1.4 Effects of a factor

Definition 1.4.1   The effects of a factor are, for a given model, the scores achieved when replacing other factors by their average nominal value, i.e. by 0. The influence of a factor is the difference between the extreme effects.

Proposition 1.4.2   Let $X$ be the column obtained from $f^{*}$ when replacing by 0 everything that is not related to this factor. Then product $A\, .\, X$ is a column containing the different effects of this factor (evenly repeated for a balanced design and unevenly repeated otherwise).

Example 1.4.3   Concerning the DOE under study, we obtain :

1. Tests where Sealing=plastic (code $1$) are, on the average, leading to a T value greater by $2.69°C$ than the constant term ($55.52°C$), while tests where Sealing=brass (code $-1$) are, on the average, leading to a T value lower by $2.69°C$ than the constant term. Hence $\Delta_{sealing}\approx5.4°C$.
2. Relative to Connection, the effects are $-2.54°C$ (crimped, code $-1,-1$), $+2.96°C$ (screwed, code $1,0$) and $-0.42°C$ (welded, code $0,1$). Hence $\Delta_{connection}\approx5.5°C$.
3. Relative to Engine, the effects are $+1.33°C$ (brass, code $-1,-1$), $-1.42°C$ (silver, code $1,0$) and $+0.08°C$ (CuNiSn, code $0,1$). Hence $\Delta_{engine}\approx2.7°C$.
4. Relative to Wire, the vector $X$ is given by :
   $$\raisebox{0.5 em}{\normalfont\textsf{t}}{X}=\left(0,0,0,0,0,0,\,4.31,\,-3.69,\,-4.69,\,0,0,0\right)$$
   and the effects $4.06°C$ (wire 1.5 X, code $-1,-1,-1$), $+4.31°C$ (wire 1.5, code $1,0,0$) and so on. Hence $\Delta_{wire}\approx9.0°C$.
5. Finally, the Pins effects are $-0.94°C$, $-2.44°C$, $+2.81°C$ and $+0.56°C$. Hence $\Delta_{pins}\approx5.3°C$.

Remark 1.4.4   When a DOE is not balanced for a given factor, the weighted average of the effects of this factor is not zero. Effects are added to the constant term, not to the mean.

Example 1.4.5   Here, factors 1,4,5 are balanced and each average effect vanishes. Concerning the other two, we have :

$$Connection : (-2.54)\times4+\left(-0.42\right)\times8+\left(2.96\right)\times4 = -1.67$$

$$Engine : \left(-1.42\right)\times4+\left(0.08\right)\times8+\left(1.33\right)\times4 = +0.33$$

This is the reason why the constant term $55.52$ differs from the average $55.44$ of experimental data. As it should be, $-1.67+0.33=\left(55.44-55.52\right)\times16$ holds.

Definition 1.4.6   The "average effects graph" is what is obtained by placing side by side the effects of all the factors (1.4).

FIG. 1.4: Average effects

Scilab 1.4.7   FIG. 1.4 has been drawn using command draw_influences, LISTING .