Randomized Complete Block Design (RCBD) - Purdue University

Randomized Complete Block Design

(RCBD)

Bruce A Craig

Department of Statistics Purdue University

STAT 514

Topic 11

1

Blocking Design

A nuisance factor is an factor that effects the response y but is not of interest to the researcher When planning an experiment must always consider the possibility of nuisance factors If unknown nuisance factor, randomization provides protection from bias but error variance will be inflated If known (and measurable) but uncontrollable use ANCOVA If known and controllable, we use a blocking design Extension of a paired t-test where pairs are the blocks

STAT 514

Topic 11

2

Randomized Complete Block Design (RCBD)

Arrange b blocks, each containing a "similar" EUs Randomly assign a treatments to the EUs in block The linear statistical model is

yij = ? + i + j + ij

i = 1, 2, . . . , a j = 1, 2, . . . , b

i - ith treatment effect j - jth block effect ij N(0, 2)

Model includes additional additive block effect

STAT 514

Topic 11

3

Partitioning the SS

Rewrite observation as:

yij = y .. + (y i. - y ..) + (y .j - y ..) + (yij - y i. - y .j + y ..)

= ?^ +

^i

+

^j

+

^ij

Can partition SST =

(yij - y ..)2 into

b (y i. - y ..)2+ a SSTreatment +

(y .j - y ..)2 +

SSBlock

+

(yij - y i. - y .j + y ..)2 SSE

Under H0, all SS/2 independent 2 Ratio of SS will be F distributed

STAT 514

Topic 11

4

Hypothesis Testing

Can show (in the fixed case):

E(MSE)=2 E(MSTreatment) = 2 + b i2/(a - 1) E(MSBlock) = 2 + a j2/(b - 1)

Use F -test to test equality of treatment effects

F0

=

SS Treatment /(a - 1) SSE/((a - 1)(b - 1))

Could also use F -test for inference on block effects but...

Usually not of interest (i.e., you chose to block for a reason) Blocks not randomized to experimental units Best to view F0 and its P-value as a measure of blocking success

STAT 514

Topic 11

5

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