proc plot; plot Mean_Illness*Dose=Dose; run;

Transcription

1 options pageno=min nodate formdlim='-'; Title 'Illness Related to Dose of Therapeutic Drug'; run; data Lotus; input Dose N; Do I=1 to N; Input Illness output; end; cards; proc sort; by Dose; proc means noprint; output out=sol mean=mean_illness; var Illness; by Dose; proc plot; plot Mean_Illness*Dose=Dose; run;

2 Clearly the effect of dose is not linear. Illness Related to Dose of Therapeutic Drug Plot of Mean_Illness*Dose. Symbol is value of Dose. Mean_Illness 105 ˆ ˆ 95 ˆ 3 90 ˆ 85 ˆ ˆ Šƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒ

3 Dose Title 'One-Way ANOVA With Welch Test, Pairwise Comparisons, and Orthogonal Contrasts'; run; proc GLM data=lotus; class Dose; model Illness = Dose / ss1 EFFECTSIZE alpha=0.1; means Dose / snk tukey regwq; means Dose / hovtest=obrien hovtest=levene welch; contrast 'Placebo vs Drug' dose ; contrast 'Lo vs Hi' dose ; contrast '10 vs 20' dose ; contrast '30 vs 40' dose ; run; By default, SAS uses the last created data set. In this case, that would be the means in Sol but we need the scores, so we point back to Lotus. One-Way ANOVA With Welch Test, Pairwise Comparisons, and Orthogonal Contrasts The GLM Procedure Class Level Information Class Levels Values Dose Number of Observations Read 100 Number of Observations Used 100

4 The GLM Procedure Dependent Variable: Illness Source DF Sum of Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE Illness Mean The R-Square here is more commonly known as eta-squared. It is simply the ratio of the model SS to the corrected total SS. Overall Noncentrality Min Var Unbiased Estimate Low MSE Estimate % Confidence Limits (48.022,119.29) Proportion of Variation Accounted for Eta-Square 0.47 Omega-Square % Confidence Limits (0.32,0.54) Omega-squared is a less biased estimator of population 2 than is sample eta-squared. Regardless of which point estimate you report, the confidence interval is the same. I have used a confidence coefficient of (1-2 ) rather than (1 - ) so that the confidence interval will be consistent with the test of significance. If I were to use (1- ) it would be possible that the F test would be significant but the confidence interval would include the value 0. Source DF Type I SS Mean Square F Value Pr > F Semipartial Eta-Square Total Variation Accounted For Semipartial Omega-Square Conservative 90% Confidence Limits Dose < Partial Variation Accounted For Partial Eta-Square Partial Omega-Square 90% Confidence Limits

5 The plot above is a schematic plot. I find it a very useful way to show groups differences in location, dispersion, and shape. It is not included in a plain-text listing, but is included in the htm output.

6 Student-Newman-Keuls Test for Illness Note: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 95 Error Mean Square Number of Means Critical Range Means with the same letter are not significantly different. SNK Grouping Mean N Dose A A A B C C C Notice that when you have asked for pairwise comparisons you do not get group standard deviations. SNK shows that illness with the 30 mg/kg dose is significantly less than with 0 or 40 mg/kg and significantly more than with 10 or 20 mg/kg.

7 The GLM Procedure Ryan-Einot-Gabriel-Welsch Multiple Range Test for Illness Note: This test controls the Type I experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 95 Error Mean Square Number of Means Critical Range Means with the same letter are not significantly different. REGWQ Grouping Mean N Dose A A A B C C C

8 The GLM Procedure Tukey's Studentized Range (HSD) Test for Illness Note: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than REGWQ. Alpha 0.05 Error Degrees of Freedom 95 Error Mean Square Critical Value of Studentized Range Minimum Significant Difference Means with the same letter are not significantly different. Tukey Grouping Mean N Dose A A A B B C B C C Critical values for difference between means: The larger the critical value, the more conservative the test (the less the power). Notice that as you move from the outer layer to the inner layers the SNK is more powerful than the REGWQ and the REGWQ more powerful than the HSD. Number of Means Spanned Technique SNK REGWQ Tukey HSD

9 The GLM Procedure Levene's Test for Homogeneity of Illness Variance ANOVA of Squared Deviations from Group Means Source DF Sum of Squares Mean Square F Value Pr > F Dose Error O'Brien's Test for Homogeneity of Illness Variance ANOVA of O'Brien's Spread Variable, W = 0.5 Source DF Sum of Squares Mean Square F Value Pr > F Dose Error I recommend these tests only if you have a genuine interest in whether or not the groups differs in variance. Do not use these tests to determine whether or not the differences in variances are so great that you need to use a Welch test. When sample sizes are large, these tests will find significant differences even when the departure from homogeneity of variance is so small as to be of no concern. When sample sizes are small, these tests may not be significant even with the variances differ so much that one should use a Welch test. Welch's ANOVA for Illness Source DF F Value Pr > F Dose <.0001 Error

10 Since it did not include a request for pairwise comparisons, the second MEANS statement does include group standard deviations. Level of Dose N Mean Illness Std Dev

11 Dependent Variable: Illness Contrast DF Contrast SS Mean Square Placebo vs Drug The GLM Procedure F Value Pr > F Total Variation Accounted For Semipartial Eta-Square Semipartial Omega- Square Conservative 90% Confidence Limits < Lo vs Hi < vs vs Partial Variation Accounted For Partial Eta-Square Partial Omega-Square 90% Confidence Limits The semipartial coefficients tell us the portion of the total sum of squares that is explained by one contrast. For example, for Low versus High, The partial coefficients excludes from the denominator variance that is explained by other contrasts. That is, of the variance that is not explained by any of the other contrasts, what proportion is explained by this contrast. For example, for Low versus High, 2 SScontrast p Please do note that the (1-2 ) confidence SS SS contrast error interval for partial 2 will be consistent with the F test, but the (1-2 ) confidence interval for semipartial 2 may not.

12 Title 'One-Way ANOVA with Trend Contrasts'; run; proc GLM data=lotus; class Dose; model Illness = Dose / ss1 EFFECTSIZE alpha=0.1; CONTRAST 'Linear' Dose ; CONTRAST 'Quadratic' Dose ; CONTRAST 'Cubic' Dose ; CONTRAST 'Quartic' Dose ; run; One-Way ANOVA with Trend Contrasts The GLM Procedure Class Level Information Class Levels Values Dose Number of Observations Read 100 Number of Observations Used 100

13 The GLM Procedure Dependent Variable: Illness Source DF Sum of Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE Illness Mean Overall Noncentrality Min Var Unbiased Estimate Low MSE Estimate % Confidence Limits (48.022,119.29) Proportion of Variation Accounted for Eta-Square 0.47 Omega-Square % Confidence Limits (0.32,0.54) Contrast DF Contrast SS Mean Square F Value Pr > F Total Variation Accounted For Semipartial Eta-Square Semipartial Omega- Square Conservative 90% Confidence Limits Linear Quadratic < Cubic Quartic Partial Variation Accounted For Partial Eta-Square Partial Omega-Square 90% Confidence Limits

14 As noted earlier, the (1-2 ) confidence interval for partial 2 will be consistent with the F test, but the (1-2 ) confidence interval for semipartial 2 may not. Look at the statistics for the cubic effect. The F test is significant, and the confidence interval for partial 2 excludes zero, but the confidence interval for semipartial 2 includes zero. Title 'Polynomial Regression'; run; data Polynomial; set Lotus; Quadratic=Dose*Dose; Cubic=Dose**3; Quartic=Dose**4; proc GLM data=polynomial; model Illness = Dose Quadratic Cubic Quartic / ss1 EFFECTSIZE alpha=0.1; run; Polynomial Regression The GLM Procedure Number of Observations Read 100 Number of Observations Used 100

15 Polynomial Regression The GLM Procedure Dependent Variable: Illness Source DF Sum of Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE Illness Mean Overall Noncentrality Min Var Unbiased Estimate Low MSE Estimate % Confidence Limits (48.022,119.29) Proportion of Variation Accounted for Eta-Square 0.47 Omega-Square % Confidence Limits (0.32,0.54) Source DF Type I SS Mean Square F Value Pr > F Total Variation Accounted For Semipartial Eta-Square Semipartial Omega- Square Conservative 90% Confidence Limits Dose Quadratic < Cubic Quartic Partial Variation Accounted For Partial Eta-Square Partial Omega-Square 90% Confidence Limits

16 Ignore the table below, these tests are based on Type III sums of squares and are not appropriate. Parameter Estimate Standard Error t Value Pr > t Intercept <.0001 Dose Quadratic Cubic Quartic title 'Polynomial Regression with Quartic Component Dropped from the Model'; run; proc GLM data=polynomial; model Illness = Dose Quadratic Cubic / ss1; run; Polynomial Regression with Quartic Component Dropped from the Model The GLM Procedure Number of Observations Read 100 Number of Observations Used 100

17 The GLM Procedure Dependent Variable: Illness Source DF Sum of Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE Illness Mean Source DF Type I SS Mean Square F Value Pr > F Dose Quadratic <.0001 Cubic Data CI;t= ; df = 95 ; n1 = 20 ; n2 = 20 ; n3 = 20 ; n4 = 20 ; n5 = 20 ; c1 = -1 ; c2 =.25 ; c3 =.25 ; c4 =.25 ; c5 =.25 ; ncp_lower = TNONCT(t,df,.975); ncp_upper = TNONCT(t,df,.025); d_lower = ncp_lower*sqrt(c1*c1/n1 + c2*c2/n2 + c3*c3/n3 + c4*c4/n4 + c5*c5/n5); d_upper = ncp_upper*sqrt(c1*c1/n1 + c2*c2/n2 + c3*c3/n3 + c4*c4/n4 + c5*c5/n5); output; run; proc print; var d_lower d_upper; title 'Confidence Interval on Standardized Contrast'; run; Confidence Interval on Standardized Contrast Obs d_lower d_upper proc gplot data=sol; symbol1 interpol=join width=4 value=triangle height=2 color=blue; plot Mean_Illness*Dose; run; quit;

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