STAT 370: Probability and Statistics for Engineers

1. STAT 370: Probability and Statistics for Engineers Donald Martin

2. Factorial data analysis Treatments may have a special structure (not just treatment 1, treatment 2, etc.) Ex: Paint experiment I may define 4 treatments: Brand A-latex, Brand A-oil-based, Brand B-latex and Brand B-oil-based These 4 treatments have a special structure: they are combinations of levels of brand and type of paint. We already know how to compare the 4 treatments using numeric and graphical summaries. More interesting�how can we look at the effect of brand only or type only and do these effects also interact, or are they �additive�?

3. Factors and levels The variables under study which define the treatment structure are called factors (a major independent variable) Levels of a factor are the specific values of the factor used in the experiment Example: paint Factors = brand, type Levels = (A,B) and (latex, oil)� The treatments are combinations of levels of factors used in the experiment

4. Factorial designs Sometimes we use a numbering notation to describe the factorial designs (2X2 above for the painting experiment). The number of numbers is the number of factors, the actual values give the number of levels. 2X3X4 means three factors, one with 2 levels, one with 3 levels, the other with 4 (could be written as 4X3X2). The number of treatment groups is obtained by multiplying.

5. Full factorial design In general, we don�t have to use ALL possible combinations (or may not be able to) But if ALL the combinations are used then it is called a Full Factorial Experiment Ex: If Brand A-latex, Brand B-latex and Brand A-oil are the only treatments, the experiment is NOT a full factorial experiment. We will focus Full Factorial designs in this class

6. Assumption For the analyses that follow, we�ll mostly leave behind the issues we�ve been discussing Assume that the experiment is well-designed in terms of controlled variables, potential lurking variables, and randomization Replication will be apparent during analysis Thus any effects we see are due only to the factors being studied.

7. Why we need more tools? The overall stats (e.g., mean and standard deviation) don�t really tell us about the treatment effects Why not use side-by-side boxplots? Typically we don�t have enough replications (and there may be too many treatments)

8. Table of means Use table of treatment means to start to get a sense of the treatment effects. Qualitatively: if the treatment means are different, there appears to be a treatment effect. The interesting thing about factorial data is to investigate effects of each factor separately and to investigate whether they act independently of each other or interact.

9. Notation = the sample mean response when factor A is at level i and factor B is at level j = the average sample mean when factor A is at level i� = the average sample mean when factor B is at level j = the grand average sample mean

11. Interaction plots Graphical representation of treatment means Choose one factor for the X-axis (doesn�t matter which one) Treatment means placed on the Y-axis Plot treatment means Connect treatment means that have the same level of the OTHER factor

13. Interaction plots What do we look for in an interaction plot? Effects of each factor and interaction (A qualitative look) The distance between the traces represents the effect of the factor NOT on the x-axis The other factor�s effect is represented by the change in height across the trace�not so easy to see, so often make both interaction plots. Which factor seems to have a bigger effect? (overall and remember, qualitative!) How do we describe this numerically? Fitted effects

14. Fitted effects Fitted Simple effects (effect of treatments) Difference between cell averages and overall mean : Fitted Main Effects (effect of levels of factors) Compare row and column means to overall average to get effect of level on response The fitted main effect for factor A at its i th level is , and B at j th Main effects sum to zero for a balanced design (same number of replicates for each treatment)

15. Three ways to check interaction If the size or direction of the effect of one factor is different for different levels of the other factor, then there is interaction. If not, then there is no interaction; thus you can talk about the effect of one factor without mentioning the other factor. Graphically, no interaction corresponds to parallel traces . So the effect of changing levels of factor A is exactly the same for each level of factor B, thus the lines are (piecewise) parallel. If the lines are not piecewise parallel, then there is interaction. We can compute interactions , a measure of departure from parallelism of lines

16. Table of simple effects for cholesterol data

17. Table of interactions for cholesterol data

18. Data: Treatment means The factors in this example do not interact! If we cover up the entry in the second row, second column, what should it be so that there is no interaction?

19. What would a model with no interaction look like? The actual data might not be as perfect as that of the last slide, so that if there is an indication of a slight interaction, it doesn�t mean that there really is one. Coming from the other direction, if the data indicates that there is a little interaction, it could very well be noise that we see. This goes for the main effects as well as interaction effects.

20. Quantification of effects We need a way to quantify what effects are statistically different from what we would expect under a model with no effects. We could entertain several models: �One factor only� �Both factors� �Row, column and interaction effects�

21. Golf ball example In the golf ball example (in-class 10), we may expect that there are compression effects on flight distances. Are there also evening effects? Are the compression effects different on different evenings? (meaning that there is interaction between compression effects and days)?