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Lecture

2k factorials.pdf

12 Pages
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Department
Statistics
Course Code
STAT368
Professor
Douglas Wiens

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133 18. 2 factorials • All of this generalizes to the 2ctorial, in which factors are investigated, each at two levels. To easily write down the estimates of the e ects, and the contrasts, we start with a table of ± signs, done here for = 3. Label the rows (1), then the product of with (1). Then all products of with the terms which are already there: × (1) = × = . Then all products of with the terms which are already there. (This is the ‘standard’ order.) Now put in the signs. Start with 2 = 8 +’s under the I, then alternate ’s and +’s, then in groups of 2, nally (under C) in groups of 4 (= 2 1). Then write in the products under the interaction terms. 134 E ect I A B C AB AC BC ABC (1) + + + + + + + + + + + + + + + + + + + + + + + + + + + + • Interpretation: Assign the appropriate signs to the combinations (1). E ect estimates are [(1) + + + ] = 4 +[ + + + ] 4 [ + + + ] [(1) + + + ] = 4 all with 2in the denominator. 135 ³ ´ • These are all of the form 2 1 for a con- trast in t³e s´ms (1) ; the corresponding is 2 2 . For example ( )2 [ + + + ] [(1) + + + ] = 8 • The sums of squares are all on 1 d.f. (including , which uses the 1 d.f.usually subtracted from = 2 for the estimation of the overall mean ), so that , obtained by subtraction, is on 2 = 2 ( 1) d.f. Then, e.g., the F-ratio to test the e ect of factor A is 0= where = and = ( ). The p-value for the hypothesis of no e ect is µ ¶ 1( 1) 0 . 2 136 • Suppose = 1, so that no d.f. are available for the estimation of 2. In the 2 there was Tukey’s test for non-additivity, which relied on the assumption that the interactions were of a certain mathematically simple but statistically du- bious form (even more so for 2). A more common remedy is to not even try to estimate certain e ects - usually higher order interactions - and use the d.f. released in this way to estimate error. • A graphical way of identifying the important ef- fects which must be in the model, and those which can be dropped to facilitate error estimation, is a normal probability plot of the absolute values of the e ect estimates - a ‘half-normal’ plot. Those e ects which deviate signicantly from the qqline tend to be the important ones. • Example. Data in Table 6-10. A chemical prod- uct is produced using two levels each of tempera- ture (A), pressure (B), concentration of formalde- hyde (C) and rate (D) at which the product is stirred. Response (Y) is the ‘ltration rate’. 137 A B C D y 1 -1 -1 -1 -1 45 2 1 -1 -1 -1 71 3 -1 1 -1 -1 48 4 1 1 -1 -1 65 5 -1 -1 1 -1 68 6 1 -1 1 -1 60 7 -1 1 1 -1 80 8 1 1 1 -1 65 9 -1 -1 -1 1 43 10 1 -1 -1 1 100 11 -1 1 -1 1 45 12 1 1 -1 1 104 13 -1 -1 1 1 75 14 1 -1 1 1 86 15 -1 1 1 1 70 16 1 1 1 1 96 138 > g anova(g) Df Sum Sq Mean Sq F value Pr(>F) A 1 1870.56 1870.56 B 1 39.06 39.06 C 1 390.06 390.06 D 1 855.56 855.56 A:B 1 0.06 0.06 A:C 1 1314.06 1314.06 A:D 1 1105.56 1105.56 B:C 1 22.56 22.56 B:D 1 0.56 0.56 C:D 1 5.06 5.06 A:B:C 1 14.06 14.06 A:B:D 1 68.06 68.06 A:C:D 1 10.56 10.56 B:C:D 1 27.56 27.56 A:B:C:D 1 7.56 7.56 Residuals 0 0.00 139 > g$effects # Note that these are twice as large in absolute value as those in the text, and the signs sometimes di er. This is because of R’s denition of ‘e ect’, and makes no di erence for comparing their absolute values. (Intercept) A1 B1 C1 -280.25 -43.25 -6.25 19.75 etc. A1:B1:D1 A1:C1:D1 B1:C1:D1 A1:B1:C1:D1 -8.25 3.25 5.25 2.75 > effects qq
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