Module 3 Assessment

Step-by-Step Reasoning:

1. Understanding the Question: We want to test if the new formulation is "significantly different" from 30 minutes, which indicates a two-tailed test.
2. Null Hypothesis (H₀): Always states "no difference" or "no effect" - the dissolution time equals the standard (μ = 30 minutes).
3. Alternative Hypothesis (H₁): States what we're trying to prove - the dissolution time is different from the standard (μ ≠ 30 minutes).
4. Population vs. Sample: Hypotheses are about population parameters (μ), not sample statistics (x̄).

Why Others Are Wrong:

• A: Reverses null and alternative hypotheses
• C: These would be for one-tailed tests, not "different from"
• D: Uses sample mean (x̄) instead of population mean (μ)

Step-by-Step Reasoning:

1. Understanding Type II Error (β): Failing to reject a false null hypothesis
2. In Pharmaceutical Context: H₀ = "Batch meets specifications"
3. Type II Error Occurs When: We fail to reject H₀ (conclude batch is good) when it's actually false (batch is bad)
4. Real-World Impact: Bad product reaches consumers (consumer's risk)
5. Power Relationship: Power = 1 - β (probability of correctly detecting a bad batch)

Memory Aid:

Type I Error (α): Producer's risk - rejecting good batch
Type II Error (β): Consumer's risk - accepting bad batch

Step-by-Step Calculation:

1. Step 1: Calculate the standard error (SE)
   SE = s / √n = 3.2 / √16 = 3.2 / 4 = 0.8 mg
2. Step 2: Calculate the difference from target
   x̄ - μ₀ = 502.3 - 500 = 2.3 mg
3. Step 3: Calculate t-statistic
   t = 2.3 / 0.8 = 2.875
4. Step 4: Degrees of freedom = n - 1 = 16 - 1 = 15

Interpretation:

With t = 2.875 and df = 15, this would be significant at α = 0.05 (critical value ≈ 2.131), suggesting the tablet weight differs significantly from the target.

Step-by-Step Decision Process:

1. Identify the Design: Two independent groups (different processes)
2. Check Sample Sizes: n₁ = 12, n₂ = 15 (unequal sample sizes)
3. Check Variance Equality:
   s₁² = (2.1)² = 4.41
   s₂² = (2.8)² = 7.84
   Ratio = 7.84/4.41 = 1.78 (suggests unequal variances)
4. Rule of Thumb: If variance ratio > 1.5 or unequal n, use Welch's test
5. Test Choice: Independent samples t-test with unequal variances

Why Others Are Wrong:

• A: No pairing between observations
• B: Variances appear unequal and sample sizes differ
• D: We have two groups, not one

Step-by-Step ANOVA Interpretation:

1. ANOVA Hypotheses:
   H₀: μ₁ = μ₂ = μ₃ = μ₄ (all means are equal)
   H₁: At least one mean differs
2. Decision Rule: Reject H₀ if p-value < α
3. Compare p-value to α: 0.018 < 0.05
4. Conclusion: Reject H₀, meaning at least one batch differs
5. Next Step: Post-hoc tests needed to identify which batches differ

Important Note:

ANOVA only tells us that differences exist somewhere among the groups, not specifically which groups differ. Post-hoc tests (Tukey's HSD, Bonferroni) are needed for pairwise comparisons.

When to Use Non-Parametric Tests:

1. Violation of Normality: Data is severely skewed, has outliers, or fails normality tests
2. Ordinal Data: Data represents ranks or ordered categories
3. Small Sample Sizes: When n < 30 and normality cannot be assumed
4. Robust Alternative: Mann-Whitney is less sensitive to outliers than t-test

Mann-Whitney U Test Advantages:

• No normality assumption required
• Robust to outliers
• Works with any distribution shape
• Only requires ordinal data

Disadvantages:

• Less powerful than t-test when assumptions are met
• Tests medians, not means

Understanding Statistical Power:

1. Definition: Power = 1 - β (probability of correctly rejecting false H₀)
2. In This Context: If tablets truly differ by 5 mg, we'll detect it 80% of the time
3. Type II Error: β = 1 - Power = 1 - 0.80 = 0.20 (20% chance of missing the difference)
4. Practical Meaning: Out of 100 studies where true difference = 5 mg, 80 would find significance

Factors Affecting Power:

• Effect Size: Larger differences → Higher power
• Sample Size: Larger n → Higher power
• Significance Level: Higher α → Higher power
• Variability: Lower SD → Higher power

Correct Confidence Interval Interpretation:

1. Key Concept: CI is about the process, not the specific interval
2. Correct Thinking: "If we repeat this sampling procedure many times..."
3. 95% Refers To: The long-run frequency of intervals containing μ
4. This Specific Interval: Either contains μ or it doesn't (probability is 0 or 1)

Common Misconceptions:

• A: The parameter μ is fixed, not random
• B: This describes sampling distribution, not CI
• D: Sample mean is already observed, not probabilistic

Formula Used:

t-Test Assumptions (Complete List):

1. Normality: Data should be approximately normally distributed
   • Test: Shapiro-Wilk, Q-Q plots
   • Robust for n > 30 (Central Limit Theorem)
2. Independence: Observations should be independent
   • Check study design
   • No autocorrelation in time series data
3. Equal Variances: For two-sample t-test (homoscedasticity)
   • Test: Levene's test, F-test
   • Alternative: Welch's test if violated

What If Assumptions Are Violated?

• Non-normality: Use non-parametric tests (Mann-Whitney, Wilcoxon)
• Unequal variances: Use Welch's t-test
• Non-independence: Use appropriate correlation structure

Step-by-Step Sample Size Calculation:

1. Step 1: Find critical values
   Z_α/2 = Z_0.025 = 1.96 (for α = 0.05, two-tailed)
   Z_β = Z_0.20 = 0.84 (for Power = 80%, β = 0.20)
2. Step 2: Calculate (Z_α/2 + Z_β)²
   (1.96 + 0.84)² = (2.80)² = 7.84
3. Step 3: Apply the formula
   n = 2 × 7.84 × (4)² / (3)²
   n = 2 × 7.84 × 16 / 9
   n = 250.88 / 9 = 27.9 ≈ 28
4. Step 4: Round up for conservative estimate
   n ≈ 35 per group (accounting for dropouts and variability)

Interpretation:

With 35 tablets per group, we have adequate power to detect a 3 mg difference in weight between formulations.

Step-by-Step Test Selection:

1. Identify Data Types:
   Shift: Nominal categorical (3 categories)
   Quality: Nominal categorical (2 categories)
2. Research Question: Is there an association between shift and defect rate?
3. Data Structure: Contingency table (3×2)
4. Appropriate Test: Chi-square test of independence

Chi-Square Test Process:

1. H₀: Shift and quality are independent
2. H₁: Shift and quality are associated
3. Calculate Expected Frequencies: E = (Row Total × Column Total) / Grand Total
4. Chi-Square Statistic: χ² = Σ(O - E)²/E
5. Compare to Critical Value: df = (rows-1)(cols-1) = (3-1)(2-1) = 2

Cohen's d Calculation:

1. Step 1: Calculate mean difference
   |x̄₁ - x̄₂| = |25.3 - 28.7| = 3.4 minutes
2. Step 2: Use given pooled SD
   s_pooled = 3.25 minutes
3. Step 3: Calculate Cohen's d
   d = 3.4 / 3.25 = 1.05
4. Step 4: Interpret effect size
   d = 1.05 → Large effect (d ≥ 0.8)

Cohen's d Interpretation Guidelines:

• Small effect: d = 0.2
• Medium effect: d = 0.5
• Large effect: d = 0.8

Practical Meaning:

A Cohen's d of 1.05 indicates the formulations differ by more than one standard deviation - a clinically meaningful difference.

Test Selection Logic:

1. Study Design: Before-after (paired/dependent observations)
2. Data Type: Ordinal (pain scale 1-10)
3. Distribution: Heavily skewed (violates normality)
4. Sample Size: n = 12 (relatively small)
5. Conclusion: Use non-parametric test for paired data

Wilcoxon Signed-Rank Test:

• Purpose: Compare paired observations when normality is violated
• Process: Ranks differences (ignoring sign), then applies signs
• Assumption: Differences are symmetric around median
• Output: Tests whether median difference = 0

Why Others Are Wrong:

• A: Requires normality (violated)
• C: For independent groups, not paired
• D: For independent groups and requires normality

Multiple Comparisons Problem:

1. Number of Comparisons: With 5 groups, there are C(5,2) = 10 possible pairwise comparisons
2. Individual α: Each t-test uses α = 0.05
3. Family-wise Error Rate: Probability of at least one Type I error
   FWER ≈ 1 - (1 - 0.05)¹⁰ = 1 - 0.599 = 0.401 (40%!)
4. Problem: Very high chance of finding "significant" differences by chance alone

Post-Hoc Test Solutions:

• Tukey's HSD: Controls FWER for all pairwise comparisons
• Bonferroni: Divides α by number of comparisons (α/k)
• Dunnett's: For comparing all groups to a control
• Scheffe's: Most conservative, for any contrast

Practical Impact:

Without correction, you might incorrectly conclude formulations differ when they don't, leading to unnecessary reformulation work and costs.

FDA Bioequivalence Assessment:

1. Regulatory Requirement: 90% CI for ratio must fall entirely within (0.80, 1.25)
2. Observed CI: (0.88, 1.09)
3. Check Lower Bound: 0.88 > 0.80 ✓
4. Check Upper Bound: 1.09 < 1.25 ✓
5. Conclusion: Entire CI falls within acceptance range → Bioequivalent

Why This Approach Works:

• Two One-Sided Tests (TOST): Tests both "not too low" and "not too high"
• 90% CI Equivalence: 90% CI corresponds to two one-sided 5% tests
• Regulatory Acceptance: Approved method by FDA/EMA

Clinical Interpretation:

The generic drug's absorption (AUC) is between 88% and 109% of the brand name, well within the acceptable range for therapeutic equivalence.