# Alias structure for Analyze Binary Response for Definitive Screening Design

## Alias structure for a definitive screening design

The alias structure describes the confounding pattern that occurs in a design. Terms that are confounded are also said to be aliased.

Aliasing occurs when the design does not include all of the combinations of factor levels. In the alias structure, fully-aliased terms have coefficients equal to 1. For example, if factor A is confounded with the 3-way interaction BCD, then the estimated effect for A is the sum of the effect of A and the effect of BCD. You cannot determine whether a significant effect is because of A, because of BCD, or because of a combination of both. If you try to include fully-aliased terms in the same model, Minitab removes the fully-aliased terms that are later in the terms list.

Partial aliasing is common in screening designs. With partial aliasing, the terms in the alias structure have coefficients other than 1. Be cautious when you interpret terms with partial aliasing. The coefficient in the alias table shows how much that term biases the estimate of the term that is in the model.

### Interpretation

Use the alias structure to examine the aliased terms. For example, a study to evaluate the power of an ultrasonic cleanser has 7 factors:
• Train time (seconds): time duration for the amplitude modulation
• Degas time (seconds): time off for bubble release between modulations
• Burst time (milliseconds): time for high energy bubble bursts
• Quiet time (milliseconds): off time for the cleanser
• Center (kilohertz): center frequency away from 40 kilohertz
• Bandwidth (kilohertz): difference of the sweep from the center frequency
• Sweep time (seconds): time to complete 1 sweep

### Screening design model: Power versus Train, Burst, Quiet, Center, Sweep

Alias Structure (up to order 2) Factor Name A Train B Degas C Burst D Quiet E Center F Bandwidth G Sweep Aliases I + 0.51 AA + 0.51 BB + 0.64 CC + 0.77 DD + 0.64 EE + 0.51 FF + 0.77 AB - 0.90 AC + 0.26 AD + 0.90 AE - 0.77 AF - 0.64 BC + 0.77 BD + 0.64 BE - 0.77 BF + 0.51 BG - 0.64 CD + 0.64 CF - 0.13 CG + 0.90 DE - 0.26 DF - 0.64 EF - 0.13 EG + 0.51 FG A C D E G GG + 0.38 AA + 0.38 BB + 0.22 CC + 0.06 DD + 0.22 EE + 0.38 FF - 0.94 AB + 1.09 AC - 0.31 AD - 1.09 AE + 0.94 AF + 0.78 BC - 0.94 BD - 0.78 BE + 0.94 BF - 0.62 BG + 0.78 CD - 0.78 CF + 0.16 CG - 1.09 DE + 0.31 DF + 0.78 EF + 0.16 EG - 0.62 FG CE - 0.23 AA - 0.23 BB - 0.04 CC + 0.16 DD - 0.04 EE - 0.23 FF + 0.16 AB - 0.35 AC - 0.61 AD + 0.35 AE - 0.16 AF + 0.04 BC + 0.16 BD - 0.04 BE - 0.16 BF + 0.77 BG + 0.04 CD - 0.04 CF - 0.19 CG + 0.35 DE + 0.61 DF + 0.04 EF - 0.19 EG + 0.77 FG

The Pareto chart shows that 2 of the main effects, 1 interaction term, and 1 quadratic effect are significant at the 0.05 level. The alias structure shows that the constant term, the quadratic term, and the interaction term are partially aliased with other effects. In the screening case, you usually assume that the strongest effects associate with the changes in the response. If the experiment does not satisfactorily identify the most important terms, the partial aliasing is 1 possible cause. If the partial aliasing is the cause, then you can collect more data to learn about more of the effects independently of other effects.

## Alias structure for a definitive screening design with covariates

In the context of designed experiments, covariates are usually variables that you measure but do not control. Because covariate settings vary among the experimental runs, covariates usually cause partial aliasing of factor effects. This partial aliasing can occur even when the terms in the model are not partially aliased with each other. Be cautious when you interpret terms in a model with partial aliasing.

### Interpretation

Use the alias structure to examine the aliased terms. For example, a study to evaluate the power of an ultrasonic cleanser has 7 factors:
• Train time (seconds): time duration for the amplitude modulation
• Degas time (seconds): time off for bubble release between modulations
• Burst time (milliseconds): time for high energy bubble bursts
• Quiet time (milliseconds): off time for the cleanser
• Center (kilohertz): center frequency away from 40 kilohertz
• Bandwidth (kilohertz): difference of the sweep from the center frequency
• Sweep time (seconds): time to complete 1 sweep
Because they cannot control the temperature, the researchers allow the temperature to vary during the experiment. Then, the researchers record the temperature for each experimental run. In a model without the covariate, the main effects have no aliases.

### Screening design model: Power versus Train, Burst, Quiet, Center, Sweep

Alias Structure (up to order 2) Factor Name A Train B Degas C Burst D Quiet E Center F Bandwidth G Sweep Aliases I + 0.51 AA + 0.51 BB + 0.64 CC + 0.77 DD + 0.64 EE + 0.51 FF + 0.77 AB - 0.90 AC + 0.26 AD + 0.90 AE - 0.77 AF - 0.64 BC + 0.77 BD + 0.64 BE - 0.77 BF + 0.51 BG - 0.64 CD + 0.64 CF - 0.13 CG + 0.90 DE - 0.26 DF - 0.64 EF - 0.13 EG + 0.51 FG A C D E G GG + 0.38 AA + 0.38 BB + 0.22 CC + 0.06 DD + 0.22 EE + 0.38 FF - 0.94 AB + 1.09 AC - 0.31 AD - 1.09 AE + 0.94 AF + 0.78 BC - 0.94 BD - 0.78 BE + 0.94 BF - 0.62 BG + 0.78 CD - 0.78 CF + 0.16 CG - 1.09 DE + 0.31 DF + 0.78 EF + 0.16 EG - 0.62 FG CE - 0.23 AA - 0.23 BB - 0.04 CC + 0.16 DD - 0.04 EE - 0.23 FF + 0.16 AB - 0.35 AC - 0.61 AD + 0.35 AE - 0.16 AF + 0.04 BC + 0.16 BD - 0.04 BE - 0.16 BF + 0.77 BG + 0.04 CD - 0.04 CF - 0.19 CG + 0.35 DE + 0.61 DF + 0.04 EF - 0.19 EG + 0.77 FG

In a model with Temperature as a covariate, all the terms have partial aliases.

### Screening design model: Power versus Temperature, Train, Burst, Quiet, ...

Alias Structure (up to order 2) Factor Name A Train B Degas C Burst D Quiet E Center F Bandwidth G Sweep Aliases I + 0.19 B + 0.52 AA + 0.51 BB + 0.66 CC + 0.75 DD + 0.63 EE + 0.48 FF + 0.81 AB - 0.91 AC + 0.26 AD + 0.82 AE - 0.77 AF - 0.66 BC + 0.69 BD + 0.61 BE - 0.74 BF + 0.54 BG - 0.61 CD + 0.62 CF - 0.16 CG + 0.92 DE - 0.24 DF - 0.02 DG - 0.67 EF - 0.18 EG + 0.45 FG A - 0.08 B - 0.01 CC + 0.01 DD + 0.01 EE + 0.01 FF - 0.02 AB + 0.03 AE + 0.01 BC + 0.03 BD + 0.01 BE - 0.01 BF - 0.01 BG - 0.01 CD + 0.01 CF + 0.01 CG - 0.01 DE - 0.01 DF + 0.01 DG + 0.01 EF + 0.02 EG + 0.03 FG C + 0.19 B - 0.01 BB + 0.01 CC - 0.02 DD - 0.01 EE - 0.04 FF + 0.04 AB - 0.01 AC - 0.08 AE - 0.02 BC - 0.08 BD - 0.03 BE + 0.03 BF + 0.02 BG + 0.03 CD - 0.02 CF - 0.03 CG + 0.02 DE + 0.01 DF - 0.02 DG - 0.03 EF - 0.05 EG - 0.06 FG D - 0.03 B + 0.01 FF - 0.01 AB + 0.01 AE + 0.01 BD - 0.01 BF + 0.01 CG + 0.01 EG + 0.01 FG E - 0.40 B - 0.01 F - 0.01 AA + 0.01 BB - 0.03 CC + 0.05 DD + 0.03 EE + 0.08 FF - 0.09 AB + 0.02 AC - 0.01 AD + 0.18 AE + 0.01 AG + 0.04 BC + 0.18 BD + 0.06 BE - 0.07 BF - 0.05 BG - 0.06 CD + 0.04 CF + 0.08 CG - 0.05 DE - 0.03 DF + 0.04 DG + 0.06 EF + 0.12 EG + 0.14 FG G + 0.49 B + 0.01 F + 0.01 AA - 0.01 BB + 0.04 CC - 0.06 DD - 0.04 EE - 0.09 FF + 0.11 AB - 0.02 AC + 0.01 AD - 0.21 AE - 0.01 AG - 0.05 BC - 0.22 BD - 0.08 BE + 0.09 BF + 0.06 BG + 0.08 CD - 0.05 CF - 0.09 CG + 0.07 DE + 0.04 DF - 0.05 DG - 0.07 EF - 0.14 EG - 0.17 FG GG - 0.21 B + 0.37 AA + 0.38 BB + 0.20 CC + 0.09 DD + 0.24 EE + 0.42 FF - 0.98 AB + 1.10 AC - 0.32 AD - AE + 0.94 AF + 0.80 BC - 0.84 BD - 0.75 BE + 0.90 BF - 0.65 BG + 0.75 CD - 0.76 CF + 0.19 CG - 1.12 DE + 0.30 DF + 0.02 DG + 0.81 EF + 0.22 EG - 0.55 FG CE - 0.01 B - 0.23 AA - 0.23 BB - 0.04 CC + 0.16 DD - 0.04 EE - 0.23 FF + 0.15 AB - 0.35 AC - 0.61 AD + 0.35 AE - 0.16 AF + 0.04 BC + 0.16 BD - 0.04 BE - 0.16 BF + 0.77 BG + 0.04 CD - 0.04 CF - 0.19 CG + 0.35 DE + 0.61 DF + 0.04 EF - 0.19 EG + 0.77 FG
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