data Lesson6_ex1; 
    input mtd school score semester $; 
    datalines; 
    1 1 68.11 Fall 
    1 1 68.11 Fall 
    1 1 68.21 Fall 
    1 1 78.11 Spring 
    1 1 78.11 Spring 
    1 1 78.19 Spring 
    1 2 59.21 Fall 
    1 2 59.13 Fall 
    1 2 59.11 Fall 
    1 2 70.18 Spring 
    1 2 70.62 Spring 
    1 2 69.11 Spring 
    1 3 64.11 Fall 
    1 3 63.11 Fall 
    1 3 63.24 Fall 
    1 3 63.21 Spring 
    1 3 64.11 Spring 
    1 3 63.11 Spring 
    2 1 84.11 Fall 
    2 1 85.21 Fall 
    2 1 85.15 Fall 
    2 1 85.11 Spring 
    2 1 83.11 Spring 
    2 1 89.21 Spring 
    2 2 93.11 Fall 
    2 2 95.21 Fall 
    2 2 96.11 Fall 
    2 2 95.11 Spring 
    2 2 97.27 Spring 
    2 2 94.11 Spring 
    2 3 90.11 Fall 
    2 3 88.19 Fall 
    2 3 89.21 Fall 
    2 3 90.11 Spring 
    2 3 90.11 Spring 
    2 3 92.21 Spring 
    3 1 74.2 Fall 
    3 1 78.14 Fall 
    3 1 74.12 Fall 
    3 1 87.1 Spring 
    3 1 88.2 Spring 
    3 1 85.1 Spring 
    3 2 74.1 Fall 
    3 2 73.14 Fall 
    3 2 76.21 Fall 
    3 2 72.14 Spring 
    3 2 76.21 Spring 
    3 2 75.1 Spring 
    3 3 80.12 Fall 
    3 3 79.27 Fall 
    3 3 81.15 Fall 
    3 3 85.23 Spring 
    3 3 86.14 Spring 
    3 3 87.19 Spring 
    ; 

1. Name the fixed and random effects.

   Fixed: A with 3 levels. In the EMS column, Q(A) reveals that A is fixed and the df indicates that it has 3 levels. Note that any factor that has a quadratic form associated with it is fixed and Q(A) is the quadratic form associated with A. This actually equals \(\sum_{i=1}^3\alpha_i^2\) where \(i = 1,2, 3\) are the treatment effects and is non-zero if the treatment means are significantly different.

   Random: B is random as indicated by the presence of Var(B). The effect of factor B is studied by sampling 3 cases (see df value for B). A*B is random as any effect involving a random factor is random. The residual is also random as indicated by the presence of the Var(residual) in the EMS column.

2. Complete the Source column of the ANOVA table above.

   Use the EMS column and start from the bottom row. The bottom-most has only Var(residual) and therefore the effect on the corresponding Source is residual. The next row up has Var(A*B) in the additional term indicating that the corresponding source is A*B, etc.

   Type 3 Analysis of Variance
   Source	DF	Sum of Squares	Mean Square	Expected Mean Square	F Value	Pr > F
   A	2	4811.400959	2405.700480	Var(Residual) + 6 Var(A*B) + Q(A)	11.38	0.0224
   B	2	29.274959	14.637480	Var(Residual) + 6 Var(A*B) + 18 Var(B)	1.02	0.3700
   A*B	4	845.784785	211.446196	Var(Residual)+ 6 Var(A*B)	14.68	<.0001
   Residual	45	647.972350	14.399386	Var(Residual)	.	.

3. How many observations are included in this study?

   N-1 = 2 + 2 +4 + 45 = 53 so N = 54. 

4. How many full replicates are there?

   Let r = number of replicates. Then N = number of levels of A * number of levels of B * r = 3 * 3 * r.  Therefore, \(9 \times r = 54\) which gives \(r=6\). 

5. Write the model equation.

   \(y_{ijk}=\mu+\alpha_i+\beta_j+(\alpha\beta)_{ij}+\epsilon_{ijk},\text{ }i,j=1,2,3\text{, and }k = 1,2,\dots,6\)

6. Write the hypotheses that can be tested with the F-statistic information.

   Effect A

   Hypotheses

   \(H_0\colon\alpha_i=0 \text{ for all } i\) vs. \(H_a\colon\alpha_i \neq 0\), for at least one \(i=1,2,3\)
   Note that \(\sum_{i=1}^3\alpha_i^2\) is the non-centrality parameter of the F statistics if \(H_a\) is true.

   F-statistic

   \(\dfrac{2405.700480}{211.446196}=11.377\) with 2 and 4 degrees of freedom

   Effect B

   Hypotheses

   \(H_0\colon\sigma_\beta^2=0\) vs. \(H_a\colon \sigma_\beta^2 > 0\)

   F-statistic

   \(\dfrac{14.63480}{211.446196}=0.0692\) with 2 and 4 degrees of freedom

   Note that SAS uses the unrestricted model (mentioned in Section 6.5) which results in MSAB as the denominator of the F-test (verified by looking at the EMS expressions). 

   Effect A*B

   Hypotheses

   \(H_0\colon\sigma_{\alpha\beta}^2=0\) vs. \(H_a\colon\sigma_{\alpha\beta}^2> 0\)

   F-statistic

   \(\dfrac{211.446196}{14.399386}=14.685\) with 4 and 45 degrees of freedom