## Friday, November 4, 2016

### Comparing Joint Probabilities from Two Contingency Tables

Comparing Joint Probabilities from Two Contingency Tables

A List of Problems Using Contingency Tables:
http://www.dailymathproblem.com/p/contingency-table-problems-table-below.html

Question:   The contingency tables below involve self-perceived general health status and self-perceived mental health status for two age groups   -- young adults 35 years old and under and older adults over age 75.

 Two Contingency Tables Less Than or Equal to 35 General Health Mental Health Excellent Medium Poor Total Excellent 7,233 2,302 26 9,561 Medium 951 6,966 84 8,001 Poor 20 91 47 158 Total 8,204 9,359 157 Greater than 75 General Health Mental Health Excellent Medium Poor Total Excellent 151 213 8 372 Medium 71 967 62 1,100 Poor 1 38 28 67 Total 223 1,218 98 1,539

Use this data to estimate the joint probability of each self-reported general health and self-reported mental health combination.

Discuss ways the two contingency tables differ and are similar.

Note on data:  The contingency tables were constructed from data published in the MEPS 2014 database.   The actual MEPS questions on self-reported health and self-reported mental health have five responses.   I used a STATA program to combine information on very good health, good, and fair health into a single category called medium health.   A description of this method can be found in the following post.

Analysis:   Below are the joint and marginal probabilities associated with the two-way general health/mental health contingency tables.

 Joint and Marginal Probabilities Less Than or Equal to 35 General Health Mental Health Excellent Medium Poor Total Excellent 40.8% 13.0% 0.1% 54.0% Medium 5.4% 39.3% 0.5% 45.2% Poor 0.1% 0.5% 0.3% 0.9% Total 46.3% 52.8% 0.9% 100.0% Greater than 75 General Health Mental Health Excellent Medium Poor Total Excellent 9.8% 13.8% 0.5% 24.2% Medium 4.6% 62.8% 4.0% 71.5% Poor 0.1% 2.5% 1.8% 4.4% Total 14.5% 79.1% 6.4% 100.0%

Observations:   There are some notable similarities and differences between perceptions of health (both general and mental) between young and old adults.

Similarities:

In both cases the joint health/mental health category with the lowest frequency involved people with excellent health and poor mental health (0.1%) in each case.    (For young people there was a tie because excellent health and poor mental health also occurs 0.1% of the time.)

In both cases, the joint health/mental health category with the second lowest frequency is the group excellent mental health and poor general health.

In both cases the marginal probability of poor health was the lowest of the three marginal probabilities for the general health category.

In both cases the marginal probability of poor mental health was the lowest marginal probability for the mental health categories.

Differences:

The most common outcome for young adults is excellent health and excellent mental health (40.8%).   The most common outcome for older people is medium health and medium general health (62.8%)

There was a large decrease in the proportion of people reporting excellent health.   The decrease in people reporting excellent mental health was also substantial but smaller than for general health.

The proportion of older people with both poor health and poor mental health is 1.8%, which is 6 times the level for younger people.   Still note most older people with poor health do not also have poor mental health.

Basically fewer than 2.0% of people over 75 report both poor general and mental health.   That is pretty good.

It would be useful to use the joint general health/mental health contingency table to illustrate the calculation of concordant and discordant pairs, Kendall’s tau, and the Spearman’s rank correlation coefficient.   I will work on this post next week, possibly after the election.

Authors Note:  I have been working very hard on my health care blog and have been proposing research on the affordable care act.  Interested readers should look at this research proposal.