ODDS

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ODDS, ODDS Ratios and Relative Risk
What are ODDS?
Odds, like probability, contain the number of times the event occurred in the
numerator, BUT the odds denominator contains only the number of times the event did
not occur.
The difference between odds and probability may be better understood by thinking of
the chance of drawing an ace from a deck of 52 cards. The probability of drawing an
ace is the number of times an ace will be drawn divided by the total number of cards or
4 of 53 or 1 of 13 (7.69%). The odds, on the other hand, are the number of times an ace
will be drawn divided by the number of times it will not be drawn or 4 to 48 or 1:12.
The odds in favor of an event or a proposition are expressed as the ratio of a pair of
integers or probabilities, such that the first represents the relative likelihood that the
event will happen, and the second, the relative likelihood it won't, as in "the odds that a
randomly chosen day of the week is a Sunday are one to six," which is sometimes
written 1:6, or 1/6. In probability theory and statistics, where the variable "p" is the
probability in favor of the event, and the probability against the event is therefore 1-p,
"the odds" of the event are the quotient of the two, or
. That value may be
regarded as the relative likelihood the event will happen, expressed as a fraction (if it is
less than 1), or a multiple (if it is equal to or greater than one) of the likelihood that the
event will not happen. In the example just given, saying the odds of a Sunday are "one
to six" or, less commonly, "one-sixth" means the probability of picking a Sunday
randomly is one-sixth the probability of not picking a Sunday. While the mathematical
probability of an event has a value in the range from zero to one, "the odds" in favor of
that same event lies between zero and infinity. The odds against the event with
probability given as "p" are
.
Consider the Table below:
SMOKE?
YES
NO
Grand Total
FATHERSMOKE
YES
58
212
270
NO
20
152
172
78
364
442
Assume that “SMOKE?” is the dependent variable which we are trying to explain or
predict. “FATHERSMOKE” is the independent variable which we are using to explain
or predict “SMOKE”. The random probability that someone smokes is 78/442 or
17.65%. The probability that someone smokes, given that their father smokes, is 58/270
or 21.48%. The odds of smoking if your father smokes is 58/212 or .27 to 1. In other
words, if your father smokes, the odds that you do not smoke are 3.65:1.
Note that when reporting odds, the ratio is implied, but never actually computed in its
reduced form. Odds are expressed as “a to b”. PLEASE NOTE THAT ODDS ARE
NOT THE SAME AS PROBABILITIES – THE UNITS ARE NOT PERCENTAGES!!
What are Odds Ratios?
The odds ratio is a way of comparing whether the probability of a certain event is the
same for two groups, where the “event” is the dependent variable and the “groups” are
the values of the independent variable. In the above example, the event of interest is
whether or not someone smokes – the dependent variable. The independent variable of
interest is whether or not the father smokes.
From the above example, your odds of smoking, if your father smokes is .27:1. Your
odds of not smoking, if your father smokes is 3.65:1. Your odds of smoking if your
father does not smoke is 20/152 or .13:1. The odds of not smoking if your father does
not smoke is 7.6:1.
So…the odds ratio of smoking depending on if your father smokes is .27/.13 or 2.08.
The interpretation here is that your odds of smoking are twice as high if your father
smokes versus if your father does not smoke.
An odds ratio of 1 implies that the event is equally likely in both groups. An odds ratio
greater than one implies that the event is more likely in the first group (father smokes).
An odds ratio less than one implies that the event is less likely in the first group.
Typically, the confidence interval for the odds ratio is built to determine if 1 is a
possible outcome of the odds ratio – if it is, the likelihood of the event is not
significantly different between the two groups.
Note that odds ratios are reported as “the odds of x are 1.2 times the odds of y”. So
again, there are no percentages here! Odds ratios only make sense if you understand
the odds.
What is Relative Risk?
A more direct measure comparing the probabilities in two groups is the relative risk,
which is also known as the risk ratio. The relative risk is simply the ratio of the two
conditional probabilities.
From the previous example, if the father smokes, the relative risk of smoking is the
probability of smoking if the father smokes divided by the probability of smoking if the
father does not smoke. Mathematically, this is (58/270)/(20/172) or 21.48%/11.63% or
1.8474. In other words, if your father smokes, you are 1.85 times more likely to smoke
than if your father did not smoke.
Like the odds ratio, a relative risk equal to 1 implies that the event is equally probable in
both groups. A relative risk greater than 1 implies that the event is more likely in the
first group. A relative risk less than 1 implies that the event is less likely in the first
group.
The SAS Code to generate the 2x2 contingency table, the odds ratios, the relative risk is:
Proc freq data=smoke;
tables Fathersmoke*smoke/chisq relrisk riskdiff;
Run;
The output will look like this:
Table of FATHERSMOKE by smoke
FATHERSMOKE(FATHERSMOKE)
smoke(SMOKE?)
Frequency
Percent
Row Pct
Col Pct
1
1
2
Total
2
Total
58
212
270
13.12
47.96
61.09
21.48
78.52
74.36
58.24
20
152
172
4.52
34.39
38.91
11.63
88.37
25.64
41.76
78
364
442
17.65
82.35
100.00
Frequency Missing = 4
DF
Value
Prob
The Chi Square statistic here
Chi-Square
1
7.0195
0.0081
would indicate that there is a
Likelihood Ratio Chi-
1
7.3503
0.0067
1
6.3579
0.0117
1
7.0036
0.0081
Statistic
Square
Continuity Adj. ChiSquare
Mantel-Haenszel ChiSquare
Phi Coefficient
0.1260
Contingency
0.1250
Coefficient
Cramer's V
0.1260
significant relationship
between smoking and
whether or not your father
smokes. What this does not
tell us, is whether you are
more or less likely to smoke
if your father smokes.
Column 1 Risk Estimates
(Asymptotic) 95%
(Exact) 95%
Confidence Limits
Confidence Limits
Risk
ASE
Row 1
0.2148
0.0250
0.1658
0.2638
0.1673
0.2687
Row 2
0.1163
0.0244
0.0684
0.1642
0.0725
0.1739
Total
0.1765
0.0181
0.1409
0.2120
0.1421
0.2153
Difference
0.0985
0.0350
0.0300
0.1671
Difference is (Row 1 - Row 2)
The Column 1 Risk Estimates tell us that, of those whose smoke, 21.48% have fathers who
smoke and 11.63% do not smoke. The difference is 9.85%, where the 95% Confidence Interval
ranges from 3% to 16.71%. Since 0 is not embedded in this interval, the difference is statistically
significant.
Column 2 Risk Estimates
(Asymptotic) 95%
(Exact) 95%
Confidence Limits
Confidence Limits
Risk
ASE
Row 1
0.7852
0.0250
0.7362
0.8342
0.7313
0.8327
Row 2
0.8837
0.0244
0.8358
0.9316
0.8261
0.9275
Total
0.8235
0.0181
0.7880
0.8591
0.7847
0.8579
-0.0985
0.0350
-0.1671
-0.0300
Difference
Difference is (Row 1 - Row 2)
The Column 2 Risk Estimates tell us that, of those whose do not smoke, 78.52% have fathers
who smoke and 88.37% have fathers who do not smoke. The difference is 9.85%, where the 95%
Confidence Interval ranges from 16.71% to 3%. Since 0 is not embedded in this interval, the
difference is statistically significant.
Estimates of the Relative Risk (Row1/Row2)
Type of Study
Value
95% Confidence Limits
Case-Control (Odds Ratio)
2.0792
1.2005
3.6012
Cohort (Col1 Risk)
1.8474
1.1536
2.9585
Cohort (Col2 Risk)
0.8885
0.8180
0.9651
The Estimates of the Relative Risk tell us that the odds ratio for smoking is 2.0792 – where your
odds of smoking are two times greater if your father smokes. Note that this calculation was
generated as – odds of smoking if father smokes/odds of smoking if father does not smoke.
Mathematically this is expressed as (58/212)/(20/152). Since 1 is not embedded in the
Confidence Interval – which ranges from 1.2005 to 3.6012 – we would conclude that the odds of
smoking are statistically higher if the father smokes.
The relative risk of smoking if the father smokes, is 1.8474. This calculation was generated as
the probability of smoking if father smokes/probability of smoking if father does not smoke.
Since 1 is not embedded in the Confidence Interval – which ranges from 1.1536 to 2.9585 – we
would conclude that the relative risk of smoking statistically significant, when the father
smokes versus when he does not smoke.
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