Runners and Heart Attacks

Date: 01/14/97 at 11:54:21
From: casey beck
Subject: Statistics

Hello,

I have a statistics problem. I tried all last night to figure it out 
and I am just stumped. The problem is:

The influence of running on preventing heart attacks has been studied 
by a local runners club. The following two-way table classifies 350 
people as runners or non-runners, and whether or not they have had a 
heart attack. The variables are runner status (yes or no) and heart 
attack (yes or no). 

                              heart attack
        
                              Yes        No

         Runner          yes   12       112

                         no    36       190

A. Of the two categorical variables, runner and heart attack status, 
   which is the response variable and which is the explanatory 
   variable?

B. Calculate the marginal distributions of runner status and heart 
   attack.

C. Calculate the conditional distribution of heart attacks given 
   runner status.

D. Is there evidence for an association between runner status and 
   heart attacks? Why or why not? (Give relevant percentages to 
   support your answer.)

I hope you can help me out.  I greatly appreciate any time you put 
into it.
 
Thanks,
Casey Beck

Date: 01/29/97 at 18:24:33
From: Doctor Statman
Subject: Re: Statistics

Dear Casey,

The response variable responds to changes in the explanatory variable.
For example, running might help to explain the risk of heart attack, 
but heart attacks wouldn't help to explain exercise, under normal 
circumstances.

Marginal distributions are found by adding across rows or columns, 
that is, by looking in the margins of the table.  For example, the 
marginal probability that a person had a heart attack would be:

 (12+36)/350 = 0.137  (about 14 percent)

Conditional distributions are found by considering only one row (or
column) at a time. For example, the conditional probability that a 
person had a heart attack, given that the person is a runner, can be 
found by considering only the people who are runners:  

 12/(12+112) = 0.097  (about 10 percent)

Telling you someone is a runner (a conditional distribution) reduces 
the probability of a heart attack from the marginal distribution 
value.


Dr. Statman,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/   

Date: 02/03/97 at 07:48:54
From: Doctor Anthony
Subject: Re: Statistics

The contingency table is calculated as follows:  Find the sums of rows 
and columns and then calculate the marginal probabilities assuming 
independence of running and heart attacks.  The table is:

                          Heart Attack                          
                        Yes        No
         Runner  Yes    12        112 | 124
                  No    36        190 | 226      ACTUAL FREQUENCIES
                     -----------------------
               Totals   48        302 | 350

If you look at the totals in the margin of the table we see that out 
of 350 people, 48 had heart attacks and 302 did not.  We also see that 
124 were runners and 226 were not. Assuming independence of these 
factors, the probabilty that someone is a runner and has a heart 
attack would be:
 
   48/350 * 124/350  

The expected number would be 350 times this answer, giving:

   48/350 * 124 = 17  

Since marginal totals must agree, we can fill in the rest of the table 
for expected totals giving:
                     
                        Heart Attack
                       Yes       No
         Runner  Yes   17        107 | 124
                  No   31        195 | 226    EXPECTED FREQUENCIES
                     ----------------------
              Totals   48        302 | 350

From this we see that the expected number of heart attacks for runners 
would be 17 (assuming independence from running) whereas in actual 
fact only 12 had heart attacks. So it seems that heart attacks can be 
avoided to some extent by running.  We can do a Chi-squared test to 
see if our result is significant, and what the level of signifance is.  
I can complete the analysis, but this will require some knowledge of 
the Chi-squared test.     
 
Dr.Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/