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May 3, 2016 7:35 AM
(6150 views)

In the DOE results, why is it possible that the parameter estimate for factor X is positive and for X*X negative, or vice versa?

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The parameter estimates are estimates of the constants (b's in my exaple below) of a second order polynomial fit to the data, e.g.

Y = b2 * X^2 + b1 * X + b0.

The coefficient you mention are b1 and b2 respectively, whereas b0 would be the intercept of your model. The b2 coefficient determines the shape of the fit, that is negative b2: "frowning face curve" and positive b2: "smiley face curve". The b1 coefficient (together with the intercept, b0) determines the location of the curve. So in short the parameter estimate for factor X and X*X are two very different things describing different characteristics of the fitted curve.

BR

Jesper

Jesper

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The parameter estimates are estimates of the constants (b's in my exaple below) of a second order polynomial fit to the data, e.g.

Y = b2 * X^2 + b1 * X + b0.

The coefficient you mention are b1 and b2 respectively, whereas b0 would be the intercept of your model. The b2 coefficient determines the shape of the fit, that is negative b2: "frowning face curve" and positive b2: "smiley face curve". The b1 coefficient (together with the intercept, b0) determines the location of the curve. So in short the parameter estimate for factor X and X*X are two very different things describing different characteristics of the fitted curve.

BR

Jesper

Jesper

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Re: DOE Results

Thanks JesperJohansen. Here's more of what I found out about this topic:

The quadratic coefficient tells both the direction and steepness of the curvature (positive value indicates the curvature is upwards while a negative value indicates the curvature is downwards). This is as you have mentioned. In general, a polynomial term can make the interpretation less intuitive because the effect of changing the predictor varies depending on the value of that predictor. The second-order case is not as easy as interpreting b1 in y = b1*x + b0, where b1 is the magnitude of change for every unit increase of x.

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Re: DOE Results

It's as Jesper said. When we do a DoE and build a model, we really are getting the coefficients in the model terms that we've requested from JMP. For example, in the custom design, if I I have only three factors x1 x2 and x3, I might decide to include only specific terms in my model, based on my understanding of the physics, chemistry and mechanics of my system. I might, for example, chose to make my model like this:

y = a + b1*x1 + b2 *x2 + b3*x3 + c1*x1*x1 + d12*x1*x3

In my notation, a is the constant (intercept) term, the b terms are coefficients of the linear terms, the c terms are the quadratic terms and the d terms are the interaction terms. I've decide on a model with the linear (main effects) terms, a quadratic effect in x1, and an x1, x3 interaction. These coefficients can take on either positive or negative values. The magnitude of each parameter estimate that JMP displays tells us the relative strength of the term's contribution to the response and the sign tells us in which direction that response operates as we increase the factor settings. In the situation you describe, increasing x makes an increase in the linear contribution to the response, but makes a negative increase (decrease) in the quadratic contribution to the response. Most times, the magnitude of these coefficients will decrease as you go from first-order (linear) terms to second order (quadratic and interaction terms), but not always.