Fixed and Random Effects. Theory of Analysis of Variance Source of variationdfEMS Between...

Fixed and Random Fixed and Random EffectsEffects

Theory of Analysis of VarianceTheory of Analysis of Variance

Source of variation df EMS

Between treatments n-1 e2 + kt

2

Within treatments nk-n e2

Total nk-1

[e2 + kt

2]/e2 = 1, if kt

2 = 0

Setting Expected Mean SquaresSetting Expected Mean Squares

The expected mean square The expected mean square for a source of variation (say X) for a source of variation (say X) contains.contains.

the error term.a term in 2

x.

a variance term for other selected interactions involving X.

Coefficients for EMSCoefficients for EMS

Coefficient for error mean square is always 1

Coefficient of other expected mean squares is # reps times the product of factors levels that do not appear

in the factor name.

Expected Mean SquaresExpected Mean Squares

Which interactions to include in an Which interactions to include in an EMS?EMS?

All the factors appear in the All the factors appear in the interaction.interaction.

All the other factors in the All the other factors in the interaction are interaction are Random Effects.Random Effects.

Pooling Sums of SquaresPooling Sums of Squares

Multiple ComparisonsMultiple Comparisons

Multiple ComparisonsMultiple Comparisons

Multiple Range Tests:t-tests and LSD’s;t-tests and LSD’s;Tukey’s and Tukey’s and

Duncan’s.Duncan’s.

Orthogonal Contrasts.

One-way Analysis of VarianceOne-way Analysis of Variance

Source d.f. MSq F-val

Between Genotypes

6 931,196 9.83 ***

Within Genotypes

21 94,773

Means and RankingsMeans and Rankings

A B C D E F G

2678 2552 2128 2127 1796 1681 1316

(1) (2) (3) (4) (5) (6) (7)

Multiple t-TestMultiple t-Test

sed[x] = (22/n)

(2 x 94,773/4)

|XA - XB|/sed[x] >= tp/2

Least Significant DifferenceLeast Significant Difference

|XA - XB|/sed[x] >= tp/2

LSD = tp/2 x sed[x]

t0.025 = 2.518

LSD = 2.518 x 217.7 = 548.2

Least Significant DifferenceLeast Significant Difference

Say one of the cultivars (E) is a control check and we want to ask: are any of the others different from the check?

LSD = 2.518 x 217.7 = 548.2

XE + LSD

1796 + 548.2 = 2342.2 to 1247.80

Means and RankingsMeans and Rankings

A B C D E F G

2678 2552 2128 2127 1796 1681 1316

(1) (2) (3) (4) (5) (6) (7)

Range = 1796 + 548.2 = 2342.2 to 1247.80

Multiple LSD ComparisonsMultiple LSD Comparisons

Comparison ResultA = 2678 – LSD = 2130 A=B; A>C, D, E, F & GB = 2552 – LSD = 2004 B=C & D; B>E, F & GC = 2128 – LSD = 1580 C=D, E & F; C>GD = 2127 – LSD = 1579 D=E & F; D>GE = 1796 – LSD = 1248 E=F; E>GF = 1681 – LSD = 1132 F=G

Lower Triangular FormLower Triangular Form

A B C D E FB n.s.C * n.s.D * n.s. n.s.E * * n.s. n.s.F * * n.s. n.s. n.s.G * * * * * n.s.

LSD Multiple ComparisonsLSD Multiple Comparisons

Genotype Mean Comparison

A 2678 aB 2552 abC 2128 bcD 2127 bcdE 1796 cdeF 1681 cdefG 1316 f

Tukey’s Multiple Range TestTukey’s Multiple Range Test

se[x] = (2/n)

(94,773/4) = 153.9

W = 4.64 x 153.9 = 714.1

W = q(p,f) x se[x]


Comparison ResultA = 2678 – W = 1964.9 A=B, C & D; A>E, F & GB = 2552 – W = 1837.9 B=C & D; B>E, F & GC = 2128 – W = 1413.9 C=D, E & F; C>GD = 2127 – W = 1412.9 D=E & F; D>GE = 1796 – W = 1081.9 E=F & GF = 1681 – W = 966.9 F=G

Tukey’s Multiple ComparisonsTukey’s Multiple Comparisons

Genotype Mean Comparison

A 2678 aB 2552 abC 2128 abcD 2127 abcdE 1796 cdeF 1681 cdefG 1316 ef

Duncan’s Multiple Range TestDuncan’s Multiple Range Test

p 2 3 4 5 6 7rp 2.94 3.02 3.18 3.24 3.30 3.33


p 2 3 4 5 6 7 rp 2.94 3.02 3.18 3.24 3.30 3.33

Rp = (rp x sed[x])/2


p 2 3 4 5 6 7 rp 2.94 3.02 3.18 3.24 3.30 3.33

Rp = (rp x sed[x] /2)

p 2 3 4 5 6 7 Rp 453 476 489 499 508 513


Comparison Result A = 2678 – R7 = 2165 A=B; A>C, D, E, F & G B = 2552 – R6 = 2044 B=C & D; B>E, F & G C = 2128 – R5 = 1628 C=D, E, & F; C>G D = 2127 – R4 = 1638 D=E & F; D>G E = 1796 – R3 = 1331 E=F; E> G F = 1681 – R2 = 1229 F=G

Duncan’s Multiple ComparisonsDuncan’s Multiple Comparisons

Genotype

Mean Comparison

A 2678 a B 2552 ab C 2128 bc D 2127 bcd E 1796 cde F 1681 cdef G 1316 ef

Orthogonal ContrastsOrthogonal Contrasts

AOV Orthogonal ContrastsAOV Orthogonal Contrasts

Source d.f. S.Sq M.Sq F-valReplicateblocks

2 108.100 54.050 3.84 n.s.

Cultivars 3 268.244 89.413 6.35 *

Error 6 198.269 14.801Total 11 574.608


Cultivar A B C D

Mean 21.3ab 25.5a 13.4b 16.0ab

Consider that cultivars A and B were Consider that cultivars A and B were developed in Idaho and developed in Idaho and

C and D developed in CaliforniaC and D developed in California

Do the two Idaho cultivars have the same yield potential?

Do the two California cultivars have the same yield potential?

Are Idaho cultivars higher yielding than California cultivars?

Analysis of VarianceAnalysis of Variance


2 108.100 54.050 3.84 n.s.

Cultivars 3 268.244 89.413 6.35 *

Error 6 198.269 14.801Total 11 574.608

OrthogonalityOrthogonality

ccii = 0 = 0

[c[c1i 1i xx cc2i2i] = 0] = 0

-1 -1 +1 +1 -- ccii = 0 = 0

-1 +1 -1 +1 -- ccii = 0 = 0

+1 -1 -1 +1 -- ccii = 0 = 0

Calculating Orthogonal ContrastsCalculating Orthogonal Contrasts

d.f. (single contrast) = 1

S.Sq(contrast) = M.Sq = [ci x Yi]2/nci2]

Orthogonal Contrasts - ExampleOrthogonal Contrasts - Example

Genotype A B C D Total summed over Replicates

64.1 76.6 40.1 57.1

Contrast (1) -1 -1 +1 +1 (2) -1 +1 0 0 (3) 0 0 -1 +1

S.Sq = [ci x Yi]/[n ci2]

S.Sq(1)

[(-1)64.1+(-1)76.6+(1)40.1+(1)47.8]2/ n ci2

= 52.82/(3x4)

= 232.32

S.Sq(2)

[(-1)x64.1+(+1) x 76.6]2/(3x2)

26.04

S.Sq(3)

[(-1)x40.1+(+1) x 47.8]2/(3x2)

9.88



2 108.100 54.050 3.84 n.s.

Contrast (1) 1 232.320 232.320 16.50 ***

(2) 1 26.042 26.042 1.85 ns (3) 1 9.882 9.882 0.70 nsError 6 198.269 14.801Total 11 574.608


Five dry bean cultivars (A, B, C, D, and E).

Cultivars A and B are drought susceptible.

Cultivars C, D and E are drought resistant.

Four Replicate RCB, one locationLimited irrigation applied.



3 175.2 58.4 2.89 n.s.

Cultivars 4 728.2 182.1 9.03 **

Error 12 242.0 20.2Total 19 1145.4

Orthogonal Contrast Example #2Orthogonal Contrast Example #2Tukey’s Multiple Range TestTukey’s Multiple Range Test

Cultivar A B C D E

Mean 32.5b 31.0b 35.3b 46.5a 29.8b

Orthogonal ContrastsOrthogonal ContrastsIs there any difference in yield potential

between drought resistant and susceptible cultivars?

Is there any difference in yield potential between the two drought susceptible cultivars?

Are there any differences in yield potential between the three drought resistant cultivars?


Genotype A B C D ETotal overReplicates

130 124 141 186 119

Contrast (1) -3 -3 +2 +2 +2(2) -1 +1 0 0 0

S.Sq(1)= [(-3)130+(-3)124+(2)141+(2)186+(2)119]2 /nci

2

1302/(4x40) = 140.8

S.Sq(2)= [(-1)130+(+1)124]2 /nci

2

62/(4x2) = 4.5

S.Sq(Rem) = S.Sq(Cult)-S.Sq(1)-S.Sq(2)

728.2-140.8-4.5 = 582.9

(with 2 d.f.)



3 175.2 58.4 2.89 ns

Contrast (1) 1 140.8 140.8 6.97 **

(2) 1 4.5 4.5 0.23 ns (rem) 2 582.9 291.5 14.4 ***

Error 12 242.0 20.2

Partition Contrast(rem)Partition Contrast(rem)


130 124 141 186 119

Contrast (3) 0 0 -1 +2 -1(4) 0 0 -1 0 +1


Source d.f. S.Sq M.Sq F-val Replicate blocks

3 175.2 58.4 2.89 ns

Contrast (1) 1 140.8 140.8 6.97 ** (2) 1 4.5 4.5 0.23 ns (3) 1 522.7 522.7 25.69 *** (4) 1 60.5 60.5 2.99 ns Error 12 242.0 20.2

Alternative Contrasts !!!!Alternative Contrasts !!!!

Genotype A B C D E Total over Replicates

130 124 141 186 119

Contrast (1) -3 -3 +2 +2 +2 (2) -1 -1 -1 +4 -1

S.Sq(1)= [(-3)130+(-3)124+(2)141+(2)186+(2)119]2 /nci

2

1302/(4x40) = 140.8

S.Sq(2)= [(-1)130+(-1)124+(-1)141+(4)186+(-1)119]2 /nci

2

2302/(4x20) = 661.2

S.Sq(Rem) = S.Sq(Cult)-S.Sq(1)-S.Sq(2)

728.2-140.8-661.2 = -73.8 (Oops !!!)

(with 2 d.f.)

c1i = 0

(-3) + (-3) + (+2) + (+2) + (+2) = 0 = c2i = 0

(-1) + (-1) + (-1) + (+4) + (-1) = 0 = [c1i x c2i] = 0

(-3)(-1)+(-3)(-1)+2(-1)+2(4)+2(-1) =10 =

OrthogonalityOrthogonality

More Appropriate ContrastsMore Appropriate Contrasts


130 124 141 186 119

Contrast (1) -1 -1 -1 +4 -1(2) -1 -1 +1 0 +1(3) -1 +1 0 0 0(2) 0 0 -1 0 +1


Source d.f. S.Sq M.Sq F-val Replicate blocks

3 175.2 58.4 2.89 ns

Contrast (1) 1 661.2 661.2 32.74 *** (2) 1 2.2 2.2 0.11 ns (3) 1 4.5 4.5 0.22 ns (4) 1 60.5 60.5 2.99 ns Error 12 242.0 20.2

ConclusionsConclusions

Almost all the variation between cultivars is accounted for by the difference between cv ‘D’ and the others.

The remaining 4 cultivars are not significantly different.

Orthogonal contrast result is exactly the same are the result from Tukey’s contrasts.

ConclusionsConclusions

Important to make the “correct” orthogonal contrasts.

Important to make contrasts which have “biological sense”.

Orthogonal contrasts should be decided prior to analyses and not dependant on the data.


Four Brassica species (B. napus, B. rapa, B. juncea, and S. alba).

Ten cultivars ‘nested’ within each species.

Three insecticide treatments (Thiodan, Furidan, no insecticide).

Three replicate split-plot design.

Analysis of VarianceAnalysis of VarianceSource d.f. S.Sq M.Sq F-valReplicates 2 31.4 15.7 1.13 nsTreatment 2 489.8 244.9 19.58 ***

Error (MP) 4 50.0 12.5 0.90 nsSpecies 3 1046.2 348.7 25.05 ***

Cult w Spec 36 1714.5 47.6 3.42 ***

Spec x Treat 6 587.6 97.9 7.03 ***

CwS x Treat 108 1633.7 15,1 1.09 nsError (SP) 216 3006.3 13.9

Species and Treatment MeansSpecies and Treatment Means

Species Control Thiodan Furidan MeanB. napus 2441 3154 2976 2857b

B. rapa 2460 2740 2588 2596c

B. juncea 2933 3079 3219 3077a

S. alba 2863 2780 2820 2821b

Mean 2674b 2938a 2901a

Control Thiodan Furidan

Contrast (1)

Contrast (2)



Contrast (1) -2 +1 +1

Contrast (2) 0 -1 +1



Source d.f. S.Sq M.Sq F-val Treat (1) 1 481.3 481.3 38.47 ** Treat (2) 1 8.3 8.3 0.67 ns Error (MP) 4 50.0 12.5 0.90 ns Species x (1) 3 482.5 106.8 11.55 *** (2) 3 105.3 35.1 2.52 ns Cult x (1) 54 825.2 15.3 1.10 ns (2) 54 808.7 15.0 1.08 ns Error (SP) 216 3006.3 13.9

Species x Treatment InteractionSpecies x Treatment Interaction

2200

2400

2600

2800

3000

3200

3400


B. napus B. rapa B. juncea S. alba

Species x Contrast (1)Species x Contrast (1)

2200

2400

2600

2800

3000

3200

Control Sprayed

B. napus B. rapa B. Juncea S. alba

Species x Contrast (2)Species x Contrast (2)

2200

2400

2600

2800

3000

3200

3400

Thiodan Furidan

B. napus B. rapa B. Juncea S. alba

More Orthogonal Contrasts More Orthogonal Contrasts …… Trend AnalysesTrend Analyses

Aim of Analyses of VarianceAim of Analyses of Variance

Detect significant differences between treatment means.

Determine trends that may exist as a result of varying specific factor levels.

Example #4Example #4

Ten yellow mustard (S. alba) cultivars.Five different nitrogen application rates

(50, 75, 100, 125, and 150)



2 12875 6438 8.34 ***

Cultivar 9 14991 1666 2.16 nsNitrogen 4 24705 6176 8.00 ***

C x N 36 32809 912 1.27 nsError 98 70610 720


Genotype 50 75 100 125 150Mean 1376 1419 1600 1678 1676


130013501400145015001550160016501700

50 75 100 125 150

Nitrogen level

Seed

Yie

ld (l

b/ac

re)


Genotype 50 75 100 125 150Total overReplicates

1376 1419 1600 1678 1676

Contrast (1) -3 -1 0 +1 +3(2) +2 -1 -2 -1 +2(3) +1 -2 0 +2 -1(4) +1 -4 +6 -4 +1


Effect Contrast

Linear (1) -3 -1 0 +1 +3

Quadratic (2) +2 -1 -2 -1 +2

Cubic (3) +1 -2 0 +2 -1

Quartic (4) +1 -4 +6 -4 +1

Analysis of VarianceAnalysis of VarianceSource d.f. S.Sq M.Sq F-val

Replicates 2 12875 6438 8.34 ***

Cultivar 9 14991 1666 2.16 ns

Nitrigen (1) 1 22188 22188 28.74 ***

(2) 1 789 789 1.02 ns

(3) 1 1421 1421 1.84 ns

(4) 1 307 307 0.40 ns

C x (1) 9 10970 1219 1.69 ns

(2) 9 7015 779 1.08 ns

(3) 9 8769 974 1.35 ns

(4) 9 6054 673 0.93 ns

Error 98 70610 720

Trend AnalysesTrend Analyses

The F-value associates with a trend contrast is significant.

All higher order trend contrasts are not significant.


130013501400145015001550160016501700

50 75 100 125 150

Nitrogen level

Seed

Yie

ld (l

b/ac

re)

LinearLinear

-3

-2

-1

0

1

2

3

QuadraticQuadratic

-3

-2

-1

0

1

2

3

CubicCubic

-3

-2

-1

0

1

2

3

QuarticQuartic

-6

-4

-2

0

2

4

6

8


Two carrot cultivars (‘Orange Gold’ and ‘Bugs Delight’.

Four seeding rates (1.5, 2.0, 2.5 and 3.0 lb/acre).

Three replicates.


Seeding Rate(lb/acre)

Cultivar 1.5 2.0 2.5 3.0

Orange Gold 4.53 4.01 5.23 4.48

Bug’s Delight 3.25 3.97 5.41 6.08


Source d.f. S.Sq M.Sq F-valReplicates 2 0.3575 0.1787 0.50 nsCultivar 1 0.0122 0.0122 0.03 nsSeeding Dens 3 12.2496 4.0832 14.10 ***

C x SD 3 6.4490 2.1497 6.27 ***

Error 98 70610 720

Analysis of VarianceAnalysis of VarianceSource d.f. S.Sq M.Sq F-valReplicates 2 0.3575 0.1787 0.50 nsCultivar 1 0.0122 0.0122 0.03 nsSeeding (L) 1 9.5316 9.5316 27.82 ***

(Q) 1 0.0000 0.0000 0.00 ns (C) 1 2.7180 2.7180 7.93 **

C x (L) 1 6.2199 6.2199 18.15 ***

(Q) 1 0.0794 0.0794 0.23 ns (C) 1 0.1498 0.1498 0.44 nsError 98 70610 720


33.5

44.5

55.5

66.5

1.5 2 2.5 3

Seeding Rate

Yiel

d

Orange Gold

Bug’s Delight

End of Analyses of End of Analyses of Variance SectionVariance Section

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