5.4 - Nested Treatment Design

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:

Factor A (Region)
i
Factor B (City)
j
Average
1 2
NE   30 18  
    35 20  
  Average \(\bar{Y}_{11.}=32.5\) \(\bar{Y}_{12.}=19\) \(\bar{Y}_{1..}=25.75\)
MW   10 20  
    9 22  
  Average \(\bar{Y}_{21.}=9.5\) \(\bar{Y}_{22.}=21\) \(\bar{Y}_{2..}=15.25\)
W   18 4  
    19 6  
  Average \(\bar{Y}_{31.}=18.5\) \(\bar{Y}_{32.}=5\) \(\bar{Y}_{3..}=11.75\)
      Average \(\bar{Y}_{...}=16.83\)

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.

Region (i)1(i=1)1(j=1)1(k=1)2(k=2)3(k=1)4(k=2)5(k=1)6(k=2)7(k=1)8(k=2)9(k=1)10(k=2)11(k=1)12(k=2)2(j=2)3(j=1)4(j=2)5(j=1)6(j=2)2(i=2)3 (i=3) City (j)Sch_Dist(k)

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 \( N=12 \).

Source d.f.
Region (a - 1) = 2
City (Region) a(b - 1) = 3
Error ab(r - 1) = 6
Total N - 1 = 11

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. 

Note:

  1. There is no interaction between a nested factor and its nesting factor.
  2. The nested factors always have to be accompanied by their nesting factor. This means that the effect B does not exist and B(A) represents the effect of B within factor A.
  3. df of B(A)= df of B + df of A*B  (This is simply a mathematically correct identity and there may not be much use practically as effects B(A) and A*B cannot coexist.)
  4. The residual effect of any ANOVA model is a nested effect - the replicate effect nested within the factor level combinations. Recall that the replicates are considered homogeneous and so any variability among them serves to estimate the model error.