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STAT:50 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 4 B 0 C 9 D 4 E 4 B 7 C 4 D 0 E 7 A 6 C 8 D 8 E 6 A 7 B D 6 E A 6 B C E A 0 B 0 C 9 D proc print data=rocket; run; Obs material operator formulation rate 4 5 6 7 8 9 0 4 5 6 4 4 5 4 5 4 5 A B c D E B c D E A c D E A B D 4 0 9 4 4 7 4 0 7 6 8 8 6 7 6
7 4 E 8 4 A 9 4 4 B 0 4 5 c 5 E 5 A 5 B 4 5 4 c 5 5 5 D 6 0 0 9 proc gplot data=rocket; plot rate* (formulation material operator); run; Rate vs. Formulation: 0 A c fonnulation D Rate vs. Material: 0 + + matenal
Rate vs. Operator: ra: 0 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 5 5 4 5 4 5 Dependent Source Model Error Corrected Source formulation material operator Variable: rate Total Sum of DF Squares 548.0000000 8.0000000 4 676.0000000 DF Type III SS 4 0.0000000 4 68.0000000 4 50.0000000 Mean Square 45.6666667 0.6666667 Mean Square 8.5000000 7.0000000 7.5000000 F Value Pr > F 4.8 0.0089 F Value Pr > FI 7.7 0.005...e,.59 0.9.5 0.0404
STAT:50 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/45mph, :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 4 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: 45 8 4 4 0 6 47 5 68 4 66 9 5 64 40 7 00 58 8 98 60 05 6 5 48 5 65 8 04 4 0 4 65 SAS statements for data input and Proc GLM: data coemission; do car = to ; do route = to ; do square = to 4; input gas $ emission ; output; end; end; end; datalines; A 45 B 4 C 47 C 48 C 8 A 0 A B 5 B 4 C 6 B 5 A 0 C 68 C 66 A 64 A 65 B 4 B 9 B 40 C 8 A A 5 C 7 B 4 B 00 A 98 B 05 B 04 A 58 C 60 C 6 A 4 C 8 B A 5 C 65 run;
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 4 4 Dependent Variable: emission Sum of Source DF Squares Model 9 6960.6667 Error 6 070.8 Corrected Total 5 00.00000 Mean Square F Value Pr > F 884.4696 5.96 <.000 8.0897 Dependent Variable: emission Source DF Type III SS gas 9.666667 car 6667.66667 route 9968.666667 square.666667 Mean Square F Value Pr > F 46..4 0.06.58 8. <.000 4984. 4.0 <.000 0.555556 0.09 0.965 Least Squares Means gas A B c emission LS MEAN 44.8 5.6666667 47.0000000 UorJ o r Co.rs '.,if f, cl/j-e
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 4 A B C 4 Dependent Variable: Source Model Error Corrected Total emission DF 5 0 5 Sum of Squares 6976. 054.66667 00.00000 Mean Square. 75556 5.7 F Value Pr > F 7.4 <.000 Dependent Variable: gas car(square) route square emission 9.666667 668. 9968.666667.666667 46. 85.46667 4984. 0.555556 0.96 0.4005 5.47 0.000.6 <.000 0.07 0.9757 The GLM Procedure Least Squares Means gas emission LS MEAN A 44.8 B 5.6666667 /,un ' l<..q_ Ccd'S (f,, c 47.0000000 Car (-F)4 J.+. <[; Im sgu.qf-<.
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 4 4 The GLM Procedure Dependent Variable: emission Sum of Source DF Squares Model 5 807.8 Error 0 958.6667 Corrected Total 5 00.00000 Mean Square F Value Pr > F 04.85556. <.000 97.908 Source DF Type III SS gas 9.66667 car 6667.6667 route(square) 08. square. 66667 Mean Square F Value Pr > F 46..49 0.48.58 4.05 <.000 85.6667 4.5 <.000 0.55556 0. 0.9545 Least Squares Means emission gas LS MEAN so j JlUL ' lwit A 44.8 B 5.6666667 l5 \oj.e S br (I uc;/"-- J '' c 47.0000000 ro e 4
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 4 4 Dependent Variable: emission Source DF Model Error 4 Corrected Total 5 Sum of Squares Mean Square 8089.00000 86.8095 94.00000 8.749 00.00000 F Value Pr > F 6. 0.0005 Dependent Variable: emission Source gas car(square) route(square) square DF Type III SS Mean Square 9.66667 46. 668.. 85.4667 08. 85.6667.66667 0.55556 F Value Pr > F.05 0.74 6.0 0.008 9.99 0.000 0.08 0.979 Least Squares Means gas emission LSMEAN Ca\S CfoJ-.es 0/'/J {,(_Q A 44.8 B 5.6666667 c 47.0000000 5
Medical Example of a Latin Square 6
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Second Example of a Latin Square Canadian Medical Association Journal, October 5, 00, 8(4). 8