title: Polynomial contrasts in Completely Randomized Block design
body: <p>In the book "Design and analysis of experiments with R", the author used the following data (page 118) to explain methods to analyze Completely Randomized Block (CRB) design:</p>
  
  <pre><code>Rat d0  d0.5    d1  d1.5    d2
  1   0.6 0.8 0.82    0.81    0.5
  2   0.51    0.61    0.79    0.78    0.77
  3   0.62    0.82    0.83    0.8 0.52
  4   0.6 0.95    0.91    0.95    0.7
  5   0.92    0.82    1.04    1.13    1.03
  6   0.63    0.93    1.02    0.96    0.63
  7   0.84    0.74    0.98    0.98    1
  8   0.96    1.24    1.27    1.2 1.06
  9   1.01    1.23    1.3 1.25    1.24
  10  0.95    1.2 1.18    1.23    1.05
  </code></pre>
  
  <p>There are 10 rats (the block) that each reveived 5 dosages (d0 to d2) of durgs (treatment factor) in a randomized order. The response is the frequency of level pressing by each rat after receiving each dosage.</p>
  
  <p>The author first performed the following analysis:</p>
  
  <pre><code>library(daewr)
  &gt; mod1 &lt;- aov( rate ~ rat + dose, data = drug )
  &gt; summary(mod1)
  </code></pre>
  
  <p>The results are as below:</p>
  
  <p><a href="https://i.stack.imgur.com/3NksW.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3NksW.jpg" alt="enter image description here"></a></p>
  
  <p>Since significant effect of dose (treatment factor) is found, the author further stated that "To interpret the differences in treatment factor levels, comparisons of means should be made". Then, the following code is used:</p>
  
  <pre><code>&gt; contrasts(drug$dose) &lt;- contr.poly(5)
  &gt; mod2 &lt;- aov( rate ~ rat + dose, data = drug)
  &gt; summary.aov(mod2,split = list(dose = list("Linear" = 1,
   "Quadratic" = 2,"Cubic" = 3, "Quartic" = 4) ) )
  </code></pre>
  
  <p>The results are shown as below:</p>
  
  <p><a href="https://i.stack.imgur.com/U0Xz1.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/U0Xz1.jpg" alt="enter image description here"></a></p>
  
  <p>The author stated that "There the split option in the summary.aov is used rather than the summary.lm function that was used in Section 2.8, since we only need to see the single degree of freedom partition for the dose factor in the model."</p>
  
  <p>Although the author did not suggest, I still tried <code>&gt; summary.lm(mod2)</code> by myself wishing to see how the results differ from above. And the results are as below:</p>
  
  <p><a href="https://i.stack.imgur.com/WOAlN.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/WOAlN.jpg" alt="enter image description here"></a></p>
  
  <p>Still, the linear and quadratic components are significant.</p>
  
  <p>My questions are:</p>
  
  <ol>
  <li><p>Why does the author need to look at the polynomial component when there is significant dose effect (6.53e-07)? Put another way, what is the additional benefit of knowing the significance of the higher-order terms after knowing that dose does matter? The author seemed to explain the reason as "comparisons of mean", but I don't see any such "comparisons" based on analysis of the polynomial terms.</p></li>
  <li><p>When investigating the polynomial terms, why dose the author say "we only need to see the single degree of freedom partition for the dose factor in the model." What is the difference between output from summary.aov function with split argument and summary.lm function in terms of interpretation of the significance of the polynomial terms?</p></li>
  </ol>
  
  <p>Thank you!</p>