## 29.9 Example: Blood pressure

A US study was conducted to determine how CHD risk factors were assessed among parts of the population. Subjects were required to report to the clinic on multiple occasions.

One RQ of interest is:

Is there a mean difference in blood pressure measurements between the first and second visits?

The parameter is $$\mu_d$$, the population mean reduction in blood pressure.

Each person has a pair of diastolic blood pressure (DBP) measurements: One each from their first and second visits. The data shown below, are from 141 people. These data were shown in Sect. 23.10. The differences could be computed in one of two ways:

• The observation from the first visit, minus the observation from the second visit: the reduction in BP; or
• The observation from the second visit, minus the observation from the first visit: the increase in BP.

Either way is fine, as long as the order remains consistent, and the direction is made clear. Here, the observations from the first visit minus the observation from the second visit will be used, so that the differences represent the decrease in BP from the first to second measurement.

The appropriate graphical summary is a histogram of differences (Fig. 23.4); the numerical summary is shown in Table 29.1. Notice that having the information about the differences is essential, as the RQ is about the differences.

TABLE 29.1: The numerical summary for the diabetes data (in mm Hg). The differences are the second visit value minus the first visit value: the decreases in diastolic blood pressure from the first to second visit
Mean Standard deviation Standard error Sample size
DBP: First visit 94.48 11.473 0.966 141
DBP: Second visit 92.52 11.555 0.973 141
Decrease in DBP 1.95 8.026 0.676 141

As always (Sect. 28.2), the null hypothesis is the ‘no difference, no change, no relationship’ position, proposing that the mean difference in the population is non-zero due to sampling variation:

• $$H_0$$: $$\mu_d=0$$ (differences: $$\text{first} - \text{second}$$);
• $$H_1$$: $$\mu_d \ne 0$$.

The alternative hypothesis is two-tailed because of the wording of the RQ. As usual, assume that $$H_0$$ is true, and then the evidence is evaluated to determine if it contradicts this assertion.

The sampling distribution describes how the sample mean difference is expected to vary from sample to sample due to sampling variation, when $$\mu_d=0$$. Under certain circumstances, the sample mean differences are likely to vary with a normal distribution, with a mean of 0 (from $$H_0$$) and a standard deviation of $$\text{s.e.}(\bar{d}) = 0.676$$.

The relative value of the observed sample statistic is found by computing a $$t$$-score, using software (jamovi: Fig. 29.6; SPSS: Fig. 29.7), or manually (Eq. (27.1), using the information in Table 29.1):

\begin{align*} t &= \frac{\text{sample statistic} - \text{assumed population parameter}} {\text{standard error of the statistic}}\\ &= \frac{1.950 - 0}{0.676} = 2.885. \end{align*} Either way, the $$t$$-score is the same.

A $$P$$-value is then needed to decide if the sample is consistent with the assumption. Using the 68–95–99.7 rule, the approximate two-tailed $$P$$-value is much smaller than 0.05. Alternatively, the software output (Fig. 29.6; Fig. 29.7) reports the two-tailed $$P$$-value as $$P=0.005$$.

We conclude:

Strong evidence exists in the sample (paired $$t=2.855$$; two-tailed $$P=0.005$$) of a population mean difference between the first and second DBP readings (mean difference $$1.95$$ mm Hg higher for first reading; 95% CI from $$0.61$$ to $$3.3$$ mm Hg; $$n=141$$).

Since $$n>25$$, the results are statistically valid.

Just saying ‘there is evidence of a difference’ is insufficient; it is also important to say which measurement is, on average higher (that is, what the differences mean).

### References

Schorling JB, Roach J, Siegel M, Baturka N, Hunt DE, Guterbock TM, et al. A trial of church-based smoking cessation interventions for rural African Americans. Preventative Medicine. 1997;26(1):92–101.
Willems JP, Saunders JT, Hunt DE, Schorling JB. Prevalence of coronary heart disease risk factors among rural blacks: A community-based study. Southern Medical Journal. 1997;90(8):814–20.