LM ANOVA 2 2 Example Background Bacteria effect of temperature 10 o C amp 15 o C and relative humidity 20 40 60 80 on growth rate cellsd 120 petri dishes with a growth ID: 760367
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Slide1
Linear Models
Two-Way ANOVA
Slide2LM ANOVA 2
2
Example -- Background
Bacteria -- effect
of temperature (10
o
C & 15
o
C) and relative humidity (20%, 40%, 60%, 80%) on growth rate (cells/d
).
120
petri
dishes with a growth
medium available
Growth
chambers where all environmental variables can be controlled.
What is the response variable, factor(s), level(s), treatment(s), replicates per treatment?
Slide3LM ANOVA 2
3
Factorial or Crossed Design
Each treatment is a combination of both factors.
Relative Humidity
20%
40%
60%
80%
Temp
10
o
C
15
o
C
Slide4LM ANOVA 2
4
Factorial or Crossed Design
Advantages (over two OFAT experiments)Efficiency – each individual “gives information” about each level of BOTH factors.
Relative Humidity20%40%60%80%Temp10oC1515151515oC15151515
TempRelative Humidity10oC15oC20%40%60%80%202020202020
OFAT
Slide5LM ANOVA 2
5
Factorial or Crossed Design
Advantages (over two OFAT experiments)
Efficiency
–
each individual
“
gives
information” about each level of BOTH factors.
Power
– increased due to increased effective n.
Effect Size
– detect smaller
differences
Interaction effect
– can be detected.
Slide6LM ANOVA 2
6
Interaction Effect
Effect of one factor on the response variable differs depending on level of the other factor.
Relative Humidity
20%
40%
60%
80%
Temp
10
o
C
7
10
13
15
15
o
C
14
12
11
8
Slide7LM ANOVA 2
7
No Interaction Effect
Relative Humidity
20%
40%
60%
80%
Temp
10
o
C
7
10
13
15
15
o
C
6
9
12
14
Slide8LM ANOVA 2
8
Main Effects
Differences in “level” means for a factor“Strong” relative humidity main effect“Weak” temperature main effect.
Relative Humidity20%40%60%80%Temp10oC710131515oC691214
6.5
9.5
12.5
14.5
11.25
10.25
Slide9LM ANOVA 2
9
Main Effects
“Strong” relative humidity main effect
“Weak” temperature main effect.
Slide10LM ANOVA 2
10
Interaction Effect
Slide11LM ANOVA 2
11
Humidity and Temperature Effects
Slide12LM ANOVA 2
12
Humidity Effect Only
Slide13LM ANOVA 2
13
Temperature Effect Only
Slide14LM ANOVA 2
14
No Effects
Slide15LM ANOVA 2
15
Example #1
Interaction Effect
Factor 1 Main EffectFactor 2 Main Effect
√
√
×
Slide16LM ANOVA 2
16
Example #2
Interaction Effect
Factor 1 Main EffectFactor 2 Main Effect
×
√
×
Slide17LM ANOVA 2
17
Interaction Effect
Factor 1 Main EffectFactor 2 Main Effect
Example #3
√
Slide18LM ANOVA 2
18
Example
#4
Interaction EffectFactor 1 Main EffectFactor 2 Main Effect
×
√
×
Slide19LM ANOVA 2
19
Interaction Effect
Factor 1 Main EffectFactor 2 Main Effect
Example #5
√
Slide20LM ANOVA 2
20
Interaction Effect
Factor 1 Main EffectFactor 2 Main Effect
Example #6
√
Slide21LM ANOVA 2
21
Example #7
Interaction Effect
Factor 1 Main EffectFactor 2 Main Effect
×
√
√
Slide22LM ANOVA 2
22
Interaction Effect
Factor 1 Main EffectFactor 2 Main Effect
Example #8
√
Slide23LM ANOVA 2
23
Example #9
Interaction Effect
Factor 1 Main EffectFactor 2 Main Effect
×
×
√
Slide24LM ANOVA 2
24
Interaction Effect
Factor 1 Main EffectFactor 2 Main Effect
Example #10
√
Slide25LM ANOVA 2
25
Terminology / Symbols
One factor
is “row”
factor
r = number of
levels
Other
factor is
“column”
factor
c = number of
levels
Y
ijk
=
response
variable
for
k
th
individual in
i
th
level of row factor and
j
th
level of column
factor
for
simplicity, assume n is same for all
i,j
Slide26LM ANOVA 2
26
Terminology / Symbols
Column Factor12…cRow Factor1…2…………………r……
`Y11.
`Y12.
`Y1c.
`Y21.
`Y22.
`Y2c.
`Yr1.
`Yr2.
`Yrc.
`Y.1.
`Y.c.
`Y.2.
`Y1..
`Y2..
`Yr..
`Y...
Treatment means
Level means
Grand mean
Slide27LM ANOVA 2
27
2-Way ANOVA Purpose
D
etermine significance
of
interaction and, if appropriate, two
main
effects.
Are differences
in
means
“different enough” given sampling
variability?
Slide28LM ANOVA 2
28
2-Way ANOVA Calculations
MS
Within
is variability about
ultimate full
model
MS
Total
is variability about
ultimate simple
model
if
MS
Among
is large relative to
MS
Within
then
ultimate full
model is
warranted
i.e
., some
difference in treatment means
implies differences due to row factor, column factor, or interaction between the two
SS
Among
=
SS
Row
+
SS
Col
+
SS
Interaction
If
MS
Row
is large relative to
MS
Within
then a difference due to the row factor is indicated
Similar argument for column and interaction effects
Slide29LM ANOVA 2
29
2-Way ANOVA Calculations
SS
Among
= SS
Row
+ SS
Column
+ SS
Interaction
Slide30LM ANOVA 2
30
2-Way ANOVA Calculations
SS
Row
=
cn
(
)
å
=
-
r
1
i
2
...
..
i
Y
Y
SS
Column
=
rn
(
)
å
=
-
c
1
i
2
...
. .
j
Y
Y
Column Factor
1
2…cRow Factor1…2…………………r……
`Y11.
`Y12.
`Y1c.
`Y21.
`Y22.
`Y2c.
`Yr1.
`Yr2.
`Yrc.
`Y.1.
`Y.c.
`Y.2.
`Y1..
`Y2..
`Yr..
`
Y
...
Slide31LM ANOVA 2
31
Two-Way ANOVA Table
Source
df
SS
MS
F
.
Row
r-1
SS
Row
SS
Row
/[r-1]
MS
Row
/
MS
Within
Column
c-1
SS
Col
SS
Col
/[c-1]
MS
Col
/
MS
Within
Inter
(r-1)(c-1
)
SS
Int
SS
Int
/[(r-1)(c-1)]
MS
Int
/
MS
Within
Within
rc
(n-1)
SS
Within
SS
Within
/[
rc
(n-1)]
Total rcn-1
SS
Total
Slide32LM ANOVA 2
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Example
What is the optimal temperature (27,35,43oC) and concentration (0.6,0.8,1.0,1.2,1.4% by weight) of the nutrient, tryptone, for culturing the Staphylococcus aureus bacterium. Each treatment was repeated twice. The number of bacteria was recorded in millions CFU/mL (CFU=Colony Forming Units).
Slide33LM ANOVA 2
33
Example -- Bacteria
Concentration0.60.81.01.21.4Temp2710231056160267131153358816117023019916943134166136209164162108144155235164161
What kind of effects are apparent?
Slide34LM ANOVA 2
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Example -- Bacteria
What kind of effects are apparent?
Slide35LM ANOVA 2
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Example -- ANOVA
Analysis of Variance TableResponse: cells Df Sum Sq Mean Sq F value Pr(>F) ftemp 2 1313 656 0.8557 0.44473 fconc 4 51596 12899 16.8154 2.041e-05ftemp:fconc 8 14703 1838 2.3958 0.06886 Residuals 15 11507 767
Weak Interaction; Nonsignificant
Significant concentration effect
Nonsignificant
temperature effect
Slide36LM ANOVA 2
36
Example -- ANOVA
Slide37LM ANOVA 2
37
Example -- ANOVA
Linear Hypotheses: Estimate Std. Error t value p value 0.8 - 0.6 == 0 36.00 15.99 2.251 0.2144 1 - 0.6 == 0 46.67 15.99 2.918 0.0680 . 1.2 - 0.6 == 0 126.83 15.99 7.932 <0.01 ***1.4 - 0.6 == 0 56.17 15.99 3.512 0.0226 * 1 - 0.8 == 0 10.67 15.99 0.667 0.9611 1.2 - 0.8 == 0 90.83 15.99 5.680 <0.01 ***1.4 - 0.8 == 0 20.17 15.99 1.261 0.7195 1.2 - 1 == 0 80.17 15.99 5.013 <0.01 ** 1.4 - 1 == 0 9.50 15.99 0.594 0.9737 1.4 - 1.2 == 0 -70.67 15.99 -4.419 <0.01 **
0.6 0.8 1.0 1.4 1.2
Slide38LM ANOVA 2
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Example -- ANOVA
a
ab
ab
c
b
0.6 0.8 1.0 1.4 1.2
a
a
Slide39Review Handout – Example 1
lm()anova()glht()fitPlot()addSigLetters()
LM ANOVA 2
39
Slide40LM ANOVA 2
40
Assumptions and Checking in R
Same as for
the one-way
ANOVA
Slide41LM ANOVA 2
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Example
Measured soil phosphorous levels in plots near Sydney, Australia.
Each plot was characterized by type of soil (shale- or sandstone-derived) and “topographic” location (valley, north, south, or hillside).
Data in
SoilPhosphorous.txt
Does mean soil phosphorous level differ by soil type or topographic location?
Is there an interaction effect?