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