现在讲一下“confounder(混淆者)”和“collider(对撞者)”。简单讲，混淆变量就是那些同时会影响政策处理变量和结果变量的变量，在英文中也叫common factors。这样的例子很多，比如研究接受职业培训项目是否会提高个体找到工作的概率？其中，可能的混淆变量包括但不限于：年龄和性别等。因为，像年龄会同时影响个体是否参与职业培训项目以及是否能找到工作。要识别出政策效应，咱们要做的就是尽量分离出混淆因素，因此，在实证中需要做的就是控制住混淆变量。具体地，咱们需要将混淆变量作为控制变量放到回归方程中。那如果有些混淆变量不可观测，此时，该如何减轻他们对政策效应估计的影响呢？尽量找到代理变量，比如用教育水平作为个体能力的代理变量。

接下来讲一讲collider。他与混淆变量是相反的，但也对政策效应估计产生重要影响。collider是对撞的意思，即政策处理变量和结果变量都会影响这个变量。比如，在上面的例子中，收入可能就是一个collider，因为参与职业培训和找工作的概率都会影响到个体的收入。此时，咱们若在个体工作回归方程中误加收入变量作为控制变量，那会导致政策效应估计出现错误(collider bias)。有时候，甚至会让两个本不相关的变量或者负(正)相关的变量，产生相关关系或正(负)相关关系。其中，以伯克森悖论(Berkson's Paradox）最为著名。也就是说，凡是collider变量，咱们都不要把他们放到回归方程中去。

伯克森悖论，指的是两个本来无关的变量之间体现出貌似强烈的相关关系。举个例子来说，假设某学校在招收学生时，要求学生要么学习成绩好，要么体育成绩好。所有的报考学生需要参加两门考试：文化（语数外），和体育（跑跳投）。最后，学校仅录取在任一考试中考到90分以上的报考学生。所以能够被学校录取的学生，要么在文化考试中考到90分以上，或者在体育考试中考到90分以上，或者在两门考试中都考到90分以上。现在如果我们分析这些被入取学生的成绩分布，会发现一个学生的学习成绩，和体育成绩是负相关的。因为那些体育成绩最好的学生（比如体育100分），他们的文化平均分为50分（假设他们的文化考试呈现正态分布）。而体育成绩最差的学生（比如体育成绩10分），其文化平均成绩为95分（因为只有超过90分的学生才被录取）。因此，分析人员可能会得出结论：体育越好，文化成绩越差。文化成绩越好，体育越差。但这个结论显然是错误的。

Mediator versus Moderator variables 

The classic reference on this topic is Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173-1182. 

Most of what is written here comes directly from this classic paper.

· Moderator variables - "In general terms, a moderator is a qualitative (e.g., sex, race, class) or quantitative (e.g., level of reward) variable that affects the direction and/or strength of the relation between an independent or predictor variable and a dependent or criterion variable. Specifically within a correlational analysis framework, a moderator is a third variable that affects the zero-order correlation between two other variables. ... In the more familiar analysis of variance (ANOVA) terms, a basic moderator effect can be represented as an interaction between a focal independent variable and a factor that specifies the appropriate conditions for its operation." p. 1174

· Mediator variables - "In general, a given variable may be said to function as a mediator to the extent that it accounts for the relation between the predictor and the criterion. Mediators explain how external physical events take on internal psychological significance. Whereas moderator variables specify when certain effects will hold, mediators speak to how or why such effects occur." p. 1176 

     The general test for mediation is to examine the relation between the predictor and the criterion variables, the relation between the predictor and the mediator variables, and the relation between the mediator and criterion variables. All of these correlations should be significant. The relation between predictor and criterion should be reduced (to zero in the case of total mediation) after controlling the relation between the mediator and criterion variables.

      Another way to think about this issue is that a moderator variable is one that influences the strength of a relationship between two other variables, and a mediator variable is one that explains the relationship between the two other variables. As an example, let's consider the relation between social class (SES) and frequency of breast self-exams (BSE). Age might be a moderator variable, in that the relation between SES and BSE could be stronger for older women and less strong or nonexistent for younger women. Education might be a mediator variable in that it explains why there is a relation between SES and BSE. When you remove the effect of education, the relation between SES and BSE disappears.

Understanding moderation is one of those topics in statistics that is so much harder than it needs to be.

Here are three suggestions to make it just a little easier.

1. Realize that moderation just means an interaction

I have spoken with a number of researchers who are surprised to learn that moderation is just another term for interaction.

Perhaps it’s because moderation often appears with discussions of mediation. Or because we tend to think of interaction as being part of ANOVA, but not regression.

In any case, both an interaction and moderation mean the same thing: the effect of one predictor on a response variable is different at different values of the second predictor.

When we speak of moderation, we usually call the first predictor an independent variable, and the second the moderator.  In other words, we’re really just interested in the effect of the independent variable on the dependent variable, and this effect is different at different values of the moderator.  We’re never interested in the effect of the moderator on the dependent variable.

When we speak of interaction, we don’t usually distinguish between independent variable and the moderator. Either predictor could be considered to “moderate” the effect of the other.

Mathematically, there is no distinction. You don’t have to interpret one variable as the independent variable and the other as the moderator.  But it can help interpretation to think of them that way.

2. Graph the means and/or predicted values

Moderation effects are difficult to interpret without a graph. It helps to see what is the effect of the independent value at different values of the moderator.

If the independent variable is categorical, we measure its effect through mean differences, and those differences are easiest to see with plots of the means.

Moderation says that those mean differences are not the same at every value of the moderator. It can be hard to discern a pattern in how they differ without seeing it.

For example, the mean difference may get larger as the moderator increases. Or it may flip signs.

If the independent variable is continuous, we measure its effect through a slope of the regression line.  So you want to plot the predicted values of those regression lines.  Moderation says that the slope of the regression line is different at every value of the moderator. (Yes, that one regression equation really represents many different lines—one for every possible value of the moderator).

Once again, a positive slope may get larger (or smaller) as the moderator increases.  Or it too can flip signs, going from a positive slope at low values of the moderator to a negative slope at high values.

But this is difficult to see without a graph.

If the moderator itself is continuous, you could potentially choose an infinite number of values at which to plot the effect of the independent variable.  Not only would that take a while, the graph would be such a mess, you couldn’t see any patterns.

Luckily, plotting the effects of the independent variable at only a few values of the moderator are usually needed to see patterns.

3. Choose the values of continuous moderators intentionally

There are conventions to help you choose the best values of the moderator for plotting predicted values.  But these conventions don’t always work in every situation.

For example, one convention suggested by Cohen and Cohen and popularized by Aiken and West is to use three values of the moderator: the mean, the value one standard deviation above, and the value one standard deviation below the mean.

Most of the time, this is a great convention and it works very well.

One situation where it doesn’t is if the moderator is positively skewed. The value one standard deviation below the mean can be beyond the range of the data. In that case, using the minimum or some other small value of the moderator may be a better choice.

Likewise, sometimes very specific values of the moderator are particularly meaningful. For example, in years of education, values of 12 and 16 generally indicate high school and college graduation.

If years of education was the moderator, plotting effects of the independent variable when education equaled 12 makes a lot of sense, even if the mean is 12.57.

It’s not that using 12.57 is wrong. But spending a little time thinking about a more appropriate value can make interpretation, and therefore communication to your audience, easier.

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