powerpoint presentation - analysis of variance (anova)
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Powerpoint presentation - Analysis of Variance (ANOVA)TRANSCRIPT
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
:: color, composition, and layout
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
:: color, composition, and layout
Sample Mean and Variance
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
:: color, composition, and layout
Sample Mean and Variance
66310
676361.Mean =
+++=
⋯
0422
9
663676636366361 222
.
).().().(Var
=
−++−+−=
⋯
7261
67
63
68
65
61
55
61
63
204210
0422.
.sem ==
).,.(
..
semx
865461
22663
=
±=
±=µ
:: color, composition, and layout
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.
:: color, composition, and layout
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
:: color, composition, and layout
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
=
−++−+−=
⋯
:: color, composition, and layout
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
:: color, composition, and layout
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
:: color, composition, and layout
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.
:: color, composition, and layout
Estimate of the treatment mean: iX
r
EMSSEM =
0725
3510.
.SEM ==
Standard error of a treatment mean:
Standard error of the mean
:: color, composition, and layout
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
:: color, composition, and layout
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.
:: color, composition, and layout
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.
:: color, composition, and layout
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
:: color, composition, and layout
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?
:: color, composition, and layout
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
:: color, composition, and layout
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.
:: color, composition, and layout
�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
:: color, composition, and layout
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
:: color, composition, and layout
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
:: color, composition, and layout
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
:: color, composition, and layout
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
:: color, composition, and layout
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
:: color, composition, and layout
SED for RCB factorial
sedType of comparison
Compare 2 B means
Compare 2 A means
Compare 2 AxB meansr
EMS2
rb
EMS2
ra
EMS2
:: color, composition, and layout
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
:: color, composition, and layout
Example: Split-plotMainplot-3 N-rates; Subplot-2 varieties; 3 reps
4Error (a)
2Block
2Nitrogen (N)
dfSource of Variation
Blk 1
Blk 2
Blk 3
N3 N1 N2
N1 N3 N2
N3 N2 N1
:: color, composition, and layout
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)
dfSource of Variation
Blk 1
Blk 2
Blk 3
N3 N1 N2
N1 N3 N2
N3 N2 N1
:: color, composition, and layout
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)
dfSource of Variation
Blk 1
Blk 2
Blk 3
N3 N1 N2
N1 N3 N2
N3 N2 N1(1x2) x 3 =
2Block
2Nitrogen (N)
dfSource of Variation
:: color, composition, and layout
Split-plot
Source of Variation df
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
:: color, composition, and layout
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
:: color, composition, and layout
Satterthwaite degrees of freedom
[ ][ ]
b
b
a
a
baab
df
Eb
df
E
EbEdf
2
2
)1(
)1(2
−+
−+=
:: color, composition, and layout
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
:: color, composition, and layout
Example: Strip-plot
4Error (a)
2Block
2Nitrogen (N)
dfSource of Variation
VF: 3 N-rates; HF: 2 Varieties; 3 reps
Blk 1
Blk 2
Blk 3
N3 N1 N2
N2 N3 N1
N1 N2 N3
:: color, composition, and layout
Example: Strip-plot
2Error (b)
4Error (a)
2Block
1Variety (V)
2Nitrogen (N)
dfSource of Variation
VF: 3 N-rates; HF: 2 Varieties; 3 reps
V2
V1
V2
V1
V2
V1
Blk 1
Blk 2
Blk 3
:: color, composition, and layout
Example: Strip-plot
2Error (b)
Total
2NxV
4Error (a)
2Block
4Error (c)
1Variety (V)
2Nitrogen (N)
dfSource of Variation
VF: 3 N-rates; HF: 2 Varieties; 3 reps
V2
V1
V2
V1
V2
V1
Blk 1
Blk 2
Blk 3
N3 N1 N2
N2 N3 N1
N1 N2 N3
17
:: color, composition, and layout
Strip-plot
Source of Variation df
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
:: color, composition, and layout
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
:: color, composition, and layout
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(
−+
−+=
back