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Experimental Design Models with Random Components

The General Random Model

Equation (1) can be expressed in matrix terms as follows:

Y = lp + Xx + e    (2)

where 1 is an N( = an)xl vectors of ones, x is a vector with components x„ i = 1,2,...a, and X is an Nxa design matrix, consisting entirely of 0,1 indicator variables. The covarianoe matrix of the vector Y is

E = o²XX' + a²I_N    (3)

In generał, if there are k random components in the model (main effects and interactions, for example), then (1) becomes

Y    = lp + X|Tj + X₂i₂ +... + X_kTk + e    (4)

where each xj is distributed N(0,a²I), the X| are partitions of the original design matrix corresponding to the experimental factors, and the covariance matrix of the observations is

E = o²X,X,' + ct²X₂X₂' + ... + o²X_łlX_k' + o²^    (5)

The expected mean squares from the ANOVA for a random model contain the variance components ai², a₂², ..., a_k², er². There are several methods for obtaining these expected mean squares. The direct method consists of applying the expected value operator to each mean square with the observations treated as random variables. Most experimental design books illustrate this method. Another method is the tabular algorithm, which is usually presented as a easier replacement for the direct approach. (Montgomery, 1991, illustrates both these methods in detail.) Hartley (1967) describes the method of synthesis, which unlike the other two methods, is suitable for Computer implementation. Hartley's method is based on the fact that every sum of squares can be written as a quadratic form, say

SS = Y'AY    (6)

where A is a matrix of constants, and the expected value of this ąuadratic form is

(7)

E(Y'AY) = tr(AE)