Proportional odds model

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# Proportional odds model
### Dr. Olanrewaju Michael Akande
### Oct 3, 2019

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## Announcements
  
- New TA: 
  + [Siqi Fu](https://stat.duke.edu/people/siqi-fu)
  + Office hours: Thur 3:00 - 5:00pm

## Outline

- Questions from last lecture

- In-class analysis for multinomial logistic regression

- Proportional odds model

- In-class analysis for proportional odds model

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# Proportional odds model

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## Ordinal responses

- Suppose the categories of our response variable has a natural ordering.

- Let's use data from Example 6.2.2 from Alan Agresti's [An Introduction to Categorical Data Analysis, Second Edition](https://find.library.duke.edu/catalog/DUKE005142588) to demonstrate this.

- This data is from a General Social Survey. Clearly, political ideology has a five-point ordinal scale, ranging from very liberal to very conservative.

.mini[
<table>
  <tr>
    <th> </th>
    <th> </th>
    <th colspan="5">Political Ideology</th>
  </tr>
  <tr>
    <th colspan="2"></th>
    <td style="text-align:center">Very Liberal</td>
    <td style="text-align:center">Slightly Liberal</td>
    <td style="text-align:center">Moderate</td>
    <td style="text-align:center">Slightly Conservative</td>
    <td style="text-align:center">Very Conservative</td>
  </tr>
  <tr>
    <th rowspan="2">Female</th>
    <td height="50px">Democratic</td>
    <td style="text-align:center">44</td>
    <td style="text-align:center">47</td>
    <td style="text-align:center">118</td>
    <td style="text-align:center">23</td>
    <td style="text-align:center">32</td>
  </tr>
  <tr>
    <td height="50px">Republican</td>
    <td style="text-align:center">18</td>
    <td style="text-align:center">28</td>
    <td style="text-align:center">86</td>
    <td style="text-align:center">39</td>
    <td style="text-align:center">48</td>
  </tr>
  <tr>
    <th rowspan="2">Male</th>
    <td height="50px">Democratic</td>
    <td style="text-align:center">36</td>
    <td style="text-align:center">34</td>
    <td style="text-align:center">53</td>
    <td style="text-align:center">18</td>
    <td style="text-align:center">23</td>
  </tr>
  <tr>
    <td height="50px">Republican</td>
    <td style="text-align:center">12</td>
    <td style="text-align:center">18</td>
    <td style="text-align:center">62</td>
    <td style="text-align:center">45</td>
    <td style="text-align:center">51</td>
  </tr>
</table>
]

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## Cumulative logits

- When we have ordinal response with categories `$1, 2, \ldots, J$`, we still want to estimate
.block[
.small[
`$$\Pr[y_i = 1 | \boldsymbol{x}_i] = \pi_{i1}, \ \Pr[y_i = 2 | \boldsymbol{x}_i] = \pi_{i2}, \ \ldots, \ \Pr[y_i = J | \boldsymbol{x}_i] = \pi_{iJ}.$$`
]
]

- However, we need to use models that can reflect the ordering
.block[
.small[
`$$\Pr[y_i\leq 1 | \boldsymbol{x}_i] \leq \Pr[y_i\leq 2 | \boldsymbol{x}_i] \leq \ldots \leq \Pr[y_i\leq J | \boldsymbol{x}_i] = 1.$$`
]
]

*Notice that the ordering of probabilities is not for the actual marginal probabilities, but rather the cumulative probabilities.*

- The multinomial logistic regression we covered in the last class does not enforce this.

- Instead, we can focus on building models for the cumulative logits, that is, models for 
.block[
.small[
`$$\textrm{log}\left(\dfrac{\Pr[y_i \leq j | \boldsymbol{x}_i]}{\Pr[y_i > j | \boldsymbol{x}_i]}\right) = \textrm{log}\left(\dfrac{\pi_{i1} + \pi_{i2} + \ldots + \pi_{ij}}{\pi_{i(j+1)} + \pi_{i(j+2)} + \ldots + \pi_{iJ}}\right), \ \ \ j = 1, \ldots, J-1.$$`
]
]