Ecological Archives E085-108-A3

Pierre Legendre, Mark R. T. Dale, Marie -Josée Fortin, Philippe Casgrain, and Jessica Gurevitch. 2004. Effects of spatial structures on the results of field experiments. Ecology 85:3202–3214.

Appendix C. Power estimates for n = 81 (Table C1) and n = 36 experimental units (Table C2).

Table C1. Selected estimates of power for designs with 81 experimental units. Power values are the slopes of the probit-transformed rejection rates as a function of the square root of the effect sizes; larger slopes mean greater power. Mean powers are only based upon the values of F and F1.

Experimental designs

#1

Random positions

#2

Completely randomized lattice

#3

36-U blocks separated in the field (with replic.)

#4

9-U blocks adjacent in the field (with replic.)

#5

9-U blocks arranged in a grid (with replic.)

#6

3-U blocks randomly positioned (unreplicated)

F -statistics

F

F

F1 (F2)

F1 (F2)

F1 (F2)

F1

Spatial autocor.

Envir. variable

Mean power

0

0

4.20

4.19

4.19 (2.65)

4.39 (3.98)

4.23 (3.75)

4.10

4.217

0

1

2.71

2.80

2.86 (1.84)

2.75 (2.59)

2.78 (2.54)

2.71

2.768

0

2

2.91

2.81

2.84 (1.86)

2.75 (2.59)

2.80 (2.56)

2.71

2.803

0

3

2.85

2.68

2.51 (1.60)

2.66 (2.55)

2.73 (2.50)

2.73

2.693

4

0

2.07

2.41

2.41 (1.47)

2.80 (2.50)

2.91 (2.48)

2.52

2.520

4

1

1.71

2.05

1.79 (1.32)

2.05 (1.75)

2.16 (2.09)

1.95

1.952

4

2

1.81

2.00

1.98 (1.32)

2.06 (1.76)

2.13 (1.95)

2.12

2.017

4

3

1.80

1.83

1.76 (1.05)

1.97 (1.85)

2.14 (1.97)

2.23

1.955

16

0

2.20

3.23

3.26 (2.19)

3.69 (3.33)

3.61 (3.23)

3.47

3.243

16

1

1.82

2.47

2.41 (1.65)

2.45 (2.26)

2.60 (2.43)

2.53

2.380

16

2

1.96

2.36

2.38 (1.54)

2.59 (2.36)

2.57 (2.41)

2.54

2.400

16

3

2.10

2.27

2.20 (1.48)

2.44 (2.27)

2.54 (2.26)

2.37

2.320

40

0

2.41

3.72

3.81 (2.47)

4.07 (3.68)

3.94 (3.48)

3.79

3.623

40

1

1.73

2.56

2.61 (1.44)

2.69 (2.45)

2.73 (2.57)

2.56

2.480

40

2

1.98

2.58

2.62 (1.68)

2.69 (2.50)

2.66 (2.50)

2.61

2.523

40

3

2.02

2.42

2.18 (1.49)

2.54 (2.48)

2.64 (2.45)

2.69

2.415

Mean power

2.268

2.649

2.613

2.787

2.823

2.727

2.644


Experimental designs	#1 Random positions	#2 Completely randomized lattice	#3 36-U blocks separated in the field (with replic.)	#4 9-U blocks adjacent in the field (with replic.)	#5 9-U blocks arranged in a grid (with replic.)	#6 3-U blocks randomly positioned (unreplicated)
F -statistics	F	F	F1 (F2)	F1 (F2)	F1 (F2)	F1
Spatial autocor.	Envir. variable							Mean power

0	0	4.20	4.19	4.19 (2.65)	4.39 (3.98)	4.23 (3.75)	4.10	4.217
0	1	2.71	2.80	2.86 (1.84)	2.75 (2.59)	2.78 (2.54)	2.71	2.768
0	2	2.91	2.81	2.84 (1.86)	2.75 (2.59)	2.80 (2.56)	2.71	2.803
0	3	2.85	2.68	2.51 (1.60)	2.66 (2.55)	2.73 (2.50)	2.73	2.693

4	0	2.07	2.41	2.41 (1.47)	2.80 (2.50)	2.91 (2.48)	2.52	2.520
4	1	1.71	2.05	1.79 (1.32)	2.05 (1.75)	2.16 (2.09)	1.95	1.952
4	2	1.81	2.00	1.98 (1.32)	2.06 (1.76)	2.13 (1.95)	2.12	2.017
4	3	1.80	1.83	1.76 (1.05)	1.97 (1.85)	2.14 (1.97)	2.23	1.955

16	0	2.20	3.23	3.26 (2.19)	3.69 (3.33)	3.61 (3.23)	3.47	3.243
16	1	1.82	2.47	2.41 (1.65)	2.45 (2.26)	2.60 (2.43)	2.53	2.380
16	2	1.96	2.36	2.38 (1.54)	2.59 (2.36)	2.57 (2.41)	2.54	2.400
16	3	2.10	2.27	2.20 (1.48)	2.44 (2.27)	2.54 (2.26)	2.37	2.320

40	0	2.41	3.72	3.81 (2.47)	4.07 (3.68)	3.94 (3.48)	3.79	3.623
40	1	1.73	2.56	2.61 (1.44)	2.69 (2.45)	2.73 (2.57)	2.56	2.480
40	2	1.98	2.58	2.62 (1.68)	2.69 (2.50)	2.66 (2.50)	2.61	2.523
40	3	2.02	2.42	2.18 (1.49)	2.54 (2.48)	2.64 (2.45)	2.69	2.415

Mean power	2.268	2.649	2.613	2.787	2.823	2.727	2.644

Notes: U = experimental unit; replic. = replication. "Spatial autocor." is the range of the spherical variogram models for generation of spatial autocorrelation in the environmental and response variables. "Envir. variable": environmental variable types, coded 0–3: see Simulation setup.

Table C2. Selected estimates of power for designs with 36 experimental units. For n = 36, columns 2 and 3 are identical. Power values are the slopes of the probit-transformed rejection rates as a function of the square root of the effect sizes; larger slopes mean greater power. Mean powers are only based upon the values of F and F1.

Experimental designs

#1

Random positions

#2

Completely randomized lattice

#3

36-U blocks separated in the field (with replic.)

#4

9-U blocks adjacent in the field (with replic.)

#5

9-U blocks arranged in a grid (with replic.)

#6

3-U blocks randomly positioned (unreplicated)

F statistics

F

F

F

F1 (F2)

F1 (F2)

F1

Spatial autocor.

Envir. variable

Mean power

0

0

2.42

2.42

2.42

2.61 (1.66)

2.39 (1.86)

2.16

2.403

0

1

1.75

1.38

1.38

1.59 (1.19)

1.35 (1.02)

1.54

1.498

0

2

1.89

1.38

1.38

1.59 (1.19)

1.37 (1.02)

1.56

1.528

0

3

1.07

1.69

1.69

1.64 (1.35)

1.37 (1.04)

1.44

1.483

4

0

1.04

1.46

1.46

1.54 (0.99)

1.54 (1.14)

1.46

1.417

4

1

0.98

1.06

1.06

1.16 (0.70)

1.12 (0.78)

1.19

1.095

4

2

0.97

1.22

1.22

1.22 (0.81)

1.10 (0.86)

1.25

1.163

4

3

0.72

1.24

1.24

1.30 (0.97)

1.11 (0.79)

1.07

1.113

16

0

1.16

1.87

1.87

2.06 (1.55)

2.13 (1.65)

1.91

1.833

16

1

1.10

1.30

1.30

1.35 (1.11)

1.34 (0.99)

1.38

1.295

16

2

1.20

1.22

1.22

1.54 (1.02)

1.35 (0.98)

1.47

1.333

16

3

0.56

1.54

1.54

1.39 (1.02)

1.38 (0.91)

1.40

1.302

40

0

1.07

2.25

2.25

2.25 (1.60)

2.26 (1.74)

2.13

2.035

40

1

1.05

1.42

1.42

1.49 (1.28)

1.34 (0.99)

1.54

1.377

40

2

1.12

1.44

1.44

1.41 (1.13)

1.38 (1.05)

1.55

1.390

40

3

0.75

1.50

1.50

1.54 (1.19)

1.35 (1.00)

1.33

1.328

Mean power

1.178

1.524

1.524

1.605

1.493

1.524

1.475


Experimental designs	#1 Random positions	#2 Completely randomized lattice	#3 36-U blocks separated in the field (with replic.)	#4 9-U blocks adjacent in the field (with replic.)	#5 9-U blocks arranged in a grid (with replic.)	#6 3-U blocks randomly positioned (unreplicated)
F statistics	F	F	F	F1 (F2)	F1 (F2)	F1
Spatial autocor.	Envir. variable							Mean power

0	0	2.42	2.42	2.42	2.61 (1.66)	2.39 (1.86)	2.16	2.403
0	1	1.75	1.38	1.38	1.59 (1.19)	1.35 (1.02)	1.54	1.498
0	2	1.89	1.38	1.38	1.59 (1.19)	1.37 (1.02)	1.56	1.528
0	3	1.07	1.69	1.69	1.64 (1.35)	1.37 (1.04)	1.44	1.483

4	0	1.04	1.46	1.46	1.54 (0.99)	1.54 (1.14)	1.46	1.417
4	1	0.98	1.06	1.06	1.16 (0.70)	1.12 (0.78)	1.19	1.095
4	2	0.97	1.22	1.22	1.22 (0.81)	1.10 (0.86)	1.25	1.163
4	3	0.72	1.24	1.24	1.30 (0.97)	1.11 (0.79)	1.07	1.113

16	0	1.16	1.87	1.87	2.06 (1.55)	2.13 (1.65)	1.91	1.833
16	1	1.10	1.30	1.30	1.35 (1.11)	1.34 (0.99)	1.38	1.295
16	2	1.20	1.22	1.22	1.54 (1.02)	1.35 (0.98)	1.47	1.333
16	3	0.56	1.54	1.54	1.39 (1.02)	1.38 (0.91)	1.40	1.302

40	0	1.07	2.25	2.25	2.25 (1.60)	2.26 (1.74)	2.13	2.035
40	1	1.05	1.42	1.42	1.49 (1.28)	1.34 (0.99)	1.54	1.377
40	2	1.12	1.44	1.44	1.41 (1.13)	1.38 (1.05)	1.55	1.390
40	3	0.75	1.50	1.50	1.54 (1.19)	1.35 (1.00)	1.33	1.328

Mean power	1.178	1.524	1.524	1.605	1.493	1.524	1.475

[Back to E085-108]