# 统计代写|线性回归分析代写linear regression analysis代考|Building an MLR Model

## 统计代写|线性回归分析代写linear regression analysis代考|Building an MLR Model

Building a multiple linear regression (MLR) model from data is one of the most challenging regression problems. The “final full model” will have response variable $Y=t(Z)$, a constant $x_1$, and predictor variables $x_2=$ $t_2\left(w_2, \ldots, w_r\right), \ldots, x_p=t_p\left(w_2, \ldots, w_r\right)$ where the initial data consists of $Z, w_2, \ldots, w_r$. Choosing $t, t_2, \ldots, t_p$ so that the final full model is a useful MLR approximation to the data can be difficult.

Model building is an iterative process. Given the problem and data but no model, the model building process can often be aided by graphs that help visualize the relationships between the different variables in the data. Then a statistical model can be proposed. This model can be fit and inference performed. Then diagnostics from the fit can be used to check the assumptions of the model. If the assumptions are not met, then an alternative model can be selected. The fit from the new model is obtained, and the cycle is repeated. This chapter provides some tools for building a good full model.

Warning: Researchers often have a single data set and tend to expect statistics to provide far more information from the single data set than is reasonable. MLR is an extremely useful tool, but MLR is at its best when the final full model is known before collecting and examining the data. However, it is very common for researchers to build their final full model by using the iterative process until the final model “fits the data well.” Researchers should not expect that all or even many of their research questions can be answered from such a full model. If the final MLR full model is built from a single data set in order to fit that data set well, then typically inference from that model will not be valid. The model may be useful for describing the data, but may perform very poorly for prediction of a future response. The model may suggest that some predictors are much more important than others, but a model that is chosen prior to collecting and examining the data is generally much more useful for prediction and inference. A single data set is a great place to start an analysis, but can be a terrible way to end the analysis.

## 统计代写|线性回归分析代写linear regression analysis代考|Predictor Transformations

As a general rule, inferring about the distribution of $Y \mid \boldsymbol{X}$ from a lower dimensional plot should be avoided when there are strong nonlinearities among the predictors.
Cook and Weisberg (1999b, p. 34)
Predictor transformations are used to remove gross nonlinearities in the predictors, and this technique is often very useful. Power transformations are particularly effective, and the techniques of this section are often useful for general regression problems, not just for multiple linear regression. A power transformation has the form $x=t_\lambda(w)=w^\lambda$ for $\lambda \neq 0$ and $x=t_0(w)=$ $\log (w)$ for $\lambda=0$. Often $\lambda \in \Lambda_L$ where
$$\Lambda_L={-1,-1 / 2,-1 / 3,0,1 / 3,1 / 2,1}$$
is called the ladder of powers. Often when a power transformation is needed, a transformation that goes “down the ladder,” e.g. from $\lambda=1$ to $\lambda=0$ will be useful. If the transformation goes too far down the ladder, e.g. if $\lambda=0$ is selected when $\lambda=1 / 2$ is needed, then it will be necessary to go back “up the ladder.” Additional powers such as \pm 2 and \pm 3 can always be added.
Definition 3.1. A scatterplot of $x$ versus $Y$ is used to visualize the conditional distribution of $Y \mid x$. A scatterplot matrix is an array of scatterplots. It is used to examine the marginal relationships of the predictors and response.

In this section we will only make a scatterplot matrix of the predictors. Often nine or ten variables can be placed in a scatterplot matrix. The names of the variables appear on the diagonal of the scatterplot matrix. The software Arc gives two numbers, the minimum and maximum of the variable, along with the name of the variable. The $R$ software labels the values of each variable in two places, see Example 3.2 below. Let one of the variables be $W$. All of the marginal plots above and below $W$ have $W$ on the horizontal axis. All of the marginal plots to the left and the right of $W$ have $W$ on the vertical axis.

# 线性回归代考

## 统计代写|线性回归分析代写linear regression analysis代考|Building an MLR Model

$t_2\left(w_2, \ldots, w_r\right), \ldots, x_p=t_p\left(w_2, \ldots, w_r\right)$ 其中初始数据包括 $Z, w_2, \ldots, w_r$. 选择 $t, t_2, \ldots, t_p$ 因此，最终的完整模型是对数据有用的 MLR 近似值可能很困难。

## 统计代写|线性回归分析代写linear regression analysis代考|Predictor Transformations

Cook 和 Weisberg (1999b, p. 34)

$$\Lambda_L=-1,-1 / 2,-1 / 3,0,1 / 3,1 / 2,1$$

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