---

Testing of Hypothesis: Complete Guide

---

1. What is a Hypothesis?

"In a statistical hypothesis, statements are made about the relationship among parameters of one or more populations."

• Barash, Cullen, and Stoelting's Clinical Anesthesia, 9e

---

2. Basic Concepts in Hypothesis Testing

2.1 The Null Hypothesis (H₀)

• Example: "There is no difference in blood pressure between the treatment group and the control group."
• Algebraically: H₀: μ₁ = μ₂

"A comparative study commences with a fundamental presumption of 'no difference,' known as the null hypothesis. The null hypothesis posits that the difference between the two treatments being compared is zero."

• Rockwood and Green's Fractures in Adults, 10th ed.

2.2 The Alternative Hypothesis (H₁ or Hₐ)

• Example: "The treatment group has lower blood pressure than the control group."
• Algebraically: H₁: μ₁ ≠ μ₂ (two-tailed) or H₁: μ₁ < μ₂ (one-tailed)

• Two-tailed (non-directional): Tests for any difference, either greater or less (μ₁ ≠ μ₂). More conservative.
• One-tailed (directional): Tests for a difference in one specific direction only (μ₁ > μ₂ or μ₁ < μ₂).

2.3 Level of Significance (α)

"In the medical literature, this threshold is typically set at a value of less than 5% (i.e., p < .05)."

• Rockwood and Green's Fractures in Adults, 10th ed.

2.4 The P-Value

• If p < α: Reject H₀ - result is "statistically significant"
• If p ≥ α: Fail to reject H₀ - result is "not statistically significant"

"The level of significance is the probability level considered too low to warrant support of the null hypothesis being tested. If sample values are sufficiently unlikely to have occurred by chance (i.e., the probability of the sample test statistic is less than the chosen level of significance), the null hypothesis is rejected."

• Barash, Cullen, and Stoelting's Clinical Anesthesia, 9e

2.5 Test Statistic

• Z-statistic (large samples, known σ)
• t-statistic (small samples, unknown σ)
• Chi-square (χ²) (categorical data)
• F-statistic (comparing variances/ANOVA)
• Fisher's Exact Test (small sample categorical comparisons)

2.6 Type I and Type II Errors

• Type I Error (α): Rejecting a true null hypothesis (false positive). Probability = α (the significance level).
• Type II Error (β): Failing to reject a false null hypothesis (false negative).
• Power (1 - β): The probability of correctly rejecting a false H₀. A well-designed study aims for power ≥ 0.80.

"Type I errors are generally considered more serious."

• Statistics LibreTexts

2.7 Critical Region and Critical Value

2.8 Confidence Interval (CI)

---

3. Procedure of Hypothesis Testing

Step 1: State the Hypotheses

• H₀ (Null): States no effect, no difference, or no relationship
• H₁ (Alternative): States the expected effect or difference

Step 2: Choose the Level of Significance (α)

Step 3: Select the Appropriate Statistical Test

• Type of data (continuous, categorical, ordinal)
• Number of groups (two groups → t-test; multiple groups → ANOVA)
• Sample size (large → Z-test; small → t-test)
• Distribution of data (parametric vs. non-parametric)

Step 4: Collect Data and Calculate the Test Statistic

Step 5: Determine the Critical Value or P-value

• Critical value method: Compare the test statistic against the critical value from standard tables (Z-table, t-table, etc.)
• P-value method: Calculate the probability of obtaining that test statistic under H₀

Step 6: Make a Decision

• If test statistic > critical value (or p < α): Reject H₀
• If test statistic ≤ critical value (or p ≥ α): Fail to reject H₀

Step 7: State the Conclusion

• Example: "At the 5% significance level, there is sufficient evidence to conclude that the new drug significantly reduces blood pressure (t = 3.2, p = 0.002)."

---

4. Worked Example (Coin Toss)

• H₀: The coin is fair (P(heads) = 0.5)
• H₁: The coin is biased toward heads
• Test: Probability = 0.5^10 ≈ 0.001 → p < 0.001
• Decision: Since p < 0.05 (α), reject H₀
• Conclusion: The coin appears to be biased toward heads (statistically significant at p < 0.001)

This example is directly from Rockwood and Green's Fractures in Adults, 10th ed. to illustrate the intuitive nature of hypothesis testing.

---

5. Summary Table

---

---

ANOVA - Analysis of Variance

---

1. What is ANOVA?

Why not just use multiple t-tests? If you compare 3 groups using separate t-tests (A vs B, B vs C, A vs C), you run 3 tests, each with a 5% chance of a false positive. Together, the cumulative Type I error rises to ~14%. With 6 groups, you'd have 15 comparisons and ~54% false positive risk. ANOVA controls this by performing a single "omnibus" test, keeping Type I error at exactly α (e.g., 5%).

---

2. Core Logic of ANOVA

Total Variation = Between-Group Variation + Within-Group Variation
     (SST)             (SSB)                      (SSW)

• Between-Group Variation (SSB): How much the group means differ from the overall grand mean. This reflects the effect of the independent variable (treatment).
• Within-Group Variation (SSW): How much individual observations vary within each group (also called "error" or "residual"). This is natural random variation.

• F near 1: Group means are similar - evidence for H₀
• F much greater than 1: Group means differ more than expected by chance - evidence against H₀

---

3. Hypotheses in ANOVA

• H₀ (Null): All group means are equal → μ₁ = μ₂ = μ₃ = ... = μₖ
• H₁ (Alternative): At least one group mean is different from the others (not all means are equal)

---

4. Types of ANOVA

4.1 One-Way ANOVA

• Compares means of 3+ groups based on one independent variable (factor)
• Example: Comparing pain relief scores across 3 drug dosages (10 mg, 20 mg, 30 mg)
• One factor with multiple levels

4.2 Two-Way ANOVA

• Compares means based on two independent variables simultaneously
• Can detect:
  • Main effect of Factor A
  • Main effect of Factor B
  • Interaction effect between A and B (does the effect of A depend on the level of B?)
• Example: Comparing blood pressure across 3 drugs AND 2 genders simultaneously

4.3 Repeated Measures ANOVA

• Used when the same subjects are measured at multiple time points or under multiple conditions
• Controls for individual differences between subjects
• Example: Measuring anxiety scores in patients before, during, and after treatment
• Requires testing the assumption of sphericity (Mauchly's test)

4.4 MANOVA (Multivariate ANOVA)

• Extension of ANOVA for two or more dependent variables at once
• Example: Comparing both systolic AND diastolic blood pressure across groups simultaneously

---

5. Assumptions of ANOVA

If assumptions are violated: Use Welch's ANOVA (for unequal variances) or Kruskal-Wallis test (non-parametric alternative to one-way ANOVA).

---

6. Step-by-Step Procedure of ANOVA

Step 1: State Hypotheses

• H₀: μ₁ = μ₂ = μ₃ (all group means equal)
• H₁: At least one μᵢ ≠ μⱼ

Step 2: Set Significance Level

• Typically α = 0.05

Step 3: Calculate Sum of Squares

Step 4: Calculate Degrees of Freedom (df)

Step 5: Calculate Mean Squares (MS)

Step 6: Calculate the F-Statistic

Step 7: Build the ANOVA Table

Step 8: Compare F to Critical Value / Find p-value

• Look up the critical F value from the F-distribution table using df_B and df_W
• Or use software to find p-value
• If F_calculated > F_critical (or p < α): Reject H₀

Step 9: Post-Hoc Tests (if H₀ is rejected)

---

7. Post-Hoc Tests

---

8. Worked Example

---

9. How to Report ANOVA Results

Format: F(df_between, df_within) = value, p = value, η² = effect size

---

10. Effect Size in ANOVA

---

11. Non-Parametric Alternative

• One-Way ANOVA → Kruskal-Wallis Test (uses ranks instead of raw values)
• Repeated Measures ANOVA → Friedman Test

---

12. Quick Summary

---