## Wednesday, November 2, 2016

### Marginal probabilities in two-way contingency tables

Marginal probabilities in two-way contingency tables

The table below has information on the perceived health status and the perceived mental health status of respondents on the 2015 Medical Expenditures Panel Survey.  My questions in this post involve the conditional probability of perceived health status given information on mental health status and the conditional probability of mental health status given information on health status.

Question One:  Using the data in the two-way health status/mental health status contingency table find the conditional probability that a person has poor health given that the person has poor mental health.   Also, find the conditional probability that a person has poor mental health given that the person has poor general health.

You are working for an insurance company after Donald Trump repeals the affordable care act that does not want to insure people who have both poor health and poor mental health.   Information is costly and you can only afford one screening question.   Should the question involve mental health or general health?

Analysis:   To get P(Poor health/Poor Mental Health) look at the poor mental health row.

P(Poor Health/Poor Mental Health) = 190/451 = 0.421.

To get P(Poor Mental Health/Poor Health) look at the poor health column

P(Poor Mental Health/Poor Health) = 190/932  =  0.204.

An insurance company trying to eliminate people with both poor mental health and poor general health will gather as much information about mental health status because mental health is a better predictor of a person having both issues than general health.

 Two-Way Contingency Table for Perceived Health and Perceived Mental Health Perceived Health Status Perceived Mental Health Status Excellent Very Good Good Fair Poor Total Excellent 9,783 2,981 1,637 524 108 15,033 Very Good 908 5,377 1,702 549 88 8,624 Good 511 1,059 4,549 1,245 280 7,644 Fair 108 161 421 977 266 1,933 Poor 23 37 83 118 190 451 Total 11,333 9,615 8,392 3,413 932 33,685
Tabulations based on the 2015 Medical Expenditures Panel Survey.

Note:   The definition of conditional probability of two events P(A/B) and B is P(A Intersection B) divided by P(B).     The definition of conditional probability of two events P(B/A) is P(A intersection B) divided by P(A).

 Joint and Marginal Probabilities for Perceived Mental Health  Status and Perceived Health Status Perceived Health Status Perceived Mental Health Status Excellent Very Good Good Fair Poor Total Excellent 0.2904 0.0885 0.0486 0.0156 0.0032 0.4463 Very Good 0.0270 0.1596 0.0505 0.0163 0.0026 0.2560 Good 0.0152 0.0314 0.1350 0.0370 0.0083 0.2269 Fair 0.0032 0.0048 0.0125 0.0290 0.0079 0.0574 Poor 0.0007 0.0011 0.0025 0.0035 0.0056 0.0134 Total 0.3364 0.2854 0.2491 0.1013 0.0277 1.0000

Note the P(Poor Health/Poor Mental Health) is 0.0056/0.0134 = 0.418.  The difference between this number and 0.421 is due to rounding.

Note that P(Poor Mental Health/Poor Health) is 0.0056/0.0277 = 0.2038.  At three significant digits this rounds to 0.204.

Authors Note:  I am doing a lot of research on health care focused largely on the impact of the affordable care act.   I am trying to get a research grant.  Interested readers should look at my health insurance blog.  Start with the research proposal below.