Hypothesis Testing Basics

Hypothesis Testing

Definition of P-value

• The P-value is the probability of obtaining results at least as extreme as the observed results, assuming the null hypothesis is true.
• A smaller P-value indicates stronger evidence against the null hypothesis.
• Common thresholds for significance:
  • P < 0.05: typically considered statistically significant.
  • P < 0.01: strong evidence against the null hypothesis.

Type I and II Errors

• Type I Error (α):

  • Occurs when the null hypothesis is rejected when it is actually true.
  • The probability of making a Type I error is denoted by α (commonly set at 0.05).

• Type II Error (β):

  • Occurs when the null hypothesis is not rejected when it is actually false.
  • The probability of making a Type II error is denoted by β.
  • Power of the test = 1 - β, which indicates the probability of correctly rejecting a false null hypothesis.

Statistical Significance

• Statistical significance indicates that the observed effect is unlikely to have occurred by chance alone.
• It is determined through the P-value in relation to the significance level (α).
• It does not imply practical significance; results may be statistically significant but not meaningful in a real-world context.

Hypothesis Formulation

• Null Hypothesis (H0): A statement asserting no effect or no difference, serving as a baseline for comparison.
• Alternative Hypothesis (H1 or Ha): A statement that indicates the presence of an effect or a difference.
• Formulation involves:
  • Clearly defining the parameters of interest.
  • Setting appropriate hypotheses before data collection to avoid bias.

Interpretation of Results

• After performing hypothesis testing, results must be interpreted in context:
  • If P < α: reject the null hypothesis, suggesting evidence for the alternative.
  • If P ≥ α: fail to reject the null hypothesis, indicating insufficient evidence to support the alternative.
• Consideration of effect size, confidence intervals, and practical implications is crucial for meaningful interpretation.
• Results should be communicated clearly, highlighting the statistical and practical significance for stakeholders.

P-value

• Represents the probability of observing results as extreme as the actual results, under the assumption that the null hypothesis is correct.
• A smaller P-value signifies stronger evidence against the null hypothesis.
• Common significance thresholds include:
  • P < 0.05 indicates statistical significance.
  • P < 0.01 signifies strong evidence against the null hypothesis.

Type I and II Errors

• Type I Error (α):
  • Involves rejecting the null hypothesis when it is true.
  • The probability of occurrence is denoted by α and is typically set at 0.05.
• Type II Error (β):
  • Occurs when the null hypothesis is not rejected while it is false.
  • The probability of making a Type II error is represented by β.
• Power of the test is equal to 1 - β, determining the likelihood of correctly rejecting a false null hypothesis.

Statistical Significance

• Indicates the unlikelihood that observed effects are due to random chance.
• Determined by comparing the P-value to the predefined significance level (α).
• Does not equate to practical significance; results can be statistically significant yet lack real-world relevance.

Hypothesis Formulation

• Null Hypothesis (H0): States there is no effect or difference, acting as a comparison baseline.
• Alternative Hypothesis (H1 or Ha): Suggests the presence of an effect or difference.
• Proper formulation requires clear definition of parameters and avoiding bias by setting hypotheses prior to data collection.

Interpretation of Results

• Results interpretation is influenced by context:
  • If P < α, reject the null hypothesis, suggesting favor for the alternative.
  • If P ≥ α, do not reject the null hypothesis, indicating insufficient evidence for the alternative.
• Critical to consider effect size, confidence intervals, and practical implications for a meaningful conclusion.
• Clear communication of results to stakeholders should highlight both statistical and practical significance.