Understanding Proportional vs Optimum Allocation in Stratified Sampling Techniques

Sampling is a fundamental part of research and data analysis. When dealing with diverse populations, stratified sampling offers a way to ensure that different subgroups are fairly represented. But within stratified sampling, choosing how to allocate samples across strata can significantly affect the accuracy and efficiency of your results. Two common methods for this allocation are proportional allocation and optimum allocation. Understanding the differences between these approaches helps researchers design better studies and make more reliable inferences.

This post explores these two allocation methods in detail, explaining how they work, when to use each, and the practical implications for your sampling strategy.

What Is Stratified Sampling?

Stratified sampling divides a population into distinct subgroups called strata. Each stratum shares a specific characteristic, such as age group, income level, or geographic region. The goal is to ensure that the sample reflects the population’s diversity by selecting samples from each stratum.

For example, if a population consists of 60% urban residents and 40% rural residents, stratified sampling ensures that both groups are represented in the sample, rather than relying on a simple random sample that might underrepresent one group.

Once strata are defined, the next step is to decide how many samples to take from each stratum. This is where allocation methods come into play.

Proportional Allocation Explained

Proportional allocation assigns sample sizes to strata based on their proportion in the overall population. If a stratum makes up 30% of the population, it will receive 30% of the total sample.

How It Works

• Calculate the size of each stratum relative to the total population.

• Multiply the total sample size by each stratum’s proportion.

• Allocate samples accordingly.

  
  

Example

Suppose you want a sample of 1,000 people from a population of 10,000, divided into three strata:

• Stratum A: 5,000 people (50%)

• Stratum B: 3,000 people (30%)

• Stratum C: 2,000 people (20%)

  
  

Using proportional allocation:

• Stratum A gets 500 samples (50% of 1,000)

• Stratum B gets 300 samples (30% of 1,000)

• Stratum C gets 200 samples (20% of 1,000)

  
  

Advantages of Proportional Allocation

• Simplicity: Easy to calculate and implement.

• Fair representation: Reflects the population structure accurately.

• Good for homogeneous strata: Works well when variability within strata is similar.

  
  

Limitations

• Does not account for differences in variability or cost of sampling across strata.

• May not be the most efficient method if some strata are more variable or expensive to sample.

  
  

Optimum Allocation Explained

Optimum allocation, also known as Neyman allocation, aims to minimize the overall variance of the estimate for a fixed total sample size. It considers both the size of each stratum and the variability within it, as well as the cost of sampling if applicable.

How It Works

• Calculate the standard deviation (or variance) within each stratum.

• Allocate more samples to strata with higher variability.

• If sampling costs differ, allocate fewer samples to expensive strata.

• The formula balances stratum size, variability, and cost to optimize precision.

  
  

Example

Using the previous population, suppose the standard deviations for a key variable in each stratum are:

• Stratum A: 10

• Stratum B: 20

• Stratum C: 5

  
  

Assuming equal sampling costs, optimum allocation would assign more samples to Stratum B because it has the highest variability, fewer to Stratum C, and a moderate number to Stratum A.

Advantages of Optimum Allocation

• Improved precision: Reduces overall sampling error.

• Cost-effective: Can incorporate sampling costs to reduce expenses.

• Better for heterogeneous strata: Adjusts for differences in variability.

  
  

Limitations

• Requires prior knowledge of stratum variances, which may not always be available.

• More complex to calculate and implement.

• May lead to over- or under-representation if assumptions about variability are incorrect.

  
  
  
  

Researcher analyzing stratified sampling data

When to Use Proportional Allocation

Proportional allocation is a good choice when:

• You have limited information about variability within strata.

• Strata are relatively homogeneous.

• Sampling costs are similar across strata.

• You want a straightforward, easy-to-explain sampling plan.

  
  

For example, in a survey of customer satisfaction across different store locations with similar customer profiles and costs, proportional allocation ensures each location is fairly represented without unnecessary complexity.

When to Use Optimum Allocation

Optimum allocation is preferable when:

• Strata vary significantly in variability.

• Sampling costs differ across strata.

• You want to maximize precision for a fixed sample size.

• You have reliable estimates of stratum variances and costs.

  
  

For instance, in agricultural research where crop yields vary widely between regions and sampling costs differ due to terrain, optimum allocation helps focus resources where they matter most.

Practical Steps to Implement Each Method

Proportional Allocation

1. Define strata and determine their population sizes.

2. Calculate each stratum’s proportion.

3. Multiply total sample size by each proportion.

4. Round to whole numbers and adjust if necessary.

   
   

Optimum Allocation

1. Define strata and estimate population sizes.

2. Collect preliminary data or use past studies to estimate standard deviations.

3. Determine sampling costs per stratum if applicable.

4. Use the formula:

   
   

   \[

   n_h = n \times \frac{N_h \times S_h / \sqrt{C_h}}{\sum (N_h \times S_h / \sqrt{C_h})}

   \]

   
   

where:

- \( n_h \) = sample size for stratum h

- \( n \) = total sample size

- \( N_h \) = population size of stratum h

- \( S_h \) = standard deviation of stratum h

- \( C_h \) = cost per unit in stratum h

1. Adjust sample sizes to whole numbers.

   
   

Comparing the Two Methods Side by Side

| Feature | Proportional Allocation | Optimum Allocation |

|--------------------------|---------------------------------------------|---------------------------------------------|

| Basis for allocation | Population proportion | Population size, variability, and cost |

| Complexity | Simple | More complex, requires variance estimates |

| Precision | Moderate | Higher precision, especially with variable strata |

| Cost considerations | Not considered | Can incorporate sampling costs |

| Use case | Homogeneous strata, limited data | Heterogeneous strata, known variances |

Real-World Example: Health Survey

Imagine a health survey across three regions with different population sizes and health variability:

• Region 1: 10,000 people, low variability in health indicators.

• Region 2: 5,000 people, high variability.

• Region 3: 2,000 people, moderate variability.

  
  

Using proportional allocation, Region 1 would get the largest sample simply because it has the largest population. But if Region 2 has more diverse health outcomes, optimum allocation would assign more samples there to capture that diversity better, improving the survey’s overall accuracy.

Key Takeaways for Researchers

• Understand your population: Know the size and variability of each stratum.

• Choose allocation based on goals: Use proportional allocation for simplicity and fairness; use optimum allocation for precision and cost-efficiency.

• Gather preliminary data: Estimating variability is crucial for optimum allocation.

• Balance complexity and benefits: Optimum allocation can improve results but requires more effort.

• Adjust sample sizes carefully: Always round and check totals to maintain the desired sample size.