Linear Combinations of Group Means

Transcription

1 Linear Combinations of Group Means Look at the handicap example on p. 150 of the text. proc means data=mth567.disability; class handicap; var score; proc sort data=mth567.disability; by handicap; proc boxplot data=mth567.disability; plot score*handicap /boxstyle=schematicid; N HANDICAP Obs N Mean Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ AMPUTEE CRUTCHES HEARING NONE WHEELCHAIR SCORE AMPUTEE CRUTCHES HEARING NONE WHEELCHAIR HANDICAP data mth567.dis2; set mth567.disability; if handicap="none" then hand=1; 1

2 if handicap="amputee" then hand=2; if handicap="crutches" then hand=3; if handicap="hearing" then hand=4; if handicap="wheelchair" then hand=5; Give the null and the alternative hypotheses. Dependent Variable: SCORE SCORE The GLM Procedure Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square Coeff Var Root MSE SCORE Mean We reject the null hypothesis. Once we reject the null hypothesis, we start to make inferences about the individual group means. We can do this two ways, with Linear Combinations and Multiple Comparisons. Linear Comparisons: proc glm data=mth567.dis2; class hand; model score=hand /clparm; estimate 'wheel-crutch vs. amp-hearing' hand /divisor=2; Standard Parameter Estimate Error t Value Pr > t wheel-crutch vs. amp-hearing Parameter 95% Confidence Limits wheel-crutch vs. amp-hearing

3 Spock Analysis Linear Combinations So with the Spock data, we can decide if we want to compare the mean percentage of women for the Judge Spock got or the average of the means for all other judges! We make Spocks judge, group #1, and we make the other judges groups 2,3,4,5,6, 7. µ H0: µ 1 = 0 6 µ µ 7 H1: µ Let s review the Spock data: data mth567.spock2; set mth567.spock; if judge="a" then judge2=2; if judge="b" then judge2=3; if judge="c" then judge2=4; if judge="d" then judge2=5; if judge="e" then judge2=6; if judge="f" then judge2=7; if judge="spock's" then judge2=1; proc glm data=mth567.spock2; class judge2; model percent=judge2 / clparm; estimate 'Spock vs all others' judge /divisor=6; PERCENT A B C D E F SPOCK'S JUDGE Standard Parameter Estimate Error t Value Pr > t 95% Confidence Limits Spock vs all others <

4 Multiple Comparison Procedures: If the null hypothesis is rejected, the next logical question is where do the means differ? This question is important and one that statisticians have spent a great deal of time answering. proc glm data=mth567.dis2; class hand; model score=hand; means hand /lsd bon cldiff; t Tests (LSD) for SCORE NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Error Degrees of Freedom 65 Error Mean Square Critical Value of t Least Significant Difference Comparisons significant at the 0.05 level are indicated by ***. Difference hand Between 95% Confidence Comparison Means Limits *** *** *** *** *** ***

5 The GLM Procedure Bonferroni (Dunn) t Tests for SCORE NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than Tukey's for all pairwise comparisons. Error Degrees of Freedom 65 Error Mean Square Critical Value of t Minimum Significant Difference Comparisons significant at the 0.05 level are indicated by ***. proc glm data=mth567.dis2; class hand; model score=hand; means hand /lsd bon ; Difference hand Between Simultaneous 95% Comparison Means Confidence Limits *** ***

6 Means with the same letter are not significantly different. t Grouping Mean N hand A A B A B A B A C B C B C C C Bonferroni (Dunn) t Tests for SCORE NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than REGWQ. Error Degrees of Freedom 65 Error Mean Square Critical Value of t Minimum Significant Difference Means with the same letter are not significantly different. Bon Grouping Mean N hand A A B A B A B A B A B A B B Tukey s Honest Significance Difference Test (HSD) Look at the following experiment where ten guinea pigs in each of 4 groups were fed four different cardiac substances. The main outcome variable was the dosage of the substance when the guinea pigs died. The main goal of the research was to determine if there were any differences among the potencies of the four substances and, if so, to quantify those differences. The data appears below: Substance Substance Substance Substance

7 n = 40, n = 10, k= 4 i Analysis Variable : dosage N substance Obs N Mean Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Analysis Variable : dosage N Mean Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 7

8 dosage substance Dependent Variable: dosage Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total proc glm data=mth567.gpig; class substance; model dosage=substance; means substance /lsd cldiff; R-Square Coeff Var Root MSE dosage Mean Error Degrees of Freedom 36 Error Mean Square Critical Value of t Least Significant Difference Comparisons significant at the 0.05 level are indicated by ***. 8

9 Difference substance Between 95% Confidence Comparison Means Limits *** *** *** *** *** *** Note that the above least significant difference is this means that any pairwise difference in means will be significant. Can tell how far apart the group means needed to be in order to be significantly different. proc glm data=mth567.gpig; class substance; model dosage=substance; means substance /tukey cldiff; NOTE: This test controls the Type I experimentwise error rate. Error Degrees of Freedom 36 Error Mean Square Critical Value of Studentized Range Minimum Significant Difference Comparisons significant at the 0.05 level are indicated by ***. Difference Simultaneous substance Between 95% Confidence Comparison Means Limits *** *** *** *** Note that the mean difference here is in order to become significantly different. 9

10 Tukey s test was originally designed for equal sample sizes, but it has been modified to be approximated with unequal sample sizes. It is then called Tukey-Kramer, but in SAS, it is still Tukey. proc glm data=mth567.spock2; class judge2; model percent=judge2; means judge2 /tukey ; The GLM Procedure Tukey's Studentized Range (HSD) Test for PERCENT NOTE: This test controls the Type I experimentwise error rate. Error Degrees of Freedom 39 Error Mean Square Critical Value of Studentized Range More output of all differences not shown Wrap Up 4. Can do a test to see if the variances (ie standard deviations) appear to be equal in each of the groups. This is called Levene s Test for Equality of variances. proc glm data=mth567.gpig; class substance; model dosage=substance; means substance /hovtest; The GLM Procedure Levene's Test for Homogeneity of dosage Variance ANOVA of Squared Deviations from Group Means Sum of Mean Source DF Squares Square F Value Pr > F substance Error If you feel the variances are not equal, then can use the welch option for unequal variances. proc glm data=mth567.gpig; class substance; model dosage=substance; means substance /welch; 10

11 The GLM Procedure Welch's ANOVA for dosage Source DF F Value Pr > F substance Error