Two-Level Factorial Designs - The Statistics Notebook

A $2^k$ full factorial studies $k$ factors, each at two levels, and includes all $2^k$ combinations. It is useful for screening: identify important factors and interactions before investing in a more detailed experiment.

The interaction interpretation comes from Factorial Experiments & Interaction. Here, coding levels as -1 and +1 makes the calculations especially simple.

Factors	Combinations at two levels	Combinations at three levels
3	8	27
5	32	243
8	256	6,561

Multiply by the number of independent replicates per combination for the total run count. Two levels reduce the cost, but run counts still grow rapidly.

Coding Natural Units¶

For low and high numerical settings $X_L,X_H$ , code

x=\frac{X-(X_H+X_L)/2}{(X_H-X_L)/2}.

(1)

Then $X_L\mapsto-1$ , $X_H\mapsto+1$ , and their midpoint maps to 0. To recover natural units,

X=\frac{X_H+X_L}{2}+x\frac{X_H-X_L}{2}.

(2)

Standard Order & Sign Columns¶

In standard (Yates) order, A changes sign every row, B every two rows, and C every four. A letter in a treatment label means that factor is high; $(1)$ means all factors are low.

Label	A	B	C	AB	AC	BC	ABC
$(1)$	-1	-1	-1	+1	+1	+1	-1
$a$	+1	-1	-1	-1	-1	+1	+1
$b$	-1	+1	-1	-1	+1	-1	+1
$ab$	+1	+1	-1	+1	-1	-1	-1
$c$	-1	-1	+1	+1	-1	-1	+1
$ac$	+1	-1	+1	-1	+1	-1	-1
$bc$	-1	+1	+1	-1	-1	+1	-1
$abc$	+1	+1	+1	+1	+1	+1	+1

Interaction columns are products: $x_{AB}=x_Ax_B$ , $x_{ABC}=x_Ax_Bx_C$ . Add a column of +1s for the intercept.

Standard order is for bookkeeping, not execution. Randomize the actual run order. Running all low-C settings first could confound C with equipment drift, fatigue, or another time-related change.

Effects as Signed Contrasts¶

For an effect column $s$ containing $-1,+1$ , define the factorial effect as

\widehat E_s=\bar Y_{s+}-\bar Y_{s-}.

(4)

For A this compares its high and low levels, averaging over all other factors. For AB it compares the combinations with positive product signs against those with negative product signs. “AB positive” is not the same as “A and B both high.”

Let $r$ be the number of independent replicates at each corner and $N=r2^k$ . If $T_i$ is the response total at corner $i$ ,

C_s=\sum_{i=1}^{2^k}s_iT_i,\qquad \widehat E_s=\frac{C_s}{r2^{k-1}}.

(5)

Equivalently, using corner means,

\widehat E_s=\frac1{2^{k-1}}\sum_i s_i\bar Y_i.

(6)

Keep totals and means distinct: using a total-based denominator with corner means introduces an erroneous factor of $r$ .

Why Regression Coefficients Are Half the Effects¶

The full coded regression contains the intercept and all factor-product columns. For two factors,

Y=\beta_0+\beta_Ax_A+\beta_Bx_B+\beta_{AB}x_Ax_B+\varepsilon.

(7)

In a balanced full factorial,

\widehat\beta_0=\bar Y,\qquad \widehat\beta_s=\frac{\widehat E_s}{2}.

(8)

A coded factor changes by two units, from -1 to +1. This is why its averaged high-minus-low effect is twice the coefficient.

For the two-factor model, the simple A effect at B level $x_B$ is

\widehat\Delta_A(x_B)=2(\widehat\beta_A+\widehat\beta_{AB}x_B).

(9)

Consequently,

\text{difference of A simple effects}=4\widehat\beta_{AB}=2\widehat E_{AB}.

(10)

The factorial AB effect is half the difference of differences under this standard convention. Distinguish these quantities when switching between an interaction plot and a sign-table calculation.

Worked 2² Example: Effects, Coefficients & Predictions

Use the same cell means as the preceding page, now arranged in standard order:

Label	A	B	AB	Cell mean
$(1)$	-1	-1	+1	10
$a$	+1	-1	-1	12
$b$	-1	+1	-1	14
$ab$	+1	+1	+1	22

\widehat E_A=\frac{12+22}{2}-\frac{10+14}{2}=5,

(11)

\widehat E_B=\frac{14+22}{2}-\frac{10+12}{2}=7,

(12)

\widehat E_{AB}=\frac{10+22}{2}-\frac{12+14}{2}=3.

(13)

Therefore,

\widehat Y=14.5+2.5x_A+3.5x_B+1.5x_Ax_B.

(14)

At $(x_A,x_B)=(+1,-1)$ , the fitted mean is $14.5+2.5-3.5-1.5=12$ .

The simple A effects are $2(2.5-1.5)=2$ at low B and $2(2.5+1.5)=8$ at high B. Their difference is 6, whereas the conventional AB factorial effect is 3.

Evaluating the fitted equation between corners is a model-based interpolation, not a direct observation. It cannot establish that the response has no curvature.

Orthogonality & ANOVA¶

For the balanced full factorial design matrix $X$ , including all product columns,

X^\mathsf TX=NI.

(15)

Different columns have zero inner product; every column has squared length $N$ . Hence,

\widehat{\boldsymbol\beta}=\frac1N X^\mathsf T\mathbf Y,\qquad \operatorname{Var}(\widehat{\boldsymbol\beta})=\frac{\sigma^2}{N}I.

(16)

The coefficient estimates are uncorrelated under independent common-variance errors; normal errors additionally imply independence. Missing runs, unequal replication, or different weights can remove this orthogonality.

Every main effect and interaction has one df. Its sum of squares is

SS_s=\frac{C_s^2}{r2^k}=N\widehat\beta_s^2=\frac N4\widehat E_s^2.

(17)

For a fully replicated design, estimate pure error within each corner:

SS_E=\sum_{i=1}^{2^k}\sum_{j=1}^r(Y_{ij}-\bar Y_i)^2,\qquad \nu_E=2^k(r-1),\qquad MSE=SS_E/\nu_E.

(18)

Then

\operatorname{SE}(\widehat\beta_s)=\sqrt{MSE/N},\qquad \operatorname{SE}(\widehat E_s)=2\sqrt{MSE/N},

(19)

t_s=\frac{\widehat E_s}{2\sqrt{MSE/N}},\qquad F_s=\frac{SS_s}{MSE}=t_s^2.

(20)

Under the corresponding zero-effect null, use $t_{\nu_E}$ or $F_{1,\nu_E}$ . An individual effect interval is

\widehat E_s\pm t_{1-\alpha/2,\nu_E}\,2\sqrt{MSE/N}.

(21)

When examining many effects, distinguish individual tests from a family of claims and plan the multiplicity treatment accordingly.

Saturated Designs & Error Estimation¶

An unreplicated $2^k$ design has $2^k$ observations and $2^k$ coefficients in the full model, including the intercept. It is saturated:

\nu_E=2^k-2^k=0.

(23)

It fits the corner responses exactly. A perfect fit is not evidence of negligible noise or an adequate response surface.

Option	What supplies error information?	Limitation
Replicate corners independently	Within-setting variation.	Requires additional runs.
Replicate center settings	Variation at the center.	Using it at corners assumes a common error variance.
Pool scientifically negligible effects into residuals	Unmodeled contrast variation.	Depends on those effects truly being negligible.
Use relevant external variance information	Prior or separate measurements.	Must be applicable to the present process and units.

For example, a 2³ design has 8 runs. Fitting the intercept, three main effects, and three two-way interactions leaves one residual df, corresponding to ABC. This is an error estimate only under the assumption that ABC is negligible; it is not pure replication error.

Do not select a favorable error denominator simply to obtain significance. Preserve hierarchy: retain the lower-order terms needed by any retained interaction. A single marginal effect can be small even when that factor participates in an important interaction.

From Screening to a Better Experiment¶

A screening experiment identifies directions and dependencies worth investigating. It does not automatically determine the best operating conditions.

Use response units and interaction plots to judge practical importance, not only $p$ -values or marginal boxplots.
Choose low/high settings that are feasible and scientifically distinguishable.
Account for participant, batch, or day when it is part of the assignment structure. Repeated trials from one participant are not automatically independent replicates.
For quantitative factors, investigate intermediate settings and curvature when needed.
For categorical factors, additional categories are new levels; they are not numerical midpoints.
Validate a promising combination with additional independent runs before treating its observed advantage as established.

A reduced model or a design using fewer than all combinations introduces additional assumptions. The full-factorial sign-table formulas above should not be applied unchanged to an arbitrary subset of runs.