ASSUMPTIONS IN THE ANOVA Assumptions in the ANOVA and the mathematical model may not always be true in data from experiments. 1.
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