The error terms or residual effects, eij, are independent from observation to observation and are randomly and normally distributed with zero mean and the same variance ó 2. This can be expressed as eij are iN (0, ó2).

2.

Variances of different samples are homogeneous.

3.

Variances and means of different samples are not correlated, i.e., are independent.

4.

The main effects (block and treatment) are additive.

Need to ensure that the data fits the assumptions of the analysis. Know assumptions and tests for violations of the assumptions Weights, lb, of vitamin-treated and control animals in a RCBD (from Little and Hills) Block Treatment

I

II

III

IV

Total

Mean

Mice-control

0.18

0.30

0.28

0.44

1.2

0.3

Mice-vitamin

0.32

0.40

0.42

0.46

1.6

0.4

Subtotal

0.50

0.70

0.70

0.90

2.8

0.35

Chickens-control

2.0

3.0

1.8

2.8

9.6

2.40

Chickens-vitamin

2.5

3.3

2.5

3.3

11.6

2.90

Subtotal

4.5

6.3

4.3

6.1

21.2

2.65

Sheep-control

108.0

140.0

135.0

165.0

548.0

137.0

Sheep-vitamin

127.0

153.0

148.0

176.0

604.0

151.0

Subtotal

235.0

293.0

283.0

341.0

1152.0

144.0

Total

240.0

300.0

288.0

348.0

1176.0

Mean

40.0

50.0

48.0

58.0

1

49.0

ANOVA Source

df

SS

MS

F

F.05

F.01

Total

23

111,567.40

Treatment

(5)

108,713.68

21,742.74

174.43**

2.90

4.56

Species

2

108,321.16

54,160.58

434.51**

3.68

6.36

Vitamin

1

142.11

142.11

1.14

4.54

8.68

Species x Vitamin

2

250.41

125.20

1.00

3.68

6.36

Block

3

984.00

328.00

2.63

3.29

5.42

Error

15

1,869.72

124.65

Weights of mice, chickens and sheep are significantly different, which is hardly a surprise. No effect of vitamin is seen. Test whether assumptions are met. 1. Normally-distributed, Random and Independent Errors Generally deviations from the assumption of normality do not seriously affect the validity of the analysis of variance. One informal test for normality is to graph the data. This is also very useful for detecting outliers. A better test for normality is to calculate and graph the error components or residuals for each observation. This is equivalent to graphing the data after correcting for treatment and block effects. Calculation of error component:

2

Error components in vitamin experiment Block Treatment

I

II

III

IV

Total

Mice-control

8.88

-1.00

0.98

-8.86

0

Mice-vitamin

8.92

-1.00

1.02

-8.94

0

Chickens-control

8.60

-0.40

0.40

-8.60

0

Chickens-vitamin

8.60

-0.60

0.60

-8.60

0

Sheep-control

-20.00

2.00

-1.00

19.00

0

Sheep-vitamin

-15.00

1.00

-2.00

16.00

0

Totals

0

0

0

0

0

These errors appear to occur in groups, rather than randomly. Graphing the errors gives the following: Errors in vitamin experiment

Range of errors is larger for the larger means. For means over 100, that the errors increase linearly as the means increase. This hardly appears random. This data set fails the assumption of normally-distributed, random and independent errors.

3

2. Homogeneity of Variances Calculate the variance for each treatment. For treatment 1: SS(t1) = ÓX2 - (ÓX)2/r = 0.182 + ... + 0.442 - (1.2)2/4 = 0.0345 2 s (t1) = SS/df = 0.0345/3 = 0.0115 Perform Bartlett’s test for homogeneity of variance Treatment

df

s2

Coded s2

Log coded s2

Mice - C

3

0.0115

11.5

1.06

Mice - V

3

0.0035

3.5

0.54

Chick - C

3

0.3467

346.7

2.54

Chick - V

3

0.2133

213.3

2.33

Sheep - C

3

546.0

546,000

5.74

Sheep - V

3

425.3

425,000

5.63

971,875

17.84

Total Mean

18

161,979

Log of Mean

5.209

4

3. Independence of Means and Variances If variances are homogeneous and independent of the means then

Treatment

Mean

s2

s

s2/Mean

s/Mean

Mice - C

0.3

0.01147

0.107

0.04

0.36

Mice - V

0.4

0.0347

0.059

0.01

0.15

Chick - C

2.4

0.3467

0.589

0.14

0.24

Chick - V

2.9

0.2133

0.462

0.07

0.16

Sheep - C

137.0

546.0

23.367

3.98

0.17

Sheep - V

151.0

425.3

20.624

2.82

0.14

Standard deviation is closely related to the mean. This suggests that a log transformation should be used.

5

4. Additivity Terms in the mathematical model for a design are additive. In an RCB, the treatment and block effects are assumed to be additive. This means the treatment effects are the same in all blocks and the block effects are the same in all treatments. Additive effects Block

I

Trt 1

10

II (+10)

20

(+20) Trt 2

30

(+20) (+10)

40

Multiplicative effects Block

I

Trt 1

10

II (+100%)

20

(+200%) Trt 2

30

(+20) (+200%)

60

Log transformation can transform multiplicative effects into additive effects. Log Transformed effects Block

I

Trt 1

1.00

II (+0.30)

1.30

(+0.48) Trt 2

1.48

(+0.48) (+0.30)

1.78

6

Tukey’s test for additivity Block Treatment

I

II

III

IV

Mean

Trt Effect

Mice-control

0.18

0.30

0.28

0.44

0.3

-48.7

Mice-vitamin

0.32

0.40

0.42

0.46

0.4

-48.6

Chickens-control

2.0

3.0

1.8

2.8

2.40

-46.6

Chickens-vitamin

2.5

3.3

2.5

3.3

2.90

-46.1

Sheep-control

108.0

140.0

135.0

165.0

137.0

88.0

Sheep-vitamin

127.0

153.0

148.0

176.0

151.0

102.0

Mean

40.0

50.0

48.0

58.0

49.0

Block Effect

-9.0

1.0

-1.0

9.0

ANOVA Source

df

SS

MS

Error (BxT)

15

1869.72

Nonadditivity

1

1822.94

1822.94

Residual Err

14

46.78

3.34

F

545.7

F.05

4.60

Assumption of additivity is incorrect. POSSIBLE COURSES OF ACTION 1. Analyze species separately. Have valid tests for each species, but no information on interactions.

7

Species

Source

df

Mice

Block

3

0.0400

0.0133

8.31

Vitamin

1

0.0200

0.0200

12.50*

Error

3

0.0048

0.0016

Block

3

1.64

0.547

41.00**

Vitamin

1

0.50

0.500

37.5**

Error

3

0.04

0.013

Block

3

2834.0

944.7

157.4**

Vitamin

1

392.0

392.0

66.3**

Error

3

18.0

6.0

Chickens

Sheep

SS

MS

F

See a significant effect of vitamin. No test for effect of species or for interaction. 2.Transform data and re-analyze. Because standard deviation is proportional to the mean, use a log transformation. Source

df

SS

MS

F

Block

3

0.12075

0.04025

13.77**

Treatment

5

28.60738

5.72148

1959.41**

Vitamin

1

0.04860

0.04860

16.62**

Species

2

28.54926

14.27463

4883.00**

VxS

2

0.00952

0.00476

Error

15

0.04385

0.00292

1.63

Effect of species is significant. Effect of vitamin is significant. Interaction is not significant. Implications: Can use a mouse model instead of the larger and more expensive sheep. After transforming, the tests of the assumptions need to be redone to make sure the assumptions are now met.

8

Example: Bartlett’s test on transformed data Treatment

s2

df

Coded s2

Log coded s2

Mice - C

3

0.0243

24.3

1.39

Mice - V

3

0.0040

4.0

0.60

Chick - C

3

0.0118

11.8

1.07

Chick - V

3

0.0048

4.8

0.68

Sheep - C

3

0.0062

6.2

0.79

Sheep - V

3

0.0038

3.8

0.58

54.9

5.11

Total

18

Mean

9.15

Log of Mean

0.9614

The assumption of homogeneity of variance is now met. TRANSFORMATIONS 1. Log transformation Use when standard deviation is proportional to the mean main effects are multiplicative rather than additive data are whole numbers and cover a wide range of values can not use if data has negative values Examples number of insects per plot number of egg masses per plant 9

Coding if data has values