Applied Statistics and Probability for Engineers

14-4 TWO-FACTOR FACTORIAL EXPERIMENTS 511

The observation in the ijth cell for the kth replicate is denoted by yijk. In performing the

experiment, the abnobservations would be run in random order.Thus, like the single-

factor experiment studied in Chapter 13, the two-factor factorial is a completely random-

ized design.

The observations may be described by the linear statistical model

(14-1)

where is the overall mean effect, iis the effect of the ith level of factor A, jis the ef-

fect of the jth level of factor B, ()ijis the effect of the interaction between Aand B, and

(^) ijkis a random error component having a normal distribution with mean zero and variance

2. We are interested in testing the hypotheses of no main effect for factor A, no main effect
   for B, and no ABinteraction effect. As with the single-factor experiments of Chapter 13,
   the analysis of variance (ANOVA) will be used to test these hypotheses. Since there are
   two factors in the experiment, the test procedure is sometimes called the two-way analysis
   of variance.
   14-4.1 Statistical Analysis of the Fixed-Effects Model
   Suppose that Aand Bare fixed factors.That is, the alevels of factor Aand the blevels of fac-
   tor Bare specifically chosen by the experimenter, and inferences are confined to these levels
   only. In this model, it is customary to define the effects i, j, and ()ijas deviations from the
   mean, so that and
   The analysis of variancecan be used to test hypotheses about the main factor effects of
   Aand Band the ABinteraction. To present the ANOVA, we will need some symbols, some of
   which are illustrated in Table 14-3. Let yi..denote the total of the observations taken at the ith
   level of factor A; y.j.denote the total of the observations taken at the jth level of factor B; yij.
   denote the total of the observations in the ijth cell of Table 14-3; and y... denote the grand total
   of all the observations. Define and as the corresponding row, column, cell,
   and grand averages. That is,
   yi.., y.j., yij., y...
   g
   b
   g j 1 1  (^2) ij0.
   a
   g i 1 1  (^2) ij0,
   b
   g j 1 j0,
   a
   i 1 i0,
   Yijkij 1  (^2) ij ijk•
   i1, 2,p, a
   j1, 2,p, b
   k1, 2,p, n
   y...
   y...
   abn
   y... a
   a
   i 1
   (^) a
   b
   j 1
   (^) a
   n
   k 1
   yijk
   j1, 2,p, b
   yij. i1, 2,p, a
   yij.
   yij. a n
   n
   k 1
   yijk
   y.j. j1, 2,p, b
   y.j.
   y.j. a an
   a
   i 1
   (^) a
   n
   k 1
   yijk
   yi.. i1, 2,p, a
   yi..
   bn
   yi.. a
   b
   j 1
   (^) a
   n
   k 1
   yijk
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