TwoWay ANOVA Blocking is used to keep extraneous

Two-Way ANOVA Blocking is used to keep extraneous factors from masking the effects of the treatments you are interested in studying. A two-way ANOVA is used when you are interested in determining the effect of two treatments. Model: yijk = μ + τ i + βj + (τ β)ijk + εij 1 ETM 620 - 09 U

Two-Way ANOVA w/ Replication You have been called in as a consultant to help the Pratt and Whitney plant in Columbus determine the best method of applying the reflective stripe that is used to guide the Automated Guided Vehicles (AGVs) along their path. There are two ways of applying the stripe (paint and coated adhesive tape) and three types of flooring (linoleum and two types of concrete) in the facilities using the AGVs. You have set up two identical “test tracks” on each type of flooring and applied the stripe using the two methods under study. You run 3 replications in random order and count the number of tracking errors per 1000 ft of track. The results are as follows: 2 ETM 620 - 09 U

Two-Way ANOVA Example Analysis is the same as with blocking, except we are now concerned with interaction effects 3 ETM 620 - 09 U

Two-Way ANOVA 4 ETM 620 - 09 U

Your Turn Complete the ANOVA in Minitab and fill in the blanks … ANOVA Source df SS MS F P-value Stripe 1 0. 4356 _____ 2. 40 0. 147 Flooring 2 4. 48 2. 24 12. 33 0. 001 Interaction ___ 0. 9644 0. 4822 _____ 0. 111 Error ___ 2. 18 0. 1817 Total 17 8. 06 What does this mean? 5 ETM 620 - 09 U

What about interaction effects? For example, suppose a new test was run using different types of paint and adhesive, with the following results: DF SS MS F Pvalue Stripe 1 0. 10889 1. 07 0. 321 Flooring 2 1. 96 0. 98 9. 64 0. 003 Interaction 2 2. 83111 1. 41556 13. 92 0. 001 Error 12 1. 22 0. 10167 Total 17 6. 12 Source 6 ETM 620 - 09 U

Understanding interaction effects Graphical methods: graph means vs factors identify where the effect will change the result for one factor based on the value of the other. 7 ETM 620 - 09 U

General factorial experiments (> 3 factors) Example: 3 -factor experiment. The model is: Use Minitab: Balanced ANOVA (if the experiment is balianced, i. e. , equal number of observations at each treatment combination. ) General Linear Model (GLM) if the experiment is unbalanced. Allows for random effects and mixed models. 8 ETM 620 - 09 U

An example Comparison of head-up and head-down displays Manual flying mode 2 display formats 3 levels of ceiling & visibility – 5000 ft and 10 miles, 200 ft and ½ statute mile, and 0 ft and 1200 ft runway visual range 2 levels of wind direction and velocity – 090 degrees at 10 knots and 135 degrees at 21 knots Use Balanced ANOVA with visibility and wind as random effects. “Model” is input as follows: Display| Wind| Visibility To discuss – use “restricted” form of the model or not? 9 ETM 620 - 09 U

Minitab Output … Using balanced ANOVA and restricted model. 10 Source DF SS MS F Display 1 3274 ** Wind 1 420. 4 420 0. 57 0. 53 Visibility 2 234 117 0. 16 0. 86 Display*Wind 1 174 0. 82 0. 46 Display*Visibility 2 65. 07 32. 5 0. 15 0. 87 Wind*Visibility 2 1466 733 8. 58 0 Display*Wind*Visibi lity 2 422. 6 211 2. 47 0. 09 Error 108 9223 85. 4 Total 119 15278 ** Denominator of F-test is zero. P ETM 620 - 09 U

Graphical evaluation of interactions … 11 ETM 620 - 09 U

2 k factorials Factorial experiments with multiple factors in which each factor can have several levels can get expensive in terms of number of trials required, especially if replication is desired. If we choose not to replicate the experiment (i. e. , only 1 observation per combination of factor levels), we lose the ability to evaluate higher level interactions. Can get more “bang for the buck” with careful selection of two levels of each factor … i. e. , a 2 k factorial design. 12 ETM 620 - 09 U

Example: 22 factorial Look at the effect of oven temperature and reaction time on the yield (in percent) of a process … Oven Temp. Reaction Time 110° 50 min. 130° 70 min. Take 2 observations at each combination with the following result: Temp 13 Observations R. T. Interaction #1 #2 SUM (1) -1 -1 1 55. 5 54. 5 110 a 1 -1 -1 60. 2 61 121. 2 b -1 1 -1 64. 5 63. 9 128. 4 ab 1 1 1 67. 7 68. 7 136. 4 ETM 620 - 09 U

To determine the effect of Temperature, we find the midpoint between the average “high” temp yield and the average “low” temp yield, or Similarly, the effect of Time is … And the interaction effect is … 14 ETM 620 - 09 U