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5. Process Improvement
5.5. Advanced topics
5.5.9. An EDA approach to experimental design Cumulative residual standard deviation plot

Motivation: How do we use the Model to Generate Predicted Values?

Design matrix with response for two factors To illustrate the details as to how a model may be used for prediction, let us consider a simple case and generalize from it. Consider the simple Yates-order 22 full factorial design in X1 and X2, augmented with a response vector Y:
    X1 X2 Y
    - - 2
    + - 4
    - + 6
    + + 8
Geometric representation This can be represented geometrically
    diagram of design with response points
Determining the prediction equation For this case, we might consider the model
    \( Y = c + B_{1}X_{1} + B_{2}X_{2} + B_{12}X_{1}X_{2} + \epsilon \)
From the above diagram, we may deduce that the estimated factor effects are:
    c =
    the average response = \( \bar{Y} \)
    (2 + 4 + 6 + 8) / 4 = 5
    E1 =

    average change in Y as X1 goes from -1 to +1
    ((4-2) + (8-6)) / 2 = (2 + 2) / 2 = 2

    Note: the (4-2) is the change in Y (due to X1) on the lower axis; the (8-6) is the change in Y (due to X1) on the upper axis.

    E2 =

    average change in Y as X2 goes from -1 to +1
    ((6-2) + (8-4)) / 2 = (4 + 4) / 2 = 4
    E12 =

    interaction = (the less obvious) average change in Y as X1*X2 goes from -1 to +1
    ((2-4) + (8-6)) / 2 = (-2 + 2) / 2 = 0
For factors coded using +1 and -1, the least-squares estimate of a coefficient is one half of the effect estimate (Bi = Ei / 2), so the fitted model (that is, the prediction equation) is
    \( \hat{Y} = 5 + 1 X_{1} + 2 X_{2} + 0 X_{1} X_{2} \)
or with the terms rearranged in descending order of importance
      \( \hat{Y} = 5 + 2 X_{2} + X_{1} \)
Table of fitted values Substituting the values for the four design points into this equation yields the following fitted values
    X1 X2 Y \( \scriptsize \hat{Y} \)
    - - 2 2
    + - 4 4
    - + 6 6
    + + 8 8
Perfect fit This is a perfect-fit model. Such perfect-fit models will result anytime (in this orthogonal 2-level design family) we include all main effects and all interactions. Remarkably, this is true not only for k = 2 factors, but for general k.
Residuals For a given model (any model), the difference between the response value Y and the predicted value \( \hat{Y} \) is referred to as the "residual":
    residual = Y - \( \small \hat{Y} \)
The perfect-fit full-blown (all main factors and all interactions of all orders) models will have all residuals identically zero.

The perfect fit is a mathematical property that comes if we choose to use the linear model with all possible terms.

Price for perfect fit What price is paid for this perfect fit? One price is that the variance of \( \scriptsize \hat{Y} \) is increased unnecessarily. In addition, we have a non-parsimonious model. We must compute and carry the average and the coefficients of all main effects and all interactions. Including the average, there will in general be 2k coefficients to fully describe the fitting of the n = 2k points. This is very much akin to the Y = f(X) polynomial fitting of n distinct points. It is well known that this may be done "perfectly" by fitting a polynomial of degree n-1. It is comforting to know that such perfection is mathematically attainable, but in practice do we want to do this all the time or even anytime? The answer is generally "no" for two reasons:
  1. Noise: It is very common that the response data Y has noise (= error) in it. Do we want to go out of our way to fit such noise? Or do we want our model to filter out the noise and just fit the "signal"? For the latter, fewer coefficients may be in order, in the same spirit that we may forego a perfect-fitting (but jagged) 11-th degree polynomial to 12 data points, and opt out instead for an imperfect (but smoother) 3rd degree polynomial fit to the 12 points.

  2. Parsimony: For full factorial designs, to fit the n = 2k points we would need to compute 2k coefficients. We gain information by noting the magnitude and sign of such coefficients, but numerically we have n data values Y as input and n coefficients B as output, and so no numerical reduction has been achieved. We have simply used one set of n numbers (the data) to obtain another set of n numbers (the coefficients). Not all of these coefficients will be equally important. At times that importance becomes clouded by the sheer volume of the n = 2k coefficients. Parsimony suggests that our result should be simpler and more focused than our n starting points. Hence fewer retained coefficients are called for.
The net result is that in practice we almost always give up the perfect, but unwieldy, model for an imperfect, but parsimonious, model.
Imperfect fit The above calculations illustrated the computation of predicted values for the full model. On the other hand, as discussed above, it will generally be convenient for signal or parsimony purposes to deliberately omit some unimportant factors. When the analyst chooses such a model, we note that the methodology for computing predicted values \( \scriptsize \hat{Y} \) is precisely the same. In such a case, however, the resulting predicted values will in general not be identical to the original response values Y; that is, we no longer obtain a perfect fit. Thus, linear models that omit some terms will have virtually all non-zero residuals.
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