1. Topic 21: ANOVA and Linear Regression

2. Outline Review cell means and factor effects models Relationship between factor effects constraint and explanatory variables

3. Cell Means Model Y ij = μ i + ε ij –where μ i is the theoretical mean or expected value of all observations at level i –The ε ij are iid N(0, σ 2 ) –Y ij ~N(μ i, σ 2 ), independent

4. Factor Effects Model A reparameterization of the cell means model Useful way at looking at more complicated models Null hypotheses are easier to state Y ij = μ +  i + ε ij –the ε ij are iid N(0, σ 2 )

5. Parameters The cell means model has r+1 parameters – r μ’s and σ 2 The factor effects model has r+2 parameters – μ, the r  ’s, and σ 2 Build restriction on  ’s in factor effects model to remove one degree of freedom (e.g., Σ i  i = 0 or  r = 0)

6. Regression Approach We can use multiple regression to reproduce the results based on the factor effects model Y ij = μ +  i +  ij and we will restrict Σ i  i = 0

7. Coding for Explanatory Variables Σ i  i = 0 implies  r = -  1 -  2 -…-  r-1 Due to restriction, i = 1 to r-1 columns X ij = 1 if Y is observation from level i = -1 if Y is observation at level r = 0 if Y is from any other level

8. KNNL Example Recall KNNL p 687 from Topic 20 It is a bit messy because n i = 5, 5, 4, 5 The grand mean is not necessarily the same as the mean of the group means (i.e., μ = (Σ i n i μ i )/n T ) We will calculate these two values You will have an easier example in the homework (n i is constant) where they are the same value

9. Means proc means data=a1 noprint; class design; var cases; output out=a2 mean=mclass; run; proc print data=a2; run;

10. Output Obs des _TYPE_ _FREQ_ mclass 1. 0 19 18.6316 2 1 1 5 14.6000 3 2 1 5 13.4000 4 3 1 4 19.5000 5 4 1 5 27.2000 Grand sample mean…not the average of the four trt sample means shown below it

11. The mean of the means proc means data=a2 mean; where _TYPE_ eq 1; var mclass; run;

12. Output The MEANS Procedure Analysis Variable : mclass Mean ƒƒƒƒƒƒƒƒƒƒƒƒ 18.6750000 ƒƒƒƒƒƒƒƒƒƒƒƒ Not a big difference from grand sample mean in this example

13. Generate explanatory variables for REG data a1; set a1; x1=(design eq 1)-(design eq 4); x2=(design eq 2)-(design eq 4); x3=(design eq 3)-(design eq 4); proc print data=a1; run;

14. Output Obs cases design x1 x2 x3 1 11 1 1 0 0 6 12 2 0 1 0 11 23 3 0 0 1 15 27 4 -1 -1 -1

15. Output with parameters des x1 x2 x3 1 1 0 0 μ +  1 2 0 1 0 μ +  2 3 0 0 1 μ +  3 4 -1 -1 -1 μ -  1 -  2 -  3  is the result of including an intercept

16. Run the regression proc reg data=a1; model cases=x1 x2 x3; run;

17. Output Anova Source DF SS MS F P Model 3 588 196 18.59 <.0001 Error 15 158 10 Total 18 746 Same ANOVA table as GLM

18. Regression coefficients Var Est Int 18.675 mean of the means x1 -4.075 Y 1./n 1 - Int x2 -5.275 Y 2./n 2 - Int x3 0.825 Y 3./n 3 - Int 18.675-4.075 = 14.6 18.675-5.275 = 13.4 18.675+0.825 = 19.5 18.675+4.075+5.275-0.825=27.2 Get same trt means

19. Last slide Read KNNL Chapter 16 We used program topic21.sas to generate the output for today