Post on 05-Jan-2016
AdvancedAnalysesAdvancedAnalysesAdvancedAnalysesAdvancedAnalyses
Example 11.1
Treatments
Litter # Group A Group B Group C1 11.8 13.6 9.2
2 12.0 14.4 9.6
3 10.7 12.8 8.6
4 11.1 13.0 8.5
5 12.1 13.4 9.8
mean 11.54 13.44 9.14
variance 0.373 0.388 0.338
Total variance = 3.631
Five litters of mice of about the same age are selected. One member of each litter was randomly assigned to 1 of 3 diets, and their weight gain was recorded.
One-Way ANOVA to compare means
Randomized Complete Block Designs
• Sometimes experiments are set up where each observation in one of the treatments has something in common with one observation in each of the other treatments - each of these groups of related data is referred to as a block.
• The idea here is that some populations contain variance that makes the within groups variance very high.
• However, sometimes this variance can be grouped, or blocked out, in order to partition out this variance from the error variance.
Randomized Complete Block Designs
• calculate another source of variance – ‘block’
• This design thus partitions some of the variance and degrees of freedom from the error variance into the block variance
Example 11.1
Treatments
Litter # Group A Group B Group C
Block
1 11.8 13.6 9.2
2 12.0 14.4 9.6
3 10.7 12.8 8.6
4 11.1 13.0 8.5
5 12.1 13.4 9.8
mean 11.54 13.44 9.14
variance 0.373 0.388 0.338
Total variance = 3.631
Five litters of mice of about the same age are selected. One member of each litter was randomly assigned to 1 of 3 diets, and their weight gain was recorded.
Example 11.1Generalized ANOVA Table
Source of Variation
Sum of Squares
df MS F
Treatment
SSTrt k-1 SSTrt/k-1 MSTrt/MSE
Block SSB B-1 SSB/B-1
Error SSE (k-1)(B-1)
SSE/ (k-1)(B-1)
Total SSTot N-1 SSTot/N-1
K = number of groups; N = total number of observationsB = number of blocks
Let’s move to the SPSS demo:
Factorial Designs• In many situations more than one factor, or
treatment interacts to produce effects beyond the sum of the effects of the 2 acting alone.– In other words, factors may act synergistically or
antagonistically.• When some form of interaction is suspected, a
two factor ANOVA, or a Factorial design, is appropriate.
• These are similar to a blocked design, except that the “block” now represents a factor in which we have an interest and each “block” x treatment cell consists of repeated observations.
Generalized ANOVA Table
Source of Variation
Sum of Squares
df MS F
Category A SSTrt A A-1 SSTrt/k-1 MSTrt A/MSE
Category B SSTrt B B-1 SSB/B-1 MSTrt B/MSE
Interaction SSA x B (A-1)(B-1) MSA x B/MSE
Error SSE (AxB)(n-1) SSE/ (k-1)(B-1)
Total SSTot N-1 SSTot/N-1
A = number of levels in category A; B = number of levels in category B N = total number of observations, n = number of observations each category
Damage No Damage
No Yeast
0 0
15 30
12 45
25 60
40 100
Yeast
30 115
15 140
20 75
20 60
25 110
No Damage Damage
20
40
60
80
100
No Yeast
Yeast
Example (Q 11.9)• The possible influence of crowding and
sex on plasma corticosterone in a highly inbred strain of rats was investigated using a factorial design. Sex (males, nongravid females, and gravid females) and crowding (low, moderate, and high) were used as the main treatment effects.
Example (Q 11.9)Sex Low Moderate High
Males 5 121.5 253
8 122 249
13 119 260
9 130 257
15 114 280
11 129 263
Nongravid Females 12 117.7 219
19 115 222
15 121 218
20 117 220
11 118 223
18 120 225
Gravid Females 37 157 289
42 160 273
50 173 280
35 182 291
40 168 205
36 170 296
SPSSMeansFigure
At low crowding stress, gravid females have the greatest plasma corticosterone levels, followed by nongravid females, with males exhibiting the lowest levels. At high and moderate stress levels, gravid females still exhibit the greatest plasma corticosterone levels. However, at these two levels of stress, males have higher corticosterone levels than nongravid females.
Low Medium High
Males Females Gravid Females
Cort
icost
ero
ne C
once
ntr
ati
on
(X
+ 1
sd)
Nested ANOVAS
BENCH FLAT
PLANT
Variation in Growth
Nested ANOVAS
BENCH FLAT
PLANT
Variation in Growth
81 87
82 84
75 79
76 77
75 67
68 69
75 66
73 72
76 77
75 74
72 74
73 80
For Nested ANOVA:
- First, EDIT > OPTIONS > “Open syntax window @ startup” - see if it opens… if not reboot.
NOW, from taskbar, there should be a new window…syntax
Analyze > General Lin. Modeldependent var = growthrandom effects = bench flat model = bench flat main effects
RunCopy LOG box from output
Paste in syntax window… change flat to flat(bench)Get cursor to top line, before UNIANOVAPRESS green ARROW to RUN!
UNIANOVA growth BY bench flat /RANDOM=bench flat /METHOD=SSTYPE(3) /INTERCEPT=INCLUDE /CRITERIA=ALPHA(0.05) /DESIGN=bench flat(bench).
Tests of Between-Subjects EffectsDependent Variable:growthSource Type III Sum of Squares df Mean Square F Sig.
Intercept Hypothesis 136052.042 1 136052.042 753.056 .001Error 361.333 2 180.667a
bench Hypothesis 361.333 2 180.667 5.510 .099Error 98.375 3 32.792b
flat(bench) Hypothesis 98.375 3 32.792 3.754 .030Error 157.250 18 8.736c
a. MS(bench)b. MS(flat(bench))c. MS(Error)
Mixed Model ANOVAS
BENCH FLAT
PLANT
Variation in Growth
Nitrogen
Nitrogen
Potassium
Potassium
Mixed Model ANOVAS
BENCH FLAT
PLANT
Variation in Growth
81 87
82 84
75 79
76 77
75 67
68 69
75 73
66 72
76 77
75 74
80 74
72 73
Nitrogen
Nitrogen
Potassium
Potassium
UNIANOVA growth BY fert bench flat /RANDOM=bench flat /METHOD=SSTYPE(3) /INTERCEPT=INCLUDE /CRITERIA=ALPHA(0.05) /DESIGN=bench fert flat(bench) fert*flat(bench).
Run an ANOVA as before with the terms… copy LOG, and rewrite DESIGN line as follows:
EXPERIMENTAL DESIGN MATTERS – IT AFFECTS HOW YOU ANALYZE THE PATTERNS IN THE DATA