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
employersponsored insurance (ESI) in 2015 and 2013.
What does this data suggest about the possible impact of the
opening of state exchanges on smallfirm offers of health insurance?
Using data from the table below conduct the unpaired ttest
and paired ttest on the hypothesis that there was no change in the mean of the
proportion of employees receiving an offer of employersponsored health
insurance in small firms.
Provide me with the pvalues of both tests. Employ a twotailed 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

Data was obtained from AHRQ at link below: https://meps.ahrq.gov/data_stats/quick_tables_search.jsp?component=2&subcomponent=2
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)/(((0.112^{2})/51)+(0.103^{2})/51)))^{0.5}
The value of the test statistic is 2.26.
The pvalue 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/(0.0658^{2}/51)^{0.5}
The value of the tstatistic is 5.20.
The pvalue is 0.
We reject the null hypothesis of no change in average
proportion of employees with offers of employersponsored insurance at small
firms at all frequently used significance levels.
Comment: There is a lot of variability across states
in the proportion of smallfirm employees getting offers of health insurance
from their employer in both 2013 and 2015.
This crosssectional variability, some of which persists across years,
is irrelevant to the test of changes between 2013 and 2015. For this reason the paireddifference 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 employersponsored 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.
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