# Can Factorial Design provide Regression Equation like RSM?

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#### RIDZWAN RAHIM

Hi friends,

Can factorial design provide regression equation same like RSM for same factor and level?

Regard,

Ridzwan, UiTM Malaysia

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#### Barbara B

Re: Regression equation

Ridzwan,

factorial designs can provide regression equations similar to response surface designs. Only the text variable coefficients are a little bit special, but to explain that it would be helpful to have an example.

You can find the coefficients in the session window in a table which is called "estimated coefficients for [response] using uncoded units" (or something like that) at the end of the analysis.

Regards,

Barbara

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#### RIDZWAN RAHIM

Re: Regression equation

Barbara,

thanks a lot for the answer. But i'm not understand about "Only the text variable coefficients are a little bit special."

If i do fractional factorial also can get the regression coefficients barbara?

Barbara, what is the different between analysis using factorial design and response surface design?

Regards,

Ridzwan.

#### Miner

##### Forum Moderator
Another difference is the RSM will provided the quadratic terms (x^2) of the model while factorial designs do not.

B

#### Barbara B

Re: Regression equation

But i'm not understand about "Only the text variable coefficients are a little bit special."
In addition to the quadratic terms (like Miner posted) there is a difference in handling text variable coefficients compared to numeric variable coefficients.

A simple example is a 2^2 factorial design with 2 factors A (text) and B (numeric) without interaction. The regression equation is given in general as:
Y = b0 + b1*A + b2*B
with
b0: constant
b1: coefficient of A
b2: coefficient of B

If A is a factor like material ("low" level in the design: type a, "high" level in the design: type b), you'll find 1 number for a coefficient b1 for A in the session window. In the DoE analysis in Minitab this coefficient (in uncoded units) gives you the shift in Y if the high level is used (e. g. type b-material).

The coefficient for B taken from the table of uncoded coefficients tells you, how much Y is altered if B changes about +1. E. g. if factor B is temperature (low level: 20°C, high level: 50°C), the coefficient b2 will give you the change in Y if the temperature increases about +1°C (or °F, depending on your factor settings).

In the example Minitab gives you the following numbers for uncoded coefficients in the session window:
constant 4.0
material -1.5 (type b, unfortunately not directly listed in the session window)
temperature 0.2

So the regression equation for this example is given as:
Y = constant + coef(type b) + coef(temperature)*temperature

If you want to predict the outcome for material = type b and a temperature of 40°C you'll get:
Y = 4.0 - 1.5 + 0.2*40 = 4.0 - 1.5 + 8.0 = 10.5

But you don't see the expression for Y if material type a is used. And here is the trick: The sum of coefficients for a text variable is always 0 (in the DoE Minitab menu, the GLM menu and as default in the general regression menu). So if you have 1 coefficient for a factor with 2 levels, the second coefficient could be easily calculated since
coef_1 + coef_2 = 0.

For the factor material in the DoE example you have a coef_1=-1.5, so coef_2 (coefficient for material type a) has to meet the equation
coef_1 + coef_2 = 0
coef_2 = -coef_1 = -(-1.5) = +1.5
(This works in general for all text variables. You'll always get 1 coefficient less than the number of levels for a text variable in Minitab and can derive the last coefficient as all coefficients of 1 text variable must add up to 0. Somewhat tricky, but that's how Minitab works. Unfortunately. Imho this should be much more simple if all coefficients are given directly in the session window.)

If you want to predict the outcome for material type a in the example and the temperature is set to 40°C, the regression equation is given as:
Y = const + coef(type a) + coef(temperature)*temperature = 4.0 + 1.5 + 0.2*40 = 13.5

An easier way to get these equations and more complex equations with interactions of text variables or combinations of text and numeric variables for orthogonal designs is provided by Minitab R16 in the regression menu with the general regression.

You can select the factors and interactions which should be included in the analysis and add informations about which of the columns should be taken as text variables in the field below the model field (called "Categorical predictors (optional)"). The (simplified) regression equations are directly given in the session menu after running the analysis.

For the example with A: material (type a, type b) and B: temperature the session window would provide the two equations:
regression equation
material = type a Y = 5.5 + 0.2 temperature
material = type b Y = 2.5 + 0.2 temperature

"Simplified regression equation" because the constant and the coefficients for a specific level are added directly (5.5 = 4.0 + 1.5 = constant + coef(type a)).

Hope this helps Barbara

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