统计测试假设清单

Key assumptions for ordinary least squares (OLS) linear regression

Core model assumptions (needed for unbiasedness/consistency)

• Linearity in parameters and additive errors: The expected response is a linear combination of predictors; the error term adds to the mean structure (e.g., Y = Xβ + ε).
• Correct specification (no omitted relevant variables, correct functional form): The model includes all predictors that drive the outcome and uses appropriate transformations; otherwise, estimates can be biased.
• Random sampling and independence of observations: Data are drawn independently from the population. For time series or clustered data, independence is replaced by “no serial correlation” or independence within clusters conditional on X.
• Exogeneity (zero conditional mean): E[ε | X] = 0. Predictors are not correlated with the error term. This is the key requirement for unbiased and consistent OLS estimates.
• No perfect multicollinearity: Predictors are not exact linear combinations of each other; the design matrix has full column rank.

Assumptions for efficiency and standard (non-robust) inference

• Homoscedasticity: Var(ε | X) is constant. Violations (heteroscedasticity) do not bias OLS coefficients but invalidate conventional standard errors and reduce efficiency.
• No autocorrelation of errors: Cov(εi, εj | X) = 0 for i ≠ j. Violations (e.g., in time series) affect inference and efficiency.
• Normality of errors (optional, for exact small-sample tests): ε | X is normally distributed. Not required for unbiasedness; needed for exact t/F tests and small-sample confidence intervals. With large samples, inference relies on asymptotic normality and is less sensitive to non-normality.

Data quality and identifiability assumptions (often implicit but essential)

• Predictors measured without error (or measurement error is negligible): Classical OLS assumes X is measured accurately; errors-in-variables cause biased (typically attenuated) estimates.
• Sufficient variation in predictors and absence of near-collinearity: Predictors must vary and not be highly collinear to yield stable, interpretable estimates.

Context-specific additions (when applicable)

• Time series: Weak/strict exogeneity, stationarity or stability of the process, and no serial correlation in errors conditional on X.
• Panel/clustered data: Independence across clusters, appropriate handling of within-cluster correlation (e.g., clustered SE).

Bottom line

• For unbiased OLS estimates: linearity, correct specification, random sampling/independence, exogeneity, and no perfect multicollinearity.
• For valid conventional inference: add homoscedasticity and no autocorrelation; normality is needed only for exact small-sample tests. Robust or cluster-robust methods can relax homoscedasticity and independence assumptions for inference.

卡方检验的关键假设与前提

通用

• 数据类型：基于分类数据的“频数”（计数），每个观测只能归入一个且仅一个类别；类别应互斥且穷尽。
• 抽样与独立性：样本来自随机抽样或随机分配；观测之间相互独立。不存在配对、重复测量或群集相关；若存在复杂抽样/群集，应使用设计校正（如 Rao–Scott 修正）或相应模型。
• 渐近近似有效性（期望频数要求）：在原假设下，各单元格的期望频数应足够大。常用判据：
  • 传统规则：所有期望频数≥5；
  • 宽松规则：≤20%的单元格期望频数<5，且无期望频数<1。 若不满足，考虑合并类别、增大样本量、使用精确检验（如 Fisher）或蒙特卡罗方法。
• 固定分类方案：类别（行/列）在分析前预先定义，不应据数据事后调整；无结构性零（若有，需要专门处理/建模）。
• 计量一致性：使用原始计数而非比例/百分比；若只提供比例，需同时提供样本量以还原计数。

拟合优度检验（单维，检验样本是否来自给定分布）

• 原假设的类别概率已明确给定；若从样本估计了 m 个分布参数，需相应调整自由度（df = k − 1 − m）。
• 样本来自同一总体，满足多项分布框架和独立性。

列联表检验（独立性/同质性）

• 独立性检验：来自同一样本，按两个（或多个）分类变量进行交叉分类；样本内观测独立。
• 同质性检验：来自多个相互独立的样本/总体，比较其分类分布；各样本内独立，样本间也独立。
• 研究设计与边际：边际总数的确定应与所选检验类型一致（独立性 vs 同质性），但不作为分布近似的额外假设。

注意

• 对于2×2配对数据（非独立），不应用常规卡方检验，应使用 McNemar 检验。
• 当期望频数过小或存在稀疏表格，优先考虑 Fisher 精确检验、合并类别、或采用似然比卡方（G 检验）/蒙特卡罗方法；复杂抽样应进行设计效应调整。