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Augustus Byrd
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9 STAT:5 Applied Statistic II Latin Square Design (no replication) Rocket Propellant Example An experimenter is studying the effects of five different formulations of a rocket propellant on the burning rate of the propellant. Each formulation is mixed from a batch of raw material that is only large enough for five formulations to be tested. Also the formulations are prepared by several operntors, whose skill and experience may affect the burn rate. Response: Burn rate. Factors: Formulation (5 levels), Material (5 levels), and Operator( 5 levels). SAS statements for data input and Proc GLM: data rocket; do material=l to 5; do operator=l to 5; input formulation $ rate ; output; end; end; datalines; A B C 9 D E B 7 C D E 7 A 6 C 8 D 8 E 6 A 7 B D 6 E A 6 B C E A B C 9 D proc print data=rocket; run; Obs material operator formulation rate A B c D E B c D E A c D E A B D

10 7 E 8 A 9 B 5 c 5 E 5 A 5 B 5 c D 6 9 proc gplot data=rocket; plot rate* (formulation material operator); run; Rate vs. Formulation: ~ A c fonnulation D Rate vs. Material: + + matenal

11 Rate vs. Operator: ra: ~ ~ JO operator SAS statements for Proc GLM: proc glm data=rocket plot=diagnostics; class formulation material operator; model rate = formulation material operator/solution; lsmeans formulation/adjust=tukey pdiff; output out=diags r=residual p=predicted; run; The GLM Procedure Class Level Information Class Levels Values formulation 5 A B C D E material operator Dependent Source Model Error Corrected Source formulation material operator Variable: rate Total Sum of DF Squares DF Type III SS Mean Square Mean Square F Value Pr > F.8.89 F Value Pr > FI s~.,; ~ ca..._)?r(att~j e,

12 vnomf~vl ~~ Parameter Estimate Standard Error t Value Pr > ltl Intercept formulation A formulation B formulation C formulation D formulation E material material material material material 5 operator operator operator operator operator 5 8. B B B B B B -. B B B B B -5. B B B B B.6 < NOTE: The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable. The GLM Procedure Least Squares Means Adjustment for Multiple Comparisons: Tukey LS MEAN formulation rate LSMEAN Number A ~ A 8.6 B. c. D 9.8 E 6. 5 Least Squares Means for effect formulation Pr > ltl for HO: LSMean(i)=LSMean(j) Dependent Variable: rate A e, c_ i/j C D t D ~ ~A 8 ~ D c -=I- D

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14 STAT:5 Applied Statistic II Latin Square Design (replicated) Auto Carbon Monoxide (CO) Emission Example Suppose the EPA is interested in recommending a certain brand of gasoline for cars with respect to CO emission. There are three brands of gasoline they're interested in comparing. Since the amount of CO emitted depends on both the types of car (l:small/family sedan, :medium/van, :large/truck) and the speed the car is traveling in terms of types of routes (l:slow/5mph, :medium/5mph, :fast/65mph), these two factors, namely car and route, are taken into consideration during the design stage. A Latin Square design is decided upon, but with only treatment level:,; (a=), there will be very few d.f. for error with a single square. So, four replications are done. Response: emission. Factors: gas ( levels), car ( levels), and route ( levels). Possible nesting of blocking factors will be described later. Here are the Latin squares with rows as cars and columns as routes: A c B B A c c A B c c B A c B A A B c A B A c A c B B c A B B A c B A c Here are the observed CO emissions: SAS statements for data input and Proc GLM: data coemission; do car = to ; do route = to ; do square = to ; input gas $ emission ; output; end; end; end; datalines; A 5 B C 7 C 8 C 8 A A B 5 B C 6 B 5 A C 68 C 66 A 6 A 65 B B 9 B C 8 A A 5 C 7 B B A 98 B 5 B A 58 C 6 C 6 A C 8 B A 5 C 65 run;

15 SAS statements for SAME CARS AND SAME ROUTES: /*same cars and same routes*/ proc glm data=coemission; class gas car route square; model emission=gas car route square; lsmeans gas; run; The GLM Procedure Class Level Information Class Levels Values gas A B C car route square Dependent Variable: emission Sum of Source DF Squares Model Error Corrected Total 5. Mean Square F Value Pr > F < Dependent Variable: emission Source DF Type III SS gas car route square Mean Square F Value Pr > F < < Least Squares Means gas A B c emission LS MEAN UorJ s~ Co.rs ~ ~ s;wlr~ o r '.,if f, cl/j-e

16 SAS statements for DIFFERENT CARS AND SAME ROUTES: /*different cars and same routes*/ proc glm data=coemission; class gas car route square; model emission=gas car(square) route square; lsmeans gas; run; The GLM Procedure Class Level Information Class Levels Values gas car route square A B C Dependent Variable: Source Model Error Corrected Total emission DF 5 5 Sum of Squares Mean Square F Value Pr > F 7. <. Dependent Variable: gas car(square) route square emission < The GLM Procedure Least Squares Means gas emission LS MEAN A.8 B ' Ccd'S /,un I~ l<..q_ ~'~~ (f,, c 7. Car (-F) J.+. <[; Im sgu.qf-<.

17 SAS statements for SAME CARS AND DIFFERENT ROUTES: /*same cars and different routes.*/ proc glm data=coemission; class gas car route square; model emission=gas car route(square) square; lsmeans gas; run; The GLM Procedure Class Level Information Class Levels Values gas A B C car route square The GLM Procedure Dependent Variable: emission Sum of Source DF Squares Model Error Corrected Total 5. Mean Square F Value Pr > F < Source DF Type III SS gas car route(square) 8. square Mean Square F Value Pr > F < < Least Squares Means so emission gas LS MEAN A.8 ' j \oj.e S ~ lwit JlUL B (S~/) l5 ~ uc;/"-- ~J. br (I J '' c 7. ro e

18 SAS statements for DIFFERENT CARS AND DIFFERENT ROUTES: /*different cars and different routes.*/ proc glm data=coemission; class gas car route square; model emission=gas car(square) route(square) square; lsmeans gas; run; The GLM Procedure Class Level Information Class Levels Values gas A B C car route square Dependent Variable: emission Source DF Model Error Corrected Total 5 Sum of Squares Mean Square F Value Pr > F 6..5 Dependent Variable: emission Source gas car(square) route(square) square DF Type III SS Mean Square F Value Pr > F Least Squares Means gas emission LSMEAN A.8 B c 7. 6~ Ca\S ~ CfoJ-.es /'/J {,(_Q 5

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