## 29.9 Example: Blood pressure

A US study (Schorling et al. 1997; Willems et al. 1997) 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.

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.

*which*measurement is, on average higher (that is, what the differences

*mean*).