Factorial designs are used to study the joint effect of several factors on a response. In a factorial design we assume that there are a number of factors present and each factor has several levels. In a 2^{k} factorial design there are *k* factors and every factor has only two levels (e.g., high versus low or experimental versus control). Thus the total number of level combinations is 2^{k}. Consider an example of a 2^{3} factorial design where the three factors are denoted by *A*, *B*, and *C*. The eight combinations of levels are denoted by 1, *a*, *b*, *c*, *ab*, *bc*, *ca*, *abc*. Here, *a* is the main effect of the factor *A*, *ab* is the interaction effect of the factors *A* and *B*. All other level combinations have similar interpretations. The main goal of a factorial design is to study the main effects of factors involved in the design. An experimenter may also be interested in studying the two factor interaction effects or even higher-order interactions. To run a complete 2^{k} factorial design, the experimenter needs to have 2^{k} experimental conditions (e.g., groups). Sometimes, to achieve better efficiency, the design is replicated several times. To estimate the average of the main effect of factor *A* in a 2^{3} factorial design replicated *n* times, we use the following formula:

where *abc* is the total number of observations in *n* replicates with all factors at the high level, and all other symbols have similar interpretations.

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C
Experimental Design Strategies
FRACTIONAL FACTORIAL DESIGNS
Factorial designs are used to study the joint effect of several factors
on a response. In a factorial design we assume that there are a number of
factors present and each factor has several levels. In a 2k factorial design
there are k factors and every factor has only two levels (e.g., high versus
low or experimental versus control). Thus the total number of level com-
binations is 2k. Consider an example of a 23 factorial design where the
three factors are denoted by A, B, and C. The eight combinations of lev-
els are denoted by 1, a, b, c, ab, bc, ca, abc. Here, a is the main effect of
the factor A, ab is the interaction effect of the factors A and B. All other
level combinations have similar interpretations. The main goal of a facto-
rial design is to study the main effects of factors involved in the design.
An experimenter may also be interested in studying the two factor inter-
action effects or even higher-order interactions. To run a complete 2k
factorial design, the experimenter needs to have 2k experimental condi-
tions (e.g., groups). Sometimes, to achieve better efficiency, the design is
replicated several times. To estimate the average of the main effect of
factor A in a 23 factorial design replicated n times, we use the following
formula:
1
(a − 1)(b + 1)(c + 1) = 1 (abc + ab + ac − bc + a − b − c − 1) ,
A=
4n 4n
where abc is the total number of observations in n replicates with all fac-
tors at the high level, and all other symbols have similar interpretations.
109

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110 REVIEW OF THE NIOSH ROADMAP
The interaction effect of A and B is
1
(a − 1)(b − 1)(c + 1) = 1 (abc − ac − bc + ab + c − a − b + 1) .
AB =
4n 4n
The sum of squares due to factor A, denoted by SSA, is
1
(abc + ab + ac − bc + a − b − c − 1)2 .
SSA =
8n
Similarly, the sum of squares due to the AB interaction is
1
(abc − ac − bc + ab + c − a − b + 1)2 .
SSAB =
8n
The total sum of squares SST is
2
2 2 2 n
SST = ∑∑∑∑ ( yijkl − y ) .
i =1 j =1 k =1 l =1
To get the sum of squares due to error (i.e., SSE) we subtract the sum
of all the 7 factor level combinations (except 1) from SST. The degree
of freedom for any factor combination is 1 and that of SSE is 8(n – 1).
The significance of the interaction effect AB is tested by constructing the
F-statistic,
SSAB
F= .
SSE / 8(n − 1)
As we see from the previous discussion, if the number of factors k in
a 2k factorial design increases, the total number of runs in a complete
factorial design outgrows the resources of most experimenters. If the ex-
perimenter believes that higher-order interactions are negligible, the
main effects and the lower-order interactions can be estimated by run-
ning only a fraction of the complete experiment. Let us assume that in a
23 factorial design the second-order interactions are not significant and
the experimenter can provide only four conditions (i.e., experimental

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APPENDIX C 111
and/or control groups) to estimate the main effects. In order to conduct
this experiment in only four conditions to estimate the main effects we
have to select the treatment level combinations appropriately. To choose
the appropriate level of treatment combinations we first define a genera-
tor that is generally a higher-order interaction. Let our generator be ABC.
In each of the above average factor effect expressions (i.e., A, B, etc.) 1
has either a + or a – sign. Choose only those factor levels effects that
have –1 sign (those are A, B, C, and ABC). The average effects of these
factors in this ½ fractional factorial design are determined by
1
(a − b − c + abc )
A=
2
1
B = (− a + b − c + abc )
2
1
C = (− a − b + c + abc ),
2
and the sums of squares due to these factors are
1
(a − b − c + abc )2
SSA =
4
1
SSB = (− a + b − c + abc )
2
4
1
SSC = (− a − b + c + abc ).
4
No degrees of freedom are left for the error. Hence we can estimate
the main effects but we cannot test their significance. Generally, this is
not the case for ½ fractional factorial designs in which there are four or
more factors.
In terms of confounding, 23 ½ fractional replicate designs can esti-
mate main effects, but they are confounded with two-factor interactions.
24 ½ fractional replicate designs can estimate main effects that are un-
confounded by two-factor interactions; however, the two-factor interac-
tions may be confounded with other two-factor interactions. 25 ½
fractional replicate designs can estimate unconfounded main effects and
two-factor interactions, but three-factor interactions may be confounded

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112 REVIEW OF THE NIOSH ROADMAP
with two-factor interactions. Finally, 26 ½ fractional replicate designs can
estimate main effects and two-factor interactions unconfounded by three-
factor or less interactions, but three-factor interactions may be con-
founded with other three-factor interactions.
The previous examples illustrate the resolution of the fractional fac-
torial design. Resolution II designs are completely undesirable because
even the main effects are confounded with each other. Resolution III de-
signs (e.g., 23-1 which represents a ½ replicate of a 23 design) can esti-
mate main effects, but the main effects may be confounded with two-
factor interactions. Resolution IV designs can estimate both main effects
and two-factor interactions, but some of the two-factor interactions are
confounded with each other. Resolution V designs can estimate main
effects that are unconfounded by three-factor (or less) interactions, and
two-factor interactions that are unconfounded by other two-factor inter-
actions, but the two-factor interactions may be confounded with three-
factor interactions. Finally, resolution VI designs can estimate uncon-
founded main effects (four-factor or less) and two-factor interactions
(three-factor or less), and estimate three-factor interactions, but they may
be confounded by other three-factor interactions. As such, if we are in-
terested in preserving the integrity of both main effects and two-factor
interactions in a 2k fractional factorial design, we require a resolution V
or higher design. If all that we care about are the main effects, a resolu-
tion III design will allow us to estimate them, but a resolution IV design
is required if we want to both estimate and test the significance of the
main effects. Resolution IV 2k fractional factorial designs include 24-1,
26-2, 27-2, 27-3, 28-3, 28-4, 29-3, 29-4 designs, where for example, a 29-4 design
reduces the total number of experimental conditions (i.e., factor level
combinations) from 29 = 512 to a far more manageable 25 = 32 and still
permits estimates and tests of main effects that are unconfounded by
two-factor interactions. Resolution V 2k fractional factorial designs in-
clude 25-1, 28-2.
An alternative strategy is to use a resolution III fractional factorial
design to conduct a screening experiment, which would then be followed
by a more complete but lower-dimensional factorial design. For example,
Neter et al. (1996) describe a resolution III 210-6 design, involving 16 ex-
perimental conditions out of the 1,024 conditions needed for a full facto-
rial design, that was used to study the effects of six process variables and
four ingredient variables on the extent of crystallization in ice cream. The
16 conditions included in the screening study are as follows:

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APPENDIX C 113
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10
–1 –1 –1 –1 –1 –1 –1 –1 +1 +1
+1 –1 –1 –1 +1 –1 +1 +1 –1 –1
–1 +1 –1 –1 +1 +1 –1 +1 –1 –1
+1 +1 –1 –1 –1 +1 +1 –1 +1 +1
–1 –1 +1 –1 +1 +1 +1 –1 –1 +1
+1 –1 +1 –1 –1 +1 –1 +1 +1 –1
–1 +1 +1 –1 –1 –1 +1 +1 +1 –1
+1 +1 +1 –1 +1 –1 –1 –1 –1 +1
–1 –1 –1 +1 –1 +1 +1 +1 –1 +1
+1 –1 –1 +1 +1 +1 –1 –1 +1 –1
–1 +1 –1 +1 +1 –1 +1 –1 +1 –1
+1 +1 –1 +1 –1 –1 –1 +1 –1 +1
–1 –1 +1 +1 +1 –1 –1 +1 +1 +1
+1 –1 +1 +1 –1 –1 +1 –1 –1 –1
–1 +1 –1 –1 –1 –1
–1 +1 +1 +1
+1 +1 +1 +1 +1 +1 +1 +1 +1 +1
Three factors were identified as important, and these factors were
then studied in a 23 full factorial design.
The 2k-f designs that have highest possible resolution have been iden-
tified and catalogued for choices of k and f that are of general interest by
Box and colleagues (2005) and are also provided online by the National
Institute of Standards and Technology at http://www.itl.nist.gov/div898/
handbook/pri/section3/pri3347.htm.
Response Surface Methodology
The previous discussion of fractional factorial designs is based on
discrete levels of each factor (e.g., high or low, experimental or control).
In some cases, the factors of interest may be continuous variables, for
which simple dichotomization is not possible. An alternative approach
for exploring the effects of individual factors, low-level interactions, and
nonlinear relations is based on response surface methodology (RSM). In
RSM we first model the response function, which is influenced by sev-
eral variables, and then we optimize this function. Suppose that we have
a quantitative index of carcinogenicity Y in a test animal that depends on
the length-to-width ratio (x1) and size (x2) of a particular mineral particle
to which the animal is exposed. The scientific objective is to determine
the levels of x1 and x2 in order to achieve a certain value of Y, say y0. Let

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114 REVIEW OF THE NIOSH ROADMAP
us assume that the relationship between Y and (x1, x2) is modeled by a
function f: i.e., y = f(x1, x2) + ε, where ε represent noise in response y. Let
E(Y) = η = f(x1, x2). The surface represented by η = f(x1, x2) is called the
response surface. In RSM the functional form of f is unknown. We gen-
erally try a linear function or a polynomial function to model this rela-
tion. When there is a variation of this relation from laboratory to
laboratory, a mixed-effects polynomial model can be used and the
method of maximum likelihood or marginal maximum likelihood can be
used to estimate model parameters. Once the parameters have been esti-
mated, we can use the estimated response surface to evaluate the values
of x1 and x2 for a specific targeted value of y0—for example, a carcinoge-
netic threshold. A contour plot may help in this regard to estimate levels
of x1 and x2 corresponding to a particular level of carcinogenic risk. The
RMS method may not provide a reasonable solution for the true func-
tional relationship over the entire space of the independent variables x1
and x2. In that case a small region for the independent variables is chosen
and RMS is used sequentially. The interested reader is referred for fur-
ther discussion on these issues to Box and Draper (2007).
REFERENCES
Box, G. E. P., and N. Draper. 2007. Response surfaces, mixtures, and
ridge analyses, 2nd edition. Wiley Series in Probability and
Statistics. New York: John Wiley & Sons.
Box, G. E. P., J. S. Hunter, and W. G. Hunter. 2005. Statistics for
experimenters: Design, innovation, and discovery, 2nd edition. New
York: John Wiley & Sons.
Neter, J., M. H. Kutner, C. J. Nachtsheim, and W. Wasserman. 1996.
Applied linear statistical models, 4th edition. Burr Ridge, IL: Irwin.