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.



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