Understanding Effect Size Calculations for Power Analysis

Effect size is a crucial component in statistical power analysis, helping researchers determine the magnitude of a phenomenon or the strength of a relationship between variables. In this article, we will explore three practical examples of effect size calculations that serve as effective illustrations for power analysis across different research contexts.

Example 1: Assessing the Impact of a New Teaching Method on Student Performance

Context

A school district is interested in evaluating the effectiveness of a new teaching method compared to the traditional approach. The objective is to determine if the new method significantly improves student performance in mathematics.

To conduct this analysis, the researchers collect data from two groups of students: one group taught using the traditional method and another group taught using the new method. They aim to calculate the effect size to assess the impact of the new teaching method.

The researchers use Cohen’s d for this calculation, which is defined as the difference between two group means divided by the pooled standard deviation.

Calculation

1. Mean of traditional method group (M1): 70
2. Mean of new method group (M2): 80
3. Standard deviation of traditional method group (SD1): 10
4. Standard deviation of new method group (SD2): 12

5. Pooled standard deviation (SDp) can be calculated using the formula:

   

   SDp = √[(SD1² + SD2²) / 2]
   SDp = √[(10² + 12²) / 2] = 11.18

6. Cohen’s d calculation:
   Cohen’s d = (M2 - M1) / SDp
   Cohen’s d = (80 - 70) / 11.18 = 0.895

Notes

An effect size of 0.895 indicates a large effect according to Cohen’s conventions, suggesting that the new teaching method significantly improves student performance in mathematics.

Example 2: Evaluating a Drug’s Effect on Blood Pressure

Context

A pharmaceutical company is testing a new drug aimed at reducing blood pressure compared to a placebo. The researchers must determine the effect size of the drug based on clinical trial data to inform future studies.

The analysis will involve comparing the mean blood pressure levels of the treatment group to the control group. For this, the researchers will again use Cohen’s d.

Calculation

1. Mean blood pressure for placebo group (M1): 130 mmHg
2. Mean blood pressure for drug group (M2): 120 mmHg
3. Standard deviation for placebo group (SD1): 15 mmHg
4. Standard deviation for drug group (SD2): 10 mmHg

5. Pooled standard deviation (SDp) calculation:

   SDp = √[(SD1² + SD2²) / 2]
   SDp = √[(15² + 10²) / 2] = 12.25

6. Cohen’s d calculation:
   Cohen’s d = (M2 - M1) / SDp
   Cohen’s d = (120 - 130) / 12.25 = -0.816

Notes

An effect size of -0.816 suggests a large negative effect, indicating that the new drug significantly lowers blood pressure compared to the placebo.

Example 3: Analyzing the Effect of an Exercise Program on Weight Loss

Context

A fitness organization wants to evaluate the effectiveness of a 12-week exercise program on weight loss. The organization has two groups of participants: one group follows the exercise program, while the other group does not.

To quantify the impact of the program, they will calculate the effect size using Cohen’s d.

Calculation

1. Mean weight loss for the exercise group (M1): 8 kg
2. Mean weight loss for the control group (M2): 3 kg
3. Standard deviation for the exercise group (SD1): 2 kg
4. Standard deviation for the control group (SD2): 1.5 kg

5. Pooled standard deviation (SDp) calculation:

   SDp = √[(SD1² + SD2²) / 2]
   SDp = √[(2² + 1.5²) / 2] = 1.75

6. Cohen’s d calculation:
   Cohen’s d = (M1 - M2) / SDp
   Cohen’s d = (8 - 3) / 1.75 = 2.857

Notes

An effect size of 2.857 indicates an extraordinarily large effect, demonstrating that the exercise program has a significant impact on weight loss compared to the control group.

Conclusion

These examples of effect size calculations for power analysis illustrate the importance of quantifying the strength of relationships in research. By understanding and applying effect size, researchers can design more effective studies and interpret results with greater clarity.