- Introduction to probability distributions (use instructor's last name)
A. This problem describes a type of problem typically discussed in Marketing
classes, where managers are trained to understand market participation and
market share. We have simplified the number of variables and cases in the
problem to make it easier to analyze. A typical realistic problem may have
hundreds of variables and thousands of cases. Data►
- What is the probability of hospitalization given that you are male? Select
all males and count the number of patients who were hospitalized.
Calculate the probability as the ratio of males hospitalized to number
- Create a contingency table for interaction between age and gender.
Is age independent from gender.
- Is insurance independent of age? Check that the probability of
combination of insurance and age can be estimated from the product of
probability of insurance and age or use the contingency table of age and
- What is the probability associated of being more than 65 years old among hospitalized patients? Start by selecting all hospitalized patients, then count the number
among hospitalized patients who are more than 65 years old. The
likelihood or the probability of being over 65 among hospitalized
patient is the number of patients hospitalized and above 65 divided by
the number of hospitalized patients:
- What is the probability of being hospitalized given you are more than 65 years old?
This time we are switching the condition. Now we are asking for the
probability among patients who are more than 65 years old. So
select all patients who are more than 65 years old and then count the
number who are hospitalized. In contrast to the previous question
the ratio is calculated by dividing the number of patients above 65 who
were hospitalized divided by number above 65 years.
- In predicting hospitalization, what is the likelihood ratio,
LR, associated with being more than 65 years old? This is not
the same as the likelihood of being above 65 given that you are
hospitalized. It should be calculated as follows:
- What is the prior odds for hospitalization before any other information is available?
The probability of hospitalization is calculated as the number
hospitalized by the number in the sample. Prior odds is calculated as
the probability of hospitalization by one minus the probability. A
simpler way to do so, the prior odds is the ratio of number hospitalized
divided by the number not hospitalized or as:
- Analyze the data in the Table and report if any two variables are conditionally independent of each other in predicting probability of hospitalization? Consider
the pairs Gender & Age, Age & Insured, and Gender & Insured. If two
events are independent, then the likelihood ratio associated with the
combined event should be the product of the likelihood ratios of each
event. If the likelihood ratio cannot be calculated because of
division by zero, then skip that check. In using likelihood ratios to
test the independence of two variables, note that you have to test it
for all levels in the variable. So for example, if we are
the independence of age and gender, then you would test the independence
of four set of combination of variables from their components:
mind that because the number of cases are too few, many ratios cannot be calculated.
- Likelihood ratio Age>65 and Male = Likelihood ratio of Age>65 *
Likelihood ratio of Male
- Likelihood ratio Age>65 and Female = Likelihood ratio of Age>65
* Likelihood ratio of Female
- Likelihood ratio Age<=65 and Male = Likelihood ratio of Age<=65
* Likelihood ratio of Male
- Likelihood ratio Age<=65 and Female = Likelihood ratio of
Age<=65 * Likelihood ratio of Female
- Hazard functions from Bernoulli probabilities