Class Notes, Chapter 13, second part

One-Factor Experiments: General

Reading: 13.5-13.10

Design I: CRSF

13.5 Single degree of freedon comparisons

Comparisons are hypotheses - a linear combination of population means

General formula for hypotheses is:

For the main effect of treatment:

For a single degree of freedom comparison, examples and general formula:

The comparison hypothesis is tested like the main effect hypothesis

Find the Sum of Squares omega:

MS_omega = SS_omega / df_omega

F_omega = MS_omega / MS_E

so:

Anova Table, 13.7

Source	SS	df	MS	F
Treatment, A	85356	4	21339	4.30
Comparison, omega	14553	1	14553	2.93
Error, E	124021	25	4961
Total, T	209377	29

Orthogonal comparisons: the contrasts are "independent"

orthogonal set

k - 1 comparisons

all mutually orthogonal

13.6 Multiple comparisons

Note: relationship between F and t:

Experiment-wise error rate

If H₀ true, there is still a possibility of rejecting it: Type-I, or alpha, error

If many tests are done in an experiment, the probability of at least one false rejection goes up:

Methods for reducing experiment-wise error rates

I. Bonferroni Test

for j comparisons, adjust alpha using alpha/j as the new alpha-level; for example, .05/20 = .0025; experiment-wise alpha then = .049

conservative test

II. Tukey's test - adjust for doing all possible paired comparisons, compare each difference of sample means to:

the critical q value is in Table A.12 for alpha = .05

III. Dunnett's Test - compare all treatments to a control group, compare each difference between a test group mean and the control group mean to:

Data transformation

ANOVA is robust to non-normality

But sometimes the non-normality leads to unequal variances

Some distributions are known to become better in this regard if the data are transformed

examples:

lognormal distribution: take the natural log

Poisson: take the square root of the data

exponential or gamma: take the natural log

DESIGNS

Design classifications

Basic element:

(1) CR-SF

(note: could also be: Generalized Randomized Block design)

Completely Randomized Single Factor Design:

Means coding

Effects coding

Fixed Effects (Model I)

k groups, or treatments

n_i measurements (observations, data points) in each group

if all n_i equal, just use n

the "traditional" hypothesis of the main effect of treatment is

that is, all group means are equal

Residuals are calculated:

Other designs, with the purpose of increasing statistical precision

blocking - the focus today

(2) RCB-SF

Randomized Complete Block Single Factor

Fixed Effects (Model I)

a treatments

b blocks

n = 1 observation per cell

blocks help remove extraneous error

assume normal distribution, random sampling, random asignment, homogeneity of variance, plus ...

have to assume, and there must not be, any treatment x block interaction

model: epsilon_ij

(note no interaction terms in the model)

two factors: one treatment, that we do test, and one blocking, that we don't

H₀: alpha_i = 0

Residuals are calculated:

computational formulas:

Anova Table

Source	SS	df	MS	F
Treatment, A	SSA	dfA	MSA	F = MSA/MSE
Blocks, B	SSB	dfB
Error, E	SSE	dfE	MSE
Total, T	SST	dfT

(3) RB-IF

Randomized Block Incomplete Factorial; specifically, Latin Square design

Fixed Effects (Model I)

a treatments

b blocks₁

c blocks₂

n = 1 observation per cell

blocks help remove extraneous error

assume normal distribution, random sampling, random asignment, homogeneity of variance, plus ...

have to assume, and there must not be, any treatment x block interactions, block x block interactions or treatment x block x block interaction

model:

(note no interaction terms in the model)

three factors: one treatment, that we do test, and two blocking, that we don't

H₀: alpha_i = 0

Residuals are calculated:

computational formulas for the Latin Square design with n = 1, two blocking and one treatment factor, r = the number of levels of the factors:

Anova Table

Source	SS	df	MS	F
Treatment, A	SSA	dfA	MSA	F = MSA/MSE
Blocks, B	SSB	dfB
Blocks, C	SSC	dfC
Error, E	SSE	dfE	MSE
Total, T	SST	dfT

An example of a situation where a Latin Square design would be useful is:

"Suppose we are interested in the yields of 4 varieties of wheat using 4 different fertilizers over a period of 4 years." [Note: 1 treatment (wheat type), 2 blocking (fertilizers, years)] If the types of wheat are A, B, C and D, then a common way of showing the data would be:

Yields of Wheat in kg/plot

Fertilizer	1981	1982	1983	1984
f1	A: 70	B: 75	C: 68	D: 81
f2	D: 66	A: 59	B: 55	C: 63
f3	C: 59	D: 66	A: 39	B: 42
f4	B: 41	C: 57	D: 39	A: 55

Designs 4, 5 and 6 are included for those who are interested:

(4) Random Effects - SF

Random Effects Model - Single Factor

Random Effects (Model II)

a treatments - treatment levels are a random variable

assume normal distribution, independent observations, homogeneity of variance

model: Y_ij = mu + A_i + E_ij

(text uses capitals to emphasize the random variables)

one factor: one treatment

H₀: A_i = 0

computational formulas:

just like CR-SF

(5) Random Effects - RB-Random Blocks

Randomized Blocks - Random Block

Random Effects (Model II)

a treatments - treatment levels are a random variable

b blocks - block levels are a random variable

assume normal distribution, independent observations, homogeneity of variance

model: Y_ij = mu + A_i + B_j + E_ij

(text uses capitals to emphasize the random variables)

two factors: one treatment, one blocking

H₀: A_i = 0

computational formulas:

just like RCB-SF

(6) Random Effects - RB-IF

Random Incomplete Blocking - Random Blocks; specifically, Latin Square design

Random Effects (Model II)

a treatments - treatment levels are a random variable

b blocks - block levels are a random variable

c blocks - block levels are a random variable

assume normal distribution, independent observations, homogeneity of variance

model: Y_ijk = mu + A_i + B_j + C_k + E_ijk

(text uses capitals to emphasize the random variables)

three factors: one treatment, two blocking

all three must have the same number of levels

H₀: A_i = 0

computational formulas:

just like RIB-SF Latin Square