Orthogonal contrasts for a 2x2 factorial design Example p130

Transcription

1 Week 9: Orthogonal comparisons for a 2x2 factorial design. The general two-factor factorial arrangement. Interaction and additivity. ANOVA summary table, tests, CIs. Planned/post-hoc comparisons for the factors or treatments. (ch13) [pp13-153] Orthogonal contrasts for a 2x2 factorial design Example p13 Tabulated statistics: Stress, Diet Rows: Stress D1 Columns: Diet D2 High Low Cell Contents: Cholesterol : Mean Cholesterol : DATA 1

2 Data Display D1_HighStress D2_HighStress D1_LowStress D2_LowStress One-way ANOVA: D1_HighStress, D2_HighStress, D1_LowStress, D2_LowStress Factor Error.8.2 Total S =.11 R-Sq = 98.55% R-Sq(adj) = 97.7% Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev D1_HighStress (---*---) D2_HighStress (---*---) D1_LowStress (---*---) D2_LowStress (---*---) Interaction Plot (data means) for Cholesterol Stress High Low Mean D1 Diet D2 2

3 H : No interaction in the effects of the two factors H : µ µ = µ µ S1D1 S1D2 S2D1 S2D2 H : µ µ µ µ S1D1 S1D2 S2D1 S2D2 H : µ + µ µ + µ SD 1 1 S2D2 SD 1 2 S2D2 ψ = µ µ µ µ int S1D1 S1D2 S2D1 S2D2 Check that the three contrasts are orthogonal. SS (int contrast) =.25 SS(Stress) =.85 SS(Diet) =.5 = = µ SD 1 1 µ = simple effect of diet at low SD 1 2 stress level µ SD 1 1 µ = simple effect of diet at high SD 1 2 stress level 3

4 Note: When the interaction is significant, the tests for main effect hypotheses are of dubious value. P132 CN - illustrative plots are important - interaction between two factors A and B implies that both factors have effects, but the effect of factor A depend on the level of factor B present and vice versa. P133 CN - Three way interaction

5 The model: The General Two Factor Model Observation = fit + error y ijk = µ ij + ε ijk y ijk = µ + µ µ µ µ µ + ε ijk effect of level i of factor A = µ ( µ ) + ( µ ) + ( ij i + µ ) µ i effect of level j of factor B = µ µ j interaction effect at treatment (i,j) = µ ij µ µ + µ i j ε ijk ~ iid N (, σ ) Estimates y ijk = y + ( y i y) + ( y j y) + ( y ij y i y j + y) + ( y ijk y ij ) 5

6 SS decomposition SSTot = SSA + SSB +SS(AB)+SSE It can be shown that SS(AB) = SSTrt SS(A)-SS(B) 6

7 Standard hypotheses: H (1) : no interaction H (2) : there is no main effect of factor A i.e µ µ = for all i i H (3) : there is no main effect of factor B i.e µ µ = for all j j 7

8 Example (Example 3 CN p11) Data Display Row Yield Variety Nitrogen

9 Or I tabulated form: Tabulated statistics: Nitrogen, Variety Rows: Nitrogen Columns: Variety Cell Contents: Yield : DATA 9

10 One-way ANOVA: Yield versus Trt Trt Error Total Residual Plots for Yield 99 Normal Probability Plot of the Residuals 8 Residuals Versus the Fitted Values 9 Percent 5 1 Residual Residual Fitted Value 8 Histogram of the Residuals 8 Residuals Versus the Order of the Data Frequency Residual Residual Observation Order

11 Two-way ANOVA: Yield versus Nitrogen, Variety Nitrogen Variety Interaction Error Total S = 6.53 R-Sq = 92.9% R-Sq(adj) = 86.7% Residual Plots for Yield Normal Probability Plot of the Residuals Residuals Versus the Fitted Values Percent 5 1 Residual Residual Fitted Value 8 Histogram of the Residuals 8 Residuals Versus the Order of the Data Frequency Residual Residual Observation Order Anderson Darling test for normality Probability Plot of RESI1 Normal Percent Mean StDev.697 N 2 AD.63 P-Value RESI

12 Interaction Plot (data means) for Yield 9 8 Variety Mean Nitrogen 25 12

13 Compare the two ANOVA tables above One-way ANOVA: Yield versus Trt Trt Error Total S = 6.53 R-Sq = 92.9% R-Sq(adj) = 86.7% Two-way ANOVA: Yield versus Nitrogen, Variety Nitrogen Variety Interaction Error Total S = 6.53 R-Sq = 92.9% R-Sq(adj) = 86.7% 13

14 Elements of the ANOVA table Tabulated statistics: Nitrogen, Variety Rows: Nitrogen Columns: Variety All All Cell Contents: Yield : Mean Yield : Sum Yield : Standard deviation Count Yield : DATA 1

15 Further analysis Two-way ANOVA: Yield versus Nitrogen, Variety Nitrogen Linear sig Quadratic n.s. Variety vs n.s 2 vs 1 n.s (1,3) vs (2,) sig Interaction Error Total Main Effects Plot (data means) for Yield 75 7 Mean of Yield Nitrogen 25 15

16 CI for the difference between means at two levels of variety (or nitrogen) y y ± t 1 1 i j /2 s n + α n i j Ex Find a 95% CI for the difference between the means of variety 1 and variety 2 Ex Find a 95% CI for the difference between the means of the treatments (V1, 15) and (V2 and 15). 16

17 Fits and residuals from the interaction model ANOVA: Yield versus Nitrogen, Variety Factor Type Levels Values Nitrogen fixed 3 15, 2, 25 Variety fixed 1, 2, 3, Analysis of Variance for Yield Nitrogen Variety Nitrogen*Variety Error Total S = R-Sq = 92.9% R-Sq(adj) = 86.7% Data Display Row Yield Variety Nitrogen RESI1 FITS

18 Additive model ANOVA: Yield versus Nitrogen, Variety Factor Type Levels Values Nitrogen fixed 3 15, 2, 25 Variety fixed 1, 2, 3, Analysis of Variance for Yield Nitrogen Variety Error Total S = R-Sq = 86.58% R-Sq(adj) = 82.86% - Compare with the interaction model ANOVA: Yield versus Nitrogen, Variety Factor Type Levels Values Nitrogen fixed 3 15, 2, 25 Variety fixed 1, 2, 3, Analysis of Variance for Yield Nitrogen Variety Nitrogen*Variety Error Total S = R-Sq = 92.9% R-Sq(adj) = 86.7% 18

19 Residuals from the additive model Data Display Row Yield Variety Nitrogen RESI1 FITS

20 GLM approach for unbalanced designs (eg if some observations are missing) -Use indicator variables for qualitative factors and use GLM approach using regression procedure Regression Analysis: Yield versus Nitrogen, nsq,... The regression equation is Yield = Nitrogen -.31 nsq + 25 v v v v1n v2n v3n v1nsq +.63 v2nsq +.53 v3nsq S = R-Sq = 92.9% R-Sq(adj) = 86.5% Analysis of Variance Regression Residual Error Total Source DF Seq SS Nitrogen nsq v v v v1n v2n 1 5. v3n v1nsq v2nsq v3nsq Note: we should include nitrogen_sq (denoted n2 above ) when there are three levels for that factor 2

21 - Compare with ANOVA approach ANOVA: Yield versus Nitrogen, Variety Factor Type Levels Values Nitrogen fixed 3 15, 2, 25 Variety fixed 1, 2, 3, Analysis of Variance for Yield Nitrogen Variety Nitrogen*Variety Error Total