## Hierarchical Designs

All of the multitreatment designs that have been discussed so far have had crossed treatments. Treatments A and B are crossed, for example, if each level of treatment B appears once with each level of treatment A and vice versa. Treatment B is nested in treatment A if each level of treatment B appears with only one level of treatment A. The nesting of treatment B within treatment A is denoted by B(A) and is read "B within

A CRH-24(A) design B CRF 22 design

A CRH-24(A) design B CRF 22 design b j ¿>2 ¿3 Z>4 ¿i ¿2 b

Figure 1.16 Comparison of designs with nested and crossed treatments. In panel A, treatment B(A) is nested in treatment A because b1 and b2 appear only with a1 while b3 and b4 appear only with a2. In panel B, treatments A and B are crossed because each level of treatment B appears once and only once with each level of treatment A and vice versa.

Figure 1.16 Comparison of designs with nested and crossed treatments. In panel A, treatment B(A) is nested in treatment A because b1 and b2 appear only with a1 while b3 and b4 appear only with a2. In panel B, treatments A and B are crossed because each level of treatment B appears once and only once with each level of treatment A and vice versa.

Figure 1.17 Diagram of a three-treatment completely randomized hierarchical design (CRH-24(A)8(AB) design). The four schools, bi, .. . , b4, are nested in the two approaches to introducing long division, treatment A. The eight teachers, ci,. . ., c8, are nested in the schools and teaching approaches. Students are randomly assigned to the pq(j) r(jk) = (2)(2)(2) = 8 treatment combinations with the restriction that n students are assigned to each combination.

Figure 1.17 Diagram of a three-treatment completely randomized hierarchical design (CRH-24(A)8(AB) design). The four schools, bi, .. . , b4, are nested in the two approaches to introducing long division, treatment A. The eight teachers, ci,. . ., c8, are nested in the schools and teaching approaches. Students are randomly assigned to the pq(j) r(jk) = (2)(2)(2) = 8 treatment combinations with the restriction that n students are assigned to each combination.

As is often the case, the nested treatments in the drug and educational examples resemble nuisance variables. The researcher in the drug example probably would not conduct the experiment just to find out whether the dependent variable is different for the two hospitals assigned to drug ai or the hospitals assigned to a2. The important question for the researcher is whether the new drug is more effective than the currently used drug. Similarly, the educational researcher wants to know whether one approach to teaching long division is better than the other. The researcher might be interested in knowing whether some schools or teachers perform better than others, but this is not the primary focus of the research. The distinction between a treatment and a nuisance variable is in the mind of the researcher—one researcher's nuisance variable can be another researcher's treatment.

The classical model equation for the drug experiment is

Yijk = V + «j + pk( j) + ei (jk) (i = 1,...,n; j = 1,..., p; k = 1,...,q), where

Yijk is an observation for participant i in treatment levels aj and bk(j). V is the grand mean of the population means. aj is the treatment effect for population aj and is equal to v. — V. It reflects the effect of drug a. $k(j) is the treatment effect for population bk(f) and is equal to Vjk — Vj.. It reflects the effects of hospi tal bk( j) that is nested in aj.

is the within-cell error effect associated with Yijk and is equal to Yijk — v — aj — Pk(j). It reflects all effects not attributable to treatment levels aj and b.

Notice that because treatment B(A) is nested in treatment A, the model equation does not contain an A x B interaction term.

This design enables a researcher to test two null hypotheses:

(Treatment A population means are equal.)

(Treatment B(A) population means are equal.)

If the second null hypothesis is rejected, the researcher can conclude that the dependent variable is not the same for the populations represented by hospitals bi and b2, that the dependent variable is not the same for the populations represented by hospitals b3 and b4, or both. However, the test of treatment B(A) does not address the question of whether, for example, vii = V23 because hospitals b1 and b3 were assigned to different levels of treatment A.

Hierarchical Design With Crossed and Nested Treatments

In the educational example, treatments B(A) and C(AB) were both nested treatments. Hierarchical designs with three or more treatments can have both nested and crossed treatments. Consider the partial hierarchical design shown in Figure i.i8. The classical model equation for this design is

Yijki = V + aj + pk + h(k) + («p)jk + (al)jt(k) + ei (jki) (i = i,...,n; j = i,..., p; k = i,...,q; l = i,...,r), where

Yijkl is an observation for participant i in treatment levels aj, bk, and ci(k). V is the grand mean of the population means.

aj is the treatment effect for population aj and is equal to Vj.. — V. Pk is the treatment effect for population bk and is equal to v.k. — V. -yl(k) is the treatment effect for population cl(k) and is equal to Vki — Vk-(ap)jk is the interaction effect for populations aj and bk and is equal to Vjk• — Vjk — Vj k + Vj k ■ (ary)jl(k) is the interaction effect for populations aj and ci(k) and is equal to Vjkl — Vjkl — Vjkl + Vjkl'. ei (jkl) is the within-cell error effect associated with Yijkl and is equal to Yijkl — V — «j — Pk — h(k) — (ap)jk — (al)jl(k).

Figure 1.18 Diagram of a three-treatment completely randomized partial hierarchical design (CRPH-jfgr(B) design). The letter P in the designation stands for "partial" and indicates that not all of the treatments are nested. In this example, treatments A and B are crossed; treatment C(B) is nested in treatment B because c1 and c2 appear only with b1 while c3 and c4 appear only with b2. Treatment C(B) is crossed with treatment A because each level of treatment C(B) appears once and only once with each level of treatment A and vice versa.

Figure 1.18 Diagram of a three-treatment completely randomized partial hierarchical design (CRPH-jfgr(B) design). The letter P in the designation stands for "partial" and indicates that not all of the treatments are nested. In this example, treatments A and B are crossed; treatment C(B) is nested in treatment B because c1 and c2 appear only with b1 while c3 and c4 appear only with b2. Treatment C(B) is crossed with treatment A because each level of treatment C(B) appears once and only once with each level of treatment A and vice versa.

Notice that because treatment C(B) is nested in treatment B, the model equation does not contain B x C and A x B x C interaction terms.

This design enables a researcher to test five null hypotheses:

(Treatment A population means are equal.)

(Treatment B population means are equal.)

(Treatment C(B) population means are equal.)

(All A x B interaction effects equal zero.)

Ho: Mjkl - Mjkl - Mjkl + Mj'kl' = 0 for all j, k and I

(All A x C (B) interaction effects equal zero.)

blocks, and refinement of techniques for measuring a dependent variable. In this section, I describe an alternative approach to reducing error variance and minimizing the effects of nuisance variables. The approach is called analysis of co-variance (ANCOVA) and combines regression analysis and analysis of variance.

Analysis of covariance involves measuring one or more concomitant variables (also called covariates) in addition to the dependent variable. The concomitant variable represents a source of variation that was not controlled in the experiment and one that is believed to affect the dependent variable. Analysis of covariance enables a researcher to (a) remove that portion of the dependent-variable error variance that is predictable from a knowledge of the concomitant variable, thereby increasing power, and (b) adjust the dependent variable so that it is free of the linear effects attributable to the concomitant variable, thereby reducing bias.

Consider an experiment with two treatment levels a1 and a2. The dependent variable is denoted by Yij, the concomitant variable by Xj. The relationship between X and Y for a1 and a2 might look like that shown in Figure 1.19. Each participant in the experiment contributes one data point to the figure as determined by his or her Xij and Yij scores. The points form two scatter plots—one for each treatment level. These scatter plots are represented in Figure 1.19 by ellipses. Through each ellipsis a line has been drawn representing the regression of Y on X. In the typical ANCOVA model it is assumed that each regression line is a straight line and that the lines have the same slope. The size of the error variance in ANOVA is determined by the dispersion of the marginal distributions (see Figure 1.19). The size of the error variance in ANCOVA is determined by the

If the last null hypothesis is rejected, the researcher knows that treatments A and C interact at one or more levels of treatment B.

Lack of space prevents me from describing other partial hierarchical designs with different combinations of crossed and nested treatments. The interested reader is referred to the extensive treatment of these designs in Kirk (1995, chap. 11).

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