倾向得分匹配

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In the statistical analysis of observational data, propensity score matching (PSM) is a statistical matching technique that attempts to estimate the effect of a treatment, policy, or other intervention by accounting for the covariates that predict receiving the treatment. PSM attempts to reduce the bias due to confounding variables that could be found in an estimate of the treatment effect obtained from simply comparing outcomes among units that received the treatment versus those that did not. Paul R. Rosenbaum and Donald Rubin introduced the technique in 1983.

In the statistical analysis of observational data, propensity score matching (PSM) is a statistical matching technique that attempts to estimate the effect of a treatment, policy, or other intervention by accounting for the covariates that predict receiving the treatment. PSM attempts to reduce the bias due to confounding variables that could be found in an estimate of the treatment effect obtained from simply comparing outcomes among units that received the treatment versus those that did not. Paul R. Rosenbaum and Donald Rubin introduced the technique in 1983.

【机器翻译】在观察数据的统计分析中，倾向性评分匹配是一种统计匹配技术，它试图通过计算预测接受治疗的协变量来估计治疗、政策或其他干预的效果。PSM 试图减少由于混杂变量造成的偏倚，这些变量可以通过简单地比较接受治疗的单位和没有接受治疗的单位之间的结果来估计治疗效果。保罗 · 罗森鲍姆和唐纳德 · 鲁宾在1983年介绍了这项技术。

在观察数据的统计分析中，倾向性评分匹配Propensity Score Matching (PSM)是一种统计匹配技术，用来估计治疗、政策或其他干预的效果，方法是将协变量对样本“是否接受处理”的影响考虑在内。PSM试图减少由于混杂变量造成的偏倚。这些偏倚一般会在那些只对处理单元和对照单元的结果做简单对比的评估中出现。保罗·罗森鲍姆Paul R. Rosenbaum和唐纳德·鲁宾Donald Rubin在1983年介绍了这项技术。

The possibility of bias arises because a difference in the treatment outcome (such as the average treatment effect) between treated and untreated groups may be caused by a factor that predicts treatment rather than the treatment itself. In randomized experiments, the randomization enables unbiased estimation of treatment effects; for each covariate, randomization implies that treatment-groups will be balanced on average, by the law of large numbers. Unfortunately, for observational studies, the assignment of treatments to research subjects is typically not random. Matching attempts to reduce the treatment assignment bias, and mimic randomization, by creating a sample of units that received the treatment that is comparable on all observed covariates to a sample of units that did not receive the treatment.

The possibility of bias arises because a difference in the treatment outcome (such as the average treatment effect) between treated and untreated groups may be caused by a factor that predicts treatment rather than the treatment itself. In randomized experiments, the randomization enables unbiased estimation of treatment effects; for each covariate, randomization implies that treatment-groups will be balanced on average, by the law of large numbers. Unfortunately, for observational studies, the assignment of treatments to research subjects is typically not random. Matching attempts to reduce the treatment assignment bias, and mimic randomization, by creating a sample of units that received the treatment that is comparable on all observed covariates to a sample of units that did not receive the treatment.

【机器翻译】出现偏倚的可能性是因为治疗组和未治疗组之间治疗结果(如平均治疗效果)的差异可能是由预测治疗的因素而不是治疗本身造成的。在随机实验中，随机化可以对治疗效果进行无偏估计; 对于每个协变量，随机化意味着治疗组将按照大数定律在平均水平上达到平衡。不幸的是，对于观察性研究来说，对研究对象的治疗分配通常不是随机的。匹配试图减少处理分配偏差，并模拟随机化，通过创建一个样本单位接受的处理是可比的所有观察到的协变量的一个样本单位没有接受处理。

出现偏倚的原因可能是某个因素通过决定样本是否接受处理而导致了处理组和对照组的效果（如平均处理效果）差异，而不是处理本身导致了差异。在随机实验中，随机化选择样本可以做到对处理效果的无偏估计，根据大数定律，随机化意味着基于协变量的平均水平，均衡分配处理组和对照组。不幸的是，对于观察性研究来说，研究对象通常不是随机接受处理的。匹配就是要减少对象非随机接受处理产生的偏倚，并模拟随机试验，方法是从处理组和对照组中分别取样，让两组样本的全部协变量都比较接近。

For example, one may be interested to know the consequences of smoking. An observational study is required since it is unethical to randomly assign people to the treatment 'smoking.' The treatment effect estimated by simply comparing those who smoked to those who did not smoke would be biased by any factors that predict smoking (e.g.: gender and age). PSM attempts to control for these biases by making the groups receiving treatment and not-treatment comparable with respect to the control variables.

For example, one may be interested to know the consequences of smoking. An observational study is required since it is unethical to randomly assign people to the treatment 'smoking.' The treatment effect estimated by simply comparing those who smoked to those who did not smoke would be biased by any factors that predict smoking (e.g.: gender and age). PSM attempts to control for these biases by making the groups receiving treatment and not-treatment comparable with respect to the control variables.

【机器翻译】例如，人们可能有兴趣知道吸烟的后果。因为随机分配患者接受‘吸烟’治疗是不道德的，所以需要一个观察性研究简单地比较吸烟者和不吸烟者的治疗效果会受到任何预测吸烟的因素的影响(例如:。: 性别及年龄)。PSM 试图通过使接受治疗和不接受治疗的组与控制变量相比较来控制这些偏差。

例如，人们想知道吸烟的后果。但是随机分配让患者“吸烟”是不道德的，所以需要做一个观察性研究。简单对比评估吸烟者和不吸烟者会让处理效果产生偏差，它会受到能影响吸烟行为的因素的影响（例如：性别及年龄）。PSM要做的是通过让处理组和对照组的控制变量尽量相似来达到控制这些偏差的目的。

综述

PSM is for cases of causal inference and simple selection bias in non-experimental settings in which: (i) few units in the non-treatment comparison group are comparable to the treatment units; and (ii) selecting a subset of comparison units similar to the treatment unit is difficult because units must be compared across a high-dimensional set of pretreatment characteristics.

PSM is for cases of causal inference and simple selection bias in non-experimental settings in which: (i) few units in the non-treatment comparison group are comparable to the treatment units; and (ii) selecting a subset of comparison units similar to the treatment unit is difficult because units must be compared across a high-dimensional set of pretreatment characteristics.

【机器翻译】PSM 适用于非实验环境中因果推断和简单选择偏差的情况，其中: (i)非处理对照组中与处理单元可比的单元很少; (ii)选择与处理单元类似的比较单元子集很困难，因为必须跨一组高维预处理特征进行比较。

PSM适用于非实验环境中因果推断和简单选择偏差的情况，其中: (i)对照组与处理组中的类似的单元很少; (ii)选择与处理单元类似的对照单元集合很困难，因为必须对一组高维的处理前特征进行比较。

In normal matching, single characteristics that distinguish treatment and control groups are matched in an attempt to make the groups more alike. But if the two groups do not have substantial overlap, then substantial error may be introduced. For example, if only the worst cases from the untreated "comparison" group are compared to only the best cases from the treatment group, the result may be regression toward the mean, which may make the comparison group look better or worse than reality.

In normal matching, single characteristics that distinguish treatment and control groups are matched in an attempt to make the groups more alike. But if the two groups do not have substantial overlap, then substantial error may be introduced. For example, if only the worst cases from the untreated "comparison" group are compared to only the best cases from the treatment group, the result may be regression toward the mean, which may make the comparison group look better or worse than reality.

【机器翻译】在正常的匹配中，区分治疗组和对照组的单一特征被匹配，试图使这些组更加相似。但是，如果这两个组没有实质性的重叠，那么可能会引入实质性的错误。例如，如果只将来自未经治疗的对照组的最差病例与来自治疗组的最好病例进行比较，结果可能是趋中回归，这可能使对照组看起来比实际情况更好或更糟。

在正常的匹配中，对一组能够区分处理组和对照组的特征做匹配，以使两组的特征更加相似。但如果这两个组的特征没有实质性的重叠，那么可能会引入实质性的错误。例如，拿对照组最糟的病例和处理组最好的病例进行比较，结果可能倾向于回归均值，这会让对照组看起来比实际情况更好或更糟。

PSM employs a predicted probability of group membership—e.g., treatment versus control group—based on observed predictors, usually obtained from logistic regression to create a counterfactual group. Propensity scores may be used for matching or as covariates, alone or with other matching variables or covariates.

PSM employs a predicted probability of group membership—e.g., treatment versus control group—based on observed predictors, usually obtained from logistic regression to create a counterfactual group. Propensity scores may be used for matching or as covariates, alone or with other matching variables or covariates.

【机器翻译】PSM 使用了一种预测的群体成员概率---- 例如，治疗组与控制组---- 基于观察预测，通常从 Logit模型获得来创造一个反事实的群体。倾向得分可用于匹配或作为协变量，单独或与其他匹配变量或协变量。

PSM利用观察数据预测样本落入不同分组（例如，处理组与控制组）的概率，通常用Logistic回归方法，然后利用此概率创造一个反事实的群体。倾向性评分可用于匹配，也可作为协变量，可以单独使用，也可以与其他匹配变量或协变量一同使用。

一般步骤

1. Run logistic regression:

1. Run logistic regression:

1. 做Logistic回归：

• Dependent variable: Z = 1, if unit participated (i.e. is member of the treatment group); Z = 0, if unit did not participate (i.e. is member of the control group).

• 因变量：参与处理（属于处理组），则Z = 1；未参与处理（属于对照组），则Z = 0。

• Choose appropriate confounders (variables hypothesized to be associated with both treatment and outcome)

• 选择合适的混杂因素（既影响处理方式又影响处理结果的变量）

• Obtain an estimation for the propensity score: predicted probability (p) or log[p/(1 − p)].

• 计算倾向性评分的估计量：预测概率(p)或log[p/(1 − p)]。

2. Check that covariates are balanced across treatment and comparison groups within strata of the propensity score.

2. Check that covariates are balanced across treatment and comparison groups within strata of the propensity score.

2. 【机器翻译】检查协变量是平衡的治疗和比较组内的倾向分层。

2. 检查协变量的倾向性评分在处理组和对照组是否均衡

• Use standardized differences or graphs to examine distributions

• 使用标准化差异指标或者图形来检验分布情况

3. Match each participant to one or more nonparticipants on propensity score, using one of these methods:

3. Match each participant to one or more nonparticipants on propensity score, using one of these methods:

3. 【机器翻译】根据倾向得分，将每个参与者与一个或多个非参与者进行匹配，使用以下方法之一:

3. 根据倾向性评分，将每个参与者与一个或多个非参与者匹配，使用以下方法之一：

• Nearest neighbor matching

• 最近邻匹配

• Caliper matching: comparison units within a certain width of the propensity score of the treated units get matched, where the width is generally a fraction of the standard deviation of the propensity score

• 卡钳匹配：在处理单元倾向性评分的一个范围内选取对照单元，范围的宽度通常用倾向性评分的标准差乘上一个比例值

• Mahalanobis metric matching in conjunction with PSM

• 马氏度量 与PSM配合使用

• Stratification matching

• 分层匹配

• Difference-in-differences matching (kernel and local linear weights)

• 双重差分匹配（核和局部线性加权）

• Exact matching

• 精确匹配

4. Verify that covariates are balanced across treatment and comparison groups in the matched or weighted sample

4. Verify that covariates are balanced across treatment and comparison groups in the matched or weighted sample

4.【机器翻译】验证协变量是平衡的处理和对照组在匹配或加权样本

4. 对比处理组和对照组的匹配样本或加权样本，验证协变量是否均衡

5. Multivariate analysis based on new sample

5. Multivariate analysis based on new sample

5.【机器翻译】基于新样本的多变量分析

5. 基于新样本的多变量分析

• Use analyses appropriate for non-independent matched samples if more than one nonparticipant is matched to each participant

• 如果每个参与者都匹配了多个非参与者，则适当应用非独立匹配样本分析

Note: When you have multiple matches for a single treated observation, it is essential to use Weighted Least Squares rather than Ordinary Least Squares.

Note: When you have multiple matches for a single treated observation, it is essential to use Weighted Least Squares rather than Ordinary Least Squares.

【机器翻译】注意: 当你有多个匹配的单一处理的观察，它是必不可少的使用加权最小二乘而不是一般最小平方法。

注意：当一个处理样本有多个匹配时，则必须用加权最小二乘法，而不能用普通最小二乘法。

正式定义

基本设置

The basic case is of two treatments (numbered 1 and 0), with N [Independent and identically distributed random variables|i.i.d] subjects. Each subject i would respond to the treatment with [math]\displaystyle{ r_{1i} }[/math] and to the control with [math]\displaystyle{ r_{0i} }[/math]. The quantity to be estimated is the average treatment effect: [math]\displaystyle{ E[r_1]-E[r_0] }[/math]. The variable [math]\displaystyle{ Z_i }[/math] indicates if subject i got treatment (Z = 1) or control (Z = 0). Let [math]\displaystyle{ X_i }[/math] be a vector of observed pretreatment measurement (or covariate) for the ith subject. The observations of [math]\displaystyle{ X_i }[/math] are made prior to treatment assignment, but the features in [math]\displaystyle{ X_i }[/math] may not include all (or any) of the ones used to decide on the treatment assignment. The numbering of the units (i.e.: i = 1, ..., i = N) are assumed to not contain any information beyond what is contained in [math]\displaystyle{ X_i }[/math]. The following sections will omit the i index while still discussing about the stochastic behavior of some subject.

基本场景是，有两种处理方式（分别记为1和0），N个独立同分布物体。每个物体i如果接受了处理则响应为[math]\displaystyle{ r_{1i} }[/math]，接受控制则响应为[math]\displaystyle{ r_{0i} }[/math]。被估计量是平均处理效应：[math]\displaystyle{ E[r_1]-E[r_0] }[/math]。变量[math]\displaystyle{ Z_i }[/math]指示物体i接受处理(Z = 1)还是接受控制(Z = 0)。让[math]\displaystyle{ X_i }[/math]代表第i个物体处理前观测值（或者协变量）的向量。对[math]\displaystyle{ X_i }[/math]的测量发生于处理前，但是[math]\displaystyle{ X_i }[/math]中也可以不包括那些决定是否接受处理的特征。单元编号（即：i = 1, ..., i = N）不包含任何[math]\displaystyle{ X_i }[/math]所包含信息之外的的信息。以下部分在讨论某些物体的随机行为的时候将省略索引i。

强可忽略处理分配

See also: Ignorability

See also: 可忽略性

Let some subject have a vector of covariates X (i.e.: conditionally unconfounded), and some potential outcomes r₀ and r₁ under control and treatment, respectively. Treatment assignment is said to be strongly ignorable if the potential outcomes are independent of treatment (Z) conditional on background variables X. This can be written compactly as

设某个物体有向量协变量X（即：条件非混杂变量），以及潜在结果r₀和r₁分别对应着控制和处理两种情况。如果潜在结果在给定背景变量X的条件下独立于处理举动（Z），则可以说样本是否接受处理分配是强可忽略的。可简洁表述为

[math]\displaystyle{ r_0, r_1 \perp Z \mid X }[/math]

这里[math]\displaystyle{ \perp }[/math]代表统计独立.

平衡得分

A balancing score b(X) is a function of the observed covariates X such that the conditional distribution of X given b(X) is the same for treated (Z = 1) and control (Z = 0) units:

A balancing score b(X) is a function of the observed covariates X such that the conditional distribution of X given b(X) is the same for treated (Z = 1) and control (Z = 0) units:

平衡得分b(X)是观测协变量X的函数。在给定b(X)时，处理单元和控制单元的X有相同的条件分布：

[math]\displaystyle{ Z \perp X \mid b(X). }[/math]

最一般的平衡得分函数是[math]\displaystyle{ b(X) = X }[/math].

倾向性评分

A propensity score is the probability of a unit (e.g., person, classroom, school) being assigned to a particular treatment given a set of observed covariates. Propensity scores are used to reduce selection bias by equating groups based on these covariates.

Suppose that we have a binary treatment indicator Z, a response variable r, and background observed covariates X. The propensity score is defined as the conditional probability of treatment given background variables:

A propensity score is the probability of a unit (e.g., person, classroom, school) being assigned to a particular treatment given a set of observed covariates. Propensity scores are used to reduce selection bias by equating groups based on these covariates.

Suppose that we have a binary treatment indicator Z, a response variable r, and background observed covariates X. The propensity score is defined as the conditional probability of treatment given background variables:

倾向性评分是根据协变量观测值计算得出的一个单元（例如：个人，教室，学校）被指配接受特定处理的概率。通倾向性评分可用于让处理组和对照组的协变量趋同，从而减少选择偏差。

假设有一个二值处理标识Z，一个响应变量r，以及被观测的背景协变量X。倾向性评分定义为，在给定背景变量条件下单元接受处理的条件概率：

[math]\displaystyle{ e(x) \ \stackrel{\mathrm{def}}{=}\ \Pr(Z=1 \mid X=x). }[/math]

In the context of causal inference and survey methodology, propensity scores are estimated (via methods such as logistic regression, random forests, or others), using some set of covariates. These propensity scores are then used as estimators for weights to be used with Inverse probability weighting methods.

In the context of causal inference and survey methodology, propensity scores are estimated (via methods such as logistic regression, random forests, or others), using some set of covariates. These propensity scores are then used as estimators for weights to be used with Inverse probability weighting methods.

在因果推断和调查方法的范围内，通过Logistic回归、随机森林或其他方法，利用一组协变量估计倾向性评分。然后将这些倾向性评分即可作为权重估计量用于逆概率加权方法。

主要定理

The following were first presented, and proven, by Rosenbaum and Rubin in 1983:

The following were first presented, and proven, by Rosenbaum and Rubin in 1983:[1]

以下是Rosenbaum和Rubin于1983年首次提出并证明的：

• The propensity score [math]\displaystyle{ e(x) }[/math] is a balancing score.
• 倾向性评分[math]\displaystyle{ e(x) }[/math]是平衡得分。

• Any score that is 'finer' than the propensity score is a balancing score (i.e.: [math]\displaystyle{ e(X)=f(b(X)) }[/math] for some function f). The propensity score is the coarsest balancing score function, as it takes a (possibly) multidimensional object (X_i) and transforms it into one dimension (although others, obviously, also exist), while [math]\displaystyle{ b(X)=X }[/math] is the finest one.
• 任何比倾向性评分更“精细”的得分都是平衡得分（即：对于函数f，[math]\displaystyle{ e(X)=f(b(X)) }[/math]）。倾向性评分是最粗粒度的平衡得分函数，因为它把一个（可能是）多维的对象（X_i）转换成只有一维（尽管其他维度显然也存在），而[math]\displaystyle{ b(X)=X }[/math]则是最细粒度的平衡得分函数（保留全部维度）。

• If treatment assignment is strongly ignorable given X then:
• 如果对于给定的X，处理分配满足强可忽略条件，则：

• It is also strongly ignorable given any balancing function. Specifically, given the propensity score:
• 给定任何的平衡函数，具体来说，给定倾向性评分，处理分配也是强可忽略的：

[math]\displaystyle{ (r_0, r_1) \perp Z \mid e(X). }[/math]

回归

• For any value of a balancing score, the difference between the treatment and control means of the samples at hand (i.e.: [math]\displaystyle{ \bar{r}_1-\bar{r}_0 }[/math]), based on subjects that have the same value of the balancing score, can serve as an unbiased estimator of the average treatment effect: [math]\displaystyle{ E[r_1]-E[r_0] }[/math].
• 对于有相同平衡得分值的处理样本和对照样本，它们响应变量均值之差（即：[math]\displaystyle{ \bar{r}_1-\bar{r}_0 }[/math]），可以作为平均处理效应的无偏估计量：[math]\displaystyle{ E[r_1]-E[r_0] }[/math]。

• Using sample estimates of balancing scores can produce sample balance on X
• 利用平衡得分的样本估计可产生在X上均衡的样本

与充分性的关系

If we think of the value of Z as a parameter of the population that impacts the distribution of X then the balancing score serves as a sufficient statistic for Z. Furthermore, the above theorems indicate that the propensity score is a minimal sufficient statistic if thinking of Z as a parameter of X. Lastly, if treatment assignment Z is strongly ignorable given X then the propensity score is a minimal sufficient statistic for the joint distribution of [math]\displaystyle{ (r_0, r_1) }[/math].

If we think of the value of Z as a parameter of the population that impacts the distribution of X then the balancing score serves as a sufficient statistic for Z. Furthermore, the above theorems indicate that the propensity score is a minimal sufficient statistic if thinking of Z as a parameter of X. Lastly, if treatment assignment Z is strongly ignorable given X then the propensity score is a minimal sufficient statistic for the joint distribution of {\displaystyle (r_{0},r_{1})}{\displaystyle (r_{0},r_{1})}.

如果我们把Z的值想成影响X分布的群体参数，则平衡得分充当了Z的充分统计量。进一步，上述定理指出，如果把Z视为X的参数，则倾向性评分就是最小充分统计量。最后，给定X，如果Z是强可忽略的，则倾向性评分是[math]\displaystyle{ (r_0, r_1) }[/math]联合分布的最小充分统计量。

混杂变量的图检测方法

Judea Pearl has shown that there exists a simple graphical test, called the back-door criterion, which detects the presence of confounding variables. To estimate the effect of treatment, the background variables X must block all back-door paths in the graph. This blocking can be done either by adding the confounding variable as a control in regression, or by matching on the confounding variable.

Judea Pearl has shown that there exists a simple graphical test, called the back-door criterion, which detects the presence of confounding variables. To estimate the effect of treatment, the background variables X must block all back-door paths in the graph. This blocking can be done either by adding the confounding variable as a control in regression, or by matching on the confounding variable.[2]

朱迪亚·珀尔Judea Pearl已经表明存在一个简单的图检测方法，称为后门准则，它可以检测到混杂变量的存在。为了估计处理效果，背景变量X必须阻断图中的所有后门路径。通过把混杂变量加入回归的控制变量，或者在混杂变量上进行匹配可以实现后门路径的阻断。

缺点

PSM has been shown to increase model "imbalance, inefficiency, model dependence, and bias," which is not the case with most other matching methods. The insights behind the use of matching still hold but should be applied with other matching methods; propensity scores also have other productive uses in weighting and doubly robust estimation.

【机器翻译】PSM 已经被证明会增加模型的“不平衡性、低效率、模型依赖性和偏差”，这与大多数其他匹配方法不同。使用匹配的见解仍然有效，但应该与其他匹配方法一起应用; 倾向得分在加权和双重稳健估计方面也有其他有益的用途。

PSM已经被证明会加剧模型的“不平衡性、低效率、模型依赖性和偏差”，这与大多数其他匹配方法不同。匹配方法背后的见解仍然成立，但应该与其他匹配方法一起应用；倾向得分在加权和双重稳健估计方面也有其他有益的用途。

Like other matching procedures, PSM estimates an average treatment effect from observational data. The key advantages of PSM were, at the time of its introduction, that by using a linear combination of covariates for a single score, it balances treatment and control groups on a large number of covariates without losing a large number of observations. If units in the treatment and control were balanced on a large number of covariates one at a time, large numbers of observations would be needed to overcome the "dimensionality problem" whereby the introduction of a new balancing covariate increases the minimum necessary number of observations in the sample geometrically.

【机器翻译】与其他匹配程序一样，PSM 从观测数据中估计平均处理效果。在引入 PSM 的时候，它的主要优点是，通过使用一个线性组合的协变量作为一个单一的评分，它平衡了治疗组和对照组在大量的协变量上，而不会失去大量的观察数据。如果处理和控制中的单元在大量的协变量上一次平衡，就需要大量的观测数据来克服“维数问题”，即引入新的平衡协变量几何地增加样本中必要的最小观测数据。

与其他匹配过程一样，PSM也是从观测数据中估计平均处理效应。在引入PSM之时，它的主要优点是，通过使用协变量的线性组合得到一个单一评分，以大量的协变量为基础平衡了处理组和对照组，却不大量损失观测数据。如果在有众多协变量的情况下，对每一个些变量都分别做处理单元和对照单元平衡的话，就需要大量的观测数据来克服”维度问题“，即，每引入一个新的平衡协变量都会在几何上增加最小所需的观测样本数量。

One disadvantage of PSM is that it only accounts for observed (and observable) covariates and not latent characteristics. Factors that affect assignment to treatment and outcome but that cannot be observed cannot be accounted for in the matching procedure. As the procedure only controls for observed variables, any hidden bias due to latent variables may remain after matching. Another issue is that PSM requires large samples, with substantial overlap between treatment and control groups.

【机器翻译】PSM 的一个缺点是它只能解释观察到的(和可观察到的)协变量，而不能解释潜在的特征。影响治疗分配和结果但无法观察的因素不能在匹配程序中说明。由于程序只控制观察变量，任何隐藏的偏见由于潜在变量可能仍然匹配后。另一个问题是 PSM 需要大量的样本，治疗组和对照组之间有大量的重叠。

PSM的一个缺点是它只能涵盖已观测的（和可观测的）协变量，而无法涵盖潜在变量。那些能影响处理分配却不可观测的因素无法被纳入匹配过程的考量范围。由于匹配过程只控制可观测变量，那些隐藏的偏差在匹配后依然可能存在。另一个问题是PSM还要求在大量样本中，在处理组和对照组之间有大量的重叠。

General concerns with matching have also been raised by Judea Pearl, who has argued that hidden bias may actually increase because matching on observed variables may unleash bias due to dormant unobserved confounders. Similarly, Pearl has argued that bias reduction can only be assured (asymptotically) by modelling the qualitative causal relationships between treatment, outcome, observed and unobserved covariates. Confounding occurs when the experimenter is unable to control for alternative, non-causal explanations for an observed relationship between independent and dependent variables. Such control should satisfy the "backdoor criterion" of Pearl. It can also easily be implemented manually.

【机器翻译】朱迪亚 · 珀尔也提出了关于配对的普遍担忧，他认为隐性偏见实际上可能会增加，因为观察变量的配对可能会由于潜在的未观察混杂因素而释放出偏见。同样，珀尔认为，只有通过建立治疗、结果、观察和未观察协变量之间的定性因果关系模型，才能确保(渐近地)减少偏见。当实验者无法控制对独立变量和因变量之间观察到的关系的替代性、非因果性解释时，混淆就发生了。这种控制应满足珍珠的“后门规范”。它也可以很容易地手动实现。

Judea Pearl也提出了关于匹配方法的一般性担忧，他认为对可观测变量进行匹配可能会让那些原本处于休眠状态的混杂因素被释放，从而实际上可能加剧隐藏的偏差。同样，Pearl认为，只有通过对处理、结果、可观测和不可观测的协变量之间的定性因果关系进行建模，才能确保（渐进地）减少偏差。当试验者无法控制对独立变量和因变量之间观察到的关系的替代性、非因果性解释时，混杂就会发生。这样的控制应该满足Pearl的“后门准则”。它很容易手工实现。

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