24.3 Randomized Block Designs

To improve the precision of treatment comparisons, we can reduce variability among experimental units by grouping them into blocks.
Each block contains homogeneous units, reducing the impact of nuisance variation.

Key Principles of Blocking

  • Within each block, treatments are randomly assigned to units.
  • The number of units per block is a multiple of the number of factor combinations.
  • Commonly, each treatment appears once per block.

Benefits of Blocking

  • Reduction in variability of treatment effect estimates

    • Improved power for t-tests and F-tests.

    • Narrower confidence intervals.

    • Smaller mean square error (MSE).

  • Allows comparison of treatments across different conditions (captured by blocks).

Potential Downsides of Blocking

  • If blocks are not chosen well, degrees of freedom are wasted on negligible block effects.

  • This reduces df for t-tests and F-tests without reducing MSE, causing a small loss of power.

24.3.0.1 Random Block Effects with Additive Effects

The statistical model for a randomized block design:

\[ Y_{ij} = \mu_{..} + \rho_i + \tau_j + \epsilon_{ij} \]

where:

  • \(i = 1, 2, \dots, n\) (Blocks)

  • \(j = 1, 2, \dots, r\) (Treatments)

  • \(\mu_{..}\): Overall mean response (averaged across all blocks and treatments).

  • \(\rho_i\): Block effect (average difference for the \(i\)-th block), constrained such that:

    \[ \sum_i \rho_i = 0 \]

  • \(\tau_j\): Treatment effect (average across blocks), constrained such that:

    \[ \sum_j \tau_j = 0 \]

  • \(\epsilon_{ij} \sim iid N(0, \sigma^2)\): Random experimental error.

Interpretation of the Model

  • Block and treatment effects are additive.

  • The difference in average response between any two treatments is the same within each block:

    \[ (\mu_{..} + \rho_i + \tau_j) - (\mu_{..} + \rho_i + \tau_j') = \tau_j - \tau_j' \]

  • This ensures that blocking only affects variability, not treatment comparisons.

Estimators of Model Parameters

  • Overall Mean:

    \[ \hat{\mu} = \bar{Y}_{..} \]

  • Block Effects:

    \[ \hat{\rho}_i = \bar{Y}_{i.} - \bar{Y}_{..} \]

  • Treatment Effects:

    \[ \hat{\tau}_j = \bar{Y}_{.j} - \bar{Y}_{..} \]

  • Fitted Response: \[ \hat{Y}_{ij} = \bar{Y}_{..} + (\bar{Y}_{i.} - \bar{Y}_{..}) + (\bar{Y}_{.j} - \bar{Y}_{..}) \]

    Simplifies to:

    \[ \hat{Y}_{ij} = \bar{Y}_{i.} + \bar{Y}_{.j} - \bar{Y}_{..} \]

  • Residuals:

    \[ e_{ij} = Y_{ij} - \hat{Y}_{ij} = Y_{ij} - \bar{Y}_{i.} - \bar{Y}_{.j} + \bar{Y}_{..} \]

The ANOVA decomposition partitions total variability into contributions from blocks, treatments, and residual error.

ANOVA Table for Randomized Block Design
Source of Variation Sum of Squares (SS) Degrees of Freedom (df) Fixed Treatments (E(MS)) Random Treatments (E(MS))
Blocks \(r \sum_i (\bar{Y}_{i.} - \bar{Y}_{..})^2\) \(n-1\) \(\sigma^2 + r \frac{\sum \rho_i^2}{n-1}\) \(\sigma^2 + r \frac{\sum \rho_i^2}{n-1}\)
Treatments \(n \sum_j (\bar{Y}_{.j} - \bar{Y}_{..})^2\) \(r-1\) \(\sigma^2 + n \frac{\sum \tau_j^2}{r-1}\) \(\sigma^2 + n \sigma^2_{\tau}\)
Error \(\sum_i \sum_j (Y_{ij} - \bar{Y}_{i.} - \bar{Y}_{.j} + \bar{Y}_{..})^2\) \((n-1)(r-1)\) \(\sigma^2\) \(\sigma^2\)
Total \(SSTO\) \(nr-1\) - -

24.3.0.2 F-tests in Randomized Block Designs

To test for treatment effects, we use an F-test:

For fixed treatment effects:

\[ \begin{aligned} H_0: \tau_1 = \tau_2 = \dots = \tau_r = 0 \quad \text{(No treatment effect)} \\ H_a: \text{Not all } \tau_j = 0 \end{aligned} \]

For random treatment effects:

\[ \begin{aligned} H_0: \sigma^2_{\tau} = 0 \quad \text{(No variance in treatment effects)} \\ H_a: \sigma^2_{\tau} \neq 0 \end{aligned} \]

In both cases, the test statistic is:

\[ F = \frac{MSTR}{MSE} \]

Reject \(H_0\) if:

\[ F > f_{(1-\alpha; r-1, (n-1)(r-1))} \]

Why Not Use an F-Test for Blocks?

We do not test for block effects because:

  1. Blocks are assumed to be different a priori.
  2. Randomization occurs within each block, ensuring treatments are comparable.

Efficiency Gain from Blocking

To measure the improvement in precision, compare the mean square error (MSE) in a completely randomized design vs. a randomized block design.

Estimated variance in a CRD:

\[ \hat{\sigma}^2_{CR} = \frac{(n-1)MSBL + n(r-1)MSE}{nr-1} \]

Estimated variance in an RBD:

\[ \hat{\sigma}^2_{RB} = MSE \]

Relative efficiency:

\[ \frac{\hat{\sigma}^2_{CR}}{\hat{\sigma}^2_{RB}} \]

  • If greater than 1, blocking reduces experimental error.

  • The percentage reduction in required sample size for an RBD:

\[ \left( \frac{\hat{\sigma}^2_{CR}}{\hat{\sigma}^2_{RB}} - 1 \right) \times 100\% \]

Random Blocks and Mixed Models

If blocks are randomly selected, they are treated as random effects.
That is, if the experiment were repeated, a new set of blocks would be selected, with:

\[ \rho_1, \rho_2, \dots, \rho_i \sim N(0, \sigma^2_\rho) \]

The model remains:

\[ Y_{ij} = \mu_{..} + \rho_i + \tau_j + \epsilon_{ij} \]

where:

  • \(\mu_{..}\) is fixed.
  • \(\rho_i \sim iid N(0, \sigma^2_\rho)\) (random block effects).
  • \(\tau_j\) is fixed (or random, with \(\sum \tau_j = 0\)).
  • \(\epsilon_{ij} \sim iid N(0, \sigma^2)\).

24.3.0.3 Variance and Covariance Structure

For fixed treatment effects:

\[ \begin{aligned} E(Y_{ij}) &= \mu_{..} + \tau_j \\ var(Y_{ij}) &= \sigma^2_{\rho} + \sigma^2 \end{aligned} \]

Observations within the same block are correlated:

\[ cov(Y_{ij}, Y_{ij'}) = \sigma^2_{\rho}, \quad j \neq j' \]

Observations from different blocks are independent:

\[ cov(Y_{ij}, Y_{i'j'}) = 0, \quad i \neq i', j \neq j' \]

The intra-block correlation:

\[ \frac{\sigma^2_{\rho}}{\sigma^2 + \sigma^2_{\rho}} \]

Expected Mean Squares for Fixed Treatments
Source SS E(MS)
Blocks SSBL \(\sigma^2 + r \sigma^2_\rho\)
Treatments SSTR \(\sigma^2 + n \frac{\sum \tau^2_j}{r-1}\)
Error SSE \(\sigma^2\)

24.3.0.4 Random Block Effects with Interaction

When block-treatment interaction exists, we modify the model:

\[ Y_{ij} = \mu_{..} + \rho_i + \tau_j + (\rho \tau)_{ij} + \epsilon_{ij} \]

where:

  • \(\rho_i \sim iid N(0, \sigma^2_{\rho})\) (random).

  • \(\tau_j\) is fixed (\(\sum \tau_j = 0\)).

  • \((\rho \tau)_{ij} \sim N(0, \frac{r-1}{r} \sigma^2_{\rho \tau})\), constrained such that:

    \[ \sum_j (\rho \tau)_{ij} = 0, \quad \forall i \]

  • Covariance between interaction terms:

    \[ cov((\rho \tau)_{ij}, (\rho \tau)_{ij'}) = -\frac{1}{r} \sigma^2_{\rho \tau}, \quad j \neq j' \]

  • \(\epsilon_{ij} \sim iid N(0, \sigma^2)\).

Variance and Covariance with Interaction

  • Expectation:

    \[ E(Y_{ij}) = \mu_{..} + \tau_j \]

  • Total variance:

    \[ var(Y_{ij}) = \sigma^2_\rho + \frac{r-1}{r} \sigma^2_{\rho \tau} + \sigma^2 \]

  • Within-block covariance:

    \[ cov(Y_{ij}, Y_{ij'}) = \sigma^2_\rho - \frac{1}{r} \sigma^2_{\rho \tau}, \quad j \neq j' \]

  • Between-block covariance:

    \[ cov(Y_{ij}, Y_{i'j'}) = 0, \quad i \neq i', j \neq j' \]

The sum of squares and degrees of freedom for interaction model are the same as those for the additive model. The difference exists only in the expected mean squares.

ANOVA Table with Interaction Effects
Source SS df E(MS)
Blocks \(SSBL\) \(n-1\) \(\sigma^2 + r \sigma^2_\rho\)
Treatments \(SSTR\) \(r-1\) \(\sigma^2 + \sigma^2_{\rho \tau} + n \frac{\sum \tau_j^2}{r-1}\)
Error \(SSE\) \((n-1)(r-1)\) \(\sigma^2 + \sigma^2_{\rho \tau}\)
  • No exact test is possible for block effects when interaction is present (Not important if blocks are used primarily to reduce experimental error variability)
  • \(E(MSE) = \sigma^2 + \sigma^2_{\rho \tau}\) the error term variance and interaction variance \(\sigma^2_{\rho \tau}\). We can’t estimate these components separately with this model. The two are confounded.
  • If more than one observation per treatment block combination, one can consider interaction with fixed block effects, which is called generalized randomized block designs (multifactor analysis).

24.3.0.5 Tukey Test of Additivity

Tukey’s 1-degree-of-freedom test for additivity provides a formal test for interaction effects between blocks and treatments in a randomized block design.

This test can also be used in two-way ANOVA when there is only one observation per cell.

In a randomized block design, an additive model assumes:

\[ Y_{ij} = \mu_{..} + \rho_i + \tau_j + \epsilon_{ij} \]

where:

  • \(\mu_{..}\) = overall mean

  • \(\rho_i\) = block effect

  • \(\tau_j\) = treatment effect

  • \(\epsilon_{ij}\) = random error, \(iid N(0, \sigma^2)\)

To test for interaction, we introduce a less restricted interaction term:

\[ (\rho \tau)_{ij} = D \rho_i \tau_j \]

where \(D\) is a constant measuring interaction strength.

Thus, the interaction model becomes:

\[ Y_{ij} = \mu_{..} + \rho_i + \tau_j + D\rho_i \tau_j + \epsilon_{ij} \]

The least squares estimate (or MLE) of \(D\) is:

\[ \hat{D} = \frac{\sum_i \sum_j \rho_i \tau_j Y_{ij}}{\sum_i \rho_i^2 \sum_j \tau_j^2} \]

Replacing \(\rho_i\) and \(\tau_j\) with their estimates:

\[ \hat{D} = \frac{\sum_i \sum_j (\bar{Y}_{i.} - \bar{Y}_{..})(\bar{Y}_{.j} - \bar{Y}_{..}) Y_{ij}}{\sum_i (\bar{Y}_{i.} - \bar{Y}_{..})^2 \sum_j (\bar{Y}_{.j} - \bar{Y}_{..})^2} \]

The sum of squares for interaction is:

\[ SS_{int} = \sum_i \sum_j \hat{D}^2 (\bar{Y}_{i.} - \bar{Y}_{..})^2 (\bar{Y}_{.j} - \bar{Y}_{..})^2 \]

ANOVA Decomposition

The total sum of squares (SSTO) is decomposed as:

\[ SSTO = SSBL + SSTR + SS_{int} + SS_{Rem} \]

where:

  • \(SSBL\) = Sum of squares due to blocks

  • \(SSTR\) = Sum of squares due to treatments

  • \(SS_{int}\) = Interaction sum of squares

  • \(SS_{Rem}\) = Remainder sum of squares, computed as:

\[ SS_{Rem} = SSTO - SSBL - SSTR - SS_{int} \]

We test:

\[ \begin{aligned} &H_0: D = 0 \quad \text{(No interaction present)} \\ &H_a: D \neq 0 \quad \text{(Interaction of form $D \rho_i \tau_j$ present)} \end{aligned} \]

If \(D = 0\), then \(SS_{int}\) and \(SS_{Rem}\) are independent and follow:

\[ SS_{int} \sim \chi^2_1, \quad SS_{Rem} \sim \chi^2_{(rn-r-n)} \]

Thus, the F-statistic for testing interaction is:

\[ F = \frac{SS_{int} / 1}{SS_{Rem} / (rn - r - n)} \]

which follows an \(F\)-distribution:

\[ F \sim F_{(1, nr - r - n)} \]

We reject \(H_0\) if:

\[ F > f_{(1-\alpha; 1, nr - r - n)} \]