Lab · Group slopes练习 · 分组斜率
Does one line fit every faculty?一条线适合每一个学院吗？
A writing workshop has 48 students: 12 in Arts, 12 in Social Sciences, 12 in Science, and 12 in Engineering. For each student the sheet has hours in the workshop and the course grade. The teacher can report one rise for all 48, or one rise in each faculty. The heavy line is the single rise, the pooled line: one line that ignores which faculty a student is in. The thin lines are the four faculties. If Engineering rises fast and Arts is flat, that single number matches none of the four.写作工作坊有 48 名学生：文学院、社会科学学院、理学院、工程学院各 12 人。每人有两项数据：参加工作坊的时数和课程成绩。老师可以给全部 48 人报告一个总体斜率，也可以每个学院分别报告。粗线是总体斜率，也就是不区分学院、把所有人放在一起拟合的一条线；细线是四个学院各自的线。如果工程学院上升得很快、文学院几乎持平，这一个总体数字就和哪个学院都对不上。
Why it matters.为什么重要。 A workshop can look helpful on average and still be steep in one faculty and flat in another. The average is a compromise among the groups. It is not a description of any one of them.一个工作坊平均来看可能有用，但在某个学院效果很明显，在另一个学院却几乎没有作用。平均值只是各组之间的折中，并不能描述其中任何一组。
Try this试一试 Set the faculty gap near zero and watch the lines sit together. Then raise the gap. The heavy line stays in the middle while the thin lines fan out.先把学院之间的差异调到接近零，看几条线靠在一起；再把差异调大，看细线怎样散开，而粗线始终停在中间。
How to read the chart怎么读这张图
If you read this wrong如果读错了
If the teacher tells the faculty office that the heavy line means the writing workshop lifts grades about the same in every faculty, the office may fund one campus-wide program. When the thin lines fan out, Engineering can be steep and Arts flat. The heavy line is neither faculty.如果老师跟院办说，粗线说明写作工作坊在每个学院的效果都差不多，院办可能会资助一个全校统一的项目。但细线散开时，工程学院可能很陡、文学院几乎持平，粗线代表的并不是其中任何一个学院。
What this chart can and can’t tell you这张图能说明什么、不能说明什么
It lets you see one line through all 48 students sit between the four faculty lines.能说明：用全部 48 名学生拟合的一条线，可能落在四个学院的线中间，哪个都不像。
It isn’t enough for the funding note. It still has to say why faculty is the right grouping, how much a small faculty’s line should borrow from the others, and how uncertain each line is.但还不能据此写资助报告。报告还要说明为什么按学院分组、人数少的学院该在多大程度上参考其他学院的数据，以及每条线的不确定性有多大。
The math, and its limits公式、假设与局限
Each faculty has a fixed offset: −1, −0.35, 0.35, 1. A student’s slope is 0.30 plus “how far the faculties differ” times that offset, plus a little noise. The heavy line is ordinary least squares on all 48 students, with faculty ignored. The thin lines are the same fit inside each faculty. Nothing here pulls a small faculty toward the university average.每个学院有一个固定的偏移量：−1、−0.35、0.35、1。学生的斜率等于 0.30 加上“学院差异”乘以这个偏移量，再加上少量随机波动。粗线是不区分学院、把 48 名学生放在一起做的普通最小二乘；细线是每个学院内部分别做的同样拟合。这里没有做部分合并（partial pooling），也就是没有把小学院的估计往全校平均拉。