Completely Randomized Design

Reviews for later topics– Model parameterization

(estimability)– Contrasts (power analysis)

Analysis with contrasts– Orthogonal polynomial contrasts– Polynomial goodness-of-fit

Completely Randomized Design

Cell means model:

2,0~,

,,1,,,1,

Niid

njaiY

ijij

iijiij

Effects Model

Yij ˜ . i ij

Possible constraints : ii1

a

0 or a 0

ij independent, ij ~ N 0, 2

GLM for Effects Model

aa an

a

n

n

a

an

a

n

n

Y

Y

Y

Y

Y

Y

1

2

21

11

11

1

2

1

.

1

2

21

1

11

22

1 ~

1111

1111

0101

0101

0011

0011

CRD Contrasts

Balanced case (ni=n)

-A linear combination L has the form:

-A contrast is a linear combination with the additional constraint: 0

1

a

i ic

i

a

iicL

1

Cotton Fiber Example

Treatment--% cotton by weight (15%, 20%, 25%, 30%, 35%)

Response--Tensile strength

Montgomery, D. (2005) Design and Analysis of Experiments, 6th Ed. Wiley, NY.

51

51

,,

,,

kkk cc

c

Cotton Fiber Example

c 1 1, 1,0,0,0 L1 c1

' 1 2

Cotton Fiber Example

ˆ L 1 c 1 ˆ y 1. y 2.

ˆ V ˆ L 1 ˆ 2

nc1i

2

i1

a

SSL1

ˆ L 12

1

nc1i

2

i1

a

Contrast Test Statistic

T ˆ L 1

ˆ n

c1i2

i1

a

~ tn . a,

T 2 SSL1

ˆ 2F ~ F1,n . a

Under Ho:L1=0,

Unbalanced CRD Contrast SS

5

1

2

2ˆ

i i

iL

nc

LSS

Orthogonality

Contrasts are orthogonal if, for contrasts L1 and L2, we have

)caseunbalanced(0

)casebalanced(0

21

21

i

ii

ii

n

cccc

Orthogonality

The usual a-1 ANOVA contrasts are not orthogonal (though columns are linearly independent)

Orthogonality implies effect estimates are unaffected by presence/absence of other model terms

Orthogonality

Sums of squares for orthogonal contrasts are additive, allowing treatment sums of squares to be partitioned

Mathematically attractive, though not all contrasts will be interesting to the researcher

Cotton Fiber Example

Two sets of covariates (orthogonal and non-orthogonal) to test for linear and quadratic terms

Term Orth. SS Non-Orth SS

L 33.6 33.6

L|Q 33.6 364.0

Q 343.2 12.8

Q|L 343.2 343.2

L & Q 376.8 376.8

Cotton Fiber Example

For Orthogonal SS, L&Q=L+Q; Q=Q|L; L=L|Q

For Nonorthogonal SS, L&Q=L+Q|L=Q+L|QTerm Orth. SS Non-Orth SS

L 33.6 33.6

L|Q 33.6 364.0

Q 343.2 12.8

Q|L 343.2 343.2

L & Q 376.8 376.8

Orthogonal polynomial contrasts

Require quantitative factors Equal spacing of factor levels (d)

Equal ni

Usually, only the linear and quadratic contrasts are of interest

Orthogonal polynomial contrasts

Cotton Fiber Example

Orthogonal polynomial contrasts

Cotton Fiber Example

Orthogonal polynomial contrasts

F 33.62 343.21 /2

8.0623.38 (p .0001)

F 64.98 33.95 /2

8.066.137 (p .0084)

Cotton Fiber ExampleIs a L+Q model better than an intercept model?Is a L+Q model not as good as a cell means model? (Lack of Fit test)

Orthogonal polynomial contrasts

Yandell has an interesting approach to reconstructing these tests– Construct the first (linear) term– Include a quadratic term that is neither

orthogonal, nor a contrast– Do not construct higher-order contrasts

at all– Use a Type I analysis for testing