Sample Size Calculations for Two Independent Proportions Study Notes

Sample Size Calculations for Two Independent Proportions

Overview of Sample Size Determination

• Objective: Calculate sample sizes, denoted as $n1$ and $n2$, needed to detect a difference of interest between two groups with a specified power of detection.

Steps to Calculate Sample Size

1. Define Proportions: Determine the proportions that are expected under the alternative hypothesis, $H_a$:

   • $p<em>1 = p</em>{1a}$

   • $p<em>2 = p</em>{2a}$

2. Determine Power: Decide on the desired power for detecting the alternative hypothesis, often set as 0.80 (80%). This reflects the probability of correctly rejecting the null hypothesis when it is false.

3. Compute Sample Size: Utilize a statistical formula or a calculator for sample size calculation, factoring in the known effect size, denoted as $d$.

Example 1: Sample Size Calculation for PrEP Study

• Study Design: This study compares 2-year rates of HIV positive infection in high-risk populations (such as sex workers) in Mozambique.

  • Group 1: HIV prevention counseling.

  • Group 2: Pre-exposure prophylaxis treatment (PrEP).

• Research Objective: Determine the sample size required to have 80% power to detect differences between infection rates where:

  • $p_{1a} = 0.15$ (expected proportion for Group 1)

  • $p_{2a} = 0.05$ (expected proportion for Group 2)

Calculating Sample Size for PrEP Study

• Z-Values:

  • $z_{1- rac{α}{2}} = 1.96$ (for a two-tailed test with significance level $ ext{α} = 0.05$)

  • z_{1-eta} = 0.84 (for desired power of 80% or 0.80)

• Effect Size Calculation: The effect size ($d$) is computed based on the proportions, and a total sample size formula is applied.

• Total Sample Size: The research requires a total of $n = n<em>1 + n</em>2 = 282$ participants to achieve the desired statistical power.

G*Power Calculations for Two Independent Proportions (Z-Test)

• Software Configuration:

  • Test Family: z tests

  • Statistical Test: Proportions: Difference between two independent proportions

  • Type of Power Analysis: A priori to compute required sample size

  • Tails: Two-tailed test

  • α Error Probability: Set at 0.05

  • Power: Set at 0.80

  • Proportion p2: Set at 0.05

  • Proportion p1: Set at 0.15

  • Allocation Ratio (N2/N1): Set at 1 (equal allocation)

• Outcome Calculation: The setup confirms that the total sample size required equals $n = 282$ participants.

G*Power Calculations for Two Independent Proportions (Exact Test)

• Software Configuration:

  • Test Family: Exact tests

  • Statistical Test: Proportions: Inequality, two independent groups (using Fisher’s exact test)

  • Type of Power Analysis: A priori to compute required sample size

  • Tails: Two-tailed test

  • α Error Probability: Set at 0.05

  • Power: Set at 0.80

  • Proportion p1: Set at 0.15

  • Proportion p2: Set at 0.05

  • Allocation Ratio (N2/N1): Set at 1 (equal allocation)

• Outcome Calculation: Analysis indicates that a total sample size of $n = 302$ participants is necessary when using Fisher’s exact test for the proportions.

Exercise on Statistical Power

• Definition of Statistical Power: The power of a study is the probability that it will correctly reject a false null hypothesis. It is the likelihood of detecting an effect, given that there is one.

• Relationships:

  • Power, Sample Size, and Effect Size are interrelated, with the following relationships:

    • As sample size increases, power increases (assuming the effect size and alpha level are constant).

    • As effect size increases, power increases (given a constant sample size and alpha level).

    • Formulas to demonstrate power calculation in hypothesis tests can be employed for better understanding; one could reference any hypothesis test discussed within the coursework.

Power Calculation Formulas:

• For various hypothesis tests, the power calculation can yield insights into how adjustments in sample size, alpha levels, or effect sizes can affect the outcomes of the study.