1. 1 Always be contented, be grateful, be understanding and be compassionate.

2. 2 Blocking We will add a factor even if it is not of interest so that the study of the prime factors is under more homogeneous conditions. This factor is called “block”. Most of time, the block does not interact with prime factors. Popular block factors are “location”, “gender” and so on. A RxC two-factor design with one block factor is called a “randomized block design with RxC factorial structure”.

3. RBD Model (Section 15.2) 3 A randomized (complete) block design is an experimental design for comparing t treatments (or say levels) in b blocks. Treatments are randomly assigned to units within a block and without replications. The probability model of RBD is the same as two-way Anova model with no interaction term (so can conduct multiple comparisons for each factor separately)

4. 4 For example, suppose that we are studying worker absenteeism as a function of the age of the worker, and have different levels of ages: 25-30, 40-55, and 55-60. However, a worker’s gender may also affect his/her amount of absenteeism. Even though we are not particularly concerned with the impact of gender, we want to ensure that the gender factor does not pollute our conclusions about the effect of age. Moreover, it seems unlikely that “gender” interacts with “ages”. We include “gender” as a block factor.

5. O/L: Example 15.1 5 Goal: To compare the effects of 3 different insecticides on a variety of string beans. Condition: It was necessary to use 4 different plots of land. Response of interest: the number of seedlings that emerged per row.

6. Data: 6 insecticideplotseedlings 1156 1248 1366 1462 2183 2278 2394 2493 3180 3272 3383 3485

7. Minitab>>General Linear Model, response seedlings, model insecticide & plot 7 General Linear Model: seedings versus insectcide, plot Analysis of Variance for seedlings, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P insecticide 2 1832.00 1832.00 916.00 211.38 0.000 plot 3 438.00 438.00 146.00 33.69 0.000 Error 6 26.00 26.00 4.33 Total 11 2296.00 S = 2.08167 R-Sq = 98.87% R-Sq(adj) = 97.92% Unusual Observations for seedings Obs seedings Fit SE Fit Residual St Resid 11 83.0000 86.0000 1.4720 -3.0000 -2.04 R R denotes an observation with a large standardized residual.

8. 8

9. RBD with random blocks We would like to apply our conclusions on a large pool of blocks We are able to sample blocks randomly Example: Minitab unit 5 –Goal: to study the difference of 3 appraisers on their appraised values –Blocks: randomly selected 5 properties 9

10. 10 Latin Square Design (Section 15.3) Example: Three factors, A (block factor), B (block factor), and C (treatment factor), each at three levels. A possible arrangement: B 1 B 2 B 3 A 1 C 1 C 1 C 1 A C 2 C 2 C 2 A 3 C 3 C 3 C 3 2

11. 11 Notice, first, that these designs are squares: all factors are at the same number of levels, though there is no restriction on the nature of the levels themselves. Notice, that these squares are balanced: each letter (level) appears the same number of times; this insures unbiased estimates of main effects. How to do it in a square? Each treatment appears once in every column and row. Notice, that these designs are incomplete; of the 27 possible combinations of three factors each at three levels, we use only 9.

12. 12 Example: Three factors, A (block factor), B (block factor), and C (treatment factor), each at three levels, in a Latin Square design; nine combinations. B 1 B 2 B 3 A 1 C 1 C 2 C 3 A C 2 C 3 C 1 A 3 C 3 C 1 C 2 2

13. 13 Example with 4 Levels per Factor Automobiles Afour levels Tire positions Bfour levels Tire treatments Cfour levels FACTORS Lifetime of a tire (days) VARIABLE

14. 14 The Model for (Unreplicated) Latin Squares Example: (p.965 for full descriptions of model terms) Note that interaction terms are not present in the model.  Threefactors , , and  eachatmlevels,  y ijk =  +  i +  j +  k +  i=1,... m j= 1,..., m k =1,..., m Same three assumptions: normality, constant variances, and randomness. Y = (  A + B + C (+ e) AB, AC, BC, ABC

15. 15 Putting in Estimates: Total variability among yields Variability among yields associated with Rows Variability among yields associated with Columns Variability among yields associated with Inside Factor where R = orbringingy totheleft–handside, (y ijk –y... )=(y i.. –y... )+(y.j. –y )+(y..k –y... )+R, =+ + y ijk = y... +( y i.. – y... )+( y.j. – y )+( y.. k – y... )+ R y ijk –y i.. –y.j. –y k +2y...

16. 16 Actually, R Actually, R Residual (R) is an “interaction-like” term. (After all, there’s no replication!) =y ijk -y i.. -y.j. -y k +2y... =(y ijk -y... ) - (y i.. -y... ) ( y.j. - y ) (y..k -y... ), - -

17. 17 The analysis of variance (omitting the mean squares, which are the ratios of second to third entries), and expectations of mean squares: Source of variation Sum of squares Degrees of freedom Expected value of mean square Rows  m(y i.. –y... ) 2  i=1 m m–1   2 + mm Rows Columns  m(y.j. –y... ) 2  j=1 m m–1   2 + Col Inside factor  m(y..k –y... ) 2  k=1 m m–1   2 + Insidefactor by subtraction (m–1)(m–2)   2 Total   i  j (y ijk –y... ) 2  k m 2 –1 Error mm mm

18. 18 The expected values of the mean squares immediately suggest the F ratios appropriate for testing null hypotheses on rows, columns and inside factor.

19. 19 Our Example: Tire Position Auto. (Inside factor = Tire Treatment)

20. 20 General Linear Model: Lifetime versus Auto, Postn, Trtmnt Factor Type Levels Values Auto fixed 4 1 2 3 4 Postn fixed 4 1 2 3 4 Trtmnt fixed 4 1 2 3 4 Analysis of Variance for Lifetime, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Auto 3 17567 17567 5856 2.17 0.192 Postn 3 4679 4679 1560 0.58 0.650 Trtmnt 3 26722 26722 8907 3.31 0.099 Error 6 16165 16165 2694 Total 15 65132 Unusual Observations for Lifetime Obs Lifetime Fit SE Fit Residual St Resid 11 784.000 851.250 41.034 -67.250 -2.12R

21. 21 Minitab DATA ENTRY VAR1VAR2VAR3VAR4 855114 962211 848313 831412 877123 817222............ 871443

22. 22 Latin Square with REPLICATION Case One: using the same rows and columns for all Latin squares (same blocks). Case Two: using different rows and columns for different Latin squares (different blocks). Case Three: using the same rows but different columns for different Latin squares (same row blocks but different column blocks).

23. 23 Treatment Assignments for n Replications Case One: repeat the same Latin square n times. Case Two: randomly select one Latin square for each replication. Case Three: randomly select one Latin square for each replication.

24. 24 Example: n = 2, m = 4, trtmnt = A,B,C,D Case One: column row1234 1ABCD 2BCDA 3CDAB 4DABC column row1234 1ABCD 2BCDA 3CDAB 4DABC Row = 4 tire positions; column = 4 cars

25. 25 column row1234 1ABCD 2BCDA 3CDAB 4DABC column row5678 5BCDA 6ADCB 7DBAC 8CABD Case Two Row = clinics; column = patients; letter = drugs for flu

26. 26 5678 BCDA ADCB DBAC CABD Case Three column row1234 1ABCD 2BCDA 3CDAB 4DABC Row = 4 tire positions; column = 8 cars

27. 27 ANOVA for Case 1 SSB R, SSB C, SSB IF are computed the same way as before, except that the multiplier of (say for rows) m (Y i.. -Y … ) 2 becomes mn (Y i.. -Y … ) 2 and degrees of freedom for error becomes (nm 2 - 1) - 3(m - 1) = nm 2 - 3m + 2  

28. 28 ANOVA for other cases: Using Minitab in the same way can give Anova tables for all cases. 1.SS: please refer to the book, Statistical Principles of research Design and Analysis by R. Kuehl. 2.DF: # of levels – 1 for all terms except error. DF of error = total DF – the sum of the rest DF’s.

29. 29 Three or More Factors Notation: Y = response; A, B, C, … = input factors AB = interaction between A and B ABC = interaction between A, B, and C The term involving k factors has order of k: eg. AB  order 2 term ABC  order 3 term

30. 30 Full model = the model includes all factors and their interactions, denoted as (1) Two factors A|B (= A+B+AB) (2) Three factors A|B|C (= A+B+C+AB+AC+BC+ABC) (3) And so on.

31. 31 ABC B lowhigh 1012 1315 low high A C at level 1 B lowhigh 101515 low high A C at level 2 B lowhigh 1013 1310 low high A C at level 3 B lowhigh1212 low high A C at level 4 ABAB 1320 22.52.5-2.5 300-3 4000 Discussion of examples: Notice that in C at level 2 & 3 interaction is as large as or larger than main effects.

32. 32 Example: Three factors each at two levels The dependent variable is response rate of a direct mail offering. 2_to_3_design.mpj on class webpage low high A postage3rd class1st class B price$9.95$12.95 Cenvelope size#109 x 12

33. 33 Backward Model Selection 1.Fit the full model and delete the most insignificant highest order term. 2.Fit the reduced model from 1. and delete the most insignificant highest order term. 3.Repeat 2. until all remaining highest order terms are significant. 4.Repeat the same procedure (deleting the most insignificant term each time until no insignificant terms) for the 2 nd highest order, then the 3 rd highest order, …, and finally the order 1 terms. 5.Determine the final model and do assumption checking for it.

34. 34 Note. If a term is in the current model, then all lower order terms involving factors in that term must not be deleted even if they are insignificant. eg. If ABC is significant (so it is in the model), then A, B, C, AB, AC, BC cannot be deleted.

35. 35 Note. The procedure of backward model selection can be very time-consuming if the number of factors, k, is large. In such cases, we delete all insignificant terms together when we are processing the order 4 or higher terms. Examples are in Minitab unit 11.