3D ANOVA VI

Inputs/Outputs

In any ANOVA, you look for evidence that the factors or interactions among factors have a significant effect on experimental outcomes. What varies with each model is the method used to do this.

3D ANOVA Random and Fixed Effects

A factor is a random effect if it has a large population of levels about which you want to draw conclusions but such that you cannot sample from all levels. You thus pick levels at random and hope to generalize about all levels. A factor is a fixed effect if you can sample from all levels about which you want to draw conclusions.

3D ANOVA Statistical Model

Let x_pqrs equation be the s^th observation at the p^th, q^th, and r^th levels of A, B, and C respectively, where s = 0, 1, ..., L – 1. Express each observation as the sum of eight components. Thus,

where

• µ is the overall mean.
• α_p is the average effect of the p^th level of factor A.
• β_q is the average effect of the q^th level of factor B.
• γ_r is the average effect of the r^th level of factor C.
• (αβ)_pq is the two-factor interaction of the p^th level of factor A with the q^th level of factor B.
• (αγ)_pr is the two-factor interaction of the p^th level of factor A with the r^th level of factor C.
• (βγ)_qr is the two-factor interaction of the q^th level of factor B with the r^th level of factor C.
• (αβγ)_pqr is the three-factor interaction of the p^th level of factor A, the q^th level of factor B, and the r^th level of factor C.
• ε_pqrs is random fluctuation.

3D ANOVA Hypotheses

Each of the following hypotheses is a different way of saying that a factor or an interaction among factors has no effect on experimental outcomes. This VI assumes that there are no effects and then seeks evidence to contradict these assumptions. The following are the seven hypotheses:

• (A) that α_p = 0 for all levels p if factor A is fixed, and that σ_A² = 0 if factor A is random.
• (B) that β_q = 0 for all levels q if factor B is fixed, and that σ_B² = 0 if factor B is random.
• (C) that γ_r = 0 for all levels r if factor C is fixed, and that σ_C² = 0 if factor B is random.
• (AB) that (αβ)_pq = 0 for all levels p and q if factors A and B are fixed, and that σ_AB² = 0 if either factor A or B is random.
• (AC) that (αγ)_pr = 0 for all levels p and q if factors A and C are fixed, and that σ_AC² = 0 if either factor A or C is random.
• (BC) that (βγ)_qr = 0 for all levels p and q if factors B and C are fixed, and that σ_BC² = 0 if either factor B or C is random.
• (ABC) that (αβγ)_pqr = 0 for all levels p, q, and r if factors A, B, and C are fixed, and that σ_ABC² = 0 if any of factors A, B, or C is random.

3D ANOVA Assumptions

The 3D ANOVA VI makes the following assumptions:

• Assume that for each p, q, and r, ε_pqrs is Normally distributed with mean 0 and variance σ_e².
• If a factor A is fixed, assume the populations of measurements at each level of A are Normally distributed with mean α_p + µ and variance σ_A² and that all the populations at each of the levels have the same variance. In addition, assume that α_p sum to zero. Analogous assumptions are made for B and C.
• If a factor A is random, assume the effect of the level of A itself, α_p, is a random variable Normally distributed with mean 0 and variance σ_A². Analogous assumptions are made for B and C.
• If some of the factors, such as A and B, associated with the effect of an interaction are (αβ)_pq fixed, then assume that the populations of measurements at each level of A and B are Normally distributed with mean µ + α_p + β_q + (αβ)_pq and variance σ_AB². For any fixed p, the means (αβ)_pq sum to zero when summing over all q. Similarly, for any fixed q, (αβ)_pq sum to zero when summing over all p.
• If any of the factors, such as A and B, associated with the effect of an interaction (αβ)_pq are random, assume the effect is a random variable Normally distributed with mean 0 and variance σ_AB². If A is fixed but B is random, assume that for any fixed q, the means (αβ)_pq sum to zero when summing over all p. Similarly, if B is fixed but A is random, assume that for any fixed p the means (αβ)_pq sum to zero when summing over all q.
• Assume all effects taken to be random variables are mutually independent.

3D ANOVA General Method

In each of the models, the VI breaks up the total sum of squares, tss, a measure of the total variation of the data from the overall population mean, into a number of component sums of squares.

Each component in the sum tss is a measure of variation attributed to a certain factor or interaction among the factors. Here ssa is a measure of the variation due to factor A; ssb is a measure of the variation due to factor B; ssc is a measure of the variation due to factor c; ssab is a measure of the variation due to the interaction between factors A and B; and so on for ssac, ssbc, and ssabc. Also, sse is a measure of the variation due to random fluctuation. The VI divides each by its own degrees of freedom to obtain the corresponding averages msa, msb, msc, msab, msac, msbc, msabc, and mse. For example, if factor A has a strong effect on the experimental observations, then msa will be relatively large.

Testing the 3D ANOVA Hypotheses

For each hypothesis, the VI computes number f that is used to calculate the associated sig probability. For example, for the hypothesis (A), that (p = 0 for all the levels p), (fixed A), the VI computes

then

where

is an F distribution with degrees of freedom a – 1 and abc(L – 1). You then can use the probabilities sigA, sigB, sigC, sigAB, …, sig ABC to determine when you should reject the associated hypotheses (A), (B), (C), (AB), …, (ABC).

How do you know when to reject the null hypothesis? For each hypothesis, you choose a level of significance. This level of significance is how likely you want it to be that you mistakenly reject the hypothesis (a common choice is 0.05). Compare your chosen level of significance with the associated sig probability output. If the sig probability is less than your chosen level of significance, you should reject the null hypothesis. For example, if A is a random effect, your level of significance is 0.05, and sigA = 0.03, you must reject the hypothesis that σ_A² = 0 and conclude that factor A has an effect on the experimental observations.

With some models there are no appropriate tests for certain hypotheses. If such is the case, the output parameters directly involved with the testing of these hypotheses are –1.0.

3D ANOVA Formulas

Let x_pqrs be the s^th observation at the p^th, q^th, and r^th levels of A, B, and C respectively, where s = 0, 1, ..., L – 1.

Let

a = |A levels|

b = |B levels|

c = |C levels|

then