## Friday, January 27, 2017

### Paired and Unpaired Difference Tests: A Health Insurance Example

Paired and Unpaired Difference Tests:
A Health Insurance Example

Question:   The table below has information on the likelihood that employees in small firms received an offer of employer-sponsored insurance (ESI) in 2015 and 2013.

What does this data suggest about the possible impact of the opening of state exchanges on small-firm offers of health insurance?

Using data from the table below conduct the unpaired t-test and paired t-test on the hypothesis that there was no change in the mean of the proportion of employees receiving an offer of employer-sponsored health insurance in small firms.

Provide me with the p-values of both tests.   Employ a two-tailed test.

The Data:

 Statistics on the percent of employees in small firms receiving ESI  (50 states plus DC) 2015 2013 Difference Average 30.37% 35.16% -4.79% Standard Deviation 11.18% 10.26% 6.58% Min 16.70% 16.60% -20.30% 25th 24.75% 29.90% -9.70% 50th 27.80% 33.60% -6.10% 75th 33.50% 38.45% 0.05% Max 83.50% 83.30% 13.60% Count 51 51 51

A small firm is defined here as a firm with fewer than 10 employees.

The Two Statistical Tests:

The test statistic for the unpaired test is of the form:

t=(0-.04779)/(((0.1122)/51)+(0.1032)/51)))0.5

The value of the test statistic is -2.26.

The p-value can be obtained from the TDIST function in Excel with 100 degrees of freedom.  (DF for this test is sum of sample size – 2.)

TDIST(-2..26,100,2) = 0.026.

The unpaired test would reject the null hypothesis for a test with a 0.05 significance level but would fail to reject the null hypothesis for a test with a 0.01 significance level.

The test statistic for the paired test is of the form:

t=0.0479/(0.06582/51)0.5

The value of the t-statistic is -5.20.

The p-value is 0.

We reject the null hypothesis of no change in average proportion of employees with offers of employer-sponsored insurance at small firms at all frequently used significance levels.

Comment:   There is a lot of variability across states in the proportion of small-firm employees getting offers of health insurance from their employer in both 2013 and 2015.   This cross-sectional variability, some of which persists across years, is irrelevant to the test of changes between 2013 and 2015.  For this reason the paired-difference test provides a clearer view of changes in the underlying variable.

Comment:   It would be useful to use the raw data and calculate the Wilcoxon Sign test and the Kolmogorov Smirnoff test for this issue.

Concluding Remarks:  Note that 2014 is the first year of the implementation of state exchanges.     The analysis presented here found that the implementation of state exchanges coincided with a significant decrease in the average across states in the proportion of employees at small firms receiving an offer of employer-sponsored insurance.

Employees at small firms now often get their health insurance through state exchanges.   Many small employers no longer have to deal with health insurance hassles.   However, if ACA is repealed smaller firms may have no choice but to once again find health insurance for their employees.

People interested in the debate over health care may want to read my health memos blog.   Here is a recent post on a possible bipartisan alternative to the ACA.