Lab · Confounding混杂练习 · 混杂
Is the pay gap the subsidy?工资差距是补贴吗？
Someone at the labour department has 300 workers. For each person the sheet has three numbers: whether they took a retraining subsidy, monthly pay one year later, and household savings before the program. If pay rises with use, the note will recommend expanding the subsidy. Savings can raise both use and later pay, so the line can slope up even when the subsidy itself adds nothing. That shared push is confounding.劳工处的同事手上有 300 名工人。每人三列数字：有没有领再培训补贴、一年后的月薪、参加之前家里的积蓄。如果月薪随着领补贴往上走，说明里就会建议把补贴扩大。积蓄可以同时抬高“去领”和“后来的月薪”，所以补贴本身一分钱都没加上时，线仍可能往上。这种两边一起推，就叫混杂。
Why it matters.为什么要紧。 The line is later pay against whether someone took the subsidy. It can rise because savings push both, even when the subsidy itself adds nothing. A note that claims the subsidy’s own effect has to say what the savings column is, and why it is fair to adjust for it.这条线画的是后来的月薪，对着有没有领补贴。积蓄可以两边一起推，所以补贴本身没加钱时，线仍会往上。说明如果要讲的是补贴本身的作用，就得说出积蓄那一列是什么，以及为什么可以把它拿来调整。
Try this试一试 Leave the subsidy’s own effect at zero. Turn up how strongly savings push both use and later pay. Does the line still rise?把补贴本身的作用留在零。把积蓄推动领补贴和后来月薪的力度调高。线还会上升吗？
How to read the chart怎么读这张图
If you read this wrong要是读错了
If the analyst tells the person who approves the budget that the upward line means the subsidy raised pay, the office can expand a program that added nothing. With the subsidy’s own effect at zero, the tilt is household savings pushing both who takes the subsidy and later pay.如果同事告诉批预算的人，往上的线表示补贴提高了月薪，部门就可能扩大一个本身一分钱都没加上的项目。补贴本身的作用为零时，这条斜，来自家里的积蓄同时推动了“谁去领”和“后来的月薪”。
What this picture is enough for这张图够说明什么
Enough to see one line through 300 workers slope up when the subsidy itself adds nothing to pay.看得出：补贴本身没给月薪加一分钱时，穿过 300 名工人的线仍可以往上。
Not enough to recommend expanding the subsidy. The note still has to name the savings column, say why it is fair to adjust for savings, and use a design that is not this one straight line.还不够拿来建议扩大补贴。说明仍要点出积蓄那一列，说明为什么可以按积蓄来调整，并且用的不能只是这一条直线。
The equation, and where it stops式子、假设和边界
U → X → Y, and also U → Y. U is savings. X is taking the subsidy. Y is later pay. Savings, and the two random wobbles eₓ and eᵧ, are drawn independently from a standard normal. Then X = aU + eₓ and Y = τX + bU + eᵧ. The sliders set a, b, and τ. τ is the subsidy’s own effect.U 是积蓄，X 是有没有领补贴，Y 是后来的月薪。积蓄和两处随机跳动 eₓ、eᵧ，都独立地从标准正态分布里抽。然后 X = aU + eₓ，Y = τX + bU + eᵧ。滑块设定 a、b 和 τ。τ 就是补贴本身的作用。
The line that ignores savings is Y on X only. The line that uses savings is Y on X and U. In the whole population, the slope that ignores savings is τ + ab / (a² + 1). One draw of 300 workers wobbles around that number.不看积蓄的线，是只用 X 去画 Y。把积蓄放进去的线，是用 X 和 U 一起画 Y。在全体工人里，不看积蓄的斜率是 τ + ab / (a² + 1)。抽一次 300 人，会在这个数附近晃一下。
Adjusting works here only because savings are measured without error, the straight-line story is the right one, and nothing else is pushing both sides. A real study has to argue for those assumptions. Adding controls at random can create bias. This page does not include a collider (a common result of X and Y), a mediator (a step on the way from the subsidy to pay), or selection into the sheet.这里调整能行，只因为积蓄量得没有误差，直线这个故事是对的，而且没有别的东西在两边一起推。真研究得为这些假设讲出理由。随便加控制变量，可能带来偏误。这一页没有放进对撞（X 和 Y 共同造成的结果）、中介（从补贴到月薪中间的一步），也没有放进“谁会被选进这张表”。
Easy to get wrong.容易踩的坑。 Once a control is on the sheet, it is easy to treat any adjusted slope as “the causal effect.” Here the adjustment recovers τ only under the assumptions just stated. Conditioning on a common result of X and Y, or on a step between the subsidy and pay, can pull the estimate off the effect. Those pictures are not in this simulation. Another mistake is to read one draw’s slope as the population tilt ab / (a² + 1). The sliders fix that population number. The dots add wobble around it.表上一旦有了控制变量，就容易把调整后的任何斜率都当成“因果效应”。这里只有在刚才那些假设下，调整才找回 τ。如果去控制 X 和 Y 共同造成的结果，或控制补贴和月薪中间的一步，估计会偏离真正的作用。这些图不在这一页里。另一个错，是把一次抽取的斜率读成全体的倾斜 ab / (a² + 1)。滑块固定的是全体的那个数。点只是在它周围加一点跳动。