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Power and Effect Size Concepts in Statistical Analysis

Power in Statistics

  • Definition of Power

    • Power is the probability of correctly rejecting a false null hypothesis ($H_0$).
    • It is calculated as:
      Power = 1 - \beta
    • Where:
    • $\beta$ = probability of a Type II error (failing to reject a false null hypothesis).

  • Factors Affecting Power

    • Alpha Level ($\alpha$): The probability of a Type I error (rejecting a true null hypothesis).
    • Effect Size: The degree to which the phenomenon is present in the population.
    • If $H_0$ is true, Effect Size = 0.
    • If $H_0$ is false, Effect Size > 0.
    • Sample Size ($N$): Larger samples generally provide more power.
    • Variability in Data: Less variability in data increases power.

Outcomes of Statistical Tests

  • Outcomes of hypothesis testing can lead to four possible results:
    • Fail to reject $H_0$ when it is true: Correct Decision
    • Fail to reject $H_0$ when it is false: Type II Error
    • Reject $H_0$ when it is true: Type I Error
    • Reject $H_0$ when it is false: Correct Decision

Effect Size

  • Understanding Effect Size

    • Indicates how much of a difference there is between groups.
    • For example, using GRE scores:
    • Population mean ($\mu$) = 587; standard deviation ($\sigma$) = 152.
    • After training, sample mean ($M$) = 595.

  • Cohen’s $d$: A common measure of effect size calculated as:
    d = \frac{\mu{treatment} - \mu{control}}{\sigma}

    • For the example provided:
      d = \frac{595 - 587}{152} = 0.052
    • Indicates a small effect since it's only 0.05 standard deviations.

Examples of Power Analysis

  1. Example 1:

    • Population: $\mu = 80$, $\sigma = 10$; Sample Size ($n$) = 25; Expected Effect = 8 points ($\mu = 88$).
    • Effect Size: Calculated to be $d = 0.8$.
    • Power calculation resulting in approximately 97.93% chance of detecting the 8-point effect.

  2. Example 2:

    • Same population, $\mu = 80$, $\sigma = 10$; Sample Size ($n$) = 25; Expected Effect = 4 points ($\mu = 84$).
    • Lower power calculated at approximately 51.60% for detecting the 4-point effect.

  3. Example 3:

    • Population: $\mu = 80$, $\sigma = 10$; Sample Size ($n$) = 50; Expected Effect = 4 points.
    • Effect size again calculated as $d = 0.4$.
    • Resulted in higher power at approximately 81.06% chance of detecting the 4-point effect due to increased sample size.

A Priori Power Analysis

  • Purpose: To estimate the required sample size to achieve a specified power with a particular effect size and alpha level.

  • Strategies to increase power:

    1. Increase the strength of manipulation of the independent variable.
    2. Increase the number of participants.
    3. Reduce variability through more trials per participant or obtaining more accurate measurements.

  • Use of Software: G*Power can be utilized for complex power calculations, allowing better planning in study designs.