## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Boston Housing

In Chaps. 3 and 7 , linear models were used to analyse if the variations of the price (the variables were transformed in Sect. 1.9) could be explained by other variables. A reduced model was obtained in Sect. $7.3$ with the results shown in Table 7.1, with $r^2=0.763$. The model was:
\begin{aligned} X_{14}=& \beta_0+\beta_4 X_4+\beta_5 X_5+\beta_6 X_6+\beta_8 X_8+\beta_9 X_9+\beta_{10} X_{10}+\beta_{11} X_{11} \ &+\beta_{12} X_{12}+\beta_{13} X_{13} \end{aligned}
One factor $\left(X_4\right)$ was coded as a binary variable (1, if the house is close to the Charles River and 0 if it is not). Taking advantage of the ANCOVA models described above, we would like to add to a new factor built from the original quantitative variable $X_9=$ index of accessibility to radial highways. So we will transform $X_4$ as being 1 if close to the Charles River and $-1$ if not, and we will replace $X_9$ by a new factor coded $X_{15}=1$ if $X_9 \geq$ median $\left(X_9\right)$ and $X_{15}=-1$ if $X_9<\operatorname{median}\left(X_9\right)$. We also want to consider the interaction of $X_4$ with $X_{12}$ (proportion of blacks) and the interaction of $X_4$ with the new factor $X_{15}$. The results are shown in Table 8.5.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Categorical Responses

In many applications, the response variable of interest is qualitative or categorical, in the sense that the response can take its nominal value in one of, say, $K$ classes or categories. Often we observe counts $y_k$, the number of observations in category $k=1, \ldots, K$. If the total number of observations $n=\sum_{k=1}^K y_k$ is fixed and we may assume independence of the observations, we obtain a multinomial sampling process.

If we denote by $p_k$ the probability of observing the $k$ th category with $\sum_{k=1}^K p_k=$ 1, we have $\mathrm{E}\left(Y_k\right)=m_k=n p_k$. The likelihood of the sample can then be written as:
$$L=\frac{n !}{\prod_{k=1}^K y_{k} !} \prod_{k=1}^K\left(\frac{m_k}{n}\right)^{y_k} .$$

In contingency tables, the categories are defined by several qualitative variables. For example in a $\left(J \times K\right.$ ) two-way table, the observations (counts) $y_{j k}, j=1, \ldots, J$ and $k=1, \ldots, K$ are reported for row $j$ and column $k$. Here $n=\sum_{j=1}^J \sum_{k=1}^K y_{j k}$. Log-linear models introduce a linear structure on the logarithms of the expected frequencies $m_{j k}=\mathrm{E}\left(y_{j k}\right)=n p_{j k}$, with $\sum_{j=1}^J \sum_{k=1}^K p_{j k}=1$. Log-linear structures on $m_{j k}$ will impose the same structure for the $p_{j k}$, the estimation of the model will then be obtained by constrained maximum likelihood. Three-way tables $(J \times K \times L)$ may be analysed in the same way.

Sometimes additional information is available on explanatory variables $x$. In this case, the logit model will be appropriate when the categorical response is binary ( $K=2$ ). We will introduce these models when the main response of interest is binary (for instance tables $(2 \times K)$ or $(2 \times K \times L)$ ). Further, we will show how they can be adapted to the case of contingency tables. Contingency tables are also analysed by multivariate descriptive tools in Chap. $15 .$

## 统计代写|多元统计分析代写多元统计分析代考|波士顿住房

\begin{aligned} X_{14}=& \beta_0+\beta_4 X_4+\beta_5 X_5+\beta_6 X_6+\beta_8 X_8+\beta_9 X_9+\beta_{10} X_{10}+\beta_{11} X_{11} \ &+\beta_{12} X_{12}+\beta_{13} X_{13} \end{aligned}

## 统计代写|多元统计分析代写多元统计分析代考|分类反应

.

$$L=\frac{n !}{\prod_{k=1}^K y_{k} !} \prod_{k=1}^K\left(\frac{m_k}{n}\right)^{y_k} .$$

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