4441.0.55.002 - A Comparison of Volunteering Rates from the 2006 Census of Population and Housing and the 2006 General Social Survey, Jun 2012  
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D. ODDS, ODDS RATIOS AND PREDICTED PROBABILITY

ODDS

The odds of an outcome is the ratio of the expected number of times the event will occur to the expected number of times the event will not occur. Put simply, the odds are the ratio of the probability of an event occurring to the probability of no event. In our case, we can calculate the odds that our base case is a volunteer according to the following ratio:

Odds for volunteering =
Probability of being a volunteer
Probability of being a non-volunteer
=
P [Y = 1]
1 - P [Y = 1]

Applying a logistic regression to the GSS data, the odds of being a volunteer for the base case is estimated at 0.35.

ODDS RATIO

This is the ratio of two odds. In our example, the odds ratio compares the odds of volunteering by a female to the odds of volunteering by a male. Imagine a person who has all the same characteristics as the base case except that she is female. To work out the effect of being female on being a volunteer we can calculate an odds ratio using the odds for each of the cases as follows:


Odds ratio for female =
Odds that a female is a volunteer
= 1.28
Odds that a male is a volunteer


If the odds ratio equals 1, then men and women are equally likely to be volunteers. An odds ratio of less than 1 would suggest that, all other things being equal, women are less likely than men to be volunteers. In our case, the odds ratio for women is 1.28 suggesting that women are more likely to be volunteers than men. Put differently, a woman’s odds of volunteering are 28% larger than a man’s.

PREDICTED PROBABILITY OF VOLUNTEERISM

Once we obtain the odds ratio, we can convert this odds ratio into predicted probability by simple algebraic transformation. The relationship between the odds ratio and probability is as follows:


Odds = exp(b0+b1x1+...+bpxp) =
P [Y=1]
1- P [Y=1]


Simplifying the notation:


Odds =
P
1 - P


Cross multiplying and rearranging the above:


Odds - P x Odds =
P
P + P x Odds =
Odds
P (1 + Odds) =
Odds
P =
Odds
1 + Odds


Continuing from the example above, the odds for a female to be a volunteer is:

Odds for volunteering x Odds ratio for female =
0.35 x 1.28 = 0.45

This odds then can be translated into the predicted probability by:


P =
Odds
=
0.45
= 0.31, or 31%
1 + Odds
1 + 0.45