Treadmill

Description

From After Treadmill)

Heart rate measurements for 12 subjects before and after a treadmill session (baseline vs after5). Because both measurements come from the same subject, observations are paired, not independent — the appropriate test is the paired-sample version of the t-test (or its non-parametric analogue).

File: data/health_promo_hr.csv. Columns: baseline, after5.

Paired t-test

From After Treadmill)

Compute $D_i=X_i-Y_i$; test $H_0:{\mu }_D=0$ via

$$T_2=\frac{\bar{D}}{s/\sqrt{n}}\sim t_{n-1}\text{ under }{H}_0$$

R code

hr_df <- read.csv("data/health_promo_hr.csv")
before <- hr_df$baseline
after  <- hr_df$after5
t.test(before, after, paired=TRUE)

Python code

import pandas as pd
from scipy import stats
 
hr_df = pd.read_csv("data/health_promo_hr.csv")
paired_out = stats.ttest_rel(hr_df.baseline, hr_df.after5)
print(f"""
Test statistic: {paired_out.statistic:.3f}.
p-val: {paired_out.pvalue:.3f}.""")

Assumption check caveat: with only 12 subjects the case for Normality of the $D_i$ is weak. For small-sample paired data, a graphical check + the non-parametric analogue below is the safer route.

Paired/agreement diagnostic plots:

• Paired-line plot — one line per subject connecting baseline to after. Similar gradients (consistently up or consistently down) support a systematic effect; mixed signs weaken it.
• Agreement plot (after vs. before, with the $y=x$ reference line) — if points scatter around $y=x$, no mean shift.

Non-parametric paired test

From After Treadmill)

Wilcoxon Signed-Rank Test (WST) — paired-sample analogue of Wilcoxon Rank-Sum. Tests whether the median of $D_i=X_i-Y_i$ is 0. No Normality assumption.

Procedure:

1. Drop $D_i=0$.
2. Rank the remaining $|D_i|$ from 1 (smallest) to $m$ (largest).
3. $R_2$ = sum of ranks where $D_i>0$.
4. Under $H_0$, $E(R_2)=m(m+1)/4$; the continuity-corrected test statistic $W_2\sim N(0,1)$ approximately when the number of non-zero $D_i\ge 16$.

R code

wilcox.test(before, after, paired=TRUE, exact=FALSE)

Python code

wsr_out = stats.wilcoxon(hr_df.baseline, hr_df.after5,
                         correction=True, method='approx')
print(f"""Test statistic: {wsr_out.statistic:.3f}.
p-val: {wsr_out.pvalue:.3f}.""")

Small-sample caveat: with only 12 subjects here, we do not hit the $\ge 16$ threshold for the Normal approximation. The ideal is the exact version, but R/Python can’t run exact when there are ties. SAS does use the exact version — which accounts for any difference in p-values between SAS and R/Python.

Why test statistics look different across software:

• R / Python report $R_2$ directly.
• SAS reports $R_2-E(R_2)$.

For $n=12$:

$$0-\frac{12(12+1)}{4}=-39$$

reconciles a reported 0 in R with SAS’s -39.

If neither approximation is great and exact-with-ties is unavailable, fall back to the bootstrap or a permutation test (see L10 Simulation).

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See also: L7 Two-sample Hypothesis Tests · L10 Simulation