2D ANOVA VI

Inputs/Outputs

2D ANOVA Factors, Levels, and Cells

A factor is a basis for categorizing data. For example, if you count the number of sit-ups individuals can do, one basis of categorization is age. For age, you might have the following levels:

Another possible factor is weight, with the following levels:

Now, suppose that you made a series of observations to see how many sit-ups people could do. If you took a random sampling of n people, you might find the following results:

and so on.

If you plot observations as a function of factor A and factor B, they fall into cells of a matrix with factor A as rows and factor B as columns. Each cell must contain at least one observation, and each cell must contain the same number of observations.

To perform the analysis of variance, you specify an array X of observations, with values 10, 15, 20, 25, 17, and 4. The array Index A specifies the level (or category) of factor A to which each observation applies. In this case, the array would have the values 0, 1, 1, 1, 0, and 0.

The array Index B specifies the level (or category) of factor B to which each observation applies. In this case, the array would have the values 0, 0, 2, 1, 1, and 2. Finally, there are two possible levels for factor A and three possible levels for factor B, so you pass in a value of 2 for the A levels parameter and a value of 3 for the B levels parameter.

You can apply any one of the following models, where L is the specified observations per cell:

• Model 1: Fixed-effects with no interaction and one observation per cell (per specified levels x and y of the factors A and B, respectively).
• Model 2: Fixed-effects with interaction and L > 1 observations per cell.
• Model 3: Either of the fixed-effects models with interaction and L > 1 observations per cell.
• Model 4: Random-effects with interaction and L > 1 observations per cell.

2D 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.

The input parameters A levels and B levels represent the number of levels in factors A and B, respectively, as well as whether the factors are random or fixed. For example, if factor A is random, you set A levels to be negative the number of levels in factor A. Notice that if there is only one observation per cell, both A levels and B levels must be positive. That is, you use model 1.

2D ANOVA Statistical Model

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

Model 1 expresses each observation as the sum of four components.

Models 2, 3, and 4 express each observation as the sum of five components.

where

• µ is the overall mean response (the average of the mean response for all the populations).
• α_p is the effect of the p^th level of factor A (equal to µ_p – µ where µ_p is the average of the p^th level of factor A over all levels of factor B).
• β_q is the effect of the q^th level of factor B (equal to µ_q – µ where µ_q is the average of the q^th level of factor B over all levels of factor A).
• (αβ)_pq is the interaction between the p^th level of factor A and the q^th level of factor B (equal to µ_pq – (µ + α_p + β_q) where µ_pq is the population mean of the pq^th cell).
• ε_pqr is the deviation of x_pqr from the population mean response for the pq^th population.

2D 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 this assumption. The following are the three 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.
• (AB) that (αβ)_pq = 0 for all levels p and q if both factors A and B are fixed, and that σ_A² = 0 if either factor A or factor B is random. (This does not apply to model 1. In model 1 there is no interaction, and the associated output parameters are superfluous.)

2D ANOVA Assumptions

The 2D ANOVA VI makes the following assumptions:

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

2D 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 some number of component sums of squares. In model 1

whereas in models 2 through 4

Each component sum in 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, ssab is a measure of the variation due to the interaction between factors A and B, and sse is a measure of the variation due to random fluctuation. Notice that with model 1 you have no ssab term. This is what no interaction means.

The VI divides each of the values ssa, ssb, ssab, and sse by their own degrees of freedom to compute the mean square quantities msa, msb, msab, and mse. If one factor, such as factor A, has a strong effect on the experimental observations, the respective mean square quantity msa will be relatively large.

Testing the 2D ANOVA Hypotheses

For each hypothesis, the VI computes a number f that is used to calculate the associated sig probability. For example, for the hypothesis (A), that α_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 (a – 1)(b – 1). You then can use the probabilities sigA, sigB, and sigAB to determine when you should reject the associated hypotheses (A), (B), and (AB).

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 chosen level of significance is 0.05, and the output sigA is 0.03, then you must reject the hypothesis αA² = 0 and conclude that factor A has an effect on the experimental observations.

2D ANOVA Formulas

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

Let

a = |A levels|

b = |B levels|

then

dofab = (a – 1)(b – 1) if L > 1

dofab = 0 if L = 1

dofe = ab(L – 1) if L > 1

dofe = (a – 1)(b – 1) if L = 1

sigA = Prob{F_{a – 1, ab(L – 1)} > fa}          if B is fixed

sigA = Prob{F_{a – 1, (a – 1)(b – 1)} > fa}    if B is random

sigB = Prob{F_{a – 1, ab(L – 1)} > fb}          if A is fixed

sigB = Prob{F_{a – 1, (a – 1)(b – 1)} > fb}    if A is random

sigAB = Prob{F_{(a – 1)(b – 1), ab(L – 1)} > fab}