Hw 4 Solutions
1. a) The GLM Procedure
Dependent Variable: y
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 4 48568.87638 12142.21909 78.08 <.0001
Error 20 3110.18912 155.50946
C. Total 24 51679.06550
F = 78.08 with 4 and 20 df.
p-value <0.0001
We reject the null hypothesis and state that there is significant evidence that
the mean serum T3 concentration is not the same in the five stages considered in
this experiment.
1b) The GLM Procedure
Tukey's Studentized Range (HSD) Test for y
NOTE: This test controls the Type I experimentwise error rate.
Alpha 0.05
Error Degrees of Freedom 20
Error Mean Square 155.5095
Critical Value of Studentized Range 4.23186
Minimum Significant Difference 23.601
Comparisons significant at the 0.05 level are indicated by ***.
Difference
stage Between Simultaneous 95%
Comparison Means Confidence Limits
3 - 2 93.304 69.703 116.905 ***
3 - 4 95.462 71.861 119.063 ***
3 - 5 118.184 94.583 141.785 ***
3 - 1 119.782 96.181 143.383 ***
2 - 4 2.158 -21.443 25.759
2 - 5 24.880 1.279 48.481 ***
2 - 1 26.478 2.877 50.079 ***
4 - 5 22.722 -0.879 46.323
4 - 1 24.320 0.719 47.921 ***
5 - 1 1.598 -22.003 25.199
Thus, at the 0.05 level, stage 3 is significantly different from all other
stages, stage 2 is different from stages 5 and 1, and stage 4 is different from
stage 1. All other comparisons are not significantly different and may have the
same mean serum T3 concentration.
1c) proc glm;
class stage;
model y=stage;
/*Conduct one-sided tests with alternative H_A:Treatment>Control*/
means stage / alpha=0.05 dunnettu('1');
run;
The GLM Procedure
Dunnett's One-tailed t Tests for y
NOTE: This test controls the Type I experimentwise error for comparisons of all
treatments against a control.
Alpha 0.05
Error Degrees of Freedom 20
Error Mean Square 155.5095
Critical Value of Dunnett's t 2.30445
Minimum Significant Difference 18.175
Comparisons significant at the 0.05 level are indicated by ***.
Difference
stage Between Simultaneous 95%
Comparison Means Confidence Limits
3 - 1 119.782 101.607 Infinity ***
2 - 1 26.478 8.303 Infinity ***
4 - 1 24.320 6.145 Infinity ***
5 - 1 1.598 -16.577 Infinity
Stages 2, 3, and 4 have a significantly greater mean serum T3 concentration
than the control, stage 1.
1d) We can be 95% confident that Stage 3 is the only stage in the set of stages
that could have the highest mean. All other stages are significantly smaller
than stage 3.
1e) Stages 1 and 5 are in the set of stages with the smallest mean. Although
stage 1 has the smallest observed mean, the mean for stage 5 is not
significantly greater than the mean for stage 1. Stages 2, 3, and 4 are
significantly larger than the mean for stage 1. Thus we can rule these stages
out with 95% confidence.
1f) No, there is no significant evidence (p-value = 0.7279) of a difference
between the mean serum T3 concentration corresponding to the stages with the two
smallest means.
2a) No, C1 and C3 do not sum to zero.
2b) No, the sum of (the product of the contrast coefficients divided by the
number of replications per treatment) does not sum to zero. That is
sum[(ki*li)/ri] does not = 0. Since the experiment is assumed to be balanced,
the number of replications is the same for all treatments and we can just sum of
the product of the contrast coefficients.
(1*1) + (3*-1) + (-1*0) + (-1*-1) + (-1*1) = -2 does not equal 0.
2c) There are infinitely many possibilities. A simple one is (1, -1, 0, 0, 0).
(0*1) + (0*-1) + (2*0) + (-1*0) + (-1*0) = 0.