data one; input site count; cards; <data here> ; proc glm data=one; class site; model count=site; output out=two residual=ehat predicted=yhat; run; data two; set two; sqrtabsehat=sqrt(abs(ehat)); run; proc univariate plot data=two; var ehat; run; proc plot data=two; plot ehat*yhat; plot sqrtabsehat*yhat; run; data three; set one; y=log(count+1/6); run; proc glm data=three; class site; model y=site; output out=four residual=ehat predicted=yhat; run; data four; set four; sqrtabsehat=sqrt(abs(ehat)); run; proc univariate plot data=four; var ehat; run; proc plot data=four; plot ehat*yhat; plot sqrtabsehat*yhat; run; 1 Histogram 390+* . .* . . . .* . . . .* . . .* . .* .** .** .*** .******** .******************************** .******** .********** -70+****** ----+----+----+----+----+----+-* may represent up to 2 counts Variable: # 1 Boxplot * 2 * 1 * 2 * 1 * 2 4 4 5 16 64 16 20 12 0 0 0 | +--+--+ *-----* +-----+ | | ehat Normal Probability Plot 390+ * | | * * | | | | * | | | | ** | +++ | +++ | *++ | ++ | +++ * | +++ ** | +++ ** | +++ ** | +++ **** | ************* | ****++ | *******++ -70+* * ******** +++ +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 2 Plot of ehat*yhat. Legend: A = 1 obs, B = 2 obs, etc. ehat ‚ 393.28 ˆ A ‚ ‚ ‚ A ‚ A ‚ 327.28 ˆ ‚ ‚ ‚ ‚ ‚ 261.28 ˆ A ‚ ‚ ‚ ‚ ‚ 195.28 ˆ A A ‚ ‚ ‚ ‚ ‚ A 129.28 ˆ ‚ ‚ ‚ ‚ A A ‚ A A B 63.28 ˆ ‚ A A A A ‚ A ‚ A A ‚ BC A ‚ AB D A -2.72 ˆ MJ B B A ‚ HI P D A B ‚ E A ‚ G C ‚ O A ‚ G -68.72 ˆ I Šƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒ 0 10 20 30 40 50 60 70 yhat 3 Plot of sqrtabsehat*yhat. Legend: A = 1 obs, B = 2 obs, etc. sqrtabsehat ‚ ‚ 20.0 ˆ A ‚ ‚ A ‚ A ‚ 17.5 ˆ ‚ ‚ ‚ A ‚ 15.0 ˆ ‚ ‚ A A ‚ ‚ 12.5 ˆ ‚ ‚ A ‚ ‚ 10.0 ˆ ‚ ‚ A A ‚ A A F ‚ H 7.5 ˆ A E ‚ A A N ‚ A A A ‚ E B ‚ A C A 5.0 ˆ D A ‚ BB A ‚ A B A ‚ A J C B ‚ HO I A 2.5 ˆ HC B A ‚ AA A ‚ BA ‚ BA A A ‚ 0.0 ˆ ‚ Šƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒƒ 0 10 20 30 40 50 60 70 yhat 4 The normality assumption is clearly violated, but with so many observations per treatment group (25) that might not be a problem. More troubling is the clear violation of the constant variance assumptions. Thus we seek a transformation to alleviate the non-constant variance problems. Because variation in the residuals is increasing with the estimated mean, we want to try the natural log transform. Because there are some 0 counts, we need to add a small positive constant to every count before taking the natural log. log(y+1/6) works well in this case. Note that the constant variance assumption seems to be approximately satisfied after transformation. The situation regarding normality has improved. Residuals are still non-normal, but this is not a concern due to the large sample sizes within each group. 5 Stem 4 3 3 2 2 1 1 0 0 -0 -0 -1 -1 -2 -2 -3 -3 -4 Leaf 03 557788 033 5557799 0011122333444 666677789 01222344 55566666778889999 01111223333444 3333222222211111110 99999988755 4100 665 1 8888777777666666666 55555 111100000 ----+----+----+----+ Variable: # 2 6 3 7 13 9 8 17 14 19 11 4 3 1 19 5 9 Boxplot | | | | | +-----+ | | | | *--+--* | | | | +-----+ | | | | | | ehat Normal Probability Plot 4.25+ +++ * * | ****** | ** | +** | **** | ***+ | **+ | **** | **** | **** | ***+ | **+ | ++* | ++ * | +****** | ++ | ****** -4.25+* * **+ +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 6 Plot of ehat*yhat. Legend: A = 1 obs, B = 2 obs, etc. ehat ‚ ‚ 6 ˆ ‚ ‚ ‚ ‚ ‚ ‚ A 4 ˆ A ‚ A A B ‚ A A A A ‚ A ‚ A A ‚ B A C ‚ B A C A A 2 ˆ B A B ‚ B A B B ‚ A A A A ‚ A A A A ‚ B A B C A ‚ D B A A A ‚ B A C C B 0 ˆ A E C A A ‚ E D A A ‚ C ‚ B E A A ‚ A A ‚ A A ‚ B -2 ˆ ‚ A ‚ I F ‚ D ‚ ‚ E ‚ -4 ˆ E D ‚ Šˆƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒˆƒ 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 yhat 7 Plot of sqrtabsehat*yhat. Legend: A = 1 obs, B = 2 obs, etc. sqrtabsehat ‚ ‚ 2.25 ˆ ‚ ‚ ‚ A 2.00 ˆ A E D ‚ A A B ‚ A A E ‚ A A 1.75 ˆ A ‚ A D A ‚ I F B ‚ B B A A 1.50 ˆ B A A A ‚ A A B A ‚ A A A ‚ A B 1.25 ˆ B A B A A ‚ A A B ‚ A ‚ A A A A 1.00 ˆ A A A ‚ E B C B ‚ A C ‚ A A A 0.75 ˆ D B ‚ C ‚ A A A ‚ B D 0.50 ˆ A E B ‚ A B ‚ E B ‚ B 0.25 ˆ A A A ‚ ‚ A ‚ 0.00 ˆ ‚ Šƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒ 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 yhat 8