Sample Size for Regression Analysis Examples

Understanding Sample Size in Regression Analysis

Determining the appropriate sample size is crucial in regression analysis as it directly affects the power of your statistical tests. A well-defined sample size ensures that the results are reliable and generalizable to a larger population. Below are three practical examples that illustrate how to determine sample size for regression analysis in different scenarios.

Example 1: Predicting Housing Prices

In a real estate firm, analysts aim to understand how various factors affect housing prices, such as square footage, number of bedrooms, and location. They want to ensure their regression model has enough power to detect significant relationships among these variables.

To determine the sample size, they decide on the following parameters:

• Effect Size (f²): 0.15 (medium effect)
• Alpha Level (α): 0.05 (5% significance level)
• Power (1 - β): 0.80 (80% chance of detecting an effect)
• Number of Predictors (k): 3 (square footage, bedrooms, location)

Using the formula for sample size in multiple regression analysis:

total ext{Sample Size} =
rac{(Z_{1- rac{eta}{2}} + Z_{
rac{eta}{2}})^{2} imes (1+
rac{1}{k})}{(f^{2})}

Substituting the values:

• Z(0.05) = 1.96 (two-tailed)
• Z(0.20) = 0.84 (one-tailed)
• f² = 0.15
• k = 3

Calculating gives a required sample size of approximately 82.

Notes:
This example illustrates a practical application in real estate analysis, but the same principles can be applied to other fields where multiple predictors are analyzed. Adjusting the effect size assumption can lead to different sample size requirements.

Example 2: Marketing Campaign Effectiveness

A marketing team wants to evaluate the effectiveness of their latest campaign on customer retention rates. They plan to use a regression model to predict retention based on advertising spend, customer service ratings, and promotional offers.

The team sets the following parameters for their sample size calculation:

• Effect Size (f²): 0.10 (small effect)
• Alpha Level (α): 0.01 (1% significance level)
• Power (1 - β): 0.90 (90% chance of detecting an effect)
• Number of Predictors (k): 4 (advertising spend, service ratings, promotional offers)

Applying the same formula:

total ext{Sample Size} =
rac{(Z_{1- rac{eta}{2}} + Z_{
rac{eta}{2}})^{2} imes (1+
rac{1}{k})}{(f^{2})}

Calculating:

• Z(0.01) = 2.576
• Z(0.10) = 1.28
• f² = 0.10
• k = 4

The resulting sample size is approximately 116.

Notes:
This scenario emphasizes the importance of a higher power and smaller alpha level for marketing analysis, where precision can significantly impact strategic decisions. The choice of effect size will heavily influence the final sample size.

Example 3: Clinical Trial for a New Drug

In a clinical trial, researchers want to assess the effect of a new drug on reducing blood pressure. They plan to use regression analysis to account for patient age, weight, and baseline blood pressure.

To find the necessary sample size, they establish these parameters:

• Effect Size (f²): 0.25 (large effect)
• Alpha Level (α): 0.05 (5% significance level)
• Power (1 - β): 0.85 (85% chance of detecting an effect)
• Number of Predictors (k): 3 (age, weight, baseline BP)

Using the formula:

total ext{Sample Size} =
rac{(Z_{1- rac{eta}{2}} + Z_{
rac{eta}{2}})^{2} imes (1+
rac{1}{k})}{(f^{2})}

Substituting:

• Z(0.05) = 1.96
• Z(0.15) = 1.04
• f² = 0.25
• k = 3

This results in a sample size requirement of approximately 84.

Notes:
In clinical trials, a larger effect size often leads to a smaller required sample size, which can expedite the research process. It is crucial to consider ethical implications and feasibility when determining sample sizes in medical research.

Conclusion

Determining the right sample size for regression analysis is essential for obtaining reliable results. Each of these examples highlights different contexts and considerations that affect sample size calculations. By understanding these factors, researchers can make informed decisions that enhance the quality of their analysis.