Suppose we needed four batches of formaldehyde, and coulddoonly4runsperbatch. Thisisthena2 4 factorial in 2 2 blocks.

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1 factorials in 2 blocks Suppose we needed four batches of formaldehyde, and coulddoonly4runsperbatch. Thisisthena2 4 factorial in 2 2 blocks. Some more algebra: If two effects are confounded with blocks, then so is their product, which is defined by multiplication mod 2 : 0 = 2 = = E.g. = 2 =. Pick two effects to be confounded with blocks: and. Then also = is confounded. We wouldn t pick and, since =.

2 59 For the choices and we have = = = = with () a b ab c ac bc abc Block I IV II III IV I III II d ad bd abd cd acd bcd abcd Block III II IV I II III I IV Block I : () Block II : Block III : Block IV :

3 A B C D blocks ABC ACD BD y

4 6 > g <- lm(y ~blocks + A + B + C + D + A*B + A*C + A*D + B*C + C*D + A*B*D + B*C*D + A*B*C*D) > anova(g) Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) blocks A B C D A:B A:C.6.6 A:D B:C C:D A:B:D B:C:D A:B:C:D Residuals 0 0.0

5 62 Normal Q Q Plot Sample Quantiles A:C A:B C:D D B:C:D A:B:D A:B:C:D B blocks3 blocks2 A:D B:C C A blocks4 0 Theoretical Quantiles Fig Half normal plot for 2 4 factorial in 2 2 blocks. Itlookslikewecandropthemaineffect of D if we keep some of its interactions.

6 63 R will, by default, estimate a main effect if an interaction is in the model. To fit blocks, A, B, C, AB, AD, BC, CD but not D, we can add the SS and df for D to those for Error. > h <- lm(y ~blocks + A + B + C + B*C + A*B + A*D + C*D) > anova(h) Df Sum Sq Mean Sq F value Pr(>F) blocks *** A *** B *** C *** D B:C ** A:B * A:D ** C:D Residuals This would change to ( ) 5 =3 08 on 5 d.f. - not a helpful step (since was larger than ).

7 64 mean of y B mean of y D A A mean of y C mean of y D B C Fig Interaction plots. The best combination seemstobea,c,dhigh,blow.

8 Partial confounding To get an estimate of error, we have to either drop certain effects from the model, or replicate the design. If we replicate, we can either: Confoundthesameeffects with blocks in each replication - complete confounding, or Confound different effects with each replication - partial confounding. Partial confounding is often better, since we then get estimates of effects from the replications in which they are not confounded.

9 66 Example 7-3 from text. Two replicates of a 2 3 factorial are to be run, in 2 block each. Replicate : Confound ABC with blocks. So = =0for() and =for Replicate 2: Confound AB with blocks. So = + 2 =0for() and = for Rep Rep2 Block Block 2 Block 3 Block 4 () = 550 = 669 () = 604 =650 =642 = 633 = 052 = 60 =749 = 037 =635 =868 = 075 =729 =860 =063

10 67 A B C Rep Block ABC AB y I I I I I I I I II II II II II II II II When the levels of one factor (Blocks) make sense only within the levels of another factor (Replicates) we say that the first is nested within the second. A waytoindicatethisinrisas:

11 > h <- lm(y ~Rep + Block%in%Rep + A + B + C + A*B + A*C + B*C + A*B*C) > anova(h) Analysis of Variance Table 68 Response: y Df Sum Sq Mean Sq F value Pr(>F) Rep A * B C e-05 *** Rep:Block A:B A:C ** B:C A:B:C Residuals Through the partial confounding we are able to estimate all interactions. It looks like only A, C, and AC are significant.

12 69 resids resids c(a) c(c) resids resids c(b) c(block) Normal Q Q Plot resids Sample Quantiles fits Theoretical Quantiles Fig Residuals for fit to A, C, and AC only.

13 70 mean of y II I A B C Block mean of y A Factors C Fig Design and interactions. How is ( ) computed? One way is to compute in Rep I, where this effect is confounded with blocks, and similarly in

14 7 RepII,andaddthem: " # 2 ( ) +( ) = 8 = 338 = =20 25 ( ) = = in agreement with the ANOVA output. See the programme on the course web site to see how to do this calculation very easily. Another method goes back to general principles. We calculate a SS for blocks within each replicate (since blocks make sense only within the replicates): ( ) =4 X X ³ 2 = = 2 = 2 Here is the average in block of replicate, and is the overall average of that replicate, which is the only one in which that block makes sense. See the R calculation.

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