**Stat 330 Assignment 9 Partial Solutions
**

- Chapter 11 Q4: I ran SAS and, after editing, I got the following
output:
General Linear Models Procedure Dependent Variable: COVER Sum of Mean Source DF Squares Square F Value Pr > F PAINT 3 296.25000000 98.75000000 10.97 0.0075 ROLLER 2 4.66666667 2.33333333 0.26 0.7798 Error 6 54.00000000 9.00000000 Corrected Total 11 354.91666667 R-Square C.V. Root MSE COVER Mean 0.847852 6.581353 3.0000000 45.583333 Tukey's Studentized Range (HSD) Test for variable: COVER Alpha= 0.05 Confidence= 0.95 df= 6 MSE= 9 Critical Value of Studentized Range= 4.896 Minimum Significant Difference= 8.4794 Comparisons significant at the 0.05 level are indicated by '***'. Simultaneous Simultaneous Lower Difference Upper PAINT Confidence Between Confidence Comparison Limit Means Limit 1 - 2 -0.479 8.000 16.479 1 - 3 3.521 12.000 20.479 *** 1 - 4 3.854 12.333 20.813 *** 2 - 3 -4.479 4.000 12.479 2 - 4 -4.146 4.333 12.813 3 - 4 -8.146 0.333 8.813

I see a clear effect of paint brand but no visible effect of roller brand. Brand 1 is better than 3 or 4 but not definitely better than 2. However, even that difference is nearly significant. - Chapter 11 Q 6: We find
*MSA*=11.7/2=5.85,*MSB*=113.5/4=28.375 and*MSE*= 25.6/8=3.2. The*F*statistic for the hypothesis of no difference between assessors is 5.85/3.2 =1.83 which is not significant at the 5% level (the critical value is 4.46. The design is chosen to make sure that variations between values for different assessors are not due to house value differences. A design in which the different assessors assessed different houses would be much less sensitive to small differences between the assessors because the variation in value from house to house is large compared to the likely size of the variation from assessor to assessor. Note that the effect due to houses is large and statistically significant but that no one would test this hypothesis since we all know different houses have different values. - Q 14:
Source SS df MS F P A 30763 2 15381.5 3.79 0.037 B 34185.6 3 11728.5 2.81 0.061 A*B 43581.2 6 7263.5 1.79 0.144 Error 97436.8 24 4059.9 Total 205966.6 35

The interactions are not significant. The main effect of Factor A is marginally significant while that of B is marginally not so. Generally it seems likely that curing time has an effect on compressive strength and that Factor B might do too. The Tukey intervals for , and are all estimate plus or minus (2.92)(63.7)/ . (The number 63.7 is just .) NOTE: this is a typical exam type question. - Q16: I got the following from SAS. It shows no real evidence of
interactions (
*P*=0.25) and significant main effects of both formula and speed. It shows that the speed 70 gives a significantly lower yield than either the lower or higher speed. To get estimates of the main effects you need to average the columns and subtract the grand mean or average the top 9 numbers and bottom 9 numbers in the table and then subtract the grand mean. I did not produce the probability plot though I think you know how to do so with SAS.General Linear Models Procedure Dependent Variable: YIELD Sum of Mean Source DF Squares Square F Value Pr > F FORMULA 1 2253.4422222 2253.4422222 376.27 0.0001 SPEED 2 230.8144444 115.4072222 19.27 0.0002 FORMULA*SPEED 2 18.5811111 9.2905556 1.55 0.2516 Error 12 71.8666667 5.9888889 Corrected Total 17 2574.7044444 R-Square C.V. Root MSE YIELD Mean 0.972087 1.391696 2.4472206 175.84444 Tukey's Studentized Range (HSD) Test for variable: YIELD Alpha= 0.05 Confidence= 0.95 df= 12 MSE= 5.988889 Critical Value of Studentized Range= 3.773 Minimum Significant Difference= 3.7693 Comparisons significant at the 0.05 level are indicated by '***'. Simultaneous Simultaneous Lower Difference Upper SPEED Confidence Between Confidence Comparison Limit Means Limit 80 - 60 -2.719 1.050 4.819 80 - 70 4.297 8.067 11.836 *** 70 - 60 -10.786 -7.017 -3.247 ***

- Q48: I used proc glm with the statement
`model smooth = method fabric`to get the following output which shows a very clear effect of drying method. There is no need to look at the effect of fabric; as a blocking variable it would be surprising if it did not have an effect. Tukey's procedure shows that drying methods are divided into two groups: methods 1 and 3 giving significantly less smoothness than 2, 4 or 5. Note that I have rearranged the SAS output to match the form in the text.General Linear Models Procedure Dependent Variable: SMOOTH Sum of Mean Source DF Squares Square F Value Pr > F FABRIC 8 9.69600000 1.21200000 11.89 0.0001 METHOD 4 14.96222222 3.74055556 36.70 0.0001 Error 32 3.26177778 0.10193056 Corrected Total 44 27.92000000 R-Square C.V. Root MSE SMOOTH Mean 0.883174 12.94320 0.3192657 2.4666667 Tukey's Studentized Range (HSD) Test for variable: SMOOTH Alpha= 0.05 df= 32 MSE= 0.101931 Critical Value of Studentized Range= 4.086 Minimum Significant Difference= 0.4349 Means with the same letter are not significantly different. Tukey Grouping Mean N METHOD A 3.3556 9 1 A A 2.9556 9 3 B 2.0222 9 4 B B 2.0111 9 5 B B 1.9889 9 2

- Q 50: Most students will simply have done a two way anova on this
data set and found no significant effect of Sowing Rate. However, the very
high variability within plot 1 and low variability within plots 3 and 4
suggests that the assumption of constant is probably wrong. There
is a test, called Tukey's one degree of freedom test for non-additivity
which would have suggested a transformation is needed. I analyzed the
logarithms of the clover accumulations and concluded that there probably is
a difference. First the SAS code:
options pagesize=60 linesize=80; data Q50; infile 'q50.dat'; input plot rate clover; logcl=log(clover); proc glm data=Q50; class plot rate; model logcl = plot rate; means rate / tukey cldiff alpha=0.05; run;

and some of the output:General Linear Models Procedure Dependent Variable: LOGCL Sum of Mean Source DF Squares Square F Value Pr > F Model 6 24.064 4.0107 19.91 0.0001 Error 9 1.813 0.2014 Corrected Total 15 25.877 Root MSE LOGCL Mean 0.4488151 6.1196277 Source DF Type I SS Mean Square F Value Pr > F PLOT 3 16.740 5.58 27.70 0.0001 RATE 3 7.324 2.44 12.12 0.0016 Tukey's Studentized Range (HSD) Test for variable: LOGCL Alpha= 0.05 Confidence= 0.95 df= 9 MSE= 0.201435 Critical Value of Studentized Range= 4.415 Minimum Significant Difference= 0.9907 Simultaneous Simultaneous Lower Difference Upper RATE Confidence Between Confidence Comparison Limit Means Limit 13.5 - 10.2 -1.0257 -0.0350 0.9558 13.5 - 6.6 -0.3236 0.6671 1.6579 13.5 - 3.6 0.6426 1.6333 2.6241 *** 10.2 - 6.6 -0.2886 0.7021 1.6929 10.2 - 3.6 0.6776 1.6683 2.6590 *** 6.6 - 3.6 -0.0246 0.9662 1.9569

I have rearranged things. Note that the procedure analyzes means of the logarithm not of the original variable. The conclusions are that there is an effect to Sowing Rate and that the lowest level is definitely worse than either of the two highest levels at producing clover. To get the same analysis on the original scale you drop mention of logcl and put clover in the model statement.

Wed Apr 1 15:31:24 PST 1998