1. Analyses of Variance

2. Assumptions behind the ANOVA
Assumption of data being normally distributed.
Homogeneity of error variance.
Additivity of variance effects.
Data collected from a properly randomized experiment.

3. Dealing with Wrongful Data
It is usually assumed that the data collected is correct!.
Why would data not be correct?
Mis-recording, mis-classification, transcription errors, errors in data entry.
Outliers.

4. Dealing with Wrongful Data
What things can help?
Keep detailed records, on each experimental unit.
Decide beforehand what values would arouse suspision.

5. Dealing with Wrongful Data
What do you do with suspicios data?
If correct, and it is discarded, then valuable information is lost. This will bias the results.
If wrong and included, will bias results and may have extreme consequences.

6. Checking ANOVA Accurucy
Coefficient of variation: [e/]x100.
CV=(√100.9/73.75)*100=13.6%
R2 value = {[TSS-ESS]/TSS}x100.
R2 = (1654/3654)*100 = 44.7%.
Compare the effect of blocking or sub-blocking (discussed later).

7. Analyses of CRB Designs
Yij =  + ti eij

8. Analysis of Variance of CRB
Source df SS
Between treatments
k-1
[G12/n1 + G22/n2 … Gk2/nk] - CF
Within treatments
jk-k
By difference
Total
jk-1
[x112 + x … + xjk2] - CF
CF = [xij]2/jk

9. Example of Analysis of Variance
Source df SS MS F
Between genotypes 3
2598.3
866.1
4.16 **
Within genotypes 12
2501.1
208.4
Total 23
3654.5
** = 0.01 > P > 0.001

10. Analyses of RCB Designs
Yij =  + bi + tj + eij

11. Analysis of Variance of RCB
Source df SS
Blocks
r-1
[B12 + B22 + … + Br2]/t – CF
Treatments
t-1
[T12 + T22 + … + Tt2]/r – CF
Error
(r-1)(t-1)
By difference
Total
rt-1
[x112 + x … + xrt2] – CF
CF = [xij]2/rt

12. Example Treatment 1 2 3 4 Total T-1 330 288 295 313 1225 T-2 372 340
343
341
1396
T-3
359
337
373
302
1371
Totals
1061
965
1011
956
3993
CF = [3993]2/12 = 1,328,671
TSS = [ … ] – CF = 8,864

13. Example Treatment 1 2 3 4 Total T-1 330 288 295 313 1225 T-2 372 340
343
341
1396
T-3
359
337
373
302
1371
Totals
1061
965
1011
956
3993
Block SS = [ ]/3 – CF = 2,330
Treat SS = [ ]/4 – CF = 4,212

14. Analysis of Variance Source df SS MS F Blocks 3 2330 777 2.00 ns
Insecticides 2
4212
2106
5.44 *
Error 6
2322
387
Total 11
8864
* = 0.05 > P > 0.01

15. Example Treatment 1 2 3 4 Mean T-1 330 288 295 313 306 T-2 372 340 343
341
349
T-3
359
337
373
302
sed[mean] = (2e2)/4 = (2 x 387)/4 = df se[mean] = (e2)/4 = (387/4) = df

16. Analyses of Latin Designs
Yijk =  + ri + cj + tk(ij) + eijk

17. Analysis of Variance of Latin
Source df SS
Rows
t-1
[R12 + R22 + … + Rt2]/t – CF
Columns
[C12 + C22 + … + Ct2]/t – CF
Treatments
[T12 + T22 + … + Tt2]/t – CF
Error
(t-1)(t-2)
By difference
Total
t2-1
[x112 + x … + xtt2] – CF
CF = [xij]2/t2

18. Example Row/Col 1 2 3 4 Total B-1.64 D-1.21 C-1.42 A-1.34 5.62 C-1.45
5.35
A-1.67
C-0.71
B-1.66
D-1.18
5.22
D-1.56
A-1.65
C-0.66
5.17
Totals
6.35
4.39
6.14
4.47
21.36
CF = [21.36]2/16 =
TSS = [ … ] – CF =

19. Example Row/Col 1 2 3 4 Total B-1.64 D-1.21 C-1.42 A-1.34 5.62 C-1.45
5.35
A-1.67
C-0.71
B-1.66
D-1.18
5.22
D-1.56
A-1.65
C-0.66
5.17
Totals
6.35
4.39
6.14
4.47
21.36
Row SS = [ ]/4 – CF =
Col SS = [ ]/4 – CF =

20. Error = Total SS – Row SS – Col SS – Gen SS
Example
Row/Col 1 2 3 4
Total
B-1.64
D-1.21
C-1.42
A-1.34
5.62
C-1.45
A-1.18
D-1.40
B-1.29
5.35
A-1.67
C-0.71
B-1.66
D-1.18
5.22
D-1.56
A-1.65
C-0.66
5.17
Totals
6.35
4.39
6.14
4.47
21.36
Gen SS = [ ]/4 – CF =
Error = Total SS – Row SS – Col SS – Gen SS

21. * = 0.05 > P > 0.01; ** = 0.01 > P > 0.001
Analysis of Variance
Source df SS MS F
Rows 3
0.0301
0.0101
0.46 ns
Column
0.8274
0.2758
12.78 **
Genotypes
0.4268
0.1423
6.69 *
Error 6
0.1296
0.0216
Total 15
1.4139
* = 0.05 > P > 0.01; ** = 0.01 > P > 0.001

22. Efficiency of Latin Squares
cw CRB Design
[MSr + MSc + (t-1)EMS]/(t+1)EMS
[ (4-1)0.0216/(4+1)0.0216
= 3.25
Latin square in this instance increased precision by 225% over CRB
CRD would have need 2.25 x 4 = 9 replicates to be as accurate.

23. Efficiency of Latin Squares
cw RCB Design
R(RCB) = [MSr + (t-1)EMS]/(t+1)EMS
C(RCB) = [MSc + (t-1)EMS]/(t+1)EMS
R(RCB) = 0.81 : C(RCB) = 3.66
Latin square in this instance increased precision on 266% over RCB or is less precise if replicates were switched.

24. a.
I II III IV I II b.
III IV I
III c. II IV

25. +266%
-19% ☺

26. Analyses of Lattice Squares
Yijk =  + ri + baj + tak + eijk
See Table 5 & 6, Page 105 & 106

27. Analyses of Lattice Squares
Calculate sub-block totals (b) and replicate totals (R).
Calculate the treatment totals (T) and the grand total (G).
For each treatment, calculate the Bt values which is the sum of all block totals that contain the ith treatment.

28. Analyses of Lattice Squares
Calculate sub-block totals (b) and replicate totals (R).
Calculate the treatment totals (T) and the grand total (G).
For each treatment, calculate the Bt values which is the sum of all block totals that contain the ith treatment.

29. Analyses of Lattice Squares
Treatment 5 is in block 2, 5, 10, 15, and 20, so B5 = = 3411.
Note that the sum of the Bt values is G x k, where k is the block size.
For each treatment calculate:
W = kT – (k+1)Bt + G
W5 = 4(816)-(5)(3,411)+13,746 = -45

30. Lattice Square ANOVA - d.f.
Source df
Reps k 4
Trt(unadj)
k2 – 1 15
Block(adj)
Intra-Block Error
(k-1)(k2-1) 45
Trt (adj)
k2 – 1
Effective Error
Total
k2(k+1)-1 79

31. Analyses of Lattice Squares
Compute the total correction factor as:
CF = (∑xij)2/n
CF = G2/[(k2)(k+1)]
(13,746)2/(16)(5)
2,361,906

32. Analyses of Lattice Squares
Compute the total SS as:
Total SS = xij2 – CF
[ …+2252] – 2,361,906
= 58,856

33. Analyses of Lattice Squares
Compute the replicate block SS as:
Replicate SS = R2/k2 – CF
[ …+29252]/16 – 2,361,906
= 5,946

34. Analyses of Lattice Squares
Compute the unadjusted treatment SS as:
Treatment (unadj) SS = T2/(k+1)–CF
[ …+8662]/5 – 2,361,906
= 26,995

35. Analyses of Lattice Squares
Compute the adjusted block SS as:
Block (adj) SS = W2/k3(k+1) – CF
[ …+8662]/320 – 2,361,906
= 11,382

36. Analyses of Lattice Squares
Compute the intra-block error SS as:
IB error SS =
TSS–Rep SS–Treat(unadj) SS–Blk(adj) SS
58, , , ,382 = 14,533

37. Lattice Square ANOVA Calculate Mean Squares for block(adj) and IBE.
Source df SS MS
Reps 4
5,946
1,486
T(unadj) 15
26,995
1,800
Blk(adj)
11,382
759
Intra block error 45
14,533
323

38. Analyses of Lattice Squares
Compute adjusted treatment totals (T’) as:
T’i = Ti + Wi
 = [Blk(adj) MS-IBE MS]/[k2 Blk(adj) MS]

39. Analyses of Lattice Squares
Compute adjusted treatment totals (T’) as:
 = [ ]/(16)(759) =
T’ = T + W
 = [Blk(adj) MS-IBE MS]/[k2 Blk(adj) MS]

40. Analyses of Lattice Squares
Compute adjusted treatment totals (T’) as:
Note if IBE MS > Blk(adj) MS, then =zero. So no adjustment.
T’ = T + W
 = [Blk(adj) MS-IBE MS]/[k2 Blk(adj) MS]

41. Analyses of Lattice Squares
Compute adjusted treatment totals (T’) as:
Note also greatest adjustment when Blk(adj) MS large and IBE MS is small.
T’ = T + W
 = [Blk(adj) MS-IBE MS]/[k2 Blk(adj) MS]

42. Analyses of Lattice Squares
Compute adjusted treatment totals (T’) as:
T’5 = T5 + W5
T’5 = x (-45) = 814
T’ = T + W
 = [Blk(adj) MS-IBE MS]/[k2 Blk(adj) MS]

43. Analyses of Lattice Squares
Compute adjusted treatment means (M’) as:
M’ = T’/[k+1]

44. Analyses of Lattice Squares
Compute adjusted treatment SS as:
Treat (adj) SS = T’2/(k+1) – CF
[ …+8392]/5 – 2,361,906
= 24,030

45. Analyses of Lattice Squares
Compute effective error MS as:
EE MS = (Intra-block error MS)(1+k)
323[1 + 4(0.0359)]
369

46. Lattice Square ANOVA Source df SS MS F Reps 4 5,946 T(unadj) 15 26,995
Blk(adj)
11,382
Intra error 45
14,533
T(adj)
24,030
Eff. Error
16,605

47. Lattice Square ANOVA Source df SS MS F Reps 4 5,946 1,486 4.03 *
T(unadj) 15
26,995
1,800 -
Blk(adj)
11,382
759
2.35 ns
Intra error 45
14,533
323
T(adj)
24,030
1,602
4.34 **
Eff. Error
16,605
369

48. Efficiency of Lattice Design
100 x [Blk(adj)SS+Intra error SS]/k(k2-1)EMS
100 [11, ,533]/4(16)369
117%
I II III IV V
I II III
IV V

49. Lattice Square ANOVA Source df SS MS F Reps 4 5,946 1,486 4.03 *
T(unadj) 15
26,995
1,800 -
Blk(adj)
11,382
759
2.35 ns
Intra error 45
14,533
323
T(adj)
24,030
1,602
4.34 **
Eff. Error
16,605
369

50. RCB ANOVA Source df SS MS F Reps 4 5,946 1,486 3.44 * T(unadj) 15
26,995
1,800
4.25 **
Error 60
25,915
432 -

51. Lattice Square ANOVA Source df SS MS F Reps 4 5,946 1,486 4.03 *
T(unadj) 15
26,995
1,800 -
Blk(adj)
1,382 92
0.17 ns
Intra error 45
24,533
545
T(adj)
24,030
1,602
2.71 *
Eff. Error
26,605
591

52. Lattice Square ANOVA CV Lattice = 11.2%; CV RCB = 12.1%.
Range Lattice 119 to 197; Range RCB 116 to 199.
Variation between treatments is small compared to environmental error or variation.

53. Comparison of Rankings

54. ANOVA of Factorial Designs