Key to the final homework
1) Logit regression for syncopal signs on age gives the following results:
L = 2.921 - 0.106 (Age)
The intercept and coefficient on age are statistically insignificant
at the a = 0.05 level. However, to conclude that age has no effect on
syncopal signs is premature due to the fact that such a conclusion
would be based on only 8 observations. More data needs to be
collected.
The R squared given by JMP in logit regression is :
-log likelihood Model / - log likelihood C total. It is still a
measure of goodness of fit where R squared = 1 is perfect, and R
squared = 0 is no fit at all.
The value of 0.2032 is not
particularly high but given what we are trying to measure, it is not
bad.
For full marks, needed some kind of interpretation of the intercept
and/or the coefficient on age.
2) At first glance, the graph seems to suggest that as age increases,
the probability of showing syncopal signs declines. However, JMP codes
the probability of failure ie. The probability of showing no signs,
hence the opposite is true.
The logistic regression model
predicts probabilities as an S shaped function of X values and thereby
do not exceed the {0, 1} boundaries.
3) In terms of log odds, b1=0.04 means that an increase of $1000 in
land value, with all other things remaining constant, will increase
the log odds of responding "yes" to the groundwater protection plan by
0.04; b2=-0.03 means that an increase of 1 year in age, with all other
things remaining constant, will decrease the log odds of responding
"yes" by 0.03.
In terms of odds, b1=0.04 represents that, with a $1000 increase in
land value, the odds of saying "yes" are multiplied by exp{0.04}, which
is 1.0408; b2=-0.03 says that, with an increase in 1 year, the odds of
saying "yes" are multiplied by exp{-0.03}, which is 0.9704.
4) By plugging into the regression equation with the means of all the
variables except for INCOME, we get the following equation:
L( i )=-2.3032 + 0.00002 X(i3)
By choosing different values of X3, we get different corresponding L;
then transform the log likelihoods L into probabilities with :
P = 1/(1+exp{-L})
Then graph the probabilities against incomes. Form the graph you could
see that the willingness to pay increases as income
increases.
5)In a similar way, you could get the graphs easily. You should write
down the functions used in getting the loglikelihoods, and point out
in the interpretations of the graph that :
1. The willingness to pay decreases as the bid increases regardless of
income levels.
2. At all bid sizes, the probability of saying
"yes" to the plan is greater for people at $80,000 income level than
those at $50,000 income level.