In multi-factor experiments, combinations of factor levels are applied to experimental units. To illustrate this idea consider again the single-factor greenhouse experiment discussed in previous lessons. Suppose there is suspicion that the different fertilizer types may be more effective for certain species of plant. To accommodate this, the experiment can be extended to a multi-factor study by including plant species as an additional factor along with fertilizer type. This will allow for assessment as to whether or not the optimal height growth is perhaps attainable by a unique combination of fertilizer type and plant species. A treatment design that enables analysis of treatment combinations is a factorial design. Within this design, responses are observed at each level of all combinations of the factors. In this setting the factors are said to be "crossed"; thus the design is also sometimes referred to as a crossed design.

A factorial design with \(t\) factors can be defined using the notation “\(l_1 \times l_2 \times ... \times l_t\)”, where \(l_i\) is the number of levels in the \(i^{th}\) factor for \(i=1,2,...,t\). For example, a factorial design with 2 factors, A and B, where A has 4 levels and B has 3 levels,  would be a \(4 \times 3\) factorial design.

One complete replication of a factorial design with \(t\) factors requires \(l_1 \times l_2 \times ... \times l_t\) experimental units and this quantity is called the replicate size. If \(r\) is the number of complete replicates, then the total number of observations \(N\) is equal to \(r \times (l_1 \times l_2 \times... \times l_t)\). It is easy to see that with the addition of more and more crossed factors, the replicate size will increase rapidly and design modifications may have to be made to make the experiment more manageable (discussed more in later lessons).

In a factorial experiment it is important to differentiate between the lone (or main) effects of a factor on the response and the combined effects of a group of factors on the response. The main effect of factor A is the effect of A on the response ignoring the effect of all other factors. The main effect of a given factor is equivalent to the factor effect associated with the single-factor experiment using only that particular factor. The combined effect of a specific combination of \(t\) different factors is called the interaction effect (more details later). Typically, the interaction effect of most interest is the two-way interaction effect between only two of the \(t\) possible factors. Two-way interactions are typically denoted by the product of the two letters assigned to the two factors. For example, in a factorial design with 3 factors A, B, C, the two-way interaction effects are denoted \(A\times B\), \(A\times C\), and \(B\times C\) (or just \(AB\), \(AC\), and \(BC\)). Likewise, the three-way interaction effect of these 3 factors is denoted by \(A\times B\times C\).

Let us now examine how the degrees of freedom (df) values of a single-factor ANOVA can be extended to the ANOVA of a two-factor factorial design. Note that the interaction effects are additional terms that need to be included in a multi-factor ANOVA, but the ANOVA rules studied in Lesson 2 for single-factor situations still apply for the main effect of each factor. If the two factors of the design are denoted by A and B with \(a\) and \(b\) as their number of levels respectively, then the df values of the two main effects are \((a-1)\) and \((b-1).\) The df value for the two-way interaction effect is \((a-1)(b-1)\), the product of df values for A and B. The ANOVA table below gives the layout of the df values for a \(2\times 2\) factorial design with 5 complete replications. Note that in this experiment, \(r\) equals 5, and \(N\) is equal to 20.

If in the single-factor model,

\(Y_{ij}=\mu+\tau_i+\epsilon_{ij}\)

 \(\tau_i\) is effectively replaced with \(\alpha_i+\beta_j+(\alpha\beta)_{ij}\), then the resulting equation shown below will represent the model equation of a two-factor factorial design. 

\(Y_{ijk}=\mu+ \alpha_i+\beta_j+(\alpha\beta)_{ij} +\epsilon_{ijk}\)

where \( \alpha_i\) is the main effect of factor A, \(\beta_j\) is the main effect of factor B, and \((\alpha\beta)_{ij}\) is the interaction effect \((i=1,2,...a, j=1,2,...,b, k=1,2,...,r)\).

This reflects the following partitioning of treatment deviations from the grand mean:

\(\underbrace{\bar{Y}_{ij.}-\bar{Y}_{...}}_{\substack{\text{Deviation of estimated treatment mean} \\ \text{ around overall mean}}} = \underbrace{\bar{Y}_{i..}-\bar{Y}_{...}}_{\text{A main effect}} + \underbrace{\bar{Y}_{.j.}-\bar{Y}_{...}}_{\text{B main effect}} +\underbrace{\bar{Y}_{ij.}-\bar{Y}_{i..}-\bar{Y}_{.j.}+\bar{Y}_{...}}_{\text{AB interaction effect}}\)

The main effects for Factor A and Factor B are straightforward to interpret, but what exactly is an interaction effect? Delving in further, an interaction can be defined as the difference in the response to one factor at various levels of another factor. Notice that \((\alpha\beta)_{ij}\), the interaction term in the model, is multiplicative, and as a result may have a large impact on the response variable. Interactions go by different names in various fields. In medicine for example, physicians commonly ask about current medications before prescribing a new medication. They do this out of a concern for interaction effects of either interference (a canceling effect) or synergism (a compounding effect).

Graphically, in a two factor factorial design with each factor having 2 levels, the interaction can be represented by two non-parallel lines connecting means (adapted from Zar, H. Biostatistical Analysis, 5th Ed., 1999). This is because the interaction reflects the difference in response between the two different levels of one factor for both levels of the other factor. So, if there is no interaction, then this difference in response will be the same, which will result in two parallel lines graphically. Examples of several interaction plots can be seen below. Notice, parallel lines are a consistent feature in all settings with no interaction, whereas in plots depicting interaction, the lines do cross (or would cross if the lines kept going).

In the presence of multiple factors with their interactions multiple hypotheses can be tested. For a two-factor factorial design, those hypotheses are the following. 

Main effect of Factor A:

\(H_0 \colon \alpha_{1} =\alpha_{2}= \cdots =\alpha_{a} = 0\)
\(H_A \colon \text{ not all }\alpha_{i} \text{ are equal to zero}\)

Main effect of Factor B:

\(H_0 \colon \beta_{1} =\beta_{2}= \cdots =\beta_{b} = 0\)
\(H_A \colon \text{ not all }\beta_{j} \text{ are equal to zero}\)

A × B interaction effect:

\(H_0 \colon \text{ there is no interaction }\)
\(H_A \colon \text{ an interaction exists }\)

When testing these hypotheses, it is important to test for the significance of the interaction effect first. If the interaction is significant, the main effects are of no consequence - rather the differences among different factor level combinations should be looked into. The greenhouse example, extended to include a second (crossed) factor will illustrate the steps.

In a factorial design, we first look at the interactions for significance. In the case where the interaction is not significant, we can drop the term from our model and end up with an "Additive Model".

For a two-factor factorial, the model we initially consider (as we have discussed in Section 5.1) is:

\(Y_{ij}=\mu+\alpha_i+\beta_j+(\alpha\beta)_{ij} +\epsilon_{ijk}\)

If the interaction is found to be non-significant, then the model reduces to:

\(Y_{ij}=\mu+\alpha_i+\beta_j+\epsilon_{ijk}\)

Here we can see that the response variable is simply a function of adding the effects of the two factors.

When setting up a multi-factor study, sometimes it is not possible to cross the factor levels. In other words, because of the logistics of the situation, we may not be able to have each level of treatment combined with each level of another treatment. This is most easily illustrated with an example. 

Suppose a research team conducted a study to compare the activity levels of high school students across the 3 geographic regions in the United States: Northeast (NE), Midwest (MW), and the West (W). The study also included a comparison of activity levels among cities within each region. Two school districts were chosen from two major cities from each of these 3 regions and the response variable, the average number of exercise hours per week for high school students, was recorded for each school district. 

A diagram to illustrate the treatment design can be set up as follows. Here, the subscript (\(i\)) identifies the regions, and the subscript (\(j\)) indicates the cities:

The table above shows the data obtained: the grand mean, the treatment level means (or marginal means), and finally the cell means. The cell means are the averages of the two school district mean activity levels for each combination of Region and City.

This example drives home the point that the levels of the second factor (City) cannot practically be crossed with the levels of the first factor (Region) as cities are specific or unique to regions. Note that the cities are identified as 1 or 2 within each region. But it is important to note that city 1 in the Northeast is not the same as city 1 in the Midwest. The concept of nesting does come in useful to describe this type of situation and the use of parentheses is appropriate to clearly indicate the nesting of factors. To indicate that the City is nested within the factor Region, the notation "City(Region)" will be used. Here, City is the nested factor and Region is the nesting factor.

We can partition the deviations as before into the following components:

\(\underbrace{Y_{ijk}-\bar{Y}_{...}}_{\text{Total deviation}} = \underbrace{\bar{Y}_{i..}-\bar{Y}_{...}}_{\text{A main effect}} + \underbrace{\bar{Y}_{ij.}-\bar{Y}_{i..}}_{\text{Specific B effect when } \\ \text{A at the }i^{th} \text{level}} + \underbrace{Y_{ijk}-\bar{Y}_{ij.}}_{\text{Residual}}\)

Let us now examine the df values of a two-factor nested design. Note that the nested effect is an additional term that needs to be included in a multi-factor ANOVA, but the ANOVA rules studied in Lesson 2 for single-factor situations still apply for the nesting effect. If the two factors of the design are denoted by A and B(A) with a and b as their number of levels respectively, then the df value of the nesting factor A is \( (a−1) \) and the df value for the nested factor is \( a(b−1) \). The ANOVA table below gives the layout of the df values for highschool example given above where region has \( a=3 \) levels, city has \( b=2 \) levels, \( r=2 \) complete replications and \( N=12 \).

The statistical model shown below represents two-factor nested design. 

\(Y_{ijk}=\mu+\alpha_{i}+\beta_{j(i)}+\epsilon_{ijk}\)

where \(\mu\) is a constant, \(\alpha_{i}\) are constants subject to the restriction \(\sum\alpha_i=0\), \(\beta_{j(i)}\) are constants subject to the restriction \(\sum_j\beta_{j(i)}=0\) for all i, \(\epsilon_{ijk}\) are independent from \( N(0, \sigma^2) \), with \(i = 1, ... , a, j = 1, ... , b \text{ and } k = 1, ... ,r\).

In the presence of multiple factors multiple hypotheses can be tested. For a two-factor nested design, specifically the high school example presented above, those hypotheses are the following. 

For Factor A, the nesting factor:

\(H_0 \colon \mu_{\text{Northeast}}=\mu_{\text{Midwest}}=\mu_{\text{West}} \text{ vs. } H_A \colon \text{ Not all factor A level means are equal }\)

Up to this point, we have been stating hypotheses in terms of the means, but we can alternatively state the hypotheses in terms of the parameters for that treatment in the model. For example, for the nesting factor A, we could also state the null hypothesis as

\(H_0 \colon \alpha_{\text{Northeast}}=\alpha_{\text{Midwest}}=\alpha_{\text{West}}=0\),

or equivalently \(H_0 \colon \text{ all } \alpha_i=0\).

For Factor B, the nested factor:

When stating the hypotheses for Factor B, the nested effect, alternative notation has to be used. For the nested factor B, the hypotheses should differentiate between the nesting and the nested factors, because we are evaluating the nested factor within the levels of the nesting factor. So for the nested factor (City, nested within Region), we have the following hypotheses.

\(H_0: \text{ all }\beta_{j(i)} =0\) vs. \(H_A: \text{ not all }\beta_{j(i)} =0\) for \( j = 1,2 \)

The F-tests can then proceed as usual using the ANOVA results. 

Multi-factor studies can involve factor combinations in which factors are crossed and/or nested. These treatment designs are based on the extensions of the concepts discussed so far.

Consider an example where manufacturers are evaluated for total processing times. There were three factors of interest: manufacturer (1, 2 or 3), site (1 or 2), and assembly order (1, 2 or 3).

Each manufacturer can utilize each assembly order, and so these factors can be crossed. Also, each site can utilize each assembly order and so these factors also are crossed. However, the sites (1 or 2) are unique to each manufacturer, and as a result the site is nested within the manufacturer.

The statistical model contains both crossed and nested effects and is:

\(Y_{ijkl}=\mu+\alpha_i+\beta_{j(i)}+\gamma_k+(\alpha\gamma)_{ik}+(\beta\gamma)_{j(i)k}+\epsilon_{ijkl}\)

With the ANOVA table as follows:

Notice that the two main effects Manufacturer (Factor A) and Order (Factor C) along with their interaction effect are included in the model. The nested relationship of Site (Factor B) within Manufacturer is represented by the Site(Manufacturer) term and the crossed relationship between Site and Order is represented by their interaction effect.

Notice that the main effect Site and the crossed effect Site \(\times\) Manufacturer are not included in the model. This is consistent with the facts that a nested effect cannot be represented as the main effect and also that a nested effect cannot interact with its nesting effect.

In this lesson, we discussed important elements of the 'Treatment Design’, one of the two components of an ‘Experimental Design’. 

In a full factorial design, the experiment is carried out at every factor level combination. Most factorial studies do not go beyond a two-way interaction and if a two-way interaction is significant, the mean response values should be compared among different combinations of the two factors rather than among the single factor levels. In other words, the focus should be on response vs. interaction effect rather than response vs. main effects. An interaction plot is a useful graphical tool to understand the extent of interactions among factors (or treatments) with parallel lines indicating no interaction.

In a nested design, the experiment need not be conducted at every combination of levels in all factors. Given two factors in a nested design, there is a distinction between the nested and the nesting factor. The levels of the nested factor may be unique to each level of the nesting factor. Therefore, the comparison of the nested factor levels should be made within each level of the nesting factor. This fact should be kept in mind when stating null and alternative hypotheses for the nested factor(s) and also when writing programming code.

The following section expands on what you've learned in lesson 5. However, this material (5.9) is not assessed in the actual course.