Appendix A. The statistical design for ANOVA showing factors, multipliers, and formulas to determine estimated mean squares, variance components, and F ratio tests.
Factor Treatment has i =…a levels; the factor Ponds(Treatment) was replicated n times (r =…n), it served as the error term for the spatial variation; factor Period has j =…b levels; factor Time(Period) has k =…c levels. All combinations of the temporal variation effect were replicated N times (s =…N). No residual error could be calculated because only a single sample was taken within each pond at each time. There are two orthogonal factors and two nested factors. Factors Treatment, Period, and Time(Period) are fixed and Ponds nested within Treatments is random. Multipliers and formula’s to determine estimated mean squares (E(MS)), variance components and denominators for F tests (den.) are shown.
Between samples: Yi = µ + Treatmenti + Ponds(Treatment)r(i)
Within samples: Yirjk = µ + Periodj + Time(Period)k(j) + Treatment × Periodij + Treatment × Time(Period)ik(j) + Ponds(Treatment) × Periodr(i)j + Ponds(Treatment) × Time(Period)r(i) k(j)
Source of variation |
Multipliers |
E(MS) |
Den. |
Variance component |
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Spatial variation |
i |
r |
j |
k |
s |
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1. Treatmenti |
0 |
n |
- |
- |
- |
σe 2 + nσ2Treatment |
2 |
σ2Treatment = [MSTreatment-MSe] / n |
2. Ponds(Treatment) = er(i) |
1 |
1 |
- |
- |
- |
σe2 |
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Temporal variation – repeated measures |
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3. Periodj |
a |
- |
0 |
c |
N |
σE2 + cNσ2Ponds(Treatment)Period + acNσ2Period |
7 |
σ2Period = [MSPeriod – MSPonds(Treatment)Period] / caN |
4. Time(Period)k(j) |
a |
- |
1 |
0 |
N |
σE2 + Nσ2Ponds(Treatment)Time(Period) + aNσ2Time(Period) |
8 |
σ2Time(Period) = [MSTime(Period) - MSPonds(Treatment)Time(Period)] / aN |
5. Treatment × Periodij |
0 |
- |
0 |
c |
N |
σE2 + cNσ2Ponds(Treatment)Period + cNσ2TreatmentPeriod |
7 |
σ2TreatmentPeriod = [MSTreatmentPeriod - MSPonds(Treatment)Period] / cN |
6. Treatment × Time(Period)ik(j) |
0 |
- |
1 |
0 |
N |
σE2 + Nσ2Ponds(Treatment)Time(Period) +Nσ2TreatmentTime(Period) |
8 |
σ2TreatmentTime(Period) = [MSTreatmentTime(Period) - MSPonds(Treatment)Time(Period)] / N |
7. Ponds(Treatment) × Periodr(i)j |
1 |
1 |
0 |
c |
N |
σE2 + cNσ2Ponds(Treatment)Period |
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8. Ponds(Treatment) × Time(Period)r(i)k(j) |
1 |
1 |
1 |
0 |
N |
σE2 + Nσ2Ponds(Treatment)Time(Period) |
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