Factorial Design¶

Circular¶

Experimental design that sets

• which predictor vars to vary
• Over what range
• Sampling plan: With what distribution of values

Given nature of model, we can easily decide how to sample

Design	Limitation		No of measurements
One at a time	Cannot help investigate interactions
Full factorial design			\(r \prod \limits_{i=1}^F l_i\)

Full Factorial Design¶

1. Use \(l_i\) levels for factor \(F_i\)
2. It is common to normalize each factor to \([-1, 1]\): coded vars
3. Perform \(r\) complete replicates of experiment
4. Replicates are required to estimate error

Adding center point

• Center point is often the POR (plan of record) and significant data may already exist about its response
• For LR
• Center points do not affect orthogonality of design
• Center points do not change any model parameter except the intercept
• Repeated center point can be used to check linear model validity: is non-linear term required?

2-level factorial design¶

For each factor, run every combination at 2 levels: high and low labelled as \(-1, +1\)

For \(F\) factors there will be \(r \times 2^F\) experimental runs for full factorial design

With this, we can detect

• linear variations only
• interactions

We cannot detect

• Non-linear variations

This design is completely orthogonal

Fractional Factorial Design¶

Many higher order interactions may be negligible (sparsity-of-effects) principle, and hence redundant

• We can reduce number of runs by eliminating higher-order model interactions, especially the ones that are not relevant to us

Choosing subset of full factorial design

• Balanced: all combinations have same number of obs
• Orthogonal design: effects of any factor sum to zero across effects of other factors

Half-Factorial design¶

Limitation

• Aliasing: some terms may get confounded by 2-factor interactions
• Not all terms can be distinguished in 8 runs

\((x_1 x_2 = x_3 x_4), (x_1 x_3 = x_2 x_4) \implies\) collinearity

Projections¶

If one of the factors proves to have no effect on the response, the \(F\) factor half-factorial design collapses to a \(k-1\) factor full-factorial design

CCD¶

Central Composite Design

1. Take 2-level factorial design
2. Add center point with repeats: middle point b/w all factors
3. Add axial (star) points: center point except w/ one var changed to be at ± an extreme value. Do this for all vars

\(n\) level CCD more efficient than \((n+1)\) level factorial design $$ n = r(2^F + 2F + 1) $$

Types¶

Examples¶

Level	Type
2	Circumscribed
2	Face-Centered
3

Box-Behnken Design¶

1. Put a data point in the center
2. Put a data point at midpoint each edge of process space

Disadvantages¶

• Does not contain embedded factorial design: hence, cannot do pre-survey and add more points
• No corner (extreme points)

Repeated Center Points¶

• Repeated center points are not randomized
• They are run as the first and last data points

• Every spread through rest of data collection

• Help check against process instability

• All other points should have randomized order

The number of repeated center points can be set to create “uniform precision”, as \(\sigma^2_{y \text{ center}} = \sigma^2_{y \text{ corner}}\)

Sequential DOE¶

Steepest Ascent/Descent

1. Start with factorial design (linear model) about current process (POR: Plan of Record)
2. In scaled coordinates, \((0, 0, \dots, 0)\) represents center point
3. Move in direction of steepest ascent/descent
4. Find factor \(j\) with max \(\vert \beta_j \vert\)
5. Move a distance of the \(j\)th factor: \(\Delta x_j \approx 1\) (higher/lower based on judgment)
6. For every other factor, move a distance of \(\Delta x_{j'} = \dfrac{\Delta x_j \beta_{j'}}{\beta_j}\)
7. Measure response at this new point
8. Keep moving until response goes down

IDK¶

1. Start with 2-level full factorial design with repeated center points
2. Extend to central composite design if quadratic model needed

Results in 2 blocks: control number center repeated to ensure uniformity and rotatability

Mixtures¶

Factors with constraints

Consider

• \(x_j \in [0, 1] \quad \forall j \in F\)
• \(\sum_{j}^F x_j = 1\)

Simplex Design¶

Applicable for Mixtures

Each factor taking \((m+1)\) evenly space values $$ x_j = { v/m } \ v \in [0, m] \ \forall j \in F $$

Taguchi Methods¶

Statistical methods for improving manufacturing quality

1. Optimization involves use of loss function
2. Quality begins with designing a process with inherently high quality
3. Use DOE

Loss Function¶

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