1. Notation

失拟检验。Like the ANOVA test, this test is an F-test based on decomposing sums of squares. This test requires repeat observations (一个x处要有多个样本点)。

• $y_{ij}$ denotes the j-th measurement made at the i-th x-value in the data set.
• $\bar{y}_i$ denotes the average of all of the y values at the i-th x-value.
• $\hat{y_{ij}}$ denotes the predicted response for the j-th measurement made at the i-th x-value.

2. Decomposition

核心逻辑：①prediction error = ②the lack of fit of the model + ③random error

①SSE: prediction error

$$ \sum_{i}\sum_{j}(y_{ij}-\hat{y}_{ij})^2 $$

②SSLF: error due to the lack of fit of the model: how far the average observed response at each x-value is from the predicted response of each data point.

$$ \sum_{i}\sum_{j}(\bar{y}{i}-\hat{y}{ij})^2 $$

③SSPE: error due to the pure randomness from the data.

$$ \sum_{i}\sum_{j}(y_{ij}-\bar{y}_{i})^2 $$

综上，我们有：

$$ \underbrace{\sum\limits_{i=1}^c \sum\limits_{j=1}^{n_i} \left(y_{ij} - \hat{y}{ij}\right)^{2}}{\underset{\text{Error Sum of Squares}}{\text{SSE}}} = \underbrace{\sum\limits_{i=1}^c \sum\limits_{j=1}^{n_i} \left(\overline{y}{i} - \hat{y}{ij}\right)^{2}}{\underset{\text{Lack of Fit Sums of Squares}}{\text{SSLF}}} + \underbrace{\sum\limits{i=1}^c \sum\limits_{j=1}^{n_i} \left(y_{ij} - \overline{y}{i}\right)^{2}}{\underset{\text{Pure Error Sum of Squares}}{\text{SSPE}}} $$

其中c denotes the number of distinct x values you have.

【自由度】SSE为n-2，SSLF为c-2，SSPE为n-c → MSE、MSLF、MSPE。

3. F-test