Factorial Studies. Factor A I levels Factor B J levels o I*J Treatment groups o n ij values for each treatment combination

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1 Factorial tudies Factor A I levels Factor B J levels o I*J Treatment grous o n ij values for each treatment combination B1 B.. BJ A1 n 11 n 1 n 1J A n 1 n n 1J AI n I1 n I n IJ Equal relication, n relicates, in each treatment grou. Equal n ij 's Balanced design. ome of the simlified calculations won't necessaril work for unbalanced data. o Examle 1 in section 8.1 for examle is unbalanced. Examle 8.1 Factor A Joint Factor B Wood Y Joint strength BPine BOak BWalnut AButt 89, , 19 ABeveled 148, , , 44 ALa 1, , A balanced examle: ection 4. Problem 8 Y Distance of tennis ball A Charge size with I levels o.5 ml and 5. ml B Proellant with J levels o lighter fluid, gasoline and carburetor fluid n5 values for each treatment combination Proellant Lighter Carburetor Charge Fluid Gasoline Fluid

2 First lot the data!! It hels to offset the oint a bit so the don't end u on to of each other. Distance Proellant 1lighter gas carb Charge.5 Charge 5. There is some indication of larger variances for bigger values and for more a difference between low and high charge for higher values. Log scales can o Constant coefficients of variation in distances, where the standard deviations are some fraction or ercent of the mean, result in nearl constant variance in a log scale. o Effects that result in ercentage or relative changes give similar differences in the log scale. If the effect of a charge or roellant level is to increase or decrease distance b some multilicative factor, for examle the larger charge increases distance b % or multilies distance b 1.1 o Distance d c i j o c i multilicative effect of charge i o j multilicative effect of charge j o Log ( Distance) Log ( d ) + Log( ci ) + Log( j ) o The effects in Log scale add to the distance with no interactions.

3 oftware Following section 9.. in the book, we can run the factorial analsis with Excel regression. o For factor erfectl balanced data, ou can use the Excel factor data analsis. This doesn' t give residuals or other useful outut. Other more sohisticated software is reall helful for more comlicated regression or ANOVA analses. o Of the grahical user interface rogram, m favorite is A JMP. o Minitab is also nice. o Most often I use the full blown A ackage. teeer learning curve. o R software is becoming oular as well. R is the oen source, free version of. was Bell Labs' statistical ackage. The liked short names like CComlier and tatistics R is free, reall good, flexible software but usuall requires writing simle code. Another Examle: Y Time to fill a container with sru A Brand: Pure male sru, Eggo, Hansen's B Temerature: Room tem and refrigerated sru ru Time Pure Eggo Hansen's Room Fridge There is an obvious interaction between Brand and Tem. o The time difference between Pure and Eggo sru is less at room tem than refrigerated. o The difference between refrigerated and room tem ouring times is greater for Hansen's than for ure male sru. Ideall, we would like to be able to find more general conclusion for comaring Eggo and Hansen's srus. o The differences in ouring time are not the same, but we wouldn't reall exect to see this haen. o We would have anticiated larger differences in ouring times for the refrigerated, longer ouring times. o Rather than consistent differences in ouring times, we would more likel anticiate similar relative changes, meaning similar ratios or ercent changes. o With similar ratios in the original time scale We would see similar differences in the Log scale. No interaction in the log scale.

4 Ln(time) ru Project Room Fridge Pure Eggo Hansen's ince we have similar differences in the Ln scale o The ercentage or relative increase in ouring times for refrigerated srus similar for all three srus. o We are able to make more general conclusions. Often, effects in factorial studies act multilicativel rather than additivel. uose for examle that in a x factorial For factor A the effect of level is to double the resonse. For factor B the effect of level is to trile the resonse. B1 B A1 1 A 6 In this scale we have an obvious interaction. Going from B1 to B adds more to the resonse for A than for A1 Multilicative x Factorial A B 1 B But the effect of B is actuall consistent. It s just a multilicative effect rather than adding fixed amounts.

5 B changing to a log scale, we can see this consistenc. Equal ratios in the original scale corresond to equal differences in a log scale. Taking logs changes relative changes or ratios to absolute changes or differences. Log cale A B Y Ratio Log(Y) Difference : : log(y) log() + log(y) log(y) +.48 Multilicative x Factorial in Log cale Multilicative x in Log cale 1 1 B 1 B log(y) B B A 1 A Also, often data have more consistent CV's across treatment grous with widel different means. o The standard deviations in the Ln scale corresond roughl to CV's in the original scale. o o if we have similar CV's in the original scale, we have similar variances in the Ln scale, one of the assumtions in the usual ANOVA model. If we find confidence intervals for differences in mean Ln(time) o Back-transforming the interval to the original scale gives us confidence intervals for ratio of ouring time in the time scale.

6 Effect of Ignoring an Imortant Factor Ln Time 5. ru Ln Time 4.. Room Fridge. 1 4 Eggo Hansen's A t-test ignoring temerature for just Hansen's versus Eggo srus. Ln Time 5. ru Ln Time Eggo Hansen's Brand Method Mean 95% CL Mean td Dev Eggo Hansen Diff (1-) Pooled Diff (1-) atterthwaite Method Variance s DF t Value Pr > t Pooled Equal atterthwait e Unequal This analsis sas there is not a significant difference between Eggo and Hansen's. o This is obviousl not right. o The ooled standard deviation of.76 includes variabilit due to temerature. o The brand effects are lost in the inflated ooled variance.

7 Factorial analsis with Temerature and Brand. ource DF Mean quare F Value Pr > F Brand <.1 Temerature <.1 Brand*Temerature Error Corrected Total Brand Ln_time MEAN H:Mean1Mean Pr > t Eggo.51 <.1 Hansen.449 Difference Between Means 95% Confidence Limits for Mean(i)-Mean(j) Now there is a ver significant difference between Eggo and Hansen's. o The ooled standard deviation is o This is much, much smaller than the ooled standard deviation from the t-test ignoring temerature,.761. Alwas confirm our results b lotting the data! Residuals Factor B Factor A j1 j j i1 111, 11, 11, 1, 11, 1, i 11, 1, 1,, 1,, amle Means and Variances Factor B Factor A j1 j j i i 1 1

8 The fitted value for each grou is the mean for that treatment grou ˆijk ij ˆ k ijk th The residuals are Residual e Measured e ˆ ijk ijk ijk ijk ij Fitted Plotting residuals In addition to the initial lot, lot residuals to check for Equal variance Drift over time Normalit Plot residuals Versus redicted values (treatment grou means) Versus run order from 1 to I J n Also lot residuals versus A levels B levels For onl two factors, the lot of all original data oints shows rett well how for examle variances change with the factors. For factorial studies with more than factors, we can t lot all original data oints one lot, so it becomes more essential to lot residuals versus the factors. Normal lot or lots Residuals. o For all residuals together if variances aear similar throughout. o With enough data in each grou, we could also look at normal lots for each treatment grou on the same lot. Proellant Original cale Residuals vs Proellant Residual Predicted ome increasing variance. Residual Lighter Fluid Gasoline Carb Fluid ome increasing variance.

9 Residuals vs Charge Residual Charge ome increasing variance. Possibl a bit of increasing variance. Residual No aarent trends. Residuals versus Run Order Run Order tandard Normal Quantiles Normal Plot of Residuals orted Residuals Residuals aear normal Tests and Confidence Intervals Bricks Examle 5 on Page 467 Heat Treat % water Brick trength x s low , 5998, low , 588, Fast 17 84, 14, Fast , 67, As with t-tests in Chater 6, we estimate the variance, σ, with the ooled variance,. ( nij ) s ( nij 1) ( 1) ( 1) 55 + ( 1) 1 + ( 1) ( 1) + ( 1) + ( 1) + ( 1) 1 ij 787 5,646 M Error M Residual ijk is the k th value in treatment grou with Ai and Bj.

10 ( n 1) ( nij 1) ij sij ( ) ( ijk ij ) nij 1 ( nij 1) n ij 1 ( ijk ij ) ( nij 1) um of quared Residuals Residual df M Residual With equal n' s, unweighted average ij ,646 ource df M F -value Heat 1 7,8,96 7,8, Water 1 8,616 8, Heat*Water 1 76, 76,..165 Error 8,65,169 5,646 Total 11 1,474,779 o Just as in the regression ANOVA tables, the Mean quare Residual or Mean quare Error is our estimate of σ. o For factorial studies with relication, M Error ooled variance. Heat main effect 1 1 Yslow (558) + (497) Y fast (489) + (894) ˆ L Y11 + Y1 Y1 Y 156 E Lˆ 9.6 Vˆ( Lˆ) (1/ ) With equal n's this can be simlified The average of the averages is the same value as the average of all value ut together. Heat % Water Brick trength low , 5998, low , 588, Y slow Y slow

11 Y Y slow fast average of 6 values average of 6 values E Vˆ ( Y Y ) slow fast % CI 156 ±.6(9.6) 156 ± to 96 To test Ho : Heat main effect t Ha : Heat main effect *.5 < value < *.1.1 < value <. Proellants Examle Ln(Distance) Proellants in Ln cale Charge.5 Charge lighter gas carb Ln(Distance) Charge Proellant n Mean td Dev.5 Carburetor Gasoline Lighter Carburetor Gasoline Lighter Charge n Mean

12 Proellant n Mean Carburetor Gasoline Lighter ource DF Mean quare F Value Pr > F Charge <.1 Proellant <.1 Charge*Proellant Error Total tandard 95% 95% Estimate Error t -value Lower Uer Carb - Gas Carb - Lighter < Gas - Lighter < Charge 5. - Charge < Wood Joints: Examle 1 in Chater 8,, trength 1, Beveled Butt La 1 Wood 1 Pine Oak Walnut Beveled Butt La Mean Pine Oak Walnut Mean

13 Means tandard Deviations Beveled Butt La Mean of Means Beveled Butt La Pine Pine Oak Oak Walnut Walnut Mean of Means ource DF Mean quare F Value Pr > F Wood 1,776,9 888, Joint,18,745 1,9, Wood*Joint 4 468,48 117, Error 7 1,614 8,8 Total 15 4,464,996 tandard Estimate Error t Value Pr > t Oak - Pine Beveled Walnut - Beveled Oak La Walnut - La Oak La Pine - La Others Lower Uer Oak - Pine Beveled Walnut - Beveled Oak La Walnut - La Oak La Pine - La Others

14 Problem in ection Time Water Glcol Ln(Time)..5 Water Glcol Diameter Means Diam.188 Diam.14 Water Glcol Diameter tandard Deviations Diam.188 Diam.14 Water Glcol how how to find from information given above. ANOVA df M F P-value F crit Fluids E Diameter E Interaction E Error Total Where is found in the ANOVA table? Would it be a good idea to comare drain times for.188 and.14 inch tubes averaged over both liquids? What information above suorts our answer? Differences between.188 and.14 inch diameter Ln Drain Time. Water Glcol Lower Limit Uer Limit Lower Limit Uer Limit how how to find these intervals from other information above. Phsical theor suggests that drain time would be inversel roortional to the 4 th ower of drain diameter. What should the ratio of drain times be for.188 and.14 inch diameter drain tubes? Are the data for either water or glcol consistent with this model?