🔬 Module 3: Hypothesis Testing Laboratory

Master Statistical Inference & Bioequivalence Testing with Pharmaceutical Applications

Ready to Begin • 4 Hours Interactive Learning

Part 1: Hypothesis Construction Workshop

Learn to build and test hypotheses with pharmaceutical examples

30 minutes

📚 Theory: Building Hypotheses

Step-by-Step: Hypothesis Construction
  1. Research Question: Start with a clear, testable question about pharmaceutical formulations
  2. Null Hypothesis (H₀): State the assumption of no effect or no difference
  3. Alternative Hypothesis (H₁): Define what you're trying to prove
  4. Significance Level (α): Set your tolerance for Type I error (typically 0.05)
  5. Test Selection: Choose appropriate statistical test based on data type
Example: Comparing Two Tablet Formulations

Research Question: Does the new tablet formulation have the same dissolution rate as the reference?

H₀: μ_new = μ_reference (no difference in dissolution rates)

H₁: μ_new ≠ μ_reference (there is a difference)

α: 0.05 (5% chance of incorrectly rejecting H₀)

Test: Two-sample t-test (assuming normal distribution)

🧪 Practice: Hypothesis Builder

Interactive Hypothesis Constructor

Generated Hypothesis:

⚠️ Understanding Type I and Type II Errors

Step-by-Step: Error Analysis
  1. Type I Error (α): Falsely rejecting a true null hypothesis
  2. Type II Error (β): Failing to reject a false null hypothesis
  3. Power (1-β): Probability of correctly rejecting a false null hypothesis
  4. Sample Size Impact: Larger samples reduce both error types
Pharmaceutical Error Consequences

Type I Error in Bioequivalence: Concluding products are NOT bioequivalent when they actually are (rejecting good generic)

Type II Error in Bioequivalence: Concluding products ARE bioequivalent when they're not (approving inadequate generic)

Regulatory Impact: FDA sets α = 0.05 and requires Power ≥ 80% for BE studies

✅ Knowledge Check

1. In a bioequivalence study, what does the null hypothesis typically state?
A) The products are bioequivalent
B) The products have different bioavailability
C) The products are NOT bioequivalent
D) The test product is superior

Part 2: Parametric Test Suite

Master t-tests, ANOVA, and Excel implementations

45 minutes

📚 Theory: Parametric Tests

Step-by-Step: One-Sample t-Test
  1. Formula: t = (x̄ - μ₀) / (s/√n)
  2. Degrees of Freedom: df = n - 1
  3. Excel Implementation: =T.TEST(array, μ₀, tails, type)
  4. Critical Value: Use T.INV function for decision
  5. Interpretation: Compare calculated t with critical value
Testing Tablet Weight Specification

Scenario: Tablets should weigh 250mg. Sample of 10 tablets: [248, 252, 249, 251, 250, 247, 253, 249, 251, 250]

Let's solve this step-by-step:
  1. Calculate mean: x̄ = (248+252+...+250)/10 = 250.0 mg
  2. Calculate standard deviation: s = 1.89 mg
  3. Calculate t-statistic: t = (250.0 - 250) / (1.89/√10) = 0
  4. Excel formula: =T.TEST({248;252;249;251;250;247;253;249;251;250}, 250, 2, 1)
  5. Conclusion: p-value = 1.00, do not reject H₀ (tablets meet specification)

🧪 Practice: t-Test Calculator

One-Sample t-Test Calculator

📊 Two-Sample t-Tests

Comparing Two Dissolution Methods
Step-by-Step: Independent Samples t-Test
  1. Assumption Check: Normal distribution, equal variances
  2. Pooled Standard Deviation: sp = √[((n₁-1)s₁² + (n₂-1)s₂²)/(n₁+n₂-2)]
  3. t-Statistic: t = (x̄₁ - x̄₂) / (sp√(1/n₁ + 1/n₂))
  4. Excel Implementation: =T.TEST(array1, array2, 2, 2)
  5. Decision: Compare p-value with α = 0.05

Part 3: Non-Parametric Test Laboratory

Alternative methods when assumptions aren't met

30 minutes

📚 Theory: Distribution-Free Tests

When to Use Non-Parametric Tests
  1. Non-normal data: Skewed distributions, outliers present
  2. Ordinal data: Ranked measurements (taste, hardness scale)
  3. Small sample sizes: When normality cannot be assessed
  4. Robust alternatives: Less sensitive to extreme values
Mann-Whitney U Test for Sensory Evaluation

Scenario: Comparing taste acceptability scores (1-10 scale) between two formulations

Step-by-Step: Mann-Whitney U Test
  1. Combine and Rank: Merge both groups, assign ranks
  2. Sum Ranks: Calculate rank sums for each group
  3. Calculate U: U₁ = n₁n₂ + n₁(n₁+1)/2 - R₁
  4. Test Statistic: Use smaller U value
  5. Excel Alternative: Use RANK function approach

🎯 Practice: Test Selection Decision Tree

Statistical Test Selector

Part 4: Multiple Comparisons Suite

Control family-wise error in multiple testing

30 minutes

📚 Theory: Multiple Testing Problem

Why Multiple Comparisons Matter
  1. Inflation of Type I Error: More tests = higher chance of false positives
  2. Family-Wise Error Rate: Probability of at least one Type I error
  3. Bonferroni Correction: α_adjusted = α / number of comparisons
  4. Alternative Methods: Tukey's HSD, Dunnett's test for specific designs
Batch Testing with Multiple Comparisons

Scenario: Testing 5 batches for content uniformity (10 pairwise comparisons)

Bonferroni Correction Application
  1. Original α: 0.05 for each test
  2. Number of comparisons: C(5,2) = 10 pairwise tests
  3. Adjusted α: 0.05/10 = 0.005 per test
  4. Family-wise α: Remains ≤ 0.05
  5. Trade-off: Lower power but controlled overall error rate

Part 5: Bioequivalence Testing Suite

Master FDA-approved methods for proving therapeutic equivalence

45 minutes

📚 Theory: Bioequivalence Fundamentals

FDA Bioequivalence Criteria

90% Confidence Interval
Must be within 80.00% - 125.00%
Primary Parameters
AUC₀₋ₜ and Cmax
Study Design
2×2 Crossover preferred
Log Transformation
Required for ratio analysis
Step-by-Step: 90% Confidence Interval Method
  1. Log-Transform Data: ln(AUC), ln(Cmax) for each subject
  2. ANOVA for Crossover: Account for sequence, period, subject effects
  3. Calculate Difference: ln(Test) - ln(Reference) = ln(T/R)
  4. 90% CI Formula: (T-R) ± t₀.₀₅ × SE
  5. Back-Transform: e^(lower limit), e^(upper limit)
  6. Apply Criteria: Both limits must be within 80-125%
Complete Bioequivalence Analysis

Study Design: 2×2 crossover, 24 subjects, Test vs Reference formulation

Sample Data Analysis (AUC₀₋ₜ)
  1. Raw Data: AUC values for Test (T) and Reference (R)
  2. Log Transform: ln(T) = 4.89, ln(R) = 4.85 (geometric means)
  3. Difference: ln(T) - ln(R) = 0.04
  4. Standard Error: SE = 0.12
  5. 90% CI: 0.04 ± 1.714 × 0.12 = (-0.166, 0.246)
  6. Back-Transform: (e^(-0.166), e^(0.246)) = (84.7%, 127.9%)
  7. Conclusion: Upper limit exceeds 125% → NOT bioequivalent

🧪 Practice: Bioequivalence Calculator

BE 90% Confidence Interval Calculator

🔬 Two One-Sided Tests (TOST)

TOST Procedure (Schuirmann's Approach)
  1. Equivalence Hypotheses: H₀₁: μT/μR ≤ 0.80 and H₀₂: μT/μR ≥ 1.25
  2. Alternative Hypotheses: H₁₁: μT/μR > 0.80 and H₁₂: μT/μR < 1.25
  3. Test Statistics: t₁ and t₂ for each boundary
  4. Decision Rule: Reject both H₀₁ and H₀₂ at α = 0.05
  5. Equivalence to CI: TOST ≡ 90% CI within (80%, 125%)
TOST Implementation in Excel

Excel Formulas for TOST:

  • t₁ calculation: =(ln_diff - ln(0.8))/SE
  • t₂ calculation: =(ln(1.25) - ln_diff)/SE
  • p-values: =T.DIST(t1, df, TRUE) and =T.DIST(t2, df, TRUE)
  • Decision: Both p-values < 0.05 for bioequivalence

Part 6: Dissolution Profile Comparison

f2 similarity factor and model-independent methods

30 minutes

📚 Theory: Dissolution Similarity

f2 Similarity Factor Calculation
  1. Formula: f2 = 50 × log{[1 + (1/n)Σ(Rt - Tt)²]^(-0.5) × 100}
  2. Where: Rt = reference % dissolved at time t, Tt = test % dissolved
  3. Acceptance Criteria: f2 ≥ 50 indicates similarity
  4. Time Points: Use 3-4 points before 85% dissolution
  5. Difference Factor (f1): f1 = {[Σ|Rt - Tt|]/[ΣRt]} × 100
Dissolution Profile Comparison Example
Let's calculate f2 step-by-step:
Time (min) Reference (R) Test (T) (R-T)²
15 23% 25% 4
30 48% 52% 16
45 71% 69% 4
60 84% 82% 4
  1. Sum of (R-T)²: 4 + 16 + 4 + 4 = 28
  2. Average: 28/4 = 7
  3. f2 calculation: 50 × log((1 + 7)^(-0.5) × 100)
  4. Result: f2 = 64.5
  5. Conclusion: f2 > 50, profiles are similar

🧪 Practice: f2 Calculator

Dissolution Similarity Calculator

Part 7: Clinical vs Statistical Significance

Understanding practical importance beyond p-values

15 minutes

📚 Theory: Beyond p-Values

Statistical vs Clinical Significance
  1. Statistical Significance: p < 0.05 (unlikely due to chance)
  2. Clinical Significance: Meaningful difference in practice
  3. Effect Size: Magnitude of the difference (Cohen's d)
  4. Confidence Intervals: Range of plausible values
  5. Sample Size Impact: Large n can make small differences "significant"
Dissolution Rate Comparison

Scenario: New formulation dissolves 2% faster (p = 0.03, n = 1000)

  • Statistical Significance: Yes (p < 0.05)
  • Clinical Significance: Questionable (only 2% improvement)
  • Regulatory Impact: Unlikely to affect bioavailability
  • Business Decision: Cost-benefit analysis needed

Part 8: Interactive Challenges & Simulations

Test your knowledge with engaging activities

15 minutes

🎮 Challenge 1: p-Value Interpretation Quiz

What does p = 0.03 mean in a bioequivalence study?
A) 97% chance the products are bioequivalent
B) 3% chance the null hypothesis is true
C) 3% chance of observing this difference if products were bioequivalent
D) The products are definitely not bioequivalent

🎯 Challenge 2: Error Type Simulator

Adjust α and β to see their impact

Current Settings:

Power (1-β): 80%

Interpretation: Balanced approach for pharmaceutical studies

📝 Module 3 Assessment Quiz

Test your understanding with 30 multiple choice questions

15 minutes
1. In bioequivalence testing, the 90% confidence interval for the Test/Reference ratio must be within:
A) 75.00% - 133.33%
B) 80.00% - 125.00%
C) 85.00% - 115.00%
D) 90.00% - 111.11%
2. Which test is most appropriate for comparing dissolution profiles?
A) Paired t-test
B) f2 similarity factor
C) Chi-square test
D) ANOVA
3. In TOST procedure, bioequivalence is concluded when:
A) Both null hypotheses are rejected
B) One null hypothesis is rejected
C) The alternative hypothesis is rejected
D) The p-value is greater than 0.05

🏆 Mini-Project: Bioequivalence Study Analysis

Complete analysis of a 2×2 crossover bioequivalence study

45 minutes

📋 Project Description

Study Protocol
  • Design: 2×2 crossover, randomized, single-dose
  • Subjects: 24 healthy volunteers
  • Products: Test (Generic) vs Reference (Brand)
  • Parameters: AUC₀₋ₜ and Cmax
  • Washout: 7 days between periods
  • Objective: Demonstrate bioequivalence
Required Deliverables
  1. Statistical Analysis Plan (SAP): Describe analysis methods
  2. Descriptive Statistics: By sequence and period
  3. ANOVA Results: Sources of variation table
  4. 90% Confidence Intervals: For AUC and Cmax ratios
  5. Bioequivalence Conclusion: Based on FDA criteria
  6. Excel Workbook: All calculations and formulas

📊 Sample Dataset

Download Project Dataset

Click the button below to download the bioequivalence study dataset in Excel format:

Dataset Description:
  • 24 subjects (IDs: 001-024)
  • Sequence: TR (Test→Reference) or RT (Reference→Test)
  • AUC₀₋ₜ values in ng·hr/mL
  • Cmax values in ng/mL
  • Both raw and log-transformed data included

📝 Project Assessment Criteria

Component Weight Criteria
ANOVA Model 25% Correct model specification, sources of variation
CI Calculation 25% Accurate confidence interval computation
Interpretation 20% Proper bioequivalence conclusion
Visualization 15% Clear graphs and data presentation
Report Quality 15% Professional presentation, completeness