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# 统计代写|线性回归代写Linear Regression代考|Fitted Values

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## 统计代写|线性回归代写Linear Regression代考|Fitted Values

In rare problems, one may be interested in obtaining an estimate of $\mathrm{E}(Y \mid X=x)$. In the heights data, this is like asking for the population mean height of all daughters of mothers with a particular height. This quantity is estimated by the fitted value $\hat{y}=\beta_0+\beta_1 x$, and its standard error is
$$\operatorname{sefit}\left(\tilde{y}* \mid x\right)=\hat{\sigma}\left(\frac{1}{n}+\frac{\left(x_-\bar{x}\right)^2}{S X X}\right)^{1 / 2}$$

To obtain confidence intervals, it is more usual to compute a simultaneous interval for all possible values of $x$. This is the same as first computing a joint confidence region for $\beta_0$ and $\beta_1$, and from these, computing the set of all possible mean functions with slope and intercept in the joint confidence set (Section 5.5). The confidence region for the mean function is the set of all $y$ such that
\begin{aligned} & \left(\hat{\beta}_0+\hat{\beta}_1 x\right)-\operatorname{sefit}(\hat{y} \mid x)[2 F(\alpha ; 2, n-2)]^{1 / 2} \leq y \ & \leq\left(\hat{\beta}_0+\hat{\beta}_1 x\right)+\operatorname{sefit}(\hat{y} \mid x)[2 F(\alpha ; 2, n-2)]^{1 / 2} \end{aligned}
For multiple regression, replace $2 F(\alpha ; 2, n-2)$ by $p^{\prime} F\left(\alpha ; p^{\prime}, n-p^{\prime}\right)$, where $p^{\prime}$ is the number of parameters estimated in the mean function including the intercept. The simultaneous band for the fitted line for the heights data is shown in Figure 2.5 as the vertical distances between the two dotted lines. The prediction intervals are much wider than the confidence intervals. Why is this so (Problem 2.4)?

## 统计代写|线性回归代写Linear Regression代考|THE RESIDUALS

Plots of residuals versus other quantities are used to find failures of assumptions. The most common plot, especially useful in simple regression, is the plot of residuals versus the fitted values. A null plot would indicate no failure of assumptions. Curvature might indicate that the fitted mean function is inappropriate. Residuals that seem to increase or decrease in average magnitude with the fitted values might indicate nonconstant residual variance. A few relatively large residuals may be indicative of outliers, cases for which the model is somehow inappropriate.

The plot of residuals versus fitted values for the heights data is shown in Figure 2.6. This is a null plot, as it indicates no particular problems.

The fitted values and residuals for Forbes’ data are plotted in Figure 2.7. The residuals are generally small compared to the fitted values, and they do not follow any distinct pattern in Figure 2.7. The residual for case number 12 is about four times the size of the next largest residual in absolute value. This may suggest that the assumptions concerning the errors are not correct. Either $\operatorname{Var}(100 \times$ $\log ($ Pressure $) \mid$ Temp $)$ may not be constant or for case 12 , the corresponding error may have a large fixed component. Forbes may have misread or miscopied the results of his calculations for this case, which would suggest that the numbers in the data do not correspond to the actual measurements. Forbes noted this possibility himself, by marking this pair of numbers in his paper as being “evidently a mistake”, presumably because of the large observed residual.

Since we are concerned with the effects of case 12 , we could refit the data, this time without case 12 , and then examine the changes that occur in the estimates of parameters, fitted values, residual variance, and so on. This is summarized in Table 2.5 , giving estimates of parameters, their standard errors, $\hat{\sigma}^2$, and the coefficient of determination $R^2$ with and without case 12 . The estimates of parameters are essentially identical with and without case 12 . In other regression problems, deletion of a single case can change everything. The effect of case 12 on standard errors is more marked: if case 12 is deleted, standard errors are decreased by a factor of about 3.1 , and variances are decreased by a factor of about $3.1^2 \approx 10$. Inclusion of this case gives the appearance of less reliable results than would be suggested on the basis of the other 16 cases. In particular, prediction intervals of Pressure are much wider based on all the data than on the 16-case data, although the point predictions are nearly the same. The residual plot obtained when case 12 is deleted before computing indicates no obvious failures in the remaining 16 cases.

Two competing fits using the same mean function but somewhat different data are available, and they lead to slightly different conclusions, although the results of the two analyses agree more than they disagree. On the basis of the data, there is no real way to choose between the two, and we have no way of deciding which is the correct olS analysis of the data. A good approach to this problem is to describe both or, in general, all plausible alternatives.

## 统计代写|线性回归代写Linear Regression代考|Fitted Values

$$\operatorname{sefit}\left(\tilde{y}* \mid x\right)=\hat{\sigma}\left(\frac{1}{n}+\frac{\left(x_-\bar{x}\right)^2}{S X X}\right)^{1 / 2}$$

\begin{aligned} & \left(\hat{\beta}_0+\hat{\beta}_1 x\right)-\operatorname{sefit}(\hat{y} \mid x)[2 F(\alpha ; 2, n-2)]^{1 / 2} \leq y \ & \leq\left(\hat{\beta}_0+\hat{\beta}_1 x\right)+\operatorname{sefit}(\hat{y} \mid x)[2 F(\alpha ; 2, n-2)]^{1 / 2} \end{aligned}

## MATLAB代写

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