📚 Case Study Mastery: A Blood Pressure Drug Trial | 案例分析实战：降压药临床试验
This article walks you through a complete statistical case study based on a hypothetical double-blind, randomised controlled trial evaluating a new antihypertensive drug. We will illustrate the full statistical workflow, from experimental design and exploratory data analysis to hypothesis testing, confidence intervals, regression and non‑parametric alternatives, as typically required in Pre‑U CIE Further Mathematics Statistics papers.
本文将带你完成一个基于假想降压药双盲随机对照试验的完整统计案例分析。我们将演示从实验设计、探索性数据分析到假设检验、置信区间、回归和非参数替代方法的完整统计流程，完全对应 Pre‑U CIE 进阶数学统计部分的要求。
1. Introduction to the Case Study | 案例背景介绍
A pharmaceutical company has developed a new drug, ‘HypotensX’, intended to lower systolic blood pressure in patients with mild hypertension. The primary research question is: does HypotensX produce a greater mean reduction in systolic blood pressure than a placebo after eight weeks of treatment?
某制药公司研发了一种名为 ‘HypotensX’ 的新药，用于降低轻度高血压患者的收缩压。核心研究问题是：治疗八周后，HypotensX 降低收缩压的平均幅度是否显著大于安慰剂？
To answer this, a randomised, double‑blind, placebo‑controlled trial was conducted with 50 consenting patients. The primary endpoint was the change in systolic blood pressure (in mmHg) from baseline to week 8. In addition, secondary analyses consider the proportion of patients reaching the target blood pressure (<140 mmHg) and the dose‑response relationship for a subgroup receiving different doses of HypotensX.
为回答该问题，对50名知情同意患者进行了一项随机、双盲、安慰剂对照试验。主要终点是收缩压从基线到第8周的变化值（单位 mmHg）。此外，次要分析关注血压达标（<140 mmHg）的患者比例，以及接受不同剂量 HypotensX 的亚组中的剂量‑反应关系。
2. Experimental Design and Data Collection | 实验设计与数据收集
Fifty patients were randomly assigned to either the treatment group (n₁ = 25) receiving HypotensX or the placebo group (n₂ = 25). Randomisation was stratified by baseline hypertension severity to ensure balanced groups. Both patients and assessing clinicians were blinded to the allocation.
50名患者被随机分配到治疗组（n₁ = 25，接受 HypotensX）或安慰剂组（n₂ = 25）。随机化按基线高血压严重程度分层，确保组间均衡。患者和评估医生均不知晓分组情况。
The primary response variable, X, is the reduction in systolic blood pressure: baseline SBP minus SBP at week 8. Thus, positive values indicate improvement. Additional data recorded included age, body mass index and the dose received in an exploratory dose‑finding sub‑protocol.
主要响应变量 X 是收缩压的降低值：基线 SBP 减去第8周 SBP。因此正值表示改善。还记录了年龄、体重指数以及探索性剂量确定子方案中接受的剂量。
The following descriptive summary was obtained from the trial database (all values in mmHg).
从试验数据库中获得了以下描述性摘要（所有数值单位为 mmHg）。
| Group | n | Mean reduction, x̄ | Standard deviation, s |
|---|---|---|---|
| HypotensX | 25 | 15.2 | 4.8 |
| Placebo | 25 | 8.6 | 5.1 |
These summary statistics will be used throughout the subsequent inferential procedures.
这些汇总统计量将用于后续所有的推断过程。
3. Exploratory Data Analysis | 探索性数据分析
Before formal testing, we inspect the data graphically. Side‑by‑side boxplots suggest that the HypotensX group has a noticeably higher median reduction and a more compact interquartile range compared with the placebo group. One mild outlier was detected in the placebo group (reduction of 21 mmHg), but it was not deemed clinically implausible.
在正式检验之前，我们先用图形审视数据。并排箱线图显示，HypotensX 组的中位降幅明显更高，四分位距也更紧凑。安慰剂组检测到一个温和异常值（降幅21 mmHg），但临床上并非不可信。
A histogram of the reduction values for each group, overlaid with a normal density curve, shows rough symmetry. The skewness coefficients are −0.31 (HypotensX) and 0.28 (placebo), indicating no serious departure from symmetry. Summary statistics also suggest similar variability, with standard deviations 4.8 and 5.1 mmHg.
每组降低值的直方图并与正态密度曲线叠加，显示出粗略对称性。偏度系数分别为 −0.31（HypotensX）和 0.28（安慰剂），表明没有严重偏离对称。标准差 4.8 和 5.1 mmHg 也提示变异性相似。
This exploratory stage indicates that a two‑sample t‑test assuming equal variances is a reasonable candidate for the primary analysis.
探索阶段表明，假定方差相等的两样本 t 检验是主要分析的合理候选方法。
4. Checking Assumptions: Normality and Equal Variance | 假设检验：正态性与方差齐性
We formally assess normality using Shapiro‑Wilk tests on the reduction scores in each group. The test statistics give W = 0.974 (p = 0.762) for HypotensX and W = 0.968 (p = 0.590) for placebo, both non‑significant at the 5% level. Thus, the normality assumption is not violated.
我们使用 Shapiro‑Wilk 检验对每组降低值正式评估正态性。检验统计量给出 HypotensX 组 W = 0.974（p = 0.762），安慰剂组 W = 0.968（p = 0.590），均在 5% 水平上不显著。因此正态性假设未遭违反。
Levene’s test for equality of variances yields F = 0.184 with p = 0.670, indicating no evidence against homoscedasticity. Therefore we proceed with the pooled two‑sample t‑test, which assumes equal population variances.
方差齐性 Levene 检验给出 F = 0.184，p = 0.670，表明没有反对方差齐性的证据。因此我们采用假定总体方差相等的合并两样本 t 检验。
5. Two-Sample t-Test for Difference in Means | 两样本均值差异的 t 检验
We test H₀: μ₁ = μ₂ against H₁: μ₁ ≠ μ₂, where μ₁ is the true mean reduction for HypotensX and μ₂ is that for placebo. The significance level is α = 0.05.
我们检验 H₀: μ₁ = μ₂ 对 H₁: μ₁ ≠ μ₂，其中 μ₁ 为 HypotensX 的真实平均降幅，μ₂ 为安慰剂的真实平均降幅。显著性水平 α = 0.05。
sₚ = √[ ((n₁−1)s₁² + (n₂−1)s₂²) / (n₁+n₂−2) ] = √[ (24×4.8² + 24×5.1²)/48 ] = √(1177.2/48) ≈ 4.952
SE = sₚ √(1/n₁ + 1/n₂) = 4.952 × √(0.08) ≈ 1.400
t = (x̄₁ − x̄₂) / SE = (15.2 − 8.6) / 1.400 = 6.6 / 1.400 ≈ 4.714
The observed t‑statistic is 4.714 with 48 degrees of freedom. The critical value t₀.₀₂₅,₄₈ ≈ 2.011, so we reject H₀. The corresponding two‑tailed p‑value is much smaller than 0.001, indicating extremely strong evidence that the mean reduction for HypotensX differs from that of the placebo.
观测 t 统计量为 4.714，自由度为 48。临界值 t₀.₀₂₅,₄₈ ≈ 2.011，因此拒绝 H₀。相应的双尾 p 值远小于 0.001，表明有极强证据表明 HypotensX 的平均降幅与安慰剂不同。
In context, the data suggest that HypotensX produces a significantly greater reduction in systolic blood pressure, with the observed mean difference being 6.6 mmHg in favour of the new drug.
在此背景下，数据表明 HypotensX 导致的收缩压下降显著更多，观测到的平均差值 6.6 mmHg 有利于新药。
6. Confidence Interval for the Difference | 差值置信区间
A 95% confidence interval for μ₁ − μ₂ provides a range of plausible values for the true treatment effect.
μ₁ − μ₂ 的 95% 置信区间提供了真实治疗效果的可能值范围。
(x̄₁ − x̄₂) ± t₀.₀₂₅,₄₈ × SE = 6.6 ± 2.011 × 1.400 = 6.6 ± 2.815
Thus the interval is (3.78, 9.42) mmHg. Since the entire interval lies above zero, we can be 95% confident that HypotensX reduces systolic blood pressure more than placebo by between about 3.8 and 9.4 mmHg on average.
因此区间为 (3.78, 9.42) mmHg。由于整个区间位于零以上，我们可以 95% 相信 HypotensX 平均比安慰剂多降低收缩压约 3.8 至 9.4 mmHg。
7. Paired t-Test for Before-After Comparison | 治疗前后比较的配对 t 检验
As a secondary analysis, consider the change within the HypotensX group using pre‑ and post‑treatment systolic blood pressure measurements. For each of the 25 patients, let d = SBP_before − SBP_after. The mean of the differences is 15.2 mmHg with standard deviation s_d = 4.8 mmHg, identical to the reduction summary because the change is precisely the primary endpoint.
作为次要分析，使用治疗前后收缩压测量值考察 HypotensX 组内部的变化。对于每名患者，设 d = 治疗前 SBP − 治疗后 SBP。差值的均值为 15.2 mmHg，标准差 s_d = 4.8 mmHg，与降幅摘要相同，因为变化正是主要终点。
A paired t‑test evaluates whether the true mean difference is zero. The test statistic is
t = x̄d / (s_d/√n) = 15.2 / (4.8/5) = 15.2 / 0.96 = 15.83
With 24 degrees of freedom, the critical t‑value is 2.064, and the resulting p‑value is vanishingly small. This confirms a highly significant average reduction within the HypotensX group, complementing the two‑sample comparison.
自由度为 24 时，t 临界值为 2.064，所得 p 值极小。这确认了 HypotensX 组内存在高度显著的平均降幅，补充了两样本比较的结果。
8. Chi-Squared Test for Categorical Outcomes | 分类结果的卡方检验
In many clinical trials, the proportion of patients achieving a pre‑defined target (e.g. SBP < 140 mmHg) is also reported. In this trial, 20 out of 25 HypotensX patients and 8 out of 25 placebo patients achieved the target at week 8.
在许多临床试验中，也会报告达到预先定义目标（如 SBP < 140 mmHg）的患者比例。本试验中，HypotensX 组 25 人中有 20 人达标，安慰剂组 25 人中有 8 人达标。
We assess whether the probability of reaching the target differs between groups using a chi‑squared test of association. The observed frequencies are arranged in a 2×2 table.
我们使用关联性卡方检验评估两组达标概率是否不同。观测频数排列为 2×2 表格。
| Target achieved | Not achieved | Row total | |
|---|---|---|---|
| HypotensX | 20 | 5 | 25 |
| Placebo | 8 | 17 | 25 |
| Column total | 28 | 22 | 50 |
Expected frequencies under independence are computed, and the chi‑squared statistic is χ² = 11.69 with 1 degree of freedom (Yates’ correction not applied). The critical value at α = 0.05 is 3.84, and the p‑value is approximately 0.0006. We reject the null hypothesis of no association, concluding that the proportion achieving the target is significantly higher in the HypotensX group.
计算独立性假设下的期望频数，得到卡方统计量 χ² = 11.69，自由度 1（未使用 Yates 校正）。在 α = 0.05 下的临界值为 3.84，p 值约为 0.0006。我们拒绝无关联的零假设，得出 HypotensX 组达标比例显著更高的结论。
9. Simple Linear Regression: Dose-Response Relationship | 简单线性回归：剂量‑反应关系
An exploratory sub‑study administered four different daily doses of HypotensX (10, 20, 30, 40 mg) to a separate set of 20 volunteers, measuring the systolic blood pressure reduction. The scatterplot suggests a roughly linear increasing trend.
一项探索性子研究对另外 20 名志愿者给予四种不同日剂量（10, 20, 30, 40 mg），测量收缩压降幅。散点图显示大致线性增长趋势。
The fitted model is: Reduction = β₀ + β₁×Dose + ε. The least squares estimates are b₀ = 2.10 (SE = 1.20) and b₁ = 0.425 (SE = 0.048). The coefficient of determination R² = 0.82 indicates a strong linear relationship.
拟合模型为：降幅 = β₀ + β₁×剂量 + ε。最小二乘估计值为 b₀ = 2.10（标准误 1.20），b₁ = 0.425（标准误 0.048）。决定系数 R² = 0.82 表明线性关系很强。
Test of H₀: β₁ = 0 → t = b₁ / SE(b₁) = 0.425 / 0.048 ≈ 8.85
With 18 degrees of freedom, the highly significant result (p << 0.001) provides strong evidence that each 1 mg increase in daily dose is associated with an additional 0.425 mmHg reduction, on average.
自由度为 18，高度显著的结果（p << 0.001）提供了强有力证据：每日剂量每增加 1 mg，平均血压额外降低 0.425 mmHg。
10. Non-Parametric Alternative: Mann-Whitney Test | 非参数替代：Mann‑Whitney 检验
Although parametric assumptions were met, exam questions often ask for a non‑parametric equivalent as a sensitivity analysis. The Mann‑Whitney U test compares the distributions of the reduction values in the two groups without assuming normality.
虽然参数假设得以满足，但考题常要求使用非参数对等方法进行敏感性分析。Mann‑Whitney U 检验比较两组降幅值的分布，不假定正态性。
After ranking all 50 observations, the sum of ranks for the HypotensX group is R₁ = 850. The expected rank sum under H₀ is n₁(n₁+n₂+1)/2 = 25×51/2 = 637.5, leading to a large standardised U‑statistic. The resulting p‑value (two‑tailed, normal approximation with correction for ties) is still less than 0.001, confirming a significant difference.
对所有 50 个观测值排序后，HypotensX 组的秩和 R₁ = 850。零假设下的期望秩和为 n₁(n₁+n₂+1)/2 = 25×51/2 = 637.5，得到很大的标准化 U 统计量。所得双尾 p 值（使用正态近似并经结值校正）仍小于 0.001，确认了显著差异。
This consistency strengthens confidence in the trial findings.
这种一致性增强了我们对试验结论的信心。
11. Discussion and Interpretation | 讨论与解释
All inferential procedures consistently indicate that HypotensX is effective in reducing systolic blood pressure. The two‑sample t‑test gave a highly significant mean difference of 6.6 mmHg; the 95% CI excludes zero; the chi‑squared test showed a higher responder rate; and the dose‑response analysis demonstrates a clear linear relationship.
所有推断程序一致表明 HypotensX 能有效降低收缩压。两样本 t 检验给出高度显著的 6.6 mmHg 平均差值；95% 置信区间不包括零；卡方检验显示更高的应答率；剂量‑反应分析显示出明显的线性关系。
However, it is crucial to contextualise these findings. The trial had a modest sample size, so the confidence interval ranges over several mmHg. Confounding factors such as diet, exercise and concurrent medication were not fully controlled, even with randomisation. Replication in a larger, multi‑centre trial would be desirable before regulatory claims.
然而，必须结合背景解释这些发现。试验样本量不大，因此置信区间跨越数 mmHg。即使进行了随机化，饮食、运动以及合并用药等混杂因素并未完全控制。在提出监管声明之前，更大型的多中心试验重复验证是可取的。
For Pre‑U exam purposes, always relate results to the original research question, discuss limitations and consider both statistical and practical significance.
就 Pre‑U 考试而言，务必将结果联系原始研究问题，讨论局限性并同时考虑统计学显著性和实际意义。
12. Conclusion and Exam Tips | 结论与考试技巧
This case study has demonstrated a structured approach to analysing clinical trial data, covering design, EDA, assumption checks, parametric and non‑parametric tests, confidence intervals and regression – all key components of the CIE Pre‑U Statistics syllabus.
本案例展示了分析临床试验数据的结构化方法，涵盖了设计、EDA、假设检查、参数和非参数检验、置信区间和回归——所有这些均为 CIE Pre‑U 统计课程大纲的核心组成部分。
When tackling case‑study questions, remember to: clearly state hypotheses in words and symbols; justify your choice of test by checking assumptions; present findings with a test statistic, degrees of freedom and p‑value where appropriate; and offer a concise interpretation in the context of the problem. Practise computing pooled standard deviations, U‑statistics and expected frequencies until they become routine.
在应对案例研究题时，请记住：用语言和符号清晰陈述假设；通过检查假设条件来论证检验方法的选择；适当给出检验统计量、自由度和 p 值；并结合问题背景给出简明解释。反复练习计算合并标准差、U 统计量和期望频数，直到成为常规操作。
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