Unit 12: Analysis of Single Factor Experiments

Transcription

1 Unit 12: Analysis of Single Factor Experiments Statistics 571: Statistical Methods Ramón V. León 7/16/2004 Unit 12 - Stat Ramón V. León 1

2 Introduction Chapter 8: How to compare two treatments. Chapter 12: How to compare more than two treatments Limited to a single treatment factor Example of single factor experiment: Compare the flight distances of three types of golf balls differing in the shape of dimples on them: circular, fat elliptical, and thin elliptical Treatment factor: type of ball Factor levels: circular, fat elliptical, and thin elliptical Treatments: circular, fat elliptical, and thin elliptical How would an experiment with more than one treatment factor look? 7/16/2004 Unit 12 - Stat Ramón V. León 2

3 Experimental Designs Independent Samples Dependent Samples Two Treatments Independent Samples Design Matched Pair Design More Than Two Treatments Completely Randomized Design Randomized Block Design 7/16/2004 Unit 12 - Stat Ramón V. León 3

4 Completely Randomized Design Random sample drawn in each of six molding stations. Runs should be in random order to protect against time trend 7/16/2004 Unit 12 - Stat Ramón V. León 4

5 Completely Randomized Design Notation If the sample sizes are equal the design is balanced; otherwise the design is unbalanced N a = n j= 1 i 7/16/2004 Unit 12 - Stat Ramón V. León 5

6 Completely Randomized Design: Comments In a CRD the experimental units are randomly assigned to each treatment Similar data also arises in observational studies where the units are not assigned to the different groups by the investigator Stronger conclusions are possible with experimental data 7/16/2004 Unit 12 - Stat Ramón V. León 6

7 Completely Randomized Design Data Inspection Nominal Variable 7/16/2004 Unit 12 - Stat Ramón V. León 7

8 CRD Side-by-Side Box Plots Station 5 has two outliers Stations 4, 5, and 6 which are supplied by feeder 2 have a higher average as a group than stations 1, 2, and 3 that are supplied by feeder 1. Is this difference real or the result sampling variation? Weights Station 7/16/2004 Unit 12 - Stat Ramón V. León 8

9 CRD Model and Estimation Model assumption: the data on the i-th treatment are N µ σ 2 a random sample from an ( i, ) population Y = µ + ε ( i = 1,2,..., a; j = 1,2,..., n ) ij i ij i where ε N ij are independent and identically distributed (i.i.d.) 2 (0, σ ) random errors. 7/16/2004 Unit 12 - Stat Ramón V. León 9

10 CRD Model and Estimation 2 The treatment means µ i and the error variance σ are unknown parameters. The primary interest is on comparing the means Frequently, we write µ = µ + τ where µ is the "grand mean" defined as the weighted average of the µ : a a n 1 iµ µ i= i i= 1 i µ = = if ni = n are egual a n a i= 1 i and τ = µ µ is the deviation of the i-th treatment mean i i from this grand mean. i i We refer to τ as the i-th treatment effect. i i 7/16/2004 Unit 12 - Stat Ramón V. León 10

11 CRD Model and Estimation Alternative Formulation of the Model: Y = µ + τ + ε ( i= 1,2,..., a ; j = 1,2,..., n ) ij i ij i The τ are subject to the contraint: i a ( a τ ) i = = = 0 nτ if the n n are equal i= 1 i i = 1 i i So there are only a -1 linearly independent τ ' s. i 7/16/2004 Unit 12 - Stat Ramón V. León 11

12 CRD Parameter Estimates ˆ σ = s 2 2 Measure of common experimental error 7/16/2004 Unit 12 - Stat Ramón V. León 12

13 ANOVA in JMP s Fit Model Platform Note that the Station variable is nominal 7/16/2004 Unit 12 - Stat Ramón V. León 13

14 ˆ µ ˆ τ ˆ τ ˆ τ ˆ τ ˆ τ CRD Parameter Estimates How do we find the value of ˆ6 τ? 2 s 7/16/2004 Unit 12 - Stat Ramón V. León 14

15 Relationship to Dummy Variable Regression z i 1 if station i = 1 if station 6 0 otherwise i = 1,2,...,5 y = z 0.23z 0.33z z z + ε y = ˆ µ + ˆ τ z + ˆ τ z + ˆ τ z + ˆ τ 4 z 4 + ˆ τ 5 z 5 + ε 7/16/2004 Unit 12 - Stat Ramón V. León 15

16 CRD Parameter Estimates 2 s 7/16/2004 Unit 12 - Stat Ramón V. León 16

17 i CRD (1-α)-level Confidence Interval s y t µ y + t n i N a, α 2 i i N a, α 2 i However, usually we are more interested in comparing the µ with each other than estimating them separately. Fit Y by X: s n i 7/16/2004 Unit 12 - Stat Ramón V. León 17

18 Mean Diamonds in JMP Why do all the diamonds have the same height? 7/16/2004 Unit 12 - Stat Ramón V. León 18

19 H H Analysis of Variance Homogeneity Hypothesis : : µ = µ =... = µ vs. H : Not all the µ are equal a 1 i : τ = τ =... = τ = 0 vs. H : At least some τ a 1 i Note SSA = Treatment sums of squares 7/16/2004 Unit 12 - Stat Ramón V. León 19

20 Wrong ANOVA table: ANOVA in JMP (Model: Y = β + β Station + ε) 0 1 Note that the SS has the wrong number of degrees of freedom Correct ANOVA table: (Model: Y = µ + τ z + τ z + τ z + τ z + τ z + ε) /16/2004 Unit 12 - Stat Ramón V. León 20

21 Model Diagnostics: Residuals versus Fitted Value Part of Fit Model Output eij = yij yi This plot checks the assumption of constant error variance σ 2 A cone shape in this plot would suggest a log transformation of response 7/16/2004 Unit 12 - Stat Ramón V. León 21

22 Model Diagnostic: Assumption of Equal Variances (More Formal Tests) 7/16/2004 Unit 12 - Stat Ramón V. León 22

23 Model Diagnostics: Residual Versus Row (Time?) Order Fit Model Platform: A time pattern here would be confounded with a station effect. JMP table should be in the random order that the data is supposed to have been collected 7/16/2004 Unit 12 - Stat Ramón V. León 23

24 Model Diagnostics: Normal Plot of Residuals Strong indication that errors are normally distributed. 7/16/2004 Unit 12 - Stat Ramón V. León 24

25 0 1 Multiple Comparison of Means If H : µ =... = µ is rejected all that we can say is that a the treatment means are not equal. The F-test does not pinpoint which treatment means are significantly different from each other. We could test all pairwise equality hypotheses H : µ = µ y y Reject H if t = > t s 1 n + 1 n i j 0 ij ij N a, α 2 y y > t s 1 n + 1 n = i j N a, α 2 i j ( Least significant difference, LSD) i j 0ij i j 7/16/2004 Unit 12 - Stat Ramón V. León 25

26 Pairwise Equality Hypotheses Since each of the 15 pairwise test have a level α, the type I error probability of declaring at least one pairwise difference falsely significant will exceed α. Family Wise Error rate (FWE): FWE = P{Reject at least one true null hypothesis when they are true} If all six means are actually equal in the plastic container example FWE = when each LSD test is done at the 0.05 level. Fisher s protected LSD method: Use LSD method only after the F-test rejects (This method is not recommended today.) 7/16/2004 Unit 12 - Stat Ramón V. León 26

27 LSD Method in JMP Overlap Marks If the overlap marks overlap the two means are not significantly different according to the LSD criterion 7/16/2004 Unit 12 - Stat Ramón V. León 27

28 LSD Method in JMP Fit Y by X JMP platform: 7/16/2004 Unit 12 - Stat Ramón V. León 28

29 Tukey Method Recommended Method: FWE = α if the sample sizes are equal and is slightly conservative (i.e., the actual FWE is < α ) when sample sizes are unequal 7/16/2004 Unit 12 - Stat Ramón V. León 29

30 This report shows the ranked differences, from highest to lowest, with a confidence interval band overlaid on the plot. Confidence intervals that do not fully contain their corresponding bar are significantly different from each other. 7/16/2004 Unit 12 - Stat Ramón V. León 30

31 Tukey Method Confidence Intervals This is a way of construction 100(1-α)% Simultaneous Confidence Intervals (SCIs) for all pairwise difference of means 7/16/2004 Unit 12 - Stat Ramón V. León 31

32 Tukey Method Confidence Intervals Compare to the Minitab output at the bottom of Figure 12.6 of your textbook. How would you get the top output in that figure? 7/16/2004 Unit 12 - Stat Ramón V. León 32

33 Dunnett Method for Comparisons with a Control 7/16/2004 Unit 12 - Stat Ramón V. León 33

34 Dunnett Method in JMP 7/16/2004 Unit 12 - Stat Ramón V. León 34

35 Hsu Method for Comparison with the Best 7/16/2004 Unit 12 - Stat Ramón V. León 35

36 Box Plots for Teaching Method Test Score Case Equation Formula Unitary Analysis Method 7/16/2004 Unit 12 - Stat Ramón V. León 36

37 Hsu Method in JMP Explanation Next Page 7/16/2004 Unit 12 - Stat Ramón V. León 37

38 Hsu Method in JMP The Unitary Method is best Can t tell which is the worse method 7/16/2004 Unit 12 - Stat Ramón V. León 38

39 Randomized Block Design Blocking helps to reduce experimental error variation caused by difference in the experimental units by grouping them into homogeneous sets (called blocks). Treatments are randomly assigned within each block 7/16/2004 Unit 12 - Stat Ramón V. León 39

40 Randomized Block Design Model: Fixed Block Effects Y = µ + τ + β + ε ( i = 1,..., a; j = 1,..., b) ij i j ij 2 where εij are i.i.d. N(0, σ ) µ is called the grand mean τ is called the ith treatment effect i β is called the jth block effect j a i b τ = 0 and β = 0 so there are = 1 i j = 1 a 1 independent treatment effects b -1 independent block effects j 7/16/2004 Unit 12 - Stat Ramón V. León 40

41 Mystery of Degrees of Freedom Explained Counting the grand mean there are 1 + ( a-1) + ( b-1) = a+ b 1 unknown parameters. (This many degrees of freedom are needed to estimate these parameters.) There are N = ab observations (total degrees of freedom). So there are ν = ab ( a + b 1) = ( a 1)( b 1) degrees of freedom for estimating the error variation (degrees of freedom for error). 7/16/2004 Unit 12 - Stat Ramón V. León 41

42 No Interactions Between Treatments and Blocks The difference in mean responses between any two treatments is the same across all blocks µ µ = ( µ + τ + β ) ( µ + τ + β ) = τ τ ij i' j i j i' j i i' which is indepedent of the particular block j We say that there are no interactions between treatments and blocks Example: Consider the treatments to be fertilizer and the blocks to be different fields. Then no interaction implies that the difference in mean yields between any two fertilizers is the same for all fields. 7/16/2004 Unit 12 - Stat Ramón V. León 42

43 RBD Example Notice that interest is on the differences among the positions. We assume that these differences are the same for all three batches except for random error, that is, we assume no interaction between batch and position. 7/16/2004 Unit 12 - Stat Ramón V. León 43

44 JMP Analysis of Drip Loss Experiment Nominal 7/16/2004 Unit 12 - Stat Ramón V. León 44

45 JMP Analysis of Drip Loss Experiment Position and batch explain 86% of the variation in drip loss SSModel = SSTreatment + SSBlocks True because we assume no interaction between treatment and block. (See next slide.) 7/16/2004 Unit 12 - Stat Ramón V. León 45

46 JMP 4 Analysis of Drip Loss Experiment. III These two table were not the same in regression. They are equal here because the model is balanced. Also in regression the sum of the Type III sums of squares is not equal to the model sums of squares. This only true here because the model is balanced. The P-values show that there are significant position effects. We recommend ignoring the Block (Batch) test because it is not meaningful for the RBD. (Type III) Model SS = Recall: The sum of the Type I sums of squares is always equal to the model sums of squares 7/16/2004 Unit 12 - Stat Ramón V. León 46

47 Drip Loss in Meat Loaves: Residual Plots The predicted versus residual plot is part of the standard output of the Fit Model platform. The normal plot was obtained by saving the residuals and then going to the Distribution platform. 7/16/2004 Unit 12 - Stat Ramón V. León 47

48 Tukey Method for the RBD Using the Fit Model platform with batch and position in the model. That the two variables be included is important. Warning: Don t use the Fit Y by X platform to do Tukey s test as you will use the wrong number of degrees of freedom. 7/16/2004 Unit 12 - Stat Ramón V. León 48

49 Tukey Method for the RBD 7/16/2004 Unit 12 - Stat Ramón V. León 49

50 Tukey Method for the RBD 7/16/2004 Unit 12 - Stat Ramón V. León 50

51 Mixed Effects Model for the RB Design Y = µ + τ + β + ε ( i= 1,..., a; j = 1,..., b) ij i j ij 2 where εij are i.i.d. N(0, σ ) and β are i.i.d. N(0, σ ) i j j 2 B µ is called the grand mean τ is called the ith treatment effect β 's are called the block effects Independent a τ i = 0 so there are a 1 independent treatment effects i = 1 7/16/2004 Unit 12 - Stat Ramón V. León 51

52 7/16/2004 Unit 12 - Stat Ramón V. León 52

53 Compare with Results in Section , Example of your textbook The variability due to batches accounts for about 58.4% of the total variability in drip loss. 7/16/2004 Unit 12 - Stat Ramón V. León 53