Analysis of Variance ANOVA

• compare the means of multiple populations using one-way ANOVA (the completely randomised design)

• use the Tukey-Kramer procedure to determine which means are significantly different

• compare the means of multiple populations using two-way ANOVA (the randomised block design)

independent variable (or factor)

I. The

treatment

variable is

under the control

of the analyst

II.

classification

variable is an existing characteristic of the experimental subjects which is

outside the control

of the analyst

ANOVA assumptions

1.

Samples should be

independently

selected and

randomly assigned

to the levels of the treatment factor.

 Randomness and independence must be met – drawing a random sample

or assigning treatments randomly will ensure independence

2.

The variable level of interest for each population has a

normal

distribution

.

 Normality – various tests available: e.g., goodness-of-fit test, residual plot, etc.

3.

The variance associated with each variable level in the population is the same

(equal) => homogeneity of variance



Equal variances – F test

Partitioning total variation

 (v) Conclusion

Multiple Comparison Tests-

Tukey-Kramer

Steps:

1. Compute all possible pairs of differences 2. For each pair, compute the critical range:

α , C , N - C

r s

M S E

1

1

C r i t i c a l r a n g e = q

+

2

n

n













where MSE = Mean Square Within

qa,C,N-C = Table A.10 (pp.603-604), with df = (C, N-C)

3. A given pair is significantly different at  if the absolute difference,

|

X´i− ´Xj

|

, in the

sample means exceeds the critical range.

EXAMPLE

The Randomised Block Design -

ANOVA table for

two-factor design

Hypothesis Tests for the Randomised Block Design

1. Treatment effects (due to factor B)

H0: 1. = 2. = 3. =… c. => no treatment effects HA: not all means are equal => treatment effects 2. Blocking effects (due to factor A)

H0: .1 = .2 = .3 =… R => no blocking effects HA: not all means are equal => blocking effects

EXAMPLE

• A randomised block design study was undertaken to ascertain whether the perception of economic recovery in Australia differs according to political affiliation. The sample had three levels of political affiliation – Australian Labor Party (ALP), The Liberal-National Coalition, and the Greens. To control for

differences in socioeconomic class, a blocking variable that had five socioeconomic categories was used.

• The respondents were asked to give a score on a 25-point scale from 0 = economy was definitely not in recovery to 25 = the economy was definitely in complete recovery, and some value in between for more uncertain responses.

H0: m1 = m2 = m3 = m4 = m5

HA: at least one blocking mean is different (ii) Decision rules: a =.01; dfC= C-1 = 3-1 = 2

• dfR= n-1 = 5-1 = 4

• dfE= (C-1)(n-1)=2(4)=8

• Critical values: FC= F.01,2,8= 8.65; Reject H0 if Ftest > 8.65

• FR= F.01,4,8 = 7.01 Reject H0 if Ftest > 7.01

(iii)

Test Statistics

:

• Blocking effects (Columns): Ftest= SSC/df C

SSE/df E =

64.53/2

16.8/8 = 15.36

• Treatment effects (Rows): Ftest= SSR_SSE/df C

/df E =

137.6/4

16.8/8 = 16.38 (iv)

Decision

:

• Blocking: Reject H0 at the 1% level

• Treatment: Reject H0 at the 1% level

(v)

Conclusion:

• 1. At least one of the population means of the treatment levels is different from the others

i.e. there is a significant difference in the perception of economic recovery among supporters of the different political parties

• 2. Blocking effects are significant.

• i.e. socioeconomic background significantly affects one’s perception of economic recovery.

Therefore, the blocking has been advantageous in reducing the random error and improving the accuracy of the test

ANOVA

Source of Variation SS df MS F

Rows 137.6 4 34.400 16.381

Columns 64.53 2 32.267 15.365

Error 16.80 8 2.1