# 统计代写|回归分析作业代写Regression Analysis代考|ST503

## 统计代写|回归分析作业代写Regression Analysis代考|Using a Single Indicator Variable

First, notice that we snuck in a term, “level,” into the title of this section. The term “level” means “distinct value.” So, the “degree” variable in the GPA data set discussed in Chapter 6 has two levels, “M” and “P.” Another example is gender: If a survey question asks the respondent to identify their gender, the resulting Gender variable has two levels, “Female” and “Male.” In data where each observation is a firm, or company (called firmlevel data analysis), you might have publicly held and privately held firms in your data set. This information might be recorded in a variable called PUBLIC, with values “Yes” and “No.” In this case, the PUBLIC variable has two levels, “Yes” and “No.” If ethnicity is recorded in a survey, there will usually be more than two possibilities. Suppose there are 7 distinct ethnicities a person may identify with, including an “Other” category. Then the ETHNICITY variable will have 7 levels.

Consider the GPA prediction example, using only the “degree” variable to predict it. For now, let’s code “PhD” as 1 and “Master’s” as 0. (You can code it the other way, too, as $(0,1)$. We will do that right after we do it this way so we can compare results.) Call the resulting $(1,0)$ indicator variable “PHD.”
The classical regression model states as follows:
$$\mathrm{GPA} \mid \mathrm{PHD}=x \sim \mathrm{N}\left(\beta_0+\beta_1 x, \sigma^2\right)$$
Equivalently,
$$\mathrm{GPA}=\beta_0+\beta_1 \mathrm{PHD}+\varepsilon \text {, where } \varepsilon \sim \mathrm{N}\left(0, \sigma^2\right)$$
Drawing a scatterplot is always the best first step in the analysis of regression data; in this case, it appears as shown in Figure 10.1, with the least-squares line superimposed.

## 统计代写|回归分析作业代写Regression Analysis代考|Full Model versus Restricted Model F Tests

As we have mentioned repeatedly, tests of hypotheses are not the best way to evaluate models and assumptions. However, the $F$ test that was introduced in Chapter 8 is so common in the history of ANOVA, ANCOVA, and regression that we would be remiss not to mention it.

Models such as those shown in Figures $10.7$ and $10.6$ are often compared by using the F test, which is a test to compare “full” versus “restricted” classical regression models. (For models other than the classical regression model, full/restricted model comparison is more commonly done using the likelihood ratio test, which is used starting in Chapter 12 of this book.)
In the usual regression analysis, a full model typically has the form:
$$Y=\beta_0+\beta_1 X_1+\beta_2 X_2+\ldots+\beta_k X_k+\varepsilon$$
Here, the parameters $\beta_0, \beta_1, \beta_2, \ldots$, and $\beta_k$ are unconstrained; that is, each parameter can possibly take any value whatsoever between $-\infty$ and $\infty$, and the value that one $\beta$ parameter takes is not dependent on (or constrained by) the value that any other $\beta$ parameter takes.
A restricted model is the same model, but with constraints on the parameters. The most common restrictions are constraints such as $\beta_1=\beta_2=0$, although other constraints such as $\beta_2=1$, or $\beta_1-\beta_2=0$, or $\beta_0+15 \beta_2=100$ are also possible.

The separate slope model graphed in Figure $10.7$ is a full model relative to the restricted model that constrains all the interaction $\beta$ ‘s to be zero, shown in Figure 10.6. The $F$ test can be used to compare these models. To construct the $F$ test, let $\mathrm{SSE}{\mathrm{F}}$ denote the error sum of squares in the full model, and let $\mathrm{SSE}{\mathrm{R}}$ denote the error sum of squares in the restricted model. It is a mathematical fact that
$$\mathrm{SSE}{\mathrm{F}} \leq \mathrm{SSE}{\mathrm{R}}$$

This is an important point, so it bears more explanation. Recall $\operatorname{SSE}\left(b_0, b_1, \ldots, b_k\right)=$ $\sum_{i=1}^n\left{y_i-\left(b_0+b_1 x_{i 1}+\cdots+b_k x_{i k}\right)\right}^2$. Now, the least-squares algorithm tells you that SSE $_{\mathrm{F}}$ is the minimum value of $\operatorname{SSE}\left(b_0, b_1, \ldots, b_k\right)$ for all possible combinations $\left(b_0, b_1, \ldots, b_k\right)$.

In the restricted model, some of the $b$ values are constrained, e.g., to 0 . Therefore, the set of possible combinations $\left{b_0, b_1, \ldots, b_k\right}$ in the restricted model is a subset of the set of possible combinations $\left{b_0, b_1, \ldots, b_k\right}$ in the unrestricted model. Thus, $\mathrm{SSE}{\mathrm{R}}$ is the minimum of $\operatorname{SSE}\left(b_0, b_i, \ldots, b_k\right)$ over a set of values $\left{b_0, b_1, \ldots, b_k\right}$ that is a subset of the unrestricted set. The minimum of a set of values has to be as small, or smaller than the minimum of any subset, right? That fact proves $\mathrm{SSE}{\mathrm{F}} \leq \mathrm{SSE}_{\mathrm{R}}$.

For example, the minimum of the set of numbers $(7,4,3,5,0,4,4,3,5,4}$ is zero. Now, pick a subset of that set, like the first four: $(7,4,3,5)$. The minimum of the subset (the restricted set) is three, which is larger than zero.

Thus, even when the restricted model is the true model, the full model will appear to fit better (or at least no worse), because $\mathrm{SSE}{\mathrm{F}} \leq \mathrm{SSE}{\mathrm{R}}$.

## 统计代写|回归分析作业代写Regression Analysis代考|Using a Single Indicator Variable

$$\mathrm{GPA} \mid \mathrm{PHD}=x \sim \mathrm{N}\left(\beta_0+\beta_1 x, \sigma^2\right)$$

$$\mathrm{GPA}=\beta_0+\beta_1 \mathrm{PHD}+\varepsilon, \text { where } \varepsilon \sim \mathrm{N}\left(0, \sigma^2\right)$$

## 统计代写|回归分析作业代写Regression Analysis代考|Full Model versus Restricted Model F Tests

$$Y=\beta_0+\beta_1 X_1+\beta_2 X_2+\ldots+\beta_k X_k+\varepsilon$$

$$\mathrm{SSEF} \leq \mathrm{SSER}$$

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