Two (or more) factors, say A and B, with a and b levels, respectively.

Transcription

1 Factorial Designs ST 516 Two (or more) factors, say A and B, with a and b levels, respectively. A factorial design uses all ab combinations of levels of A and B, for a total of ab treatments. When both (all) factors have 2 levels, we have a 2 2 (2 k ) design. 1 / 22 Factorial Designs Definitions and Principles

2 E.g. a 2 2 experiment: Factor A Low High Factor B High Low Main effect of A is Average response for high level of A = Similarly, main effect of B is 11. Average response for low level of A = / 22 Factorial Designs Definitions and Principles

3 Interaction The simple effect of A at B is = 20, and the simple effect of A at B + is = 22. The difference between these simple effects is the interaction AB (actually, one half the difference). Graphical presentation: the interaction plot. 3 / 22 Factorial Designs Definitions and Principles

4 In R; lines are parallel if interaction is 0: twobytwo <- data.frame(a = c("-", "+", "-", "+"), B = c("+", "+", "-", "-"), y = c(30, 52, 20, 40)) interaction.plot(twobytwo$a, twobytwo$b, twobytwo$y) # or, saving some typing: with(twobytwo, interaction.plot(a, B, y)) ST 516 mean of y B + + A 4 / 22 Factorial Designs Definitions and Principles

5 The other way; lines are still parallel if interaction is 0: with(twobytwo, interaction.plot(b, A, y)) mean of y A + + B 5 / 22 Factorial Designs Definitions and Principles

6 Interaction with Two Quantitative Factors Write the general level of factor A as x 1, the general level of factor B as x 2, and the response as y. Code x 1 so that x 1 = 1 at A and x 1 = +1 at A +, and x 2 similarly. Estimate the regression model representation y = β 0 + β 1 x 1 + β 2 x 2 + β 1,2 x 1 x 2 + ɛ. 6 / 22 Factorial Designs Definitions and Principles

7 Least squares estimates of the β s can be found from the main effects and interaction. For the simple example: twobytwo$x1 <- ifelse (twobytwo$a == "+", 1, -1) twobytwo$x2 <- ifelse (twobytwo$b == "+", 1, -1) summary(lm(y ~ x1 * x2, twobytwo)) from which we get ŷ = x x x 1 x 2. A graph of ŷ versus x 1 and x 2 is called a response surface plot. 7 / 22 Factorial Designs Definitions and Principles

8 yhat ST x x2 8 / 22 Factorial Designs Definitions and Principles

9 Making the Response Surface Plot in R Use expand.grid() to set up a grid of values of x 1 and x 2, use the predict() method to evaluate ŷ on the grid, and use persp() to make a surface plot of it: ngrid <- 20 x1 <- with(twobytwo, seq(min(x1), max(x1), length = ngrid)) x2 <- with(twobytwo, seq(min(x2), max(x2), length = ngrid)) grid <- expand.grid(x1 = x1, x2 = x2) yhat <- predict(lm(y ~ x1 * x2, twobytwo), grid) yhat <- matrix(yhat, length(x1), length(x2)) persp(x1, x2, yhat, theta = 25, expand = 0.75, ticktype = "detailed") 9 / 22 Factorial Designs Definitions and Principles

10 A Two-Factor Example Battery life data; A = Material, a = 3, B = Temperature, b = 3, replications n = 4 (battery-life.txt): Material Temperature Life / 22 Factorial Designs Two-Factor Design

11 boxplot(life ~ factor(material) * factor(temperature), batterylife) / 22 Factorial Designs Two-Factor Design

12 Statistical model y i,j,k = µ + τ i + β j + (τβ) i,j + ɛ i,j,k is the response for Material i, Temperature j, replicate k. τ s, β s, and (τβ) s satisfy the usual constraints (natural or computer). Hypotheses No differences among Materials; H 0 : τ i = 0, all i; No effect of Temperature; H 0 : β j = 0, all j; No interaction; H 0 : (τβ) i,j = 0, all i and j. 12 / 22 Factorial Designs Two-Factor Design

13 R command batterylife <- read.table("data/battery-life.txt", header = TRUE) summary(aov(life ~ factor(material) * factor(temperature), batterylife)) Output Df Sum Sq Mean Sq F value Pr(>F) factor(material) ** factor(temperature) e-07 *** factor(material):factor(temperature) * Residuals Signif. codes: 0 *** ** 0.01 * / 22 Factorial Designs Two-Factor Design

14 Interaction plots Can be made in two ways: Lifetime versus Temperature, with one curve for each type of Material; Lifetime versus Material, with one curve for each level of Temperature. Same information either way, but usually easier to interpret with a quantitative factor on the X-axis. Here Temperature is quantitative, but Material is qualitative. 14 / 22 Factorial Designs Two-Factor Design

15 with(batterylife, interaction.plot(temperature, Material, Life, type = "b")) mean of Life Material Temperature 15 / 22 Factorial Designs Two-Factor Design

16 with(batterylife, interaction.plot(material, Temperature, Life, type = "b")) mean of Life Temperature Material 16 / 22 Factorial Designs Two-Factor Design

17 Pairwise comparisons TukeyHSD(aov(Life ~ factor(material) * factor(temperature), batterylife)) Output Tukey multiple comparisons of means 95% family-wise confidence level Fit: aov(formula = Life ~ factor(material) * factor(temperature), data = batterylife) $ factor(material) diff lwr upr p adj $ factor(temperature) diff lwr upr p adj / 22 Factorial Designs Two-Factor Design

18 $ factor(material):factor(temperature) diff lwr upr p adj 2:15-1: :15-1: :70-1: :70-1: :70-1: :125-1: :125-1: :125-1: :15-2: :70-2: :70-2: :70-2: :125-2: :125-2: :125-2: :70-3: :70-3: / 22 Factorial Designs Two-Factor Design

19 Residual Plots plot(aov(life ~ factor(material) * factor(temperature), batterylife)); Residuals vs Fitted 4 Residuals Fitted values aov(life ~ factor(material) * factor(temperature)) 19 / 22 Factorial Designs Two-Factor Design

20 Theoretical Quantiles Standardized residuals aov(life ~ factor(material) * factor(temperature)) Normal Q Q / 22 Factorial Designs Two-Factor Design

21 Fitted values Standardized residuals aov(life ~ factor(material) * factor(temperature)) Scale Location / 22 Factorial Designs Two-Factor Design

22 Constant Leverage: Residuals vs Factor Levels Standardized residuals factor(material) : Factor Level Combinations 22 / 22 Factorial Designs Two-Factor Design