Design, Conduct and Analysis: Division C

1.  IV.            High-Order interactions occur rarely
   1. Introduction
      1.                                                               i.      An interaction of factors requires specific mechanisms. For example, suppose that the asthmatic subjects, of the illustration in the previous rule, had shown a pollutant effect during exercise but not during rest. This would require an explanation by a respiratory physiologist, who might argue that pollutants only have an effect when the subject is stressed; for example, during exercise, “Why would that be the case?” A deeper explanation might then mention the immune system and how it responds to pollutants.
      2.                                                             ii.      The more complicated an interaction, the more complicated the mechanism of the organism’s response. It is an empirical observation that such situations are relatively rare.
   2. Rule of Thumb
      1.                                                               i.      High-order interactions occur rarely, it is not necessary to design experiments that incorporate tests for high-order interactions.
   3. Illustration
      1.                                                               i.      Consider a study of decline in memory among blacks and whites (race), males and females (sex), ages 70 to 90 (age) and varying levels of schooling (education). There are four explanatory variables (race, sex, age and education). A full model would involve four-factor interactions. Note also that decline in memory itself is a change variable. If such an interaction were found in an observational study, the investigators would almost certainly explore selection biases and other reasons for this observation. If they were satisfied that the interaction was “real”, they would begin to explore mechanisms.
   4. Basis of the Rule
      1.                                                               i.      The basis for this rule is that it is difficult to picture mechanisms associated with high-order interactions, particularly in studies involving organisms. There is parsimony of mechanisms. Another consideration is that to a first order approximation, effects of factors are additive. Finally, interactions are always expressed in a scale of measurement and there is a statistical theorem that says that a transformation of the scale can be made so that effects become additive (Scheffe, 1959). In other words, some interactions are simply associated with the scale of measurement.
   5. Discussions and Extensions
      1.                                                               i.      The existence of an interaction implies that the effects are not additive. Relative risks and odds ratios essentially estimate two-factor interactions the risk of disease among the exposed compared with this risk among the non-exposed. On the log scale, of there is no interaction the log of the relative risk and the log of the odds ratio are exactly zero, so that relative risk and odds are 1.
      2.                                                             ii.      Fractional factorial designs allow the application of the two rules of thumb listed above (Rule III and Rule IV). They take advantage of factorial structure while using the higher-order interactions for estimating the error. A classic book discussing this and other issues has been updates recently; see Cox and Reid (2000). It is a part of scientific insight to know which interactions can be ignored and which should be considered in the design of a study.
      3.                                                           iii.      The assumption that high-order interactions do not occur, commonly underlies a great deal of research; it forms a frequently unstated context. It is perhaps common because it is often correct. However, it will certainly pay to itemize the more common high-order interactions assumed to be non-existent.

1.     V.            Balanced designs allow assessment of joint effects
   1. Introduction
      1.                                                               i.      Since designed studies are under the control of the investigator one choice is the allocation of resources to the treatment conditions. It turns out that “nice” things happen when the allocations are arranged in a specific way. A sufficient condition for a balanced design is that all treatment combinations are based on the same number of observations.
   2. Rule of thumb
      1.                                                               i.      Aim for balance in the design of a study.
   3. Illustration
      1.                                                               i.      In a factorial design aim for equal numbers of study units to each treatment combination. This makes the various types of analysis equivalent n statistical packages. For example, with balanced data the Type I and Type II analysis in SAS are identical.
   4. Basis of the Rule
      1.                                                               i.      This rule is based in the resulting analysis that are more straightforward and typically, allow additive portioning of the total sums of squares. Studies that are balanced are often called orthogonal because additive portioning of the sums of squares is equivalent to an orthogonal partition of treatment response in the outcome space.
   5. Discussion and Extensions
      1.                                                               i.      The concept of balance is fairly straightforward: equal allocation of samples to treatments. However, it need not be quite so restrictive. For example, in a 2 X 2 factorial design the total sum of squares can still be partitioned additively when the cell frequencies are proportional to the marginal frequencies. To determine whether this is the case, simply calculate the chi-square statistic for contingency tables on the cell frequencies. If the chi-square statistics is exactly zero, the cell frequencies are in proportion to the marginal frequencies. Given table illustrates such a design. In this design the cell frequencies are determined by the marginal frequencies: n_ij=n_i. X n_.j/n., for i= 1, 2 and j=1, 2.

1.                                                             ii.      As indicated, balanced studies provide for additive partitioning of the total sums of squares. Thus there is no need to worry about the sequence of hypotheses that are analysed. Historically there has been a great deal of emphasis on balanced experiments because the computations were much simpler. This, of course, is less of a problem now. However, there is an additional reason for striving for balanced studies: ease of interpretation. Designs such as balanced incomplete block designs & partially balanced incomplete block designs, were devised to be relatively easy to analyse and easy to interpret. Another nice feature of balanced allocation is that ordinarily the standard errors of the estimates will be minimised given that the total sampling effort is fixed.
2.                                                           iii.      For a one-way classification the numbers per treatment can be arbitrary in that it still allows an additive partitioning into treatment and error sums of squares. However, unless the frequencies are balanced, it will not be possible to subdivide the treatment sum of squares additively (of course, if there are only two treatments the further partitioning is not an issue and the sample sizes can be unequal).
3.                                                           iv.      Balance is important for additive partitioning of sources of variability. It is of less importance for precision, as Rule IX indicated (in upcoming article). If balance is required, the imbalance is not too great, and there are a reasonable number of observations per factor combination, it should be possible to randomly discard a few observations in order to obtain balance. Is this is done several times (akin to bootstrapping) the average of the results can be taken as a good summary of the analysis.

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