Test 2 St 711, 2011, Dickey
I. (10 pts.) In the term “Balanced Incomplete Block Design” what does the word “balanced” mean, that
is, what makes an incomplete block design balanced?
II.(15 pts.) I randomly assign 3 feeds to sets of 30 cows (10 per feed). I measure Y= butterfat content in
their milk for 4 consecutive weeks in each cow thus having a repeated measures design with 120
observations. My covariance matrix R is
 20

 12
 12

 12
(1)
12 12 12 

20 12 12 
12 20 12 

12 12 20 
Find, if possible the variance of Y ________
(2)
Find, if possible, the cow variance component __________
(3)
Find, if possible the variance ________of the difference two Y values for different weeks within
the same cow.
III. (8 pts.) In the discussion of balanced incomplete block designs, we mentioned lattice designs in
which sets of incomplete blocks (like fields) with =1 combine or “resolve” into larger sets or
“superblocks” like farms with certain properties.
(A) What characterizes these larger sets?
(B) In comparing two treatments, what (if anything) is the advantage of such a lattice versus just
having the incomplete blocks?
IV. A split plot in a completely randomized design has two levels of the whole plot factor (A) and three
levels of the split plot factor (B).
(40 pts.) Here is a table of treatment totals, each being a total of 12 observations. Write down in the
ANOVA table, as many degrees of freedom and sums of squares as you can from the given data.
B low
B medium
B high
A low
24
60
36
A high
36
48
12
Source
df
Sum of Squares
A
______
________________
Error 1
______
________________
B
______
_________________
AB
______
_________________
Error 2
_______
________________
(12 pts.) Assuming sums of squares 100 for any entries you could not compute, give the calculated F test
for A main effects (F=__________ ) and for B main effects (F = _________)
(9 pts.) Using the linear orthogonal polynomial coefficients -1, 0, 1 and assuming the levels of B are
equally spaced, compute the sum of squares for the linear effect of B __________________________V. (6 pts.) Is it possible to construct a Youden square design with 6 rows, 10 columns, and 10
treatments? If so, how would you construct it (assuming you start with a 10*10 Latin Square)? If not,
explain how you know.
*******************Answers*****************
(1) Every pair of treatments appears together in blocks equally often.
(2) 20, 12, 2(8)=16
(3) The sets of incomplete blocks that form the larger superblocks contain one of each treatment, that
is, all treatments appear in each superblock (farm for example)
In the lattice case, the superblock effects, farm effects for example, drop out so we have a smaller
variance and higher power.
(4)
B low
B medium
B high
A low
24
60
36
sum 120 SS(A)= 1202/36 + 962 /36 - 2162/72=8
A high
36
48
12
sum 96
Sums
60
108
48
(216) overall mean is 3, 72 obs, 3 per whole plot
SS(B) = (602 +1082+482)/24-2162/72 = 84
SS(table) =( 242 + … + 122)/12-2162/72=120
Source
df
Sum of Squares
A
__1_____
_______8__________
Error 1
__22_____
______(100)_______
B
___2____
_______84__________
AB
___2____
_______28___________(120-8-84)
Error 2
___44____
______(100)_________
F for A is (8/1)/(100/22)=1.76 and for B it is 42/(100/44)=18.48
For linear, 48-60=-12 so 144 when squared. The denominator is (1+0+1)(24) so sum of squares is
144/48=3 for linear B effect (1 df). Note (not asked for) that F is then (144/48)/(100/44)=1.32.
(5) Columns of a Youden square form a balanced incomplete block design so we must have
r(k-1)(t-1). With t=10 treatments and 60 observations r is 6 and so 6(5)= (9). There is no integer 
that satisfies this. It cannot be done.