7.1 Concepts of statistical power and effect size

Statistical power and effect size are crucial concepts in experimental design. They help researchers determine if their studies can reliably detect meaningful effects. Power is the chance of finding a real effect, while effect size measures how big that effect is.

Understanding these concepts allows researchers to plan better studies. By calculating power and effect size, they can figure out how many participants they need and how strong their results might be. This knowledge is key for conducting effective experiments.

Power and Errors

Statistical Power and Types of Errors

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• Statistical power probability of correctly rejecting a false null hypothesis
  • Ranges from 0 to 1
  • Higher power indicates a higher likelihood of detecting a true effect if one exists
  • Influenced by factors such as sample size, effect size, and significance level (alpha)
• Type I error occurs when a null hypothesis is incorrectly rejected when it is actually true (false positive)
  • Probability of making a Type I error is denoted by the alpha level ($\alpha$)
  • Commonly set at 0.05, meaning a 5% chance of incorrectly rejecting a true null hypothesis
• Type II error occurs when a null hypothesis is not rejected when it is actually false (false negative)
  • Probability of making a Type II error is denoted by the beta level ($\beta$)
  • Complement of statistical power (1 - power)

Alpha, Beta, and Power Analysis

• Alpha level significance level at which the null hypothesis is rejected
  • Conventionally set at 0.05 in many fields
  • Lower alpha levels (0.01) reduce Type I error but increase Type II error
• Beta level probability of failing to reject a false null hypothesis
  • Directly related to statistical power (1 - power)
  • Lower beta levels increase power but require larger sample sizes
• Power analysis used to determine the sample size needed to achieve a desired level of statistical power
  • Takes into account effect size, alpha level, and desired power (often 0.80 or higher)
  • Helps researchers plan studies with sufficient power to detect meaningful effects

Effect Size Measures

Cohen's d and Eta-squared

• Effect size standardized measure of the magnitude of an effect or relationship
  • Allows comparison of effects across different studies and variables
  • Common effect size measures include Cohen's d, eta-squared, and odds ratio
• Cohen's d standardized difference between two means
  • Calculated as the difference between means divided by the pooled standard deviation
  • Interpreted as small (0.2), medium (0.5), or large (0.8) effects
• Eta-squared ($\eta^2$) proportion of variance in the dependent variable explained by an independent variable
  • Ranges from 0 to 1
  • Commonly used in ANOVA to assess the magnitude of group differences

Odds Ratio

• Odds ratio measure of effect size for binary outcomes
  • Represents the odds of an event occurring in one group compared to another
  • Calculated as the odds of an event in the treatment group divided by the odds of the event in the control group
  • An odds ratio of 1 indicates no difference between groups
  • Odds ratios greater than 1 indicate higher odds of the event in the treatment group
  • Odds ratios less than 1 indicate lower odds of the event in the treatment group (protective effect)