Solution to Practice Questions

Practice Question 1

Consider the following experiment design

a	b	c	Effect
+	-	-	A
-	+	-	B
+	-	+	AC
-	+	+	BC

1. Is the design balanced? Is is orthogonal?

• Balanced: YES
• Othogonal: NO

Check:

doe <- tribble(
    ~a, ~b, ~c,
     1,  0,  0,  
     0,  1,  0, 
     1,  0,  1, 
     0,  1,  1
)

# Check balance
doe %>% count(a)

#> # A tibble: 2 × 2
#>       a     n
#>   <dbl> <int>
#> 1     0     2
#> 2     1     2

doe %>% count(b)

#> # A tibble: 2 × 2
#>       b     n
#>   <dbl> <int>
#> 1     0     2
#> 2     1     2

doe %>% count(c)

#> # A tibble: 2 × 2
#>       c     n
#>   <dbl> <int>
#> 1     0     2
#> 2     1     2

# Check orthogonal
pairs <- doe %>% 
    mutate(
        ab = ifelse(a == b, 1, 0),
        bc = ifelse(b == c, 1, 0),
        ac = ifelse(a == c, 1, 0),
    )

pairs %>% count(ab)

#> # A tibble: 1 × 2
#>      ab     n
#>   <dbl> <int>
#> 1     0     4

pairs %>% count(bc)

#> # A tibble: 2 × 2
#>      bc     n
#>   <dbl> <int>
#> 1     0     2
#> 2     1     2

pairs %>% count(ac)

#> # A tibble: 2 × 2
#>      ac     n
#>   <dbl> <int>
#> 1     0     2
#> 2     1     2

The design is NOT orthogonal because A is always equal to -B.

2. Write out the equation to compute the main effect for a, b, and c. 

\[ ME(a) = \frac{A + AC}{2} - \frac{B + BC}{2} \]

\[ ME(b) = \frac{B + BC}{2} - \frac{A + AC}{2} \]

\[ ME(c) = \frac{AC + BC}{2} - \frac{A + B}{2} \]

3. Are any main effects confounded? If so, what are they confounded with?

In this design, A and B are confounded (A = -B)

To find other confounded effects, multiply the remaining effects by (a = -b):

• b(a = -b) –> ab = -I
• c(a = -b) –> ac = -bc
• ab(a = -b) –> b = -a
• ac(a = -b) –> c = -abc
• bc(a = -b) –> abc = -c
• abc(a = -b) –> bc = -ac