IHP 525 Module 4 Hypothesis Testing Paper Example

Testing the Program's Effect: A T-Test and a Chi-Square Test for Bayview's Blood Pressure Data

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Southern New Hampshire University

IHP 525: Biostatistics

Module Four Paper

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[Date]

Testing the Program's Effect: A T-Test and a Chi-Square Test for Bayview's Blood Pressure Data

Bayview County's community blood pressure program has two main questions to answer: did participants' systolic pressure fall more than usual-care patients', and did more of them reach controlled pressure? This paper answers both with standard hypothesis tests worked step by step, using complete-case data: 218 program participants and 231 comparison patients with six-month readings. It reports each result with a confidence interval, interprets the findings carefully and translates them into plain language.

Choosing the Tests

The first outcome, change in systolic pressure, is continuous and roughly symmetric, and the two groups are independent, so a two-sample t-test comparing means is appropriate. Because the groups have somewhat different standard deviations, Welch's unequal-variance form of the test is used. The second outcome, reaching controlled pressure, is binary, so the Pearson chi-square statistic, computed from a table with two groups and two outcomes, fits, with a confidence interval for the difference in proportions. Each test addresses a null hypothesis of no difference between groups.

With more than 200 people per group, the t-test is robust to modest departures from normality, and every expected cell count in the two-by-two table exceeds five, satisfying the usual condition for the chi-square approximation.

The Two-Sample T-Test

Program participants' systolic pressure fell by a mean of 11.2 mm Hg, with a standard deviation of 12.4; comparison patients' fell by 4.6 mm Hg, with a standard deviation of 13.1. The difference in mean change is 11.2 minus 4.6, or 6.6 mm Hg greater reduction in the program group. To find the difference's standard error, take the square root of 12.4 squared divided by 218 plus 13.1 squared divided by 231: the square root of 0.705 plus 0.743, which is the square root of 1.448, or 1.20 mm Hg. The t statistic is 6.6 divided by 1.20, or about 5.5, with roughly 447 degrees of freedom. A t value this large corresponds to a two-sided p-value well below 0.001.

Confidence Interval for the Difference

Gardner and Altman (1986) argued that reporting only whether a result is statistically significant hides how large and how precise the effect is, and they urged researchers to report confidence intervals. A 95% interval runs 6.6 plus or minus about 1.97 times 1.20, or 6.6 plus or minus 2.4, giving 4.2 to 9.0 mm Hg. The data are compatible with a program benefit anywhere from about 4 to 9 mm Hg greater than usual care. The lower end is slightly below the 5 mm Hg difference defined in advance as clinically meaningful, and the upper end well above it.

The Chi-Square Test

In the program group, 100 of 218 participants, or 45.9%, reached controlled pressure; in the comparison group, 67 of 231, or 29.0%, did. Under the null hypothesis, the expected counts are found by multiplying each row total by each column total and dividing by the grand total of 449. The expected number controlled in the program group is 218 times 167 divided by 449, or 81.1, and in the comparison group 231 times 167 divided by 449, or 85.9; the expected numbers not controlled are 136.9 and 145.1. The chi-square statistic sums the squared difference between observed and expected counts divided by expected for each cell: 4.42 plus 2.61 plus 4.17 plus 2.47, or about 13.7, with one degree of freedom, giving a p-value of about 0.0002.

Table 1. Blood Pressure Control at Six Months (Observed and Expected Counts)

Note. Chi-square = 13.7, 1 degree of freedom. Data are illustrative.

Difference in Proportions

The difference in the share controlled is 45.9% minus 29.0%, or 16.9 percentage points. Its standard error comes from the square root of 0.459 times 0.541 divided by 218 plus 0.290 times 0.710 divided by 231, or about 0.045. The 95% confidence interval is 16.9 plus or minus 1.96 times 4.5 points, or roughly 8.1 to 25.7 percentage points. The program was associated with a substantially higher control rate, though the precise size of the advantage is uncertain.

Avoiding Misinterpretation

Greenland et al. (2016) describe many common misreadings of statistical results. The p-value below 0.001 does not mean there is less than a 0.1% chance the program has no effect, and on its own it says nothing about whether the effect is big or matters; the confidence interval addresses size. Nor does statistical significance prove the program caused the difference. Because participants were not randomized, differences between groups, such as younger age and higher baseline pressure among participants, could account for part of the result, which later regression analyses will address.

A Nonsignificant Subgroup

Among the 89 women in the program with follow-up and 97 in the comparison group, the difference in mean change was 3.9 mm Hg, with a 95% confidence interval of about minus 0.2 to 8.0 and a p-value of 0.06. A program partner concluded that the program does not work for women. Altman and Bland (1995) warned against this reasoning: a nonsignificant result is not evidence of no effect, especially in a smaller subgroup with less precision. The interval includes no effect but also includes benefits as large as the overall result. The appropriate conclusion is that the effect among women is uncertain, not absent.

Plain-Language Summary

For program leaders, the findings can be summarized simply. Over six months, blood pressure fell about 6 to 7 points more for program participants than for similar clinic patients, and about 17 more people out of every 100 reached healthy blood pressure. These differences are unlikely to be due to chance alone, but because people chose to join the program, part of the difference may reflect who joined rather than what the program did. The results for women are less certain because fewer women were included.

Conclusion

Worked step by step, the t-test and chi-square test both show the program associated with meaningfully greater improvement, and the confidence intervals show how large that improvement might be. Careful interpretation avoids overstating certainty or causation and prevents a nonsignificant subgroup from being mistaken for proof of no effect.

References

Altman, D. G., & Bland, J. M. (1995). Absence of evidence is not evidence of absence. BMJ, 311(7003), 485. https://doi.org/10.1136/bmj.311.7003.485

Gardner, M. J., & Altman, D. G. (1986). Confidence intervals rather than P values: Estimation rather than hypothesis testing. BMJ, 292(6522), 746-750. https://doi.org/10.1136/bmj.292.6522.746

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