4.2. Models for Ordinal Responses

Remark(Adjacent-category logit models)

Let \[ L_j=\log\left\{ \frac{\pi_j(x)}{\pi_{j+1}(x)} \right\}=\text{logit}\{P(Y=j|Y=j \mbox{ or } Y={j+1})\}. \] These logits are a basic set equivalent to the Baseline category logit models.

Model: \[ L_j=\log\left\{ \frac{\pi_j(x)}{\pi_{j+1}(x)}\right\}=\alpha_j+\beta'x, \mbox{ }j=1,\ldots,J-1 \] assumes common effect for all logits.

Note that the baseline category logit model can also be used here such that \[\begin{eqnarray*} \log\left\{ \frac{\pi_j(x)}{\pi_{J}(x)}\right\}= L_j+L_{j+1}+\cdots+L_{J-1}=\sum_{h=j}^{J-1}\alpha_h+(J-j)\beta'x= \alpha_j^*+\beta'u_j. \end{eqnarray*}\]

즉 카테고리들간의 distance가 멀어질 때마다 \(\alpha, \beta\) effect도 계속 비례해서 증가한다는 의미이다.

Remark(Cumulative logit model)

Let \(F_j(x)=P(Y\le j|x)\), \(j=1,2,\ldots,J-1\). Cumulative logits are \[ L_j(x)=\log\left\{ \frac{F_j(x)}{1-F_j(x)} \right\}, \mbox{ }\mbox{ }j=1,\ldots,J-1. \]

Model: \[ L_j(x)=\log\left\{ \frac{F_j(x)}{1-F_j(x)} \right\}=\alpha_j+\beta'x. \]

Nominal data는 카테고리인 \(j\)에 기준이 없었기 때문에 모든 카테고리마다의 \(\beta_j\)들을 추정해야 했었다. 하지만 Ordinal data는 x에 대해서는 \(\beta\)만을 추정한다. 아래의 motivation을 확인하면 여기서 \(\alpha_j\)는 order를 의미한다(increasing on \(j\)).

Motivation

Let \(Y=\)observed responses, \(Y^*=\)underlying continuous response.

Suppose that there are “cutpoints” \(\{\alpha_j\}\) such that \(Y=j\) if \(\alpha_{j-1}<Y^*\le \alpha_j\).

Let \(F_j(x)=P(Y\le j|x)= P(Y^*\le \alpha_j |x)\).

Suppose \(Y^*=\beta'x+\epsilon\) has cdf \(G(y^*-\beta'x)\), where \(\beta'x\) is a location parameter. Then, \[ P(Y^*\le \alpha_j |x)=G(\alpha_j-\beta'x)\\ \iff G^{-1}\{ P(Y\le j |x)\}=\alpha_j-\beta'x. \] If \(\epsilon\sim\)normal dist., \(G^{-1}=\)probit;

If \(\epsilon\sim\)logistic, \(G^{-1}=\)logit(cumulative logit model).

중요 point

\(Y^*\) modeling: \(Y^*=\beta'x+\epsilon\)이라고 처음 모델을 세운다.
Cutpoint가정: 만약 \(\alpha_{j-1}<Y^*\le \alpha_j\)라면 \(Y=j\iff Y^*\le \alpha_j\)라면 \(Y\le j\)이다.
\(\epsilon\)의 cdf 분포 가정: \(Y^*=\beta'x+\epsilon\) has cdf \(G(y^*-\beta'x)\iff\) \(Y^*\) has cdf \(G(\epsilon)\), \(\eta\)는 location. 자세히 보이면 \[ P(Y^* \le \alpha_j |x) = P(\beta'x+\epsilon \le \alpha_j |x)=P(\epsilon \le \alpha_j-\beta'x |x) =G_\epsilon(\alpha_j-\beta'x)\\ \implies G^{-1}\{ P(Y\le j |x)\}=\alpha_j-\beta'x. \] 즉 \(\epsilon\)의 분포에 따라 모델이 바뀐다.

For logit link: \[ \log\left\{\frac{P(Y\le j|x_1)/P(Y> j|x_1)}{P(Y\le j|x_2)/P(Y> j|x_2)}\right\}= (\alpha_j-\beta'x_1)-(\alpha_j-\beta'x_2)= \beta'(x_2-x_1). \] 때문에 이 모델을 Proportional odds model이라고 부르기도 한다.

Remark

For multinomial counts, \(\{y_{ij};j=1,\ldots,J\}\) at setting \(x_i\) assumed independent at different \(x_i\), the likelihood has form

\[\begin{eqnarray*} \prod_{i}\left[ \prod_{j=1}^j\{ P(Y_{ij}=j|x_i )^{y_{ij}} \} \right]&=&\prod_{i}\left[ \prod_{j=1}^j\left\{ P(Y_{ij}\le j|x_i)-P(Y_{ij}\le j-1|x_i) \right\}^{y_{ij}} \right] \\ &=&\prod_{i}\left[ \prod_{j=1}^j\{F_j(x_i)-F_{j-1}(x_i)\}^{y_{ij}} \right] \end{eqnarray*}\] with \[ F_j(x_i)=\frac{\exp(\alpha_j-\beta'x_i)}{1+\exp(\alpha_j-\beta'x_i)}, \] \(F_J(x_i)=1\), and \(F_0(x_i)=0\) (\(j=1,\ldots,J\)이기 때문에, \(P(Y\le J)=1\)이고 \(P(Y\le 0)=P(Y< 1)=0\)이다).

We can use \(\{ \hat{\alpha}_j\}\), \(\hat\beta\) to get \(\hat F_j(x_i)\), \[ \hat\pi_j(x_i)=\hat F_j(x_i)-\hat F_{j-1}(x_i). \]

Using fitted values \(\{n_i\hat\pi_j(x_i)\}\), we calculate deviance, \(G^2\) and \(\chi^2\) for testing fit or conducting LR tests.

\(\implies \hat\pi_j(x_i)\)를 이용하여 likelihood를 구하고, \(\tilde\pi_j(x_i)=y_{ij}\)로 saturated model과의 log값의 차이를 통해 Deviance를 구한다.)

*\(P(Y\le j)= \frac{\exp(\alpha_j-\beta'x)}{1+\exp(\alpha_j-\beta'x)}\)에서 For \(k>j\), \(P(Y\le k)\) is shifted to left by \(\frac{\alpha_j-\alpha_k}{\beta}\) such that \[ F_k(x)=F_j\left( x+\frac{\alpha_j-\alpha_k}{\beta} \right) \]

Remark

\(G= \Phi\)(c.d.f of the standard normal distribution) gives cumulative probit model.
\(G^{-1}(u)=\log\{-\log(1-u)\}\) is complementary \(\log-\log\) link corresponds to \(G=\)extreme value cdf.

Model: \[ \log\{-\log (1-P(Y\le j)) \}=\alpha_j-\beta'x\\ \implies P(Y\le j|x_1)=\{ P(Y>j|x_2)\}^{\exp\{\beta'(x_1-x_2) \}}. \]

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