MPH 515 Week 6 Hypothesis Testing Exercise Example

Hypothesis Testing Exercise: Did the Rate of First Nonfatal Opioid Overdose Visits Among Bellamy County Residents Change Between 2023 and 2024?

Student Name

Chamberlain University

MPH 515: Epidemiology and Biostatistics for Public Health Application I

Instructor Name

Month Day, Year

Hypothesis Testing Exercise: Did the Rate of First Nonfatal Opioid Overdose Visits Among Bellamy County Residents Change Between 2023 and 2024?

The Question and the Data

Bellamy County's health department reported fewer first emergency department visits for nonfatal opioid overdose among residents in 2024 than in 2023. Its board asked whether the decline was real or could reflect chance variation. This exercise tests whether the incidence rate in the county's population changed between the two years. The data are 655 new cases among about 412,000 residents in 2023 and 586 new cases among about 415,000 residents in 2024, with person-time approximated by midyear population.

Step 1: Hypotheses About the Population

The hypotheses concern the true underlying rates in the county's population, not the observed counts.

Null hypothesis: the true incidence rate of first nonfatal opioid overdose visits among county residents was the same in 2024 as in 2023.

Alternative hypothesis: the true incidence rate in 2024 differed from that in 2023, in either direction.

A two-sided alternative was chosen before looking at the data, because an increase would have been as important to the board as a decrease.

Step 2: Significance Level Set First

The significance level was set at 0.05 before calculation. This means the analysis accepts a 5% probability of concluding that the rate changed when it did not, if the null hypothesis is true.

Step 3: The Test and Why

The outcome is a count of new cases over a known amount of person-time in each year, so a comparison of two Poisson rates is appropriate. With several hundred cases each year, a z test based on the normal approximation to the Poisson distribution is adequate. The rate ratio with its 95% confidence interval is calculated alongside the test.

Step 4: Statistic With Working Shown

Rate in 2023: 655 divided by 412,000, or 159.0 per 100,000 person-years.

Rate in 2024: 586 divided by 415,000, or 141.2 per 100,000 person-years.

Difference in rates: 159.0 less 141.2, or 17.8 per 100,000 person-years.

Standard error of the difference: the square root of the sum of each year's cases divided by its person-time squared, which for these data is 8.5 per 100,000 person-years.

Test statistic: the difference divided by its standard error, 17.8 divided by 8.5, gives z = 2.09.

Two-sided p value for z = 2.09: 0.037.

Rate ratio, 2024 compared with 2023: 141.2 divided by 159.0, or 0.89, with a 95% confidence interval of 0.79 to 0.99, calculated on the log scale from the two case counts.

Table 1

First Nonfatal Opioid Overdose Visits by Year, Bellamy County Residents

Note. Note. Annual counts are composite figures built for this exercise; person-years use midyear population.

Step 5: Decision Against the Criterion

Since 0.037 falls under the preset 0.05 threshold, equal rates in the two years are rejected as the explanation.

Step 6: Two Conclusions

Statistical conclusion: the data provide evidence, at the 0.05 level, that the incidence rate of first nonfatal opioid overdose visits among county residents was lower in 2024 than in 2023.

Practical conclusion: the rate fell by about 11%, or about 18 fewer first visits per 100,000 residents a year. The interval shows that the decline could be as small as about 1% or as large as about 21%. That is a modest change. It is consistent with real improvement, but it is also close to the boundary, and the board should not treat the decline as settled or attribute it to any single program on this evidence alone. Statistical significance here does not establish that the decline was large or that any particular cause produced it (Greenland et al., 2016).

Step 7: Errors Tied to This Decision

Because the null hypothesis was rejected, the error that could have occurred is a Type I error: concluding the rate changed when in truth it did not. The probability of this error was set at 5%. A Type II error, failing to detect a real change, is not possible for this decision, because the null was rejected. Had the result been p = 0.08, the right conclusion would not have been that the rate was unchanged, but that this analysis did not detect a change; failing to reject a null hypothesis is not evidence that it is true (Wasserstein & Lazar, 2016).

What the Test Cannot Address

The test treats the two years as random samples of an underlying rate, but surveillance data also change for nonstatistical reasons. A change in how emergency departments code chief complaints, a shift of patients to a hospital across the county line, or more overdoses being reversed by bystanders without an emergency visit could each lower the count without any change in overdose occurrence. Before the board concludes that fewer residents overdosed, the health department should check whether coding rules, hospital use patterns or naloxone distribution changed between the two years, and should compare the trend in overdose deaths over the same period.

Checking the Approximation

The z test uses a normal approximation to the Poisson distribution, which works well when counts are large. With 655 and 586 cases, both counts are far above the usual minimum of about 10 to 20 cases per group, so the approximation is sound. With small counts, such as a comparison for one planning area with a few dozen cases, an exact test based on the binomial distribution of cases between the two years, given their total, would be preferred.

A Second Way to See the Same Result

The confidence interval for the rate ratio gives the same decision as the test. The interval, 0.79 to 0.99, excludes 1.0, the value that would mean no change, which agrees with rejecting the null hypothesis at the 0.05 level. The interval adds information the p value does not: it shows the plausible size of the change, which is the information the board needs. Because the upper limit is close to 1.0, a decline of only about 1% remains compatible with the data.

Reporting It to the Board

A sentence for the board would read: the rate of first overdose visits among residents fell from 159.0 to 141.2 per 100,000 between 2023 and 2024, an 11% decline (rate ratio 0.89, 95% CI 0.79 to 0.99); chance by itself is an unlikely explanation for the drop, but the drop is modest and may partly reflect changes in coding or in where residents seek care.

References

Greenland, S., Senn, S. J., Rothman, K. J., Carlin, J. B., Poole, C., Goodman, S. N., & Altman, D. G. (2016). Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. European Journal of Epidemiology, 31(4), 337-350. https://doi.org/10.1007/s10654-016-0149-3

Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129-133. https://doi.org/10.1080/00031305.2016.1154108