## 统计代写|线性回归分析代写linear regression analysis代考|The No Intercept MLR Model

The no intercept $M L R$ model, also known as regression through the origin, is still $\boldsymbol{Y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{e}$, but there is no intercept in the model, so $\boldsymbol{X}$ does not contain a column of ones $\mathbf{1}$. Hence the intercept term $\beta_1=\beta_1(1)$ is replaced by $\beta_1 x_{i 1}$. Software gives output for this model if the “no intercept” or “intercept $=F$ ” option is selected. For the no intercept model, the assumption $E(\boldsymbol{e})=\mathbf{0}$ is important, and this assumption is rather strong.

Many of the usual MLR results still hold: $\widehat{\boldsymbol{\beta}}{O L S}=\left(\boldsymbol{X}^T \boldsymbol{X}\right)^{-1} \boldsymbol{X}^T \boldsymbol{Y}$, the vector of predicted fitted values $\widehat{\boldsymbol{Y}}=\boldsymbol{X} \hat{\boldsymbol{\beta}}{O L S}=\boldsymbol{H} \boldsymbol{Y}$ where the hat matrix $\boldsymbol{H}=\boldsymbol{X}\left(\boldsymbol{X}^T \boldsymbol{X}\right)^{-1} \boldsymbol{X}^T$ provided the inverse exists, and the vector of residuals is $\boldsymbol{r}=\boldsymbol{Y}-\widehat{\boldsymbol{Y}}$. The response plot and residual plot are made in the same way and should be made before performing inference.

The main difference in the output is the ANOVA table. The ANOVA $F$ test in Section 2.4 tests $H o: \beta_2=\cdots=\beta_p=0$. The test in this section tests $H o: \beta_1=\cdots=\beta_p=0 \equiv H_o: \boldsymbol{\beta}=\mathbf{0}$. The following definition and test follows Guttman (1982, p. 147) closely.

Definition 2.25. Assume that $\boldsymbol{Y}=\boldsymbol{X} \boldsymbol{\beta}+e$ where the $e_i$ are iid. Assume that it is desired to test $H o: \boldsymbol{\beta}=\mathbf{0}$ versus $H_A: \boldsymbol{\beta} \neq \mathbf{0}$.
a) The uncorrected total sum of squares
$$S S T=\sum_{i=1}^n Y_i^2 .$$
b) The model sum of squares
$$S S M=\sum_{i=1}^n \hat{Y}i^2 .$$ c) The residual sum of squares or error sum of squares is $$S S E=\sum{i=1}^n\left(Y_i-\hat{Y}i\right)^2=\sum{i=1}^n r_i^2 .$$

## 统计代写|线性回归分析代写linear regression analysis代考|Lack of Fit Tests

Then $M S P E=S S P E /(n-c)$ is an unbiased estimator of $\sigma^2$ when model (2.29) holds, regardless of the form of $m$. The PE in SSPE stands for “pure error.”

Now $S S L F=S S E-S S P E=\sum_{j=1}^c n_j\left(\bar{Y}_j-\hat{Y}_j\right)^2$. Notice that $\bar{Y}_j$ is an unbiased estimator of $m\left(\boldsymbol{x}_j\right)$ while $\hat{Y}_j$ is an estimator of $m$ if the MLR model is appropriate: $m\left(\boldsymbol{x}_j\right)=\boldsymbol{x}_j^T \boldsymbol{\beta}$. Hence SSLF and MSLF can be very large if the MLR model is not appropriate.

The 4 step lack of fit test is i) Ho: no evidence of MLR lack of fit, $H_A$ : there is lack of fit for the MLR model.
ii) $F_{L F}=M S L F / M S P E$.
iii) The pval $=P\left(F_{c-p, n-c}>F_{L F}\right)$.
iv) Reject Ho if pval $\leq \delta$ and state the $H_A$ claim that there is lack of fit. Otherwise, fail to reject Ho and state that there is not enough evidence to conclude that there is MLR lack of fit.

Although the lack of fit test seems clever, examining the response plot and residual plot is a much more effective method for examining whether or not the MLR model fits the data well provided that $n \geq 10 p$. A graphical version of the lack of fit test would compute the $\bar{Y}_j$ and see whether they scatter about the identity line in the response plot. When there are no replicates, the range of $\hat{Y}$ could be divided into several narrow nonoverlapping intervals called slices. Then the mean $\bar{Y}_j$ of each slice could be computed and a step function with step height $\bar{Y}_j$ at the $j$ th slice could be plotted. If the step function follows the identity line, then there is no evidence of lack of fit. However, it is easier to check whether the $Y_i$ are scattered about the identity line. Examining the residual plot is useful because it magnifies deviations from the identity line that may be difficult to see until the linear trend is removed. The lack of fit test may be sensitive to the assumption that the errors are iid $N\left(0, \sigma^2\right)$.

# 线性回归代考

## 统计代写|线性回归分析代写linear regression analysis代考|The No Intercept MLR Model

$H o: \beta_2=\cdots=\beta_p=0$. 本节测试测试 $H o: \beta_1=\cdots=\beta_p=0 \equiv H_o: \boldsymbol{\beta}=\mathbf{0}$. 以 下定义和测试紧跟 Guttman (1982, p. 147)。

a) 末修正的总平方和
$$S S T=\sum_{i=1}^n Y_i^2$$
b) 模型平方和
$$S S M=\sum_{i=1}^n \hat{Y} i^2$$
c) 残差平方和或误差平方和为
$$S S E=\sum i=1^n\left(Y_i-\hat{Y} i\right)^2=\sum i=1^n r_i^2$$

## 统计代写|线性回归分析代写linear regression analysis代考|Lack of Fit Tests

iii) $\mathrm{pval}=P\left(F_{c-p, n-c}>F_{L F}\right)$.
iv) 如果 pval 则拒绝 $H 0 \leq \delta$ 并说明 $H_A$ 声称不合适。否则，不佢绝 Ho 并声明没有足够 的证据得出 MLR 失拟的结论。

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