Collecting Data

A conclusion is only as good as the data behind it. How data are collected decides what you may conclude – whether you can generalize to a population 总体, and whether you can claim cause and effect. Poorly collected data can be worse than none.

• In an observational study 观察性研究 you measure individuals without trying to influence them. It can show association, but not causation, because lurking variables may explain the link.
• In an experiment 实验 you deliberately impose a treatment 处理 and compare responses. A well-designed experiment can establish cause and effect.

To learn about a population you take a sample 样本. Random sampling 随机抽样 removes selection bias 偏差 and lets you generalize. Common designs:

• Simple random sample (SRS) 简单随机样本: every group of the chosen size is equally likely.
• Stratified 分层: split the population into similar strata, then sample within each.
• Cluster 整群: split into clusters, randomly choose whole clusters.
• Systematic 系统: pick every $k$th individual from a random start.

Four random sampling designs: who gets selected, and how

A convenience sample 方便样本 or voluntary response sample is not random and is biased.

Worked example. To survey a school, an administrator lists all students by grade and randomly selects $20$ from each grade. This is a stratified sample – the grades are the strata – which guarantees every grade is represented, unlike an SRS that might by chance draw few from one grade.

Bias makes estimates systematically miss the truth:

• Undercoverage 覆盖不足: some groups are left out of the sampling frame.
• Nonresponse 无回应: selected people do not answer.
• Response bias 回应偏差: people answer inaccurately (bad wording, sensitive topics).

Bias is about a consistent error in one direction – increasing the sample size does not fix it.

Good experiments follow three principles:

• Comparison with a control group 对照组 (often a placebo 安慰剂).
• Random assignment 随机分配 of subjects to treatments, to balance out other variables.
• Replication 重复: enough subjects per treatment to see a real effect.

A completely randomized experiment compares a treatment group with a control group

Confounding 混杂 occurs when another variable is tied to the treatment so their effects cannot be separated; random assignment guards against it. Blinding 盲法 hides who is getting which treatment to prevent expectation effects: in a single-blind 单盲 study only one side is kept unaware (usually the subjects, or only the people assessing the result), while in a double-blind 双盲 study neither the subjects nor the researchers who interact with them know, blocking both the placebo effect and biased assessment. Blocking 区组 groups similar subjects and randomizes within each block to reduce variability.

Match the design to the goal: use a completely randomized design for uniform subjects; a randomized block design when a known variable (sex, age) affects the response; a matched-pairs design when each subject can serve as its own control. State how you would carry out the randomization.

Two questions decide the scope of a conclusion:

• Random assignment used? Then a significant difference can be attributed to the treatment (causation) – for these subjects.
• Random sampling from a population? Then results generalize to that population.

Only an experiment with random assignment supports a cause-and-effect claim; only random sampling supports generalization. Say exactly which you have.

Worked example. Researchers randomly assign $100$ volunteers to a new drug or a placebo, and the drug group improves significantly more. Because of the random assignment, the improvement can be attributed to the drug (causation) – but because the subjects were not randomly sampled, the conclusion applies only to these volunteers and does not automatically generalize to everyone.

• Distinguish an observational study (finds association) from an experiment (can show causation).
• Good sampling is random (SRS, stratified, cluster) — beware bias (voluntary response, undercoverage, nonresponse).
• Good experiments use control, randomization, and replication; blocking handles a known nuisance variable.
• Only a randomized experiment supports a cause-and-effect conclusion.
• Name the population, sample, and any confounding clearly.