Departures from Linearity · 对线性的偏离
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| influential/ˌɪnfluːˈenʃl/ | 有影响的 | yǒu yǐng xiǎng de |
| outlier/ˈaʊtlaɪə/ | 离群值 | lí qún zhí |
| high-leverage point/haɪ ˈliːvərɪdʒ pɔɪnt/ | 高杠杆点 | gāo gàng gǎn diǎn |
| transformation/trænsfɔːˈmeɪʃn/ | 变换 | biàn huàn |
Points that pull the line
- Not every point matters equally — some are influential 有影响的.
- An influential point noticeably changes the slope, intercept, or correlation if removed.
- The test is practical: fit the line with and without it and see if the answer moves.
- Two special kinds drive most of this: outliers and high-leverage points.
会拉动直线的点
- 不是每个点都同样重要——有些是有影响力的。
- 一个有影响力的点，若被移除会明显改变斜率、截距或相关系数。
- 检验很实用：把这个点包含与排除各拟合一次，看答案是否变动。
- 主要有两类点在起作用：离群值和高杠杆点。
Outliers vs. high leverage
- An outlier 离群值 has a large residual — it's far from the line in the $y$-direction.
- A high-leverage point 高杠杆点 has an extreme $x$-value — far out along the horizontal.
- Leverage is about horizontal extremeness; an outlier is about vertical miss.
- A point that is both (extreme $x$ and off the trend) is the most influential of all.
离群值与高杠杆
- 离群值有很大的残差——在 $y$ 方向上远离直线。
- 高杠杆点有极端的 $x$ 值——沿水平方向远远靠外。
- 杠杆关乎水平方向的极端；离群值关乎竖直方向的偏差。
- 一个两者兼备的点（$x$ 极端且偏离趋势）是所有点中最有影响力的。
How one point distorts
- A high-leverage point acts like a lever: it can swing the slope toward itself.
- Removing an influential point can flip a slope's steepness — or even its sign.
- It can also inflate or deflate $r$, making a relationship look stronger or weaker than it is.
- Always ask whether your conclusion rests on one stubborn point.
一个点如何扭曲
- 高杠杆点像一根杠杆：它能把斜率朝自己方向撬动。
- 移除一个有影响力的点，可能改变斜率的陡峭程度——甚至改变它的符号。
- 它还能抬高或压低 $r$，让关系显得比实际更强或更弱。
- 总要问一句：你的结论是否只靠一个顽固的点撑着。
Transformations for curves
- When the pattern is genuinely non-linear, a straight line is the wrong tool.
- A transformation 变换 (e.g. take logs, or square-root) can straighten a curved pattern.
- Fit the line to the transformed data, then transform predictions back.
- The limitation is real: forcing a line onto a curve gives biased, misleading predictions.
用变换处理曲线
- 当模式确实是非线性的，直线就是错误的工具。
- 一次变换（例如取对数或开方）可以把弯曲的模式拉直。
- 对变换后的数据拟合直线，再把预测变换回来。
- 局限是实实在在的：硬把直线套到曲线上，会给出有偏、误导的预测。
Leverage and outlier are not the same thing. Leverage is about an extreme $x$ (horizontal); an outlier is about a large residual (vertical). A point can be high-leverage yet sit near the line (little influence), or a mild outlier with ordinary $x$. The dangerous one is extreme in $x$ and off the pattern — it can single-handedly swing the slope and $r$.
杠杆和离群值不是一回事。杠杆关乎极端的 $x$（水平）；离群值关乎大的残差（竖直）。一个点可能杠杆很高却靠近直线（影响很小），也可能是 $x$ 普通的轻微离群值。危险的那种是 $x$ 极端且偏离模式——它能凭一己之力撬动斜率和 $r$。
A tidy line has slope $\approx 2$. Then one data point at a far-right $x$, sitting well below the trend, is added.
- That point has high leverage (extreme $x$) and a big residual.
- It drags the right end of the line down → the slope falls from $2$ toward $0.5$.
- Remove it and the slope springs back — it was influential.
一条整齐的线斜率 $\approx 2$。然后加入一个点，它的 $x$ 在最右端，位置远在趋势下方。
- 那个点既有高杠杆（$x$ 极端）又有很大的残差。
- 它把直线的右端往下拽 → 斜率从 $2$ 掉向 $0.5$。
- 移除它，斜率又弹回来——它是有影响力的。
An influential point changes the slope, intercept, or correlation when removed. An outlier has a large residual (vertical); a high-leverage point has an extreme $x$ (horizontal) — points that are both are most influential. For a truly non-linear pattern, a transformation can straighten the data; forcing a line on a curve is a real limitation.
有影响力的点在被移除时会改变斜率、截距或相关系数。离群值有很大的残差（竖直）；高杠杆点有极端的 $x$（水平）——两者兼备的点最有影响力。对真正非线性的模式，一次变换可以把数据拉直；硬把直线套在曲线上是实实在在的局限。
An influential point swings the line · 一个有影响力的点撬动直线
A far-out point off the trend can tilt the whole line. · 一个远在外侧、偏离趋势的点能让整条线倾斜。
A high-leverage point is one with an extreme value of... · 高杠杆点是指某个变量取值极端的点，这个变量是……
Leverage is about an extreme x-value; an outlier is a large residual. · 杠杆关乎极端的 x 值；离群值是大的残差。
A point is called influential if removing it noticeably changes the slope, intercept, or correlation. · 如果移除某点会明显改变斜率、截距或相关系数，就称它为有影响力的点。
That with/without comparison is the definition of influence. · 这种“有它/没它”的比较正是影响力的定义。
Match each special point to what makes it extreme. · 把每类特殊点与使其极端的因素配对。
Outlier = vertical miss; leverage = horizontal extreme. · 离群值 = 竖直偏差；杠杆 = 水平极端。
The scatterplot shows a clear curve. A good next step is to... · 散点图显示出明显的曲线。较好的下一步是……
A transformation (e.g. logs) can straighten a curved pattern. · 变换（如取对数）可以把弯曲的模式拉直。
Forcing a straight line onto a truly ___ relationship gives biased, misleading predictions. · 硬把直线套到真正 ___ 的关系上，会给出有偏、误导的预测（填“非线性”对应的英文一词）。
A line can't model a genuinely non-linear relationship well. · 直线无法很好地为真正非线性的关系建模。