# Test of Proportions

The Test of Proportions is a statistical test used to compare [[ibis.health.state.nm.us/resource/Glossary.html#R rates]] (such as a percentage or a death rate) between two independent populations (such as counties or two age groups). By "independent" we mean there is no overlap in the members of the two groups. For instance, this test would be too considered conservative to compare the same county at two different points in time.

 Contents 1 When to use Test of Proportions 2 Calculation 3 Interpretation References

# When is Test of Proportions used?

Before embarking on the statistical computation, first use this simple test to tell if two rates are different:
• IF: The two rates are from two independent populations (e.g., two different counties or two different age groups); AND
• IF: The two rates are each based on 20 or more events (in the numerator),
• THEN: Calculate the 95% confidence intervals.
• If the confidence intervals do NOT overlap, the difference between the two rates is considered statistically signficiant.

For instance, the graph below depicts the heart disease death rates for all 33 New Mexico counties, sorted from lowest to highest. The San Juan County rate is 148.7 deaths per 100,000 population, with a 95% confidence interval that ranges from 127.6 to 169.8. In the gragh image, the San Juan confidence interval has been projected across all the other bars on the graph, making it easy to see which of the other confidence intervals overlap with San Juan's.

Looking at the graph, you can see that there are 20 counties with heart disease death rates higher than San Juan County's rate (20 bars higher than the San Juan bar). And of those 20 counties, there are six (Eddy, Otero, Lea, Curry, Luna, and Roosevelt) whose confidence intervals do not overlap the San Juan County confidence interval. So, according to this simple test, the heart disease death rates in those six counties are significantly higher than the heart disease death rate in San Juan County. The graph also shows 26 counties for which the confidence intervals overlap the San Juan County confidence interval. In those cases, some other statistical test, such as a Test of Proportions, will be necessary to assess the difference. Keep in mind that if you wish to compare two geographic areas or two time periods that have different age distributions, you should use age-adjusted rates as the basis for your comparison.

# Calculation of Test of Proportions

This section shows the computation for a Test of Proportions for the Torrance County and San Juan County heart disease rates in the above graph.

• Proportion 1: 2006 Torrance County heart disease rate: 237.6 per 100,000
• Proportion 2: 2006 San Juan County heart disease rate: 148.7 per 100,000
• Difference between the two proportions: 88.9 per 100,000 If your two groups are not independent, this test will be too conservative. That is, the test might suggest the two rates are not significantly different when in fact they are. This test assumes that you have at least 5 health events in the numerator for each rate. If necessary, combine enough years to yield 5 events.

The difference between the two rates (88.9) must be considered in the context of the standard error of the difference between two rates (pooled standard error), computed as: If the difference between the two rates is greater than 1.96 x s.e.diff, then the difference is considered statistically significant. 1.96 X s.e.diff is called the "Critical Value" for significance of the Test of Proportions at the p.< .05 level. Or, in more plain English, it is the critical value for the result of the Test of Proportions to be statistically significant with a probability of 5% or less that the difference is due to random fluctuation or chance.

### Values for Calculation of the Pooled Standard Error for the Two Rates

County
Value of p
(Rate per 100,000)
Value of q
(100,000 minus Rate)
Value of n
(Number in population)
1. Torrance County

237.6

99,762.4

18,311

2. San Juan County

148.7

99,851.3

127,629 # Interpretation of Test of Proportions Results

In this case, the critical value of the difference for the Test of Proportions is 73.6 -- that is, a difference of 73.6 or greater is needed to reach statistical significance at the p.< .05 level.

The difference between our two county rates was 88.9, which is greater than 73.6, the critical value for the difference according to the Test of Proportions. Therefore, even though their confidence intervals overlapped on the graph, the Test of Proportions indicates that the rate of death from heart disease in Torrance County is significantly greater than that in San Juan County, and the difference was statistically significant at the p.< .05 level.