# 统计代写|回归分析作业代写Regression Analysis代考|Parameter Interpretation in Interaction Models

#### Doug I. Jones

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## 统计代写|回归分析作业代写Regression Analysis代考|Parameter Interpretation in Interaction Models

Parameter interpretation in interaction models is complicated by the fact that the $X$ variables in the model are functionally related to each other. Thus, the standard “increase $X_1$ by +1 while holding all other terms fixed” interpretation cannot be used, because you cannot increase $X_1$ by +1 while simultaneously keeping both $X_2$ and $X_1 X_2$ fixed.

However, you can increase $X_1$ while holding $X_2$ fixed. Consider two groups of potentially observable data, one where $X_1=x_1$ and $X_2=x_2$, and the other where $X_1=x_1+1$ and $X_2=x_2$. The interpretation of the parameters then follows logically from the model, as in all other cases you have seen (and will see): If the no interaction model is truly the data-generating process, then the following conclusions are mathematically true.
\begin{aligned} & \text { Group 1: } X_1=x_1, X_2=x_2: \text { Here, } \mathrm{E}\left(Y \mid X_1=x_1, X_2=x_2\right)=\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_1 x_2 \ & \text { Group 2: } X_1=x_1+1, X_2=x_2: \text { Here, } \ & \qquad \begin{aligned} \mathrm{E}\left(Y \mid X_1=x_1, X_2=x_2\right) & =\beta_0+\beta_1\left(x_1+1\right)+\beta_2 x_2+\beta_3\left(x_1+1\right) x_2 \ & =\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_1 x_2+\left(\beta_1+\beta_3 x_2\right) \ & =(\text { Group } 1 \text { mean })+\left(\beta_1+\beta_3 x_2\right) \end{aligned} \end{aligned}
Thus, in the interaction model, the coefficient $\beta_1$ does not measure the effect of increasing $X_1$ when $X_2$ is held constant because there is a different effect for every possible $X_2=x_2$ (namely, $\beta_1+\beta_3 x_2$ ). You can interpret $\beta_1$ as the increase in $\mathrm{E}(Y)$ per unit increase in $X_1$ when $X_2=0$, but now we are back to the interpretation that is practically useless when the $X_2$ data do not cover 0.0. Still, just like the intercept term $\beta_0$ is necessary in regression, regardless of its interpretability or “significance,” the $\beta_1$ term is necessary in the interaction model, regardless of its interpretability or “significance.” See Section 9.4 of this chapters for elaboration on this concept.

## 统计代写|回归分析作业代写Regression Analysis代考|Effect of Misanthropy on Support for Animal Rights: The Moderating Effect of Idealism

The paper “Misanthropy, idealism, and attitudes toward animals,” (Wuensch et al. 2002) develops theory and analyzes data showing that animal rights support is related to misanthropy for non-idealistic students, but that animal rights support is not strongly related to misanthropy for idealist students. They developed and validated a survey instrument to measure three variables, $Y=$ Support for Animal Rights, $X_1=$ Misanthropy, and $X_2=$ Idealism, as averages of responses (typically in the 1,2,3,4,5 scale) to three sets of questions on the survey. As averages, these data are more continuous than the 1,2,3,4,5 scale itself, and hence better modeled using normal distributions than are the actual 1,2,3,4,5 data.

The model that allows the effect of $X_1$ on $Y$ to depend on the value of $X_2$ (i.e., the moderation model) is the interaction model
$$Y_i \mid X_{i 1}=x_{i 1}, X_{i 2}=x_{i 2} \sim_{\text {ind }} \mathrm{N}\left(\beta_0+\beta_1 x_{i 1}+\beta_2 x_{i 2}+\beta_3 x_{i 1} x_{i 2}, \sigma^2\right)$$

# 回归分析代写

## 统计代写|回归分析作业代写Regression Analysis代考|Parameter Interpretation in Interaction Models

\begin{aligned} & \text { Group 1: } X_1=x_1, X_2=x_2: \text { Here, } \mathrm{E}\left(Y \mid X_1=x_1, X_2=x_2\right)=\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_1 x_2 \ & \text { Group 2: } X_1=x_1+1, X_2=x_2: \text { Here, } \ & \qquad \begin{aligned} \mathrm{E}\left(Y \mid X_1=x_1, X_2=x_2\right) & =\beta_0+\beta_1\left(x_1+1\right)+\beta_2 x_2+\beta_3\left(x_1+1\right) x_2 \ & =\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_1 x_2+\left(\beta_1+\beta_3 x_2\right) \ & =(\text { Group } 1 \text { mean })+\left(\beta_1+\beta_3 x_2\right) \end{aligned} \end{aligned}

## 统计代写|回归分析作业代写Regression Analysis代考|Effect of Misanthropy on Support for Animal Rights: The Moderating Effect of Idealism

$$Y_i \mid X_{i 1}=x_{i 1}, X_{i 2}=x_{i 2} \sim_{\text {ind }} \mathrm{N}\left(\beta_0+\beta_1 x_{i 1}+\beta_2 x_{i 2}+\beta_3 x_{i 1} x_{i 2}, \sigma^2\right)$$

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