# Doe: Screening Using Fractional Factorials

**Standard Screening Using Fractional Factorials (cont.):**

** Algorithm ****Standard screening using fractional factorials**

**Pre-step.**Define the factors and ranges, i.e., the highs, **H**, and lows, **L**, for all factors.

**Step 1****.**Form your experimental array by selecting the first m columns of the array with the selected number of runs n. The remaining n – m – 1 columns are unused.

**Step 2****.**For each factor, if it is continuous, scale the experimental design using the ranges selected by the experimenter. Ds **i,j =** **Lj 0.5(Hj – Lj)(Di,j 1) for i = 1,…,n and j = 1,…,m.** Otherwise, if it is categorical simply assign the two levels, the one associated with “low” to –1 and the level with “high” to 1.

**Step 3****.**Build and test the prototypes according to **D**s. Record the test measurements for the responses from the n runs in the n dimensional vector **Y**.

**Step 4****.**Form the so-called “design” matrix by adding a column of 1s, **1**, to the left hand side of the entire n × (n – 1) selected design **D**, i.e., **X **= (**1**|**D**). Then, for each of the q responses calculate the regression coefficients βest = **AY**, where **A **is the (**X**′**X**)–1**X**′. Always use the same **A **matrix regardless of the number of factors and the ranges.

**Step 5.**(Optional) Plot the prediction model, yest(**x**), for prototype system output yest (**x**) = βest, 1 βest, 2 x1 … βest,m xm (12.1) as a function of xj varied from –1 to 1 for j = 1, …, m, with the other factors held constant at zero. These are called “**main effects plots**” and can be generated by standard software such as Minitab or using Sagata software. A high absolute value of the slope, βest, j, provides some evidence that the factor, j, has an important effect on the average response in question.

**Step 6****.**Calculate s0 using s0 = median {|βest, 2|… |β est, n|} (12.2) where the symbols “||” stand for the absolute values. Let S be the set of non-negative numbers |βest, 2|… |β est, n| in S with values less than 2.5s0 for r = 1… q. Next, calculate

PSE = 1.5 × median {numbers in S} (12.3) and tLenth, j = |βest,j 1|/PSE for j = 1, …, m. (12.4)

**Step 7****.**If tLenth, j > tLenth Critical, α, n, then declare that factor j has a significant effect for response for j = 1… m. The critical values, tLenth critical, α, n, were provided by Ye et al. (2001). The critical values are designed

to control the experiment wise error rate (EER) and the less conservative individual error rate (IER).

**Step 8****.**If one level has been shown to offer significantly better average performance for at least one criterion of interest, then use that information subjectively in your engineered system optimization. Otherwise, consider adding more data and/or take the fact that evidence does not exist that the level change helps into account in system design.

1. The phrase “**within subjects variable**” refers to a factor in an experiment in which a single subject or group is tested for all levels of that factor.

a. For example, if tests all tests are performed by one person, then all factors are within subject variables.

2. The phrase “**between subject variables**” refers to factors for which a different group of subjects is used for each level in the experimental plan.

a. For example if each test was performed by a different person, then all factors would be between subject variables.

3. A “**within subjects design**” is an experimental plan involving only within subject variables and a “**between subjects design**” is a plan involving only between subject variables.

4. This terminology is often used in human factors and biological experimentation and can be useful for looking up advanced analysis procedures.

5. Also, small differences shown on main effects plots can provide useful evidence about factors not declared significant. First, if the average differences are small, adjusting the level settings based on other considerations besides the average response might make sense, *e.g.*, to save cost or reduce environmental impacts. Further, recent research suggests that Type II errors may be extremely common and that treating to even small differences on main effect plots (*i.e.*, small “effects”) as effective “proof” might be advisable.