SAS code for Levene s Test and Weighted Least Squares DM LOG; CLEAR; OUT; CLEAR; ; ODS GRAPHICS ON; OPTIONS NODATE NONUMBER;

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1 Weighted WeightedLeast Least Squares Squares and and Transformation Transformation Example Example 97 8

2 SAS code for Levene s Test and Weighted Least Squares M LOG; LEAR; OUT; LEAR; ; OS GRAPHIS ON; OS PRINTER PF file= :\OURSES\ST54\WGTLS2.PF ; OPTIONS NOATE NONUMER; ATA in; O treatmnt = A_.5, A_., A_.5, _.5, _., _.5, _.5, _., _.5 ; O rep = to 4; OUTPUT; EN; EN; cards; ; PRO GLM ATA=in PLOTS=(IAGNOSTIS); LASS treatmnt; MOEL = treatmnt / SS3; MEANS treatmnt / HOVTEST=LEVENE; S treatmnt / AJ=ON LINES; TITLE LEVENE S TEST EXAMPLE FOR TWO-FATOR FATORIAL ; RUN; ATA in; O trtplant = A,, ; O rate =.5,.,.5; O rep = to 4; OUTPUT; EN; EN; EN; cards; ; PRO SORT ATA=IN; Y trtplant rate; PRO MEANS ATA=in noprint; Y trtplant rate; VAR ; OUTPUT OUT=wset VAR=var_ppm; ATA wset; SET wset; wgt = /var_ppm; ROP _FREQ TYPE_; PRO PRINT ATA=wset; TITLE SAMPLE VARIANES AN WEIGHTS FOR EAH TREATMENT ; ATA in; MERGE in wset; Y trtplant rate; PRO GLM ATA=in PLOTS=(IAGNOSTIS); WEIGHT wgt; LASS trtplant rate ; MOEL = trtplant rate / SS3; S trtplant*rate / AJUST=ON LINES; TITLE WEIGHTE LEAST SQUARES EXAMPLE FOR TWO-FATOR FATORIAL ; RUN; 82

3 LEVENE'S TEST EXAMPLE FOR TWO-FATOR FATORIAL ependent Variable: Source F Sum of Squares Mean Square F Value Pr > F Model <. Error orrected Total R-Square oeff Var Root MSE Mean LEVENE'S Source TEST F EXAMPLE Type III SS FOR Mean TWO-FATOR Square F Value FATORIAL Pr > F treatmnt <. Source Levene's Test for Homogeneity of Variance ANOVA of Squared eviations from Group Means F Sum of Squares Mean Square F Value Pr > F treatmnt Error LEVENE'S TEST 27 EXAMPLE FOR TWO-FATOR 572. FATORIAL 8 istribution of F 6.75 Prob > F < A_.5 A_. A_.5 _.5 _. _.5 _.5 _. _.5 treatmnt 83

4 ependent Variable: 5 Fit iagnostics for 2 2 RStudent RStudent Predicted Value Predicted Value Leverage WEIGHTE LEAST SQUARES EXAMPLE FOR TWO-FATOR.5 FATORIAL Least 2 Squares 4 Means Quantile Adjustment for Multiple Predicted omparisons: Value onferroni Observation ook's.2.5. Percent Interaction Fit Mean Plot for Proportion Less Observations Parameters Error F MSE R-Square Adj R-Square A trtplant rate.5.5 We will now look at a weighted least squares analysis. ased on the residual vs predicted values plot, it is evident that there is a homogeneity of variance problem across the 9 factorial treatment combinations. Later, we will address the question: Will a transformation fix the problem? by finding a transformation using the empirical method. 84

5 SAMPLE VARIANES AN WEIGHTS FOR EAH TREATMENT Obs trtplant rate var_ppm wgt A A A WEIGHTE LEAST SQUARES EXAMPLE FOR TWO-FATOR FATORIAL ependent Variable: Weight: wgt Source F Sum of Squares Mean Square F Value Pr > F Model <. Error orrected Total R-Square oeff Var Root MSE Mean Source F Type III SS Mean Square F Value Pr > F trtplant <. rate <. trtplant*rate <. 85

6 LEVENE'S TEST EXAMPLE FOR TWO-FATOR FATORIAL LEVENE'S TEST EXAMPLE FOR TWO-FATOR FATORIAL Adjustment for Multiple omparisons: Adjustment for Multiple onferroni omparisons: onferroni treatmnt treatmnt A_ A_ A_ A_ A_ A_ _ _ _ _ _ _ _ _ _.5 9. _ _ WEIGHTE LEAST SQUARES EXAMPLE FOR TWO-FATOR FATORIAL _ Let s compare the results of onferroni s Least Squares 9 Means MP for effect from treatmnt the unweighted and weighted least Pr > t for H: LSMean(i)=LSMean(j) squares analyses. ependent Variable: Adjustment Least Squares for Multiple i/j Means omparisons: for effect 2 treatmnt 3 onferroni Pr > t for H: LSMean(i)=LSMean(j)..34. <. <.... ependent 2. Variable:...2 < trtplant rate i/j < A < <. <.... A 5 < <. <. <. < < A.5 6 < <. <. <. 3 <. <. <. < < <..3< < < < <..6 <..463 <. <. <. < <. <. <. <. < <. 7 <. < <. 2.9 < < < <. <... for effect trtplant*rate Pr > t for H: LSMean(i)=LSMean(j) ependent Variable: i/j <. <. <. < < < <....4 <. <. <. <. 5 < <. <. <. <. 6 <. <. <. <. <. <. <. <. 7 < <. <. < <. <. < <. <. <... 86

7 ifferences for alpha=.5 (onferroni Adjustment) Not significant Significant WEIGHTE LEAST SQUARES EXAMPLE FOR TWO-FATOR FATORIAL onferroni omparison Lines for of treatmnt LS-means with the same The letter GLM are Procedure not significantly different. Least Squares Means Adjustment for Multiple omparisons: treatmnt onferroni A 72.6 _.5 6 omparisons for trtplant*rate _. 5 LEVENE'S TEST EXAMPLE FOR TWO-FATOR FATORIAL A_ Adjustment for Multiple omparisons: 3.75 _.5 onferroni 4 onferroni omparison Lines for of treatmnt E 3.45 A_. 2 LS-means with the same letter are not significantly different. 4 E A.5 F E A_.5 treatmnt F E F E 2.9 _. 8 2 F E.5 F.5 E A A _ A A.5 F F ifferences for alpha=.5 9. (onferroni _.5 Adjustment) 7 Not significant Significant A.5 onferroni omparison Lines for of trtplant*rate LS-means with the same letter are not significantly different. trtplant rate A WEIGHTE LEAST SQUARES EXAMPLE FOR TWO-FATOR FATORIAL A.5 3 Adjustment for Multiple omparisons: onferroni onferroni omparison Lines for of trtplant*rate 4 LS-means with the same letter are not significantly different A 2 trtplant rate The LINES display does not reflect A all significant.5comparisons. The following additional pairs are significantly different: (4,) (,9) (,7) The LINES display does not reflect all significant comparisons. The following additional pairs are significantly different: (4,) (,9) (,7) 87

8 SAS code for a transformed response based on the empirical method M LOG; LEAR; OUT; LEAR; ; OS GRAPHIS ON; OS PRINTER PF file= :\OURSES\ST54\TRANS2.PF ; OPTIONS NOATE NONUMER; ATA in; O trtplant = A,, ; O rate =.5,.,.5; O rep = to 4; OUTPUT; EN; EN; EN; cards; ; **********************************************************; *** Find the transformation using the empirical method ***; **********************************************************; PRO SORT ATA=IN; Y trtplant rate; PRO MEANS ATA=in noprint; Y trtplant rate; VAR ; OUTPUT OUT=yset MEAN=mean ST=std; ATA yset; SET yset; logstd =LOG(std); logmean=log(mean); PRO PRINT ATA=yset; VAR mean std logstd logmean; TITLE EMPIRIAL SELETION OF ALPHA ; PRO GLM ATA=yset; MOEL logstd=logmean / SS3 solution; TITLE ANOVA TO FIN EMPIRIAL SELETION OF ALPHA ; *************************************; *** ANOVA AFTER A TRANSFORMATION ***; *************************************; ATA in; SET in;; transppm = **(-.6); PRO GLM ATA=in PLOTS=(IAGNOSTIS); LASS trtplant rate; MOEL transppm = trtplant rate / SS3; MEANS trtplant rate; S trtplant*rate / AJUST=ON LINES; TITLE TWO-FATOR FATORIAL WITH TRANSFORMATION ; RUN; 88

9 SAS output for transformed response based on empirical method EMPIRIAL SELETION OF ALPHA Obs mean std logstd logmean ANOVA TO FIN EMPIRIAL SELETION OF ALPHA ependent Variable: logstd Source F Sum of Squares Mean Square F Value Pr > F Model Error orrected Total R-Square oeff Var Root MSE logstd Mean Source F Type III SS Mean Square F Value Pr > F logmean Parameter Estimate Standard Error t Value Pr > t Intercept logmean α.6 λ (.6) =.6. 89

10 TWO-FATOR FATORIAL WITH TRANSFORMATION ependent Variable: transppm Source F Sum of Squares Mean Square F Value Pr > F Model <. Error orrected Total R-Square oeff Var Root MSE transppm Mean Source F Type III SS Mean Square F Value Pr > F trtplant <. ratetwo-fator 2 FATORIAL WITH TRANSFORMATION <. trtplant*rate ependent Variable: transppm Fit iagnostics for transppm RStudent 2 RStudent Predicted Value Predicted Value Leverage transppm.4.2. ook's Quantile Predicted Value 2 3 Observation 4 Fit Mean Percent Observations 36 Parameters 9 Error F 27 MSE. R-Square.933 Adj R-Square Proportion Less 9

11 TWO-FATOR FATORIAL WITH TRANSFORMATION.5 A Interaction Plot for transppm TWO-FATOR FATORIAL WITH TRANSFORMATION A Least Squares.5 Means Adjustment for Multiple A.5 omparisons: onferroni transppm rate trtplant rate transppm A A A A.5.5 A A A trtplant ifferences for alpha=.5 (onferroni Adjustment) Not significant Significant for effect trtplant*rate Pr > t for H: LSMean(i)=LSMean(j) onferroni omparison Lines for of trtplant*rate LS-means with the same letter are not significantly different. transppm trtplant rate ependent Variable: transppm i/j <. < <. < <. <. <. <. A A <. <..27 <. 5 < <. <. <. 6 <. <. <. <..766 <. <. <. A <. <. <. <. <... A TWO-FATOR FATORIAL WITH TRANSFORMATION A <..27 <. < <. <. <. < A.5 Adjustment for Multiple omparisons: onferroni onferroni omparison Lines for Least A Squares Means of trtplant*rate 2 LS-means with the same letter are not significantly different transppm.5 4 trtplant rate E A.5 3 E F E F F

4.8 Alternate Analysis as a Oneway ANOVA

4.8 Alternate Analysis as a Oneway ANOVA Suppose we have data from a two-factor factorial design. The following method can be used to perform a multiple comparison test to compare treatment means as well