Chapter 12
I’ve been interested in developing a sports-based Stat200 course. Our department naturally is
concerned with the interest students would have in such a course as well as the make-up of it. To
address this issue I posed a survey to all Stat200 students registered at UP. Approximately 45%
of those registered responded. The survey was very simple with two of the questions being,
“What is your gender” and “Would you prefer the current format, either format or a sports format
delivery of Stat200?” The format responses for current/either were combined as the focus was on
specifically prefer versus not specifically prefer the sports course.
(NOTE: to start just look at the counts and inquire as to what students believe. Then point out
that to make comparisons, unless the counts are equal for each condition, is difficult unless we
have percentages.
Tabulated statistics: Gender, Format
Rows: Gender
Columns: Format
Current
Sports
All
Female
321
66.46
259.8
162
33.54
223.2
483
100.00
483.0
Male
111
34.69
172.2
209
65.31
147.8
320
100.00
320.0
All
432
53.80
432.0
371
46.20
371.0
803
100.00
803.0
Cell Contents:
Count
% of Row
Expected count
Pearson Chi-Square = 78.171, DF = 1, P-Value = 0.000
Commonly the explanatory variable is placed on the row and the response variable on the
vertical. So for this data Gender would be explaining the attitude toward monetary support.
Conditional percents for rows are comprised by the taking a cell total divided by the row total;
for columns this conditional percent is taken by cell total divided by column total. For example,
of Females the conditional percent who said “Yes” to the sports format is 162/483 = 33.54%; for
Males this conditional percent who said “Yes” is 209/320 = 65.31%. As to columns, of those that
said “Sports” the conditional percent that were Female is 162/371 = 43.67%
Probability, Risk, and Odds
If we randomly selected a student, what is probability that the student said “Sports”? 371/803 =
0.462
What is the risk that a randomly selected student said “Sports?” Again, 371/803 = 0.462
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What is the proportion of those who said “Sports”? Again, 371/803 = 0.462
What is the percent of those who said “Sports”? Similarly 371/803 = 0.462 times 100% 46.2%
As we can see, these four are equivalently saying the same thing just using different phrasing.
What are the odds a person says sports? This is now 371/ 432 which is approximately 0.88 to 1.
Questions to class:
What is probability that a Male says “Sports”? 209/320 = 0.6531
What is the risk that a Male says “Sports”? 209/320 = 0.6531
How about the proportion and percentage? Proportion also 0.6531 while the percentage is
65.31%
What are the odds that a Male says Yes? 209/111 = 1.89 or approximately 2 to 1. How are the
odds interpreted? We would say then that the odds of a Male saying “Sports” is 2 to 1 or 2 of
every 3 would opt for Sports.
Relative Risk is when two risks are compared. For instance the relative risk that Females say
Sports to Males that say Sports would be calculated by the ratio of risk for Females saying Sports
to the risk of Males saying Sports. The Female-Sports risk is 162/483 and the Male-Sports risk is
209/320. This would make the relative risk (162/483)/( 209/320). This computes to
0.3354/0.6531 = 0.51 The interpretation is that the probability that a student prefers Sports
format is 0.51 times higher for Females than for Males. Conversely, Males are two times more
likely (i.e. invert the relative risk) to prefer the Sports format than Females.
Baseline risk is when to which another risk is compared to in a relative risk. That is it is the risk
in the denominator of the relative risk. In the previous example, Female risk of Sports was
compare to the Male risk of Sports putting the male risk in the denominator. This makes the
baseline risk the risk of Males saying Sports.
Odds ratio is similar to relative risk except it is the ratio of two odds. The odds ratio that
Females say Sports to Males say Sports is (162/321)/ (209/111) = 7.4 meaning the odds of
Females saying Sports is about 0.27 times the odds of Males saying Sports. Conversely, (i.e.
invert the odds ratio) the odds a Male prefers the sports format is 3.7 times the odds a Female
prefers the sports format.
Important To Interpret
When reading a report that provides a risk it is extremely important to know or be given the
baseline risk. For instance, say a study reported that women who binge drink are 3 times more
likely to develop liver disease than women who do not drink. This may alarm some females,
understandably. But what if the risk of getting liver disease for women who do not binge drink
(i.e. the baseline risk) was 0.001 or 1 out of a 1000 women who do not binge drink are likely to
develop liver disease. This would mean that risk of females who do binge drink developing liver
disease is 3/1000 or 0.003 Not that alarming!
To calculate:
Odds are the (number of interest with trait)/(number of interest without trait)
Risk is the (number of interest with trait)/(over total number of interest)
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