Homework 1 St 711           Due date listed on web.
(just place it on the front desk when you come to class)

A proper randomization scheme would make the probability of the first 
two units getting the same treatment the same as the probability that 
the last two get the same.  

1.  What is the probability of tossing exactly 4 heads (and 4 tails) in 
8 tosses of a fair coin? (you will see later why I asked this) 

2.  Suppose my treatment units are 10 cows on which I will run my 
experiment, getting one observation per cow.   I have 2 treatments, 
perhaps 2 ways of doing the morning milking of a cow, and I will 
assign them as follows.  For cow 1, I toss a coin.  If it comes
up heads, I use treatment 1 and if tails, treatment 2.  I do the 
same on all subsequent cows until I have used one of the 
treatments 5 times. Then I will do the rest of the trials 
with the other treatment.  

Run this program to see how the treatments are assigned in 100 repeats 
of this randomization scheme.  Note that cowj will be the treatment 
assigned to cow j. 

Data random; array cow(10); keep cow1-cow10; 

do trial = 1 to 100; 
sum1=0; sum2=0; 
do unit = 1 to 10; 
  if ranuni(1635409) > 0.5 then do; 
  cow(unit)=1; sum1=sum1+1; end; 
  else do; cow(unit)=2; sum2=sum2+1; end; 
  if sum1 = 5 then do i=unit to 9; cow(i+1)=2; unit=10; end; 
  if sum2 = 5 then do i=unit to 9; cow(i+1)=1; unit=10; end;  
end; output; 
end; 

proc print data=random(obs=10); run; 


How many times (out of 100) did units 1 and 2 get the same treatment?  
How many would you expect under a proper randomization scheme?  
How many times (out of 100) did units 9 and 10 get the same treatment?
(Hint on the above "proper randomization" question: For 10 cows
there are 10!/(5!5!) equally likely assignments into 2 groups 
of 5.  There as many of these starting with HH as there are ways
of  assigning the remaining 3H and 5T letters to 8 cows - 
namely 8!/(3!5!).  )

Hint: You can print everything out and count by hand or you can 
note that X=(A=B); is a logical function returning X=1 when A and 
B are equal and X=0 otherwise. You could use PROC MEANS with this 
idea to count up how many times some condition holds. 

3. What is the probability that I will have to toss the coin when I 
get to unit (cow) 9?
If I have stopped tossing when I get to cow 9 what is the probability 
that units (cows) 9 and 10 will get different treatments? 
If I am tossing for cow 9 what is the probability that units 9 and 10 
will get different treatments?

4. Using the above, compute the probability that cows 9 and 10 will get 
get the same treatment.  How many, then, out of 100 do you expect to 
get the same treatment?  How many did you observe? 

5. In the program, 1635409 is referred to as a "seed" for the random 
number generator.  Maybe our results just occured by accident.  Change
the seed to 15349 and comment on how the 2 counts (first 2 same, last 2 
same) change.  You might also run a few more seeds just to see how 
volatile these numbers are (just write up the 15349 result).

6. In a binomial situation with 100 trials where p is the
probability of some event and f is the observed proportion of times
that event occurred, p(1-p)/100 is the variance of f and 
Z=(f-p)/s is approximately N(0,1) where s is the square root of p(1-p)/100. 
Using the first seed 1635409, find the Zs that go with our two probabilities
(1 & 2 the same, 9 & 10 the same)  Do either of these have |Z|>1.96? Here
f is the fraction of times the event occurred in the simulation and p 
the theoretical probability of occurrence. For example in part 4, you computed 
one of these, the theoretical probablity that cows 9 and 10 are on the same
treatment. 


7. Assigning treatments as described here is not uncommon.  Briefly 
summarize in a few sentences what you have learned about the validity of 
this randomization scheme. 

Note: there may be nothing particularly important about the order
in which the cows enter the milking barn, but then again there may.  The
last cows may be in some ways weaker or more lethargic than those that
enter first (which may or may not affect the milking).  The point of 
randomization is that it guards against any such gradient whereas if 
we do not randomize, we must guarantee that we've thought of all 
possible such effects and none of them matters. 

Optional (not collected)  You might redo question 6 letting p be the expected 
probability under a proper randomization.  In that way you are doing a 
computer experiment to test the hypothesis that the scheme is a valid 
randomization scheme.  You might also try analytically or using the 
computer to figure out how many cows it would take to make the probability 
of the last two getting the same treatment some prechosen value like 0.99.