r - include error terms in linear regression model

Question

I was wondering if there is a way to include error terms for a linear regression model like:

r = lm(y ~ x1+x2)

Answer 1

该代码r = lm(y ~ x1+x2)意味着我们将 y 建模为 x1 和 x2 的线性函数。由于模型不会完美，因此会有一个残差项（即模型无法拟合的剩余项）。

在数学中，正如 Rob Hyndman 在评论中指出的那样，y = a + b1*x1 + b2*x2 + e、 其中和是常数，并且a是您的残差（假定为正态分布）。b1b2e

看一个具体的例子，考虑 R 附带的 iris 数据。

model1 <- lm(Sepal.Length ~ Sepal.Width + Petal.Length + Petal.Width, data=iris)

现在我们可以从模型中提取常数（等价于a, b1,b2在这种情况下b3也是如此）。

> coefficients(model1)
(Intercept)  Sepal.Width Petal.Length  Petal.Width 
1.8559975    0.6508372    0.7091320   -0.5564827

已为模型中使用的每一行数据计算了残差。

> residuals(model1)
           1             2             3             4             5       
0.0845842387  0.2100028184 -0.0492514176 -0.2259940935 -0.0804994772
# etc. There are 150 residuals and 150 rows in the iris dataset.

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（编辑：将摘要信息剪切为不相关。）

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编辑：

您在评论中提到的Error值在 aov 的帮助页面上进行了解释。

If the formula contains a single ‘Error’ term, this is used to
specify error strata, and appropriate models are fitted within
each error stratum.

比较以下内容（改编自?aov页面。）

> utils::data(npk, package="MASS")
> aov(yield ~  N*P*K, npk)
Call:
   aov(formula = yield ~ N * P * K, data = npk)

Terms:
                       N        P        K      N:P      N:K      P:K    N:P:K Residuals
Sum of Squares  189.2817   8.4017  95.2017  21.2817  33.1350   0.4817  37.0017  491.5800
Deg. of Freedom        1        1        1        1        1        1        1        16

Residual standard error: 5.542901 
Estimated effects may be unbalanced

> aov(yield ~  N*P*K + Error(block), npk)
Call:
aov(formula = yield ~ N * P * K + Error(block), data = npk)

Grand Mean: 54.875 

Stratum 1: block

Terms:
                    N:P:K Residuals
Sum of Squares   37.00167 306.29333
Deg. of Freedom         1         4

Residual standard error: 8.750619 
Estimated effects are balanced

Stratum 2: Within

Terms:
                        N         P         K       N:P       N:K       P:K Residuals
Sum of Squares  189.28167   8.40167  95.20167  21.28167  33.13500   0.48167 185.28667
Deg. of Freedom         1         1         1         1         1         1        12

Residual standard error: 3.929447 
1 out of 7 effects not estimable
Estimated effects may be unbalanced