powerpoint presentation - analysis of variance (anova)

Presentation Title Goes Here…presentation subtitle.

Analysis of Variance

and Mean Comparison

Violeta Bartolome

Senior Associate Scientist-Biometrics

Crop Research Informatics Laboratory

International Rice Research Institute

:: color, composition, and layout

Measures of Central Tendency and

Dispersion

• Measure of Central Tendency – describe the middle or center of

distribution

o Mean – sum of the observations divided by the number of

observations.

o Median – number wherein half are above and half are below

it.

o Mode – most frequently occurring value.

• Measure of Dispersion – describe how data varies or are spread

out.

o Variance or standard deviation

o range


Sample Mean and Variance

6010

606060=

+++=

∑=

⋯

n

xMean i

09

60606060

122

2

=−++−

=

−

−∑=

)()(

n

)xx(Var i

⋯

60

60

60

60

60

60

60

60

60

60

010

0===

n

varsem



5910

605858=

+++=

⋯

Mean

111

9

596059585960 222

.

)()()(Var

=

−++−+−=

⋯

60

60

60

60

60

58

58

58

58

58

33010

111.

.sem ==

).,.(

.

semx

33596758

33059

=

±=

±=µ



66310

676361.Mean =

+++=

⋯

0422

9

663676636366361 222

.

).().().(Var

=

−++−+−=

⋯

7261

67

63

68

65

61

55

61

63

204210

0422.

.sem ==

).,.(

..

semx

865461

22663

=

±=

±=µ


Analysis of Variance

• Used to identify sources of variability from one or

more potential sources called treatments or factors.

• Test variability due to treatments is real and not due

to random error.


0.000.00n-1=10-1=9Total

0.000.009-1=8Error

0.000.00t-1=2-1=1Treatment

One-way ANOVA (CRD)

)(

0

MSTotalVariationTotal

Var

=

=

MSSS df

Source of

Variation

Objective is to identify how much of

this variation is explained by

treatment and how much variation is

unexplained.

6060

60

60

60

60

Treatment 2

60

60

60

60

Treatment 1


1.1110.00n-1=10-1=9Total

0.000.009-1=8Error

10.0010.00t-1=2-1=1Treatment

One-way ANOVA (CRD)

MSSS df

Source of

Variation6058

60

60

60

60

Treatment 2

58

58

58

58

Treatment 1

111

9

596059585960 222

.

)()()(Var

=

−++−+−=

⋯


One way ANOVA

7261

Treatment 2Treatment 1

67

63

68

65

61

55

61

63

Is the variability between treatments different from random error?

22.04198.40n-1=10-1=9Total

10.3582.809-1=8Error

0422

9

663676636366361 222

.

).().().(Var

=

−++−+−=

⋯

115.60115.60t-1=2-1=1Treatment

MSSSdfSource of

Variation


Ho: Variability among treatments is not different from random

error. ⇒⇒⇒⇒⇒⇒⇒⇒ There are no differences among treatment means.There are no differences among treatment means.

Objective is to reject Ho so that we can conclude that

differences exist among treatments.

Ho is rejected if F-value is significant

P ≥ .05 ⇒⇒⇒⇒⇒⇒⇒⇒ F is not significant

.01 ≤ P < .05 ⇒⇒⇒⇒⇒⇒⇒⇒ F is significant at 5%F is significant at 5% ⇒⇒⇒⇒⇒⇒⇒⇒ **

P < .01 ⇒⇒⇒⇒⇒⇒⇒⇒ F is significant at 1%F is significant at 1% ⇒⇒⇒⇒⇒⇒⇒⇒ ****

EMS

TrMSF=

Hypothesis Testing


0.0102

P

11.17*

F

22.04

10.35

115.60

MS

115.60t-1=2-1=1Treatment

198.40n-1=10-1=9Total

SS

9-1=8

df

81.80Error

Source of

Variation

* - significant at 5% level.


Estimate of the treatment mean: iX

r

EMSSEM =

0725

3510.

.SEM ==

Standard error of a treatment mean:

Standard error of the mean


Estimate of the difference between 2 treatment means:

ji XX −

Standard error of the diff. bet. 2 means: r

EMSSED

2=

Least Significant Difference: LSD=t*SED

Rule: If difference is greater than the LSD

value then the 2 means are significantly

different.

Standard error of the difference


Example

Difference bet. means of Difference bet. means of TrtTrt 1 and 1 and TrtTrt 2:2:

67.0 – 60.2 = 6.8

0325

35102.

).(SED ==

LSD (5%)=2.306*2.03=4.68

Conclusion: Since 6.8 is greater than 4.68 then

the means of the two treatments are significantly

different.


Rule on the Use of LSD

•Use only when F-test for treatment effect is significant

•Number of treatments to be compared is less than 6

Why?

Probability of committing type I error increases with the

number of treatments to be compared.


Type I Error

RejectCan not Reject

�Type II ErrorHo is False

Type I Error�Ho is True

Hypothesis

Note: Ho: µi=µj

Family-wise error rate (FWER)=probability of committing Type I Error

= 1-(1-α)N where N=the number of pairwise comparison.

For t=10, N=45 → FWER=1-(1-.05)45=.90


Using LSD for multiple pairwise comparison

7.255

11.104

9.863

9.782

7.441

MeanTreatment

LSD.05=3.34

How do we put letters to compare

means?


Two way ANOVA - RCB

.1659

Treatment

Block

.009721.61

2.87

115.60115.60t-1=2-1=1

5.35

15.3561.40b-1=5-1=4

22.04198.40n-1=10-1=9Total

(b-1)(t-1)

=4*1=4

21.40Error

ProbFMSSS df

Source of

Variation

Note: formula for sed is the

same as CRD.

61

55

61

63

61

Trt 1

721

Trt 2Block

67

63

68

65

5

4

3

2


Factorial ExperimentsFactorial Experiments

Example :Example : Given the ff. treatment combinations

for a 2x2 factorial experiment

Treatment Combinations

Treatment No. Variety Nitrogen Rate

(kg/ha)

1 V1 N1(0 kg/ha)

2 V1 N2(60 kg/ha)

3 V2 N1(0 kg/ha)

4 V2 N2(60 kg/ha)

Two or more factors are tested simultaneously.


�Most important objective is to test interaction

effects between factors.

�� Interaction Interaction occurs when effects of the levels

of one factor changes with the levels of

another factor

Factorial ExperimentsFactorial Experiments


No InteractionNo Interaction

Variety N1(0 kg/ha) N2(60 kg/ha) Average

V1 1.00 3.00 2.00

V2 2.00 4.00 3.00

Average 1.50 3.50

Interaction PresentInteraction Present

Variety N1(0 kg/ha) N2(60 kg/ha) Average

V1 1.00 1.00 1.00

V2 2.00 4.00 3.00

Average 1.50 2.50

2.00

2.00

2.00

1.00 1.00 1.00

0

2.00

1.00

1.00 3.00 2.00


Yield(kg/ha)

0

1

2

3

4

5

N1 N2

V2

V1

((((a)))) Yield(kg/ha)

0

1

2

3

4

5

N1 N2

((((b))))

V2

V1

Yield(kg/ha)

0

1

2

3

4

5

N1 N2

((((d))))

V2

V1

Which illustrate(s) the presence of interaction?

Yield(kg/ha)

0

1

2

3

4

5

N1 N2

((((c))))

V2

V1


Rule in Factorial Experiments

When interaction is present comparing

means averaged over the levels of the other

factor is meaningless.

Mean

1-1B2

-11B1

MeanA2A1

0

0

0 0


Example: RCB factorialTRTS: 2 varieties, 3 Nitrogen rates, 3 reps

N3V2N3V1N1V1

N3V2N3V1N2V2

N2V1N3V2N3V1

N1V2N2V2N2V1

N2V1N1V1N1V2

N1V2N1V1N2V2

Blk 1

Blk 2

Blk 3

17Total

10Error

2NxV

2Nitrogen (N)

1Variety (V)

2Block

dfSource of Variation


Factorial RCB

Source of Variation df

Block r-1

Factor A (A) a-1

Factor B (B) b-1

A x B (a-1)(b-1)

Error (r-1)(ab-1)

Total rab-1


SED for RCB factorial

sedType of comparison

Compare 2 B means

Compare 2 A means

Compare 2 AxB meansr

EMS2

rb

EMS2

ra

EMS2


Example: Split-plotMainplot-3 N-rates; Subplot-2 varieties; 3 reps

V1V1V2

V2V1V2

V1V2V1

V2V2V1

V1V2V1

V2V1V2

Blk 1

Blk 2

Blk 3

N3 N1 N2

N1 N3 N2

N3 N2 N1


Example: Split-plotMainplot-3 N-rates; Subplot-2 varieties; 3 reps

4Error (a)

2Block

2Nitrogen (N)


Blk 1

Blk 2

Blk 3

N3 N1 N2

N1 N3 N2

N3 N2 N1


Example: Split-plot

V1V1V2

V2V1V2

V1V2V1

V2V2V1

V1V2V1

V2V1V2

Mainplot-3 N-rates; Subplot-2 varieties; 3 reps

2NxV

4Error (a)

2Block

Error (b)

1Variety (V)

2Nitrogen (N)


Blk 1

Blk 2

Blk 3

N3 N1 N2

N1 N3 N2

N3 N2 N1


Example: Split-plot

V1V1V2

V2V1V2

V1V2V1

V2V2V1

V1V2V1

V2V1V2

Mainplot-3 N-rates; Subplot-2 varieties; 3 reps

17Total

2NxV

4Error (a)

2Block

6Error (b)

1Variety (V)

2Nitrogen (N)


Blk 1

Blk 2

Blk 3

N3 N1 N2

N1 N3 N2

N3 N2 N1(1x2) x 3 =

2Block

2Nitrogen (N)



Split-plot


Block r-1

Factor A (A) a-1

Error (a) (r-1)(a-1)

Factor B (B) b-1

A x B (a-1)(b-1)

Error (b) a(r-1)(b-1)

Total rab-1


Type of pair comparison

sed t-value

Number Between

1 Two main-plot means

(averaged over all subplot

treatments)

2 Two subplot means

(averaged over all main-plot

treatments)

3 Two subplot means at the

same main-plot treatment

4 Two main-plot means at the

same or different subplot

treatment

SEDs for Split-plot

),( bdfαtinv

),( bdfαtinv

rb

E2 a ),( adfαtinv

ra

E2 b

r

E2 b

( )[ ]rb

EE1b2 ab+− ),( abdfαtinv


Satterthwaite degrees of freedom

[ ][ ]

b

b

a

a

baab

df

Eb

df

E

EbEdf

2

2

)1(

)1(2

−+

−+=


Example: Strip-plot

V2

V1

V2

V1

V2

V1

VF: 3 N-rates; HF: 2 Varieties; 3 reps

Blk 1

Blk 2

Blk 3

N3 N1 N2

N2 N3 N1

N1 N2 N3


Example: Strip-plot

4Error (a)

2Block

2Nitrogen (N)



Blk 1

Blk 2

Blk 3

N3 N1 N2

N2 N3 N1

N1 N2 N3


Example: Strip-plot

2Error (b)

4Error (a)

2Block

1Variety (V)

2Nitrogen (N)



V2

V1

V2

V1

V2

V1

Blk 1

Blk 2

Blk 3


Example: Strip-plot

2Error (b)

Total

2NxV

4Error (a)

2Block

4Error (c)

1Variety (V)

2Nitrogen (N)



V2

V1

V2

V1

V2

V1

Blk 1

Blk 2

Blk 3

N3 N1 N2

N2 N3 N1

N1 N2 N3

17


Strip-plot


Block r-1

Horizontal Factor (H) h-1

Error (a) (r-1)(h-1)

Vertical Factor (V) v-1

Error (b) (r-1)(v-1)

H x V (h-1)(v-1)

Error (c) (r-1)(h-1)(v-1)

Total rhv-1


Type of pair comparisonsed t-value

Number Between

1 Two horizontal means

(averaged over all vertical

treatments)

2 Two vertical means

(averaged over all horizontal

treatments)

3 Two vertical means at the

same horizontal treatment

4 Two horizontal means at the

same vertical treatment

SEDs for Strip-plot

),( adfαtinv

),( bdfαtinv

),( bcdfαtinv

rb

E2 a

ra

E2 b

( )[ ]ra

EE1a2 bc+−

( )[ ]rb

EE1b2 ac+−),( acdfαtinv


Satterthwaite degrees of freedom

[ ][ ]

c

c

a

a

caac

df

Eb

df

E

EbEdf

2

2

)1(

)1(2

−+

−+=

[ ][ ]

dfc

Eca

dfb

Eb

EcaEbdfbc

22

2

)1(

)1(

−+

−+=

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powerpoint presentation - analysis of variance (anova)

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