Screening with ten factors: warpage in injection moulding

Step 2 / Choosing the design

Four candidates, one criterion: keep the main effects clean

For ten factors DoEStat offers several screening designs. The table puts four of them side by side, each with four centre runs. Two figures decide: the power, that is the probability of detecting an effect of two standard deviations, and the largest correlation between a main effect and a two-factor interaction. If that correlation is not zero, an interaction can mimic or mask a main effect.

The Plackett–Burman design is the most economical, but its main effects are entangled with interactions (correlation 0.33), and with five degrees of freedom for error its power stays below 80%. The regular 2¹⁰⁻⁵ fraction is statistically the strongest but costs ten more runs. In between are two designs that keep main effects clear of interactions: the definitive screening design and the minimum-run resolution IV design. The min-run design is chosen: two levels per factor suit the question, and 15 degrees of freedom for error leave room to add interactions to the model later.

The price of 22 instead of 32 runs: the main effects are no longer exactly uncorrelated among themselves (largest correlation 0.27, variance inflation at most 1.16). That costs a little precision but distorts nothing, because the analysis estimates all ten jointly.

Decision: minimum-run resolution IV design with 22 runs, plus four centre runs — 26 instead of 1,024.

Step 4 / Main effects

Four factors stand out, six do not

First the simplest model: ten main effects. The effect of a factor is the average change in warpage when it is moved from its low to its high level.

The picture is clear: holding pressure (−0.61 mm), mould temperature (+0.44 mm), cooling time (−0.30 mm) and melt temperature (+0.21 mm) are active, all with p < 0.002. The other six effects are smaller than 0.04 mm and cannot be told from chance (p ≥ 0.48).

That does not finish the analysis. The residual standard deviation of this model is 0.118 mm — almost three times what the centre runs show as process scatter — and the lack-of-fit test responds (p = 0.039). The model is missing something.

Step 6 / Curvature

The centre runs say: the surface is not flat

A two-level design can only fit straight lines. Whether that is enough is what the centre runs check: if the model holds, they should lie on its prediction for the centre. The model predicts 1.281 mm there (95% confidence interval 1.259 to 1.303); the measured mean is 1.215 mm. The centre is better than the average of the corners — the surface sags.

In DoEStat this is tested by adding a quadratic term to the model. It is significant: 0.078 ± 0.024 mm, p = 0.004. What matters is what this test does not say: in a two-level design with centre runs the squares of all factors are one and the same column. The design knows that something is curved, but not which factor.

Conclusion: for the corners of the region the model is sound. Looking for the optimum in the interior takes a follow-up design with more than two levels — for four factors instead of ten. How that works is shown in example 2 and example 4.

Cross-check

What is really in the data

Because the measurements are simulated, the analysis can be measured against the truth. The model behind the data, in coded units and with a standard deviation of 0.035 mm:

Warpage = 1.20 + 0.11·A + 0.21·B − 0.30·D − 0.15·F + 0.10·B·D + 0.09·D² + 0.02·E − 0.015·H

All five estimated coefficients hit their true value within the confidence interval. The constant is listed as 1.29 because D² equals 1 at the corners and the two-level design cannot separate the two. The test found the curvature (estimated 0.078, true 0.09) — that it comes from the holding pressure it could not know. Two effects the analysis did not find: holding time (0.02) and screw speed (−0.015) do act, but more weakly than the process scatter. That is not a flaw of the design; it is the limit of what 26 runs resolve.