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:

Yij=μ..+ρi+τj+ϵij

where:

  • i=1,2,,n (Blocks)

  • j=1,2,,r (Treatments)

  • μ..: Overall mean response (averaged across all blocks and treatments).

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

    iρi=0

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

    jτj=0

  • ϵijiidN(0,σ2): 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:

    (μ..+ρi+τj)(μ..+ρi+τj)=τjτj

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


Estimators of Model Parameters

  • Overall Mean:

    ˆμ=ˉY..

  • Block Effects:

    ˆρi=ˉYi.ˉY..

  • Treatment Effects:

    ˆτj=ˉY.jˉY..


  • Fitted Response: ˆYij=ˉY..+(ˉYi.ˉY..)+(ˉY.jˉY..)

    Simplifies to:

    ˆYij=ˉYi.+ˉY.jˉY..

  • Residuals:

    eij=YijˆYij=YijˉYi.ˉY.j+ˉY..


24.3.0.2 ANOVA Table for Randomized Block Design

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

Source of Variation Sum of Squares (SS) Degrees of Freedom (df) Fixed Treatments (E(MS)) Random Treatments (E(MS))
Blocks ri(ˉYi.ˉY..)2 n1 σ2+rρ2in1 σ2+rρ2in1
Treatments nj(ˉY.jˉY..)2 r1 σ2+nτ2jr1 σ2+nσ2τ
Error ij(YijˉYi.ˉY.j+ˉY..)2 (n1)(r1) σ2 σ2
Total SSTO nr1 - -

24.3.0.3 F-tests in Randomized Block Designs

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

For fixed treatment effects:

H0:τ1=τ2==τr=0(No treatment effect)Ha:Not all τj=0

For random treatment effects:

H0:σ2τ=0(No variance in treatment effects)Ha:σ2τ0

In both cases, the test statistic is:

F=MSTRMSE

Reject H0 if:

F>f(1α;r1,(n1)(r1))


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:

ˆσ2CR=(n1)MSBL+n(r1)MSEnr1

Estimated variance in an RBD:

ˆσ2RB=MSE

Relative efficiency:

ˆσ2CRˆσ2RB

  • If greater than 1, blocking reduces experimental error.

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

(ˆσ2CRˆσ2RB1)×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:

ρ1,ρ2,,ρiN(0,σ2ρ)

The model remains:

Yij=μ..+ρi+τj+ϵij

where:

  • μ.. is fixed.
  • ρiiidN(0,σ2ρ) (random block effects).
  • τj is fixed (or random, with τj=0).
  • ϵijiidN(0,σ2).

24.3.0.4 Variance and Covariance Structure

For fixed treatment effects:

E(Yij)=μ..+τjvar(Yij)=σ2ρ+σ2

Observations within the same block are correlated:

cov(Yij,Yij)=σ2ρ,jj

Observations from different blocks are independent:

cov(Yij,Yij)=0,ii,jj

The intra-block correlation:

σ2ρσ2+σ2ρ


Expected Mean Squares for Fixed Treatments

Source SS E(MS)
Blocks SSBL σ2+rσ2ρ
Treatments SSTR σ2+nτ2jr1
Error SSE σ2

24.3.0.5 Random Block Effects with Interaction

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

Yij=μ..+ρi+τj+(ρτ)ij+ϵij

where:

  • ρiiidN(0,σ2ρ) (random).

  • τj is fixed (τj=0).

  • (ρτ)ijN(0,r1rσ2ρτ), constrained such that:

    j(ρτ)ij=0,i

  • Covariance between interaction terms:

    cov((ρτ)ij,(ρτ)ij)=1rσ2ρτ,jj

  • ϵijiidN(0,σ2).


Variance and Covariance with Interaction

  • Expectation:

    E(Yij)=μ..+τj

  • Total variance:

    var(Yij)=σ2ρ+r1rσ2ρτ+σ2

  • Within-block covariance:

    cov(Yij,Yij)=σ2ρ1rσ2ρτ,jj

  • Between-block covariance:

    cov(Yij,Yij)=0,ii,jj

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.


24.3.0.6 ANOVA Table with Interaction Effects

Source SS df E(MS)
Blocks SSBL n1 σ2+rσ2ρ
Treatments SSTR r1 σ2+σ2ρτ+nτ2jr1
Error SSE (n1)(r1) σ2+σ2ρτ

  • 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)=σ2+σ2ρτ the error term variance and interaction variance σ2ρτ. 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).

24.3.0.7 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:

Yij=μ..+ρi+τj+ϵij

where:

  • μ.. = overall mean

  • ρi = block effect

  • τj = treatment effect

  • ϵij = random error, iidN(0,σ2)

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

(ρτ)ij=Dρiτj

where D is a constant measuring interaction strength.

Thus, the interaction model becomes:

Yij=μ..+ρi+τj+Dρiτj+ϵij

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

ˆD=ijρiτjYijiρ2ijτ2j

Replacing ρi and τj with their estimates:

ˆD=ij(ˉYi.ˉY..)(ˉY.jˉY..)Yiji(ˉYi.ˉY..)2j(ˉY.jˉY..)2

The sum of squares for interaction is:

SSint=ijˆD2(ˉYi.ˉY..)2(ˉY.jˉY..)2


ANOVA Decomposition

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

SSTO=SSBL+SSTR+SSint+SSRem

where:

  • SSBL = Sum of squares due to blocks

  • SSTR = Sum of squares due to treatments

  • SSint = Interaction sum of squares

  • SSRem = Remainder sum of squares, computed as:

SSRem=SSTOSSBLSSTRSSint


We test:

H0:D=0(No interaction present)Ha:D0(Interaction of form Dρiτj present)

If D=0, then SSint and SSRem are independent and follow:

SSintχ21,SSRemχ2(rnrn)

Thus, the F-statistic for testing interaction is:

F=SSint/1SSRem/(rnrn)

which follows an F-distribution:

FF(1,nrrn)

We reject H0 if:

F>f(1α;1,nrrn)