Two Way Anova Example Hypothesis Statement

I think it's important to clearly separate the hypothesis and its corresponding test. For the following, I assume a balanced, between-subjects CRF-$pq$ design (equal cell sizes, Kirk's notation: Completely Randomized Factorial design).

$Y_{ijk}$ is observation $i$ in treatment $j$ of factor $A$ and treatment $k$ of factor $B$ with $1 \leq i \leq n$, $1 \leq j \leq p$ and $1 \leq k \leq q$. The model is $Y_{ijk} = \mu_{jk} + \epsilon_{i(jk)}, \quad \epsilon_{i(jk)} \sim N(0, \sigma_{\epsilon}^2)$

Design: $\begin{array}{r|ccccc|l} ~ & B 1 & \ldots & B k & \ldots & B q & ~\\\hline A 1 & \mu_{11} & \ldots & \mu_{1k} & \ldots & \mu_{1q} & \mu_{1.}\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ A j & \mu_{j1} & \ldots & \mu_{jk} & \ldots & \mu_{jq} & \mu_{j.}\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ A p & \mu_{p1} & \ldots & \mu_{pk} & \ldots & \mu_{pq} & \mu_{p.}\\\hline ~ & \mu_{.1} & \ldots & \mu_{.k} & \ldots & \mu_{.q} & \mu \end{array}$

$\mu_{jk}$ is the expected value in cell $jk$, $\epsilon_{i(jk)}$ is the error associated with the measurement of person $i$ in that cell. The $()$ notation indicates that the indices $jk$ are fixed for any given person $i$ because that person is observed in only one condition. A few definitions for the effects:

$\mu_{j.} = \frac{1}{q} \sum_{k=1}^{q} \mu_{jk}$ (average expected value for treatment $j$ of factor $A$)

$\mu_{.k} = \frac{1}{p} \sum_{j=1}^{p} \mu_{jk}$ (average expected value for treatment $k$ of factor $B$)

$\alpha_{j} = \mu_{j.} - \mu$ (effect of treatment $j$ of factor $A$, $\sum_{j=1}^{p} \alpha_{j} = 0$)

$\beta_{k} = \mu_{.k} - \mu$ (effect of treatment $k$ of factor $B$, $\sum_{k=1}^{q} \beta_{k} = 0$)

$(\alpha \beta)_{jk} = \mu_{jk} - (\mu + \alpha_{j} + \beta_{k}) = \mu_{jk} - \mu_{j.} - \mu_{.k} + \mu$
(interaction effect for the combination of treatment $j$ of factor $A$ with treatment $k$ of factor $B$, $\sum_{j=1}^{p} (\alpha \beta)_{jk} = 0 \, \wedge \, \sum_{k=1}^{q} (\alpha \beta)_{jk} = 0)$

$\alpha_{j}^{(k)} = \mu_{jk} - \mu_{.k}$
(conditional main effect for treatment $j$ of factor $A$ within fixed treatment $k$ of factor $B$, $\sum_{j=1}^{p} \alpha_{j}^{(k)} = 0 \, \wedge \, \frac{1}{q} \sum_{k=1}^{q} \alpha_{j}^{(k)} = \alpha_{j} \quad \forall \, j, k)$

$\beta_{k}^{(j)} = \mu_{jk} - \mu_{j.}$
(conditional main effect for treatment $k$ of factor $B$ within fixed treatment $j$ of factor $A$, $\sum_{k=1}^{q} \beta_{k}^{(j)} = 0 \, \wedge \, \frac{1}{p} \sum_{j=1}^{p} \beta_{k}^{(j)} = \beta_{k} \quad \forall \, j, k)$

With these definitions, the model can also be written as: $Y_{ijk} = \mu + \alpha_{j} + \beta_{k} + (\alpha \beta)_{jk} + \epsilon_{i(jk)}$

This allows us to express the null hypothesis of no interaction in several equivalent ways:

  1. $H_{0_{I}}: \sum_{j}\sum_{k} (\alpha \beta)^{2}_{jk} = 0$
    (all individual interaction terms are $0$, such that $\mu_{jk} = \mu + \alpha_{j} + \beta_{k} \, \forall j, k$. This means that treatment effects of both factors - as defined above - are additive everywhere.)

  2. $H_{0_{I}}: \alpha_{j}^{(k)} - \alpha_{j}^{(k')} = 0 \quad \forall \, j \, \wedge \, \forall \, k, k' \quad (k \neq k')$
    (all conditional main effects for any treatment $j$ of factor $A$ are the same, and therefore equal $\alpha_{j}$. This is essentially Dason's answer.)

  3. $H_{0_{I}}: \beta_{k}^{(j)} - \beta_{k}^{(j')} = 0 \quad \forall \, j, j' \, \wedge \, \forall \, k \quad (j \neq j')$
    (all conditional main effects for any treatment $k$ of factor $B$ are the same, and therefore equal $\beta_{k}$.)

  4. $H_{0_{I}}$: In a diagramm which shows the expected values $\mu_{jk}$ with the levels of factor $A$ on the $x$-axis and the levels of factor $B$ drawn as separate lines, the $q$ different lines are parallel.

Stats: Two-Way ANOVA


The two-way analysis of variance is an extension to the one-way analysis of variance. There are two independent variables (hence the name two-way).

Assumptions

  • The populations from which the samples were obtained must be normally or approximately normally distributed.
  • The samples must be independent.
  • The variances of the populations must be equal.
  • The groups must have the same sample size.

Hypotheses

There are three sets of hypothesis with the two-way ANOVA.

The null hypotheses for each of the sets are given below.

  1. The population means of the first factor are equal. This is like the one-way ANOVA for the row factor.
  2. The population means of the second factor are equal. This is like the one-way ANOVA for the column factor.
  3. There is no interaction between the two factors. This is similar to performing a test for independence with contingency tables.

Factors

The two independent variables in a two-way ANOVA are called factors. The idea is that there are two variables, factors, which affect the dependent variable. Each factor will have two or more levels within it, and the degrees of freedom for each factor is one less than the number of levels.

Treatment Groups

Treatement Groups are formed by making all possible combinations of the two factors. For example, if the first factor has 3 levels and the second factor has 2 levels, then there will be 3x2=6 different treatment groups.

As an example, let's assume we're planting corn. The type of seed and type of fertilizer are the two factors we're considering in this example. This example has 15 treatment groups. There are 3-1=2 degrees of freedom for the type of seed, and 5-1=4 degrees of freedom for the type of fertilizer. There are 2*4 = 8 degrees of freedom for the interaction between the type of seed and type of fertilizer.

The data that actually appears in the table are samples. In this case, 2 samples from each treatment group were taken.

 Fert IFert IIFert IIIFert IVFert V
Seed A-402106, 11095, 10094, 107103, 104100, 102
Seed B-894110, 11298, 99100, 101108, 112105, 107
Seed C-95294, 9786, 8798, 9999, 10194, 98

Main Effect

The main effect involves the independent variables one at a time. The interaction is ignored for this part. Just the rows or just the columns are used, not mixed. This is the part which is similar to the one-way analysis of variance. Each of the variances calculated to analyze the main effects are like the between variances

Interaction Effect

The interaction effect is the effect that one factor has on the other factor. The degrees of freedom here is the product of the two degrees of freedom for each factor.

Within Variation

The Within variation is the sum of squares within each treatment group. You have one less than the sample size (remember all treatment groups must have the same sample size for a two-way ANOVA) for each treatment group. The total number of treatment groups is the product of the number of levels for each factor. The within variance is the within variation divided by its degrees of freedom.

The within group is also called the error.

F-Tests

There is an F-test for each of the hypotheses, and the F-test is the mean square for each main effect and the interaction effect divided by the within variance. The numerator degrees of freedom come from each effect, and the denominator degrees of freedom is the degrees of freedom for the within variance in each case.

Two-Way ANOVA Table

It is assumed that main effect A has a levels (and A = a-1 df), main effect B has b levels (and B = b-1 df), n is the sample size of each treatment, and N = abn is the total sample size. Notice the overall degrees of freedom is once again one less than the total sample size.

SourceSSdfMSF
Main Effect AgivenA,
a-1
SS / dfMS(A) / MS(W)
Main Effect BgivenB,
b-1
SS / dfMS(B) / MS(W)
Interaction EffectgivenA*B,
(a-1)(b-1)
SS / dfMS(A*B) / MS(W)
WithingivenN - ab,
ab(n-1)
SS / df 
Totalsum of othersN - 1,
abn - 1
  

Summary

The following results are calculated using the Quattro Pro spreadsheet. It provides the p-value and the critical values are for alpha = 0.05.

Source of VariationSSdfMSFP-valueF-crit
Seed512.8667 2 256.4333 28.283 0.000008 3.682
Fertilizer449.4667 4 112.3667 12.393 0.000119 3.056
Interaction143.1333 8 17.8917 1.973 0.122090 2.641
Within136.0000 15 9.0667    
Total1241.4667 29     

From the above results, we can see that the main effects are both significant, but the interaction between them isn't. That is, the types of seed aren't all equal, and the types of fertilizer aren't all equal, but the type of seed doesn't interact with the type of fertilizer.




Error in Bluman Textbook

The two-way ANOVA, Example 13-9, in the Bluman text has the incorrect values in it. The student would have no way of knowing this because the book doesn't explain how to calculate the values.

Here is the correct table:

Source of VariationSSdfMSF
Sample3.920 1 3.920 4.752
Column9.680 1 9.680 11.733
Interaction54.080 1 54.080 65.552
Within3.300 4 0.825  
Total70.980 7   

The student will be responsible for finishing the table, not for coming up with the sum of squares which go into the table in the first place.


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