger population. The primary goal of a sample survey is to obtain estimates about the population parameters (like mean, proportion, total) with a certain degree of accuracy, while being more cost-effective and time-efficient than a census (collecting data from every individual).

The process involves several key stages: defining the population, choosing a sampling method, collecting data, analyzing the data, and drawing inferences about the population. The success of a sample survey hinges on the representativeness of the sample. If the sample is biased, the conclusions drawn about the population will be inaccurate.

Sampling Frame

A sampling frame is a list or map of all the units in the population from which a sample is to be drawn. It serves as the operational definition of the population for the purpose of sampling. A good sampling frame is crucial for the success of a survey.

Ideally, a sampling frame should be:

• Complete: It should include all units of the target population.
• Accurate: The information about the units (e.g., names, addresses) should be correct.
• Up-to-date: It should reflect the current status of the population.
• Unique: Each unit in the population should be listed only once.
• Free from duplication: No unit should appear more than once.

However, in practice, sampling frames often have deficiencies. Common problems include:

• Coverage Errors:
  • Undercoverage: Some population units are excluded from the frame. For example, a telephone directory does not include people with unlisted numbers or those who do not own a phone.
  • Overcoverage: The frame includes units that are not part of the target population. For example, a list of registered voters might include individuals who have moved out of the area.
• Inaccuracy: The information in the frame is incorrect (e.g., outdated addresses).
• Duplication: Some units are listed multiple times.

The quality of the sampling frame directly impacts the potential biases in the survey results. Researchers must be aware of these limitations and attempt to mitigate them through appropriate sampling designs and adjustments during analysis.

Sampling with and Without Replacement

These terms refer to how units are treated once they have been selected for the sample.

Sampling With Replacement (WR)

In sampling with replacement, once a unit is selected for the sample, it is returned to the population before the next unit is selected. This means that a unit can be selected more than once. The probability of selecting any given unit remains constant for each draw. This method is often used in theoretical statistics because it simplifies calculations, especially when dealing with independent observations.

Example: Imagine a bag with 10 balls, numbered 1 to 10. If you draw a ball, note its number, and then put it back in the bag before drawing the next ball, this is sampling with replacement. The ball numbered '5' could be drawn multiple times.

Sampling Without Replacement (WOR)

In sampling without replacement, once a unit is selected, it is not returned to the population. Therefore, each unit can be selected at most once. This is the more common method used in practical surveys because it ensures that each selected unit provides unique information. The probability of selecting a unit changes with each draw as the population size decreases.

Example: Using the same bag of 10 balls, if you draw a ball, note its number, and set it aside without putting it back, this is sampling without replacement. Once ball '5' is drawn, it cannot be drawn again.

For large populations, the difference between sampling with and without replacement has a negligible effect on the results. However, for smaller populations, the distinction is more significant. Most real-world sample surveys use sampling without replacement.

Stratified Sampling

Stratified sampling is a probability sampling technique where the population is divided into distinct subgroups, or strata, based on shared characteristics. These strata are mutually exclusive and collectively exhaustive, meaning every member of the population belongs to exactly one stratum. After stratifying the population, a separate random sample is drawn from each stratum.

Purpose: The primary goal of stratification is to ensure that specific subgroups within the population are adequately represented in the sample. This is particularly useful when certain strata are known to be different from others in terms of the variable of interest, or when researchers want to obtain precise estimates for each stratum.

Steps Involved:

1. Divide the population into homogeneous strata (groups) based on relevant characteristics (e.g., age, income, geographic location, gender).
2. Within each stratum, select a random sample using a method like simple random sampling or systematic sampling.
3. Combine the samples from all strata to form the overall sample.

Types of Stratified Sampling:

• Proportionate Stratified Sampling: The sample size for each stratum is proportional to its size in the population. If a stratum constitutes 20% of the population, it will also constitute 20% of the sample. This method is often preferred as it reflects the population composition directly.
• Disproportionate Stratified Sampling: The sample size for each stratum is not proportional to its population size. This is used when some strata are of particular interest, or when the variability within certain strata is much higher than others, requiring larger sample sizes from those strata to achieve desired precision.

Advantages:

• Increases precision of estimates compared to simple random sampling, especially if strata are homogeneous within and heterogeneous between.
• Ensures representation of key subgroups.
• Allows for separate analysis of each stratum.

Disadvantages:

• Requires prior knowledge of the population to define strata.
• Can be complex and costly to implement if many strata are involved or if stratification variables are difficult to obtain.

Example: A university wants to survey student satisfaction. They might stratify students by faculty (e.g., Arts, Science, Engineering, Medicine). Within each faculty, they then draw a random sample of students. This ensures that each faculty is represented in the survey, preventing a situation where, by chance, a faculty with very different opinions might be underrepresented if simple random sampling were used.

Two-Stage Sampling

Two-stage sampling is a cluster sampling technique where the sampling process occurs in two distinct stages. First, a sample of primary units (clusters) is selected from the population. Second, within each selected primary unit, a sample of secondary units is selected.

Steps Involved:

1. First Stage: Divide the population into a number of primary sampling units (PSUs). Select a sample of PSUs using a probability sampling method (e.g., simple random sampling or systematic sampling).
2. Second Stage: Within each selected PSU, divide it into secondary sampling units (SSUs). Select a sample of SSUs from each selected PSU using a probability sampling method.

Example: To conduct a survey on household income in a large country:

• Stage 1: Select a random sample of districts (PSUs) across the country.
• Stage 2: Within each selected district, select a random sample of villages or neighborhoods (SSUs).
• Finally, within each selected village/neighborhood, select a random sample of households.

Advantages:

• More practical and cost-effective than simple random sampling for geographically dispersed populations.
• Reduces the need for a complete list of all elementary units in the population, only requiring a list of PSUs and then lists within selected PSUs.
• Can provide reasonable estimates even with imperfect lists at the second stage.

Disadvantages:

• Generally less statistically efficient (i.e., results in larger sampling errors) than simple random sampling or stratified sampling, especially if the PSUs are heterogeneous.
• Analysis can be more complex.

Note: This can be extended to multi-stage sampling (three-stage, four-stage, etc.) by adding more levels of sampling.

Systematic Sampling

Systematic sampling is a probability sampling method where units are selected from an ordered list at regular intervals. It's often simpler to implement than simple random sampling, especially when dealing with large populations or when a random number generator is not readily available.

Steps Involved:

1. Obtain a list of all units in the population (the sampling frame), ordered in some way.
2. Determine the desired sample size, n.
3. Calculate the sampling interval, k, by dividing the population size, N, by the sample size, n (k = N/n). If N/n is not an integer, it's common to round k to the nearest integer.
4. Choose a random starting point, r, between 1 and k (inclusive).
5. Select the units at intervals of k starting from the random start: r, r+k, r+2k, r+3k, ... until n units are selected.

Example: Suppose you have a list of 1000 employees (N=1000) and you want to select a sample of 100 employees (n=100).

• Calculate the sampling interval: k = 1000 / 100 = 10.
• Choose a random start between 1 and 10. Let's say you choose 5 (r=5).
• The sample will consist of employees numbered: 5, 15, 25, 35, ..., 995.

Advantages:

• Simpler to execute than simple random sampling, especially for large populations.
• Often provides a good spread of the sample across the entire list, potentially leading to more representative samples than SRS if the list has a periodic pattern that aligns poorly with SRS.

Disadvantages:

• If the sampling frame has a periodic pattern that coincides with the sampling interval, the sample can be biased. For example, if every 10th person on a list is a supervisor, and your interval is 10, you might end up with all supervisors or no supervisors.
• Requires a complete and ordered list of the population.

Cluster Sampling

Cluster sampling is a probability sampling method where the population is divided into groups called clusters. A sample of clusters is then selected, and all or a sample of the units within the selected clusters are included in the survey. Clusters are typically naturally occurring groups, such as geographic areas (counties, cities, neighborhoods), schools, or households.

Purpose: It is often used when it is impractical or too expensive to obtain a complete list of all elementary units in the population, but a list of clusters is available. It's also effective for geographically dispersed populations.

Steps Involved:

1. Divide the population into clusters.
2. Randomly select a sample of clusters.
3. Include all the elementary units within the selected clusters in the sample (this is called single-stage cluster sampling). Alternatively, a further sample of elementary units can be selected from within the chosen clusters (this becomes two-stage cluster sampling, as discussed earlier).

Example: A researcher wants to study the reading habits of primary school children in a state.

• Clusters: Schools in the state are the clusters.
• Sampling: A random sample of schools is selected.
• Data Collection: All children within the selected schools are surveyed.

Advantages:

• Cost-effective and logistically simpler than simple random sampling, especially for large, geographically dispersed populations.
• Does not require a complete list of all elementary units, only a list of clusters.

Disadvantages:

• Generally less efficient (higher sampling error) than simple random sampling or stratified sampling because units within a cluster tend to be more similar to each other than units selected randomly from the entire population (intraclass correlation).
• Requires careful definition of clusters to avoid bias.

Regression Estimates

Regression estimates are used in sample surveys to improve the precision of estimates for a population total or mean. This technique utilizes a relationship between the variable of interest (the study variable, Y) and another variable (an auxiliary or predictor variable, X) that is correlated with Y and for which population totals or means are known or easily obtainable.

The auxiliary variable X should ideally be highly correlated with the study variable Y, and its population total (or mean) should be known from a previous census or a reliable source.

The Basic Idea: If we know the relationship between Y and X from the sample, we can use the known population total of X to predict the population total of Y more accurately than using the sample mean of Y alone.

Let:

• Y = Study variable (e.g., current year's crop yield)
• X = Auxiliary variable (e.g., previous year's crop yield, satellite imagery data)
• N = Population size
• n = Sample size
• y_i, x_i = values of Y and X for the i-th unit in the sample
• $\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i$ = sample mean of Y
• $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$ = sample mean of X
• $Y_{pop}$ = Population total of Y
• $X_{pop}$ = Population total of X (known)

The linear regression line of Y on X from the sample data is typically expressed as: $y = \bar{y} + b(x - \bar{x})$ where $b$ is the sample regression coefficient, calculated as: $b = \frac{\sum_{i=1}^{n} (y_i - \bar{y})(x_i - \bar{x})}{\sum_{i=1}^{n} (x_i - \bar{x})^2}$

The regression estimate of the population total $Y_{pop}$ is given by: $\hat{Y}_{reg} = N[\bar{y} + b(\bar{X} - \bar{x})]$ where $\bar{X} = X_{pop} / N$ is the population mean of X.

If the sample is a simple random sample, the estimate of the population total Y is often written as: $\hat{Y}_{reg} = \sum_{i=1}^{n} y_i + b(X_{pop} - \sum_{i=1}^{n} x_i)$ This formula essentially adjusts the sum of the sample observations ($\sum y_i$) based on the difference between the known population total of X ($X_{pop}$) and the observed sample total of X ($\sum x_i$), using the regression slope $b$.

Advantages:

• Can significantly reduce the sampling error (increase precision) if there is a strong positive correlation between Y and X, and the regression line accurately describes the relationship.
• Utilizes readily available information (population total of X).

Disadvantages:

• Requires a strong correlation between Y and X.
• The auxiliary variable X must be available for all units in the population, and its total/mean must be known accurately.
• The calculation of the regression coefficient $b$ can be complex.
• Assumes a linear relationship between Y and X.

Ratio Estimates

Ratio estimates are another technique used in sample surveys to improve the precision of estimates by utilizing auxiliary information. Similar to regression estimates, they rely on the correlation between the study variable (Y) and an auxiliary variable (X) for which the population total is known. However, ratio estimation assumes that the relationship between Y and X is proportional and passes through the origin (i.e., when X=0, Y=0).

The Basic Idea: The ratio of the study variable to the auxiliary variable (Y/X) is assumed to be constant across the population. By estimating this ratio from the sample, we can use the known population total of X to estimate the population total of Y.

Using the same notation as for regression estimates:

• Y = Study variable
• X = Auxiliary variable
• N = Population size
• n = Sample size
• y_i, x_i = values of Y and X for the i-th unit in the sample
• $\bar{y}, \bar{x}$ = sample means of Y and X
• $Y_{pop}$ = Population total of Y
• $X_{pop}$ = Population total of X (known)

The ratio of the sample means is calculated as: $r = \frac{\bar{y}}{\bar{x}} = \frac{\sum_{i=1}^{n} y_i / n}{\sum_{i=1}^{n} x_i / n} = \frac{\sum_{i=1}^{n} y_i}{\sum_{i=1}^{n} x_i}$

The ratio estimate of the population total $Y_{pop}$ is given by: $\hat{Y}_{ratio} = r \times X_{pop} = \frac{\bar{y}}{\bar{x}} \times X_{pop}$

This formula essentially estimates the population total of Y by multiplying the known population total of X by the ratio of the sample mean of Y to the sample mean of X.

When is Ratio Estimation Appropriate?

• When the ratio $Y/X$ is approximately constant for all units in the population.
• When the relationship between Y and X passes through the origin (i.e., if X=0, then Y=0).
• When the auxiliary variable X is highly correlated with Y.

Example: Suppose we want to estimate the total number of fish caught by fishermen in a season (Y), and we have data on the number of fishing trips made by each fisherman (X). We know the total number of fishing trips made by all fishermen in the season ($X_{pop}$).

• We take a sample of fishermen, record their total catch (y_i) and the number of trips they made (x_i).
• We calculate the sample ratio: $r = (\sum y_i) / (\sum x_i)$.
• The estimated total catch for the season is $\hat{Y}_{ratio} = r \times X_{pop}$.

Advantages:

• Often more precise than estimates based on simple random sampling, especially when the ratio is stable and X is strongly correlated with Y.
• Simple to calculate and understand.
• Utilizes readily available auxiliary information.

Disadvantages:

• Can be biased if the relationship between Y and X does not pass through the origin or is not proportional.
• Less efficient than regression estimates if the relationship is linear but does not pass through the origin.
• Requires the population total of X to be known accurately.