🌾 Randomized Complete Block Design (RCBD)

ANOVA + Tukey HSD

📖 What is RCBD? A design that groups experimental units into blocks (homogeneous groups) to remove variability due to nuisance factors. Each block contains all treatments once, allowing us to compare treatments after accounting for block differences.

📐 Statistical Model: Y_ij = μ + τ_i + β_j + ε_ij

where τ_i = treatment effect, β_j = block effect, ε_ij ~ N(0, σ²).

📊 1. Define Design & Data

Treatments (t)

Blocks (b)

📋 Response Values (Y_ij)

📐 Sum of Squares Formulas

Method 1: Correction Factor (CF)

CF = G² / N
where G = grand total, N = t × b

SSTotal = ΣY² − CF

SSTreat = [Σ(Tᵢ²)/b] − CF

SSBlock = [Σ(Bⱼ²)/t] − CF

SSError = SSTotal − SSTreat − SSBlock

Method 2: Direct Deviation (No Correction Factor)

SSTotal = ΣᵢΣⱼ (Y_ij − Ȳ_..)²

SSTreat = b × Σᵢ (Ȳ_i. − Ȳ_..)²

SSBlock = t × Σⱼ (Ȳ_.j − Ȳ_..)²

SSError = ΣᵢΣⱼ (Y_ij − Ȳ_i. − Ȳ_.j + Ȳ_..)²

✔️ Where Ȳ_.. = grand mean, Ȳ_i. = treatment mean, Ȳ_.j = block mean.

📊

Statistical Note: Tukey's Family-wise Confidence Level

✓ The confidence level (1−α) for Tukey's HSD intervals is family-wise (simultaneous), NOT per-comparison.

✓ Each individual CI has coverage > 1−α (e.g., >95% for α=0.05).

✓ All CIs together have exactly (1−α)×100% confidence that every interval contains its true difference.

✓ No further adjustment (e.g., Bonferroni) is needed — Tukey's method already controls the family-wise error rate (FWER) at α.

📖 5. How to Interpret Results

🎯 ANOVA Table Interpretation

p-value for Treatments: If p < 0.05, reject H₀ → at least one treatment mean differs significantly.
p-value for Blocks: If p < 0.05, blocking was effective at reducing variability.

📊 Effect Sizes (Partial Eta Squared - η²ₚ)

η²ₚ = SS_effect / (SS_effect + SS_error)

Small: η²ₚ ≈ 0.01 (explains ~1% of variance)
Medium: η²ₚ ≈ 0.06 (explains ~6% of variance)
Large: η²ₚ ≈ 0.14 (explains ≥14% of variance)

📏 Cohen's d (Pairwise Comparisons)

d = (M₁ - M₂) / √(MSE)

Small: d ≈ 0.2 (difference of 0.2 standard deviations)
Medium: d ≈ 0.5 (moderate, visible effect)
Large: d ≈ 0.8 (substantial practical significance)

🔍 Interpreting Confidence Intervals (Tukey HSD)

CI = (Ȳᵢ − Ȳⱼ) ± HSD, where HSD = q × √(MSE / b)

If the interval does NOT contain zero → The difference is statistically significant at the chosen α level. We are (1−α)×100% confident that the true difference between treatment means lies within this range.
If the interval contains zero → The difference is not statistically significant. We cannot rule out that the true difference is zero.
Example: If 95% CI for (A − B) is (2.5, 7.3), we are 95% confident that Treatment A is between 2.5 and 7.3 units higher than Treatment B. Since 0 is not in the interval, the difference is significant.
Family-wise interpretation: With 95% family-wise confidence, if you repeat the experiment many times, 95% of the time all confidence intervals will simultaneously capture their respective true differences.

💡 A statistically significant difference (CI does not contain 0) tells you that an effect exists, but the confidence interval width tells you how precise your estimate is. Narrower intervals indicate more precise estimates.

📚 Tukey's Critical Values (q) Table — α = 0.05 ?

Error df (ν)	t=2	t=3	t=4	t=5	t=6	t=7	t=8

ANOVA + Tukey HSD

📊 1. Define Design & Data

🏷️ Treatment Names

📌 Block Labels

📋 Response Values (Y_ij)

📐 Sum of Squares Formulas

Method 1: Correction Factor (CF)

Method 2: Direct Deviation (No Correction Factor)

📐 2. Step‑by‑Step (Correction Factor Method)

📈 3. ANOVA Table with p-values & Effect Sizes

🔍 4. Treatment Means & Tukey HSD

📏 Tukey's HSD = q_α,(t,ν) × √(MSE / b)

Statistical Note: Tukey's Family-wise Confidence Level

📖 5. How to Interpret Results

🎯 ANOVA Table Interpretation

📊 Effect Sizes (Partial Eta Squared - η²ₚ)

📏 Cohen's d (Pairwise Comparisons)

🔍 Interpreting Confidence Intervals (Tukey HSD)

📚 Tukey's Critical Values (q) Table — α = 0.05 ?

ANOVA + Tukey HSD

📊 1. Define Design & Data

🏷️ Treatment Names

📌 Block Labels

📋 Response Values (Yij)

📐 Sum of Squares Formulas

Method 1: Correction Factor (CF)

Method 2: Direct Deviation (No Correction Factor)

📐 2. Step‑by‑Step (Correction Factor Method)

📈 3. ANOVA Table with p-values & Effect Sizes

🔍 4. Treatment Means & Tukey HSD

📏 Tukey's HSD = qα,(t,ν) × √(MSE / b)

Statistical Note: Tukey's Family-wise Confidence Level

📖 5. How to Interpret Results

🎯 ANOVA Table Interpretation

📊 Effect Sizes (Partial Eta Squared - η²ₚ)

📏 Cohen's d (Pairwise Comparisons)

🔍 Interpreting Confidence Intervals (Tukey HSD)

📚 Tukey's Critical Values (q) Table — α = 0.05 ?

📋 Response Values (Y_ij)

📏 Tukey's HSD = q_α,(t,ν) × √(MSE / b)