Understand Logistic Regression

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# Understand Logistic Regression
### 王　超辰 | Chaochen Wang
### 2019-07-08 17:00~18:00 <span class="citation">@CSS</span>

---

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# Objectives

- Use a logistic regression model to compare the log odds of disease in two groups `$\Rightarrow$` estimate a crude odds ratio for **a binary exposure**.

- Use a logistic regression model to compare the odds of disease for a categorical exposure with **2 or more levels**.

- Use a logistic regression model to examine the association between an exposure and an outcome **adjusting for confounders**, assuming no effect modification.

---
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## Odds and Odds ratio (OR)

- The odds of a disease is defined as:  `$$\mathbf{Odds} = \frac{n_{D+}}{n_{D-}}$$` <br> where `$D+$` indicates individuals **with disease (such as lung cancer)** <br> `$D-$` indicates individuals **without the disease**

- The ratio of the odds between groups is a **measure for comparing the frequency of disease:** `$$\mathbf{OR} = \frac{\mathbf{Odds}_{E+}}{\mathbf{Odds}_{E-}}$$` <br> where `$E+$` indicates individuals **with exposure (such as smoking)** <br> `$E-$` indicates individuals **without the exposure**

---
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## Logistic regression model in a nutshell

$$
`\begin{aligned}
\mathbf{Odds}_{E+} & = \mathbf{Odds}_{E-} \times \mathbf{OR} \\
\Rightarrow \log(\mathbf{Odds}_{E+}) & = \log(\mathbf{Odds}_{E-}) + \log(\mathbf{OR})
\end{aligned}`
$$

Where, <br> `$\log(\mathbf{Odds}_{E-})$` is called the **"Baseline log odds in the unexposed group"** <br> `$\log(\mathbf{OR})$` is referred as the **"log odds ratio of the exposure"**

This is an example of a logistic regression model.

---
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# The end

---
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## Logistic regression model

**It models the log odds of the disease:**

- Among unexposed individuals `$(E-)$`: <br> log odds of disease = Baseline

- Among exposed individuals `$(E+)$`: <br> log odds of disease = Baseline + Exposure

Where, **"Exposure" is the log odds ratio** and the model is **additive**.

So, in any kind of statistical software, logistic regression model does everything on the log odds scale. We/the software can (manually/automatically) convert it back to odds ratio scale, the thing we use to report.

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# Why log odds of the disease?

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## Why log odds of the disease?

- Linear regression is appropriate for outcome variables like birth weight/height/BMI etc., which are continuous and follows a [normal distribution](https://ja.wikipedia.org/wiki/%E6%AD%A3%E8%A6%8F%E5%88%86%E5%B8%83).

- When the outcome variables are binary response (died/alive, infected/uninfected, diseased/healthy, succeeded/failed, etc.), it is easier to model the log odds. Because log odds can take any value, positive or negative. But percentages of died/infected/diseased/succeeded are constrained between 0 and 1.

- All logs are base `$e$`.

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background-image: url("./pic/LogReg_1.png")
background-position: 50% 50%
background-size: contain

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# Example

- [Data](https://github.com/winterwang/logistic-reg-CSS/raw/gh-pages/dataset/mortality.dta) from a study conducted in a rural area of northern Nigeria in the late 1980s to early 1990s.

- This cohort study with fixed follow-up time included persons aged `$\geqslant$` 15 years who all underwent an eye examination by ophthalmic (眼科) nurses.

- Individuals were classified as visually impaired or with normal vision.

- Communities were followed-up over a period of 3 years and deaths were identified.

- We will study the relationship between **the outcome: dead (=1) or alive (=0)**, and **two exposure factors**: 
    - visual impairment as a binary variable (0=unimpaired, 1=visually impaired (including blind))
    - microfilarial infection grouped into 4 levels (negative, <10, 10-49, ≥50 mf/mg).
    
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## Estimate the odds, log odds, and the odds ratio "by hand"

```r
# install the package if needed by install.packages("haven")
library(haven)
mortality <- read_dta("dataset/mortality.dta") 
# change your path accordingly
# install the package if needed by install.packages("Epi")
library(Epi)
tab <- stat.table(index = list(Died = died, 
                               "Visually impaired" = vimp),
                   contents = list(count(),percent(died)), 
                  data = mortality, margins = T)
print(tab, digits = 2)
```

```
##  -------------------------------- 
##         ----Visually impaired---- 
##  Died          0       1   Total  
##  -------------------------------- 
##  0       3874.00  287.00 4161.00  
##            97.56   87.77   96.81  
##                                   
##  1         97.00   40.00  137.00  
##             2.44   12.23    3.19  
##                                   
##                                   
##  Total   3971.00  327.00 4298.00  
##           100.00  100.00  100.00  
##  --------------------------------
```

---
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### Exercise 1:

- Fill in the **risk, odds and log odds** of death in the visually impaired and unimpaired individuals.

|          | Unimpaired | Visually impaired |  Total   |
| :------- | :--------: | :---------------: | :------: |
| Risk     |            |                   | `$3.19\%$` |
| Odds     |            |                   | `$0.0329$` |
| Log odds |            |                   | `$-3.414$` |

- What is the Odds Ratio?
- What is the log(odds ratio)?

---
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### Exercise 1: the answers

- Fill in the **risk, odds and log odds** of death in the visually impaired and unimpaired individuals.

|          |      Unimpaired      |  Visually impaired   |  Total   |
| :------- | :------------------: | :------------------: | :------: |
| Risk     |       `$2.44\%$`       |      `$12.23\%$`       | `$3.19\%$` |
| Odds     |  `$97/3874 = 0.0250$`  |  `$40/287 = 0.1394$`   | `$0.0329$` |
| Log odds | `$\log(0.0250)=-3.68$` | `$\log(0.1394)=-1.97$` | `$-3.414$` |

- What is the Odds Ratio? `$$0.1394/0.0250 = 5.576$$`
- What is the log(odds ratio)? `$$\log(5.576) = -1.97 - (-3.68) = 1.71$$`

---
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### Fit the model in R (1)

```r
vimp_logm0 <- glm(died ~ vimp, data = mortality, 
                  family = binomial(link = "logit"))
summary(vimp_logm0)
```

```
## Call:
## glm(formula = died ~ vimp, family = binomial(link = "logit"), 
##     data = mortality)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -0.5108  -0.2224  -0.2224  -0.2224   2.7247  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## `(Intercept)  -3.6873     0.1028 -35.870   <2e-16 ***`
## `vimp          1.7167     0.1976   8.687   <2e-16 ***`
## ---
## Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1213.8  on 4297  degrees of freedom
## Residual deviance: 1154.7  on 4296  degrees of freedom
## AIC: 1158.7
## 
## Number of Fisher Scoring iterations: 6
```

???
- The first column notes the names of the explanatory variable `vimp`. 
- The first row `(Intercept)` refers to the baseline (log odds in the baseline group)
- The second column, labelled `Estimate` is the estimates of the logOdds (unimpaired) logOR for visual impairment

---
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### Recall that the logistic regression model is written as:

`$$\log(\mathbf{Odds}_{E+})  = \log(\mathbf{Odds}_{E-}) + \log(\mathbf{OR})$$`

Therefore, we can write down our model:

`$$\log(\mathbf{Odds})  = -3.687 + 1.717 \times \mathbf{Vimp}$$`

Hence, the log odds:

- for those with **unimpaired vision**: `$\log(\mathbf{Odds}) = -3.687 + 1.717 \times 0 = -3.687$`;

- for those with **visually impaired**: `$\log(\mathbf{Odds}) = -3.687 + 1.717 \times 1 = -1.970$`

The odds ratio (OR) = `$\exp(1.717) = 5.567$`

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### Look at the output again

```
## Call:
## glm(formula = died ~ vimp, family = binomial(link = "logit"), 
##     data = mortality)
## {ommitted}
## Coefficients:
##             Estimate `Std. Error` z value Pr(>|z|)    
## (Intercept)  -3.6873     `0.1028` -35.870   <2e-16 ***
## vimp          1.7167     `0.1976`   8.687   <2e-16 ***
## {ommitted}
```

- `Std.Error` can be used to calculate the 95% confidence interval (CI): `$$\log\mathbf{OR} \pm 1.96\times\mathbf{S.E}$$`

Therefore,

- 95%CI for logOR of `vimp`: `$1.7167 \pm 1.96 \times 0.1976 = (1.329, 2.104)$`

- 95%CI for OR of `vimp`: `$(\exp(1.329) = 3.78, \exp(2.104) = 8.19)$`

---
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### Fit the model in R (2), turn logOR to OR

```r
# install the package if needed by install.packages("epiDisplay")
library(epiDisplay)
logistic.display(vimp_logm0, decimal = 3)
```

```
## 
## Logistic regression predicting died 
##  
##               `OR(95%CI)`            P(Wald's test) P(LR-test)
## `vimp: 1 vs 0  5.566 (3.779,8.199)  < 0.001        < 0.001`   
##                                                             
## Log-likelihood = -577.36596
## No. of observations = 4298
## AIC value = 1158.73192
```

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## Logistic model for comparison of more than two exposure groups

When the predictor (explanatory) variable has more than 2 levels, all OR are **relative to the baseline group**.

```r
tab <- stat.table(index = list(Died = died, 
                               "Microfilarial infection mf/mg (grouped)" = mfgrp),
                   contents = list(count(),percent(died)), 
                  data = mortality, margins = T)
print(tab, digits = 2)
```

```
##  --------------------------------------------------- 
##         --Microfilarial infection mf/mg (grouped)--- 
##  Died    Uninfected    < 10   10-49     50+   Total  
##  --------------------------------------------------- 
##  0          1274.00 1609.00  996.00  193.00 4161.00  
##               97.77   96.29   96.79   95.54   96.81  
##                                                      
##  1            29.00   62.00   33.00    9.00  137.00  
##                2.23    3.71    3.21    4.46    3.19  
##                                                      
##                                                      
##  Total      1303.00 1671.00 1029.00  202.00 4298.00  
##              100.00  100.00  100.00  100.00  100.00  
##  ---------------------------------------------------
```

---
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### Exercise 2:

- Complete this table by calculating the **odds, log odds, odds ratios** (compared to microfilarial group uninfected [0]) and log odds ratios as required.

|                | uninfected |  `$<10$`   | `$10\sim 49$` | `$\geqslant50$` |
| :------------- | :--------: | :------: | :---------: | :-----------: |
| Risk           |  `$2.23\%$`  | `$3.71\%$` |  `$3.21\%$`   |   `$4.46\%$`    |
| Odds           |  `$0.0228$`  | `$0.0385$` |  `$0.0331$`   |   `$0.0466$`    |
| Odds ratio     |    `$1$`     | `$1.693$`  |   `$1.456$`   |               |
| Log odds       |  `$-3.783$`  |          |             |   `$-3.065$`    |
| Log Odds ratio |    `$0$`     | `$0.526$`  |   `$0.375$`   |               |

---
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### Exercise 2: the answers

- Complete this table by calculating the **odds, log odds, odds ratios** (compared to microfilarial group uninfected [0]) and log odds ratios as required.

|                | uninfected |          `$<10$`           |       `$10\sim 49$`       |      `$\geqslant50$`       |
| :------------- | :--------: | :----------------------: | :---------------------: | :----------------------: |
| Risk           |  `$2.23\%$`  |         `$3.71\%$`         |        `$3.21\%$`         |         `$4.46\%$`         |
| Odds           |  `$0.0228$`  |         `$0.0385$`         |        `$0.0331$`         |         `$0.0466$`         |
| Odds ratio     |    `$1$`     |         `$1.693$`          |         `$1.456$`         | `$0.0466/0.0228$` `$=2.044$` |
| Log odds       |  `$-3.781$`  | `$\log(0.0385)$` `$=-3.257$` | `$\log(0.0466)$` `$-3.066$` |         `$-3.066$`         |
| Log Odds ratio |    `$0$`     |         `$0.526$`          |         `$0.375$`         | `$\log(2.044)$` `$= 0.715$`  |

---
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### Fit the model in R (3)

```r
vimp_logm1 <- glm(died ~ mfgrp, data = mortality, 
                  family = binomial(link = "logit"))
summary(vimp_logm1)
```

```
##Call:
##glm(formula = died ~ mfgrp, family = binomial(link = "logit"), 
##    data = mortality)
##
##Deviance Residuals: 
##    Min       1Q   Median       3Q      Max  
##-0.3019  -0.2750  -0.2553  -0.2122   2.7587  
##
##Coefficients:
##            Estimate Std. Error z value Pr(>|z|)    
##`(Intercept)  -3.7826     0.1878 -20.143   <2e-16 ***`
##`mfgrp< 10     0.5264     0.2281   2.308   0.0210 *`  
##`mfgrp10-49    0.3754     0.2580   1.455   0.1457`    
##`mfgrp50+      0.7172     0.3893   1.842   0.0655 .`  
##---
##Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
##
##(Dispersion parameter for binomial family taken to be 1)
##
##    Null deviance: 1180.4  on 4204  degrees of freedom
##Residual deviance: 1173.7  on 4201  degrees of freedom
##  (93 observations deleted due to missingness)
##AIC: 1181.7
##
##Number of Fisher Scoring iterations: 6
```

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### Dummy variables for the categorical variables

```
## id      mfgrp  mfgrp_< 10 mfgrp_10-49 mortality_50+
## 1  Uninfected           0           0             0
## 2        < 10           1           0             0
## 3        < 10           1           0             0
## 4       10-49           0           1             0
## 5        < 10           1           0             0
## 6        <NA>           0           0             0
## 14 Uninfected           0           0             0
## 15       <NA>           0           0             0
## 16       < 10           1           0             0
## 17       < 10           1           0             0
## 18       <NA>           0           0             0
## 19       < 10           1           0             0
## 22       < 10           1           0             0
## 23 Uninfected           0           0             0
## 24       < 10           1           0             0
## 29       <NA>           0           0             0
## 30      10-49           0           1             0
## 31 Uninfected           0           0             0
## 33      10-49           0           1             0
## 34 Uninfected           0           0             0
## 35      10-49           0           1             0
## 38        50+           0           0             1
```

---
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### Recall that the logistic regression model is written as:

`$$\log(\mathbf{Odds}_{E+})  = \log(\mathbf{Odds}_{E-}) + \log(\mathbf{OR})$$`

Therefore, we can write down our model as:

`$\log(\mathbf{Odds})  = -3.7826 + 0.5264 \times$` `mfgrp_< 10` `$+ 0.3754\times$` `mfgrp_10-49` `$+ 0.7172\times$` `mortality_50+`

---
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### Fit the model in R (4), turn logOR to OR

```r
logistic.display(vimp_logm1, decimal = 3)
```

```
##Logistic regression predicting died 
## 
##                       `OR(95%CI)            P(Wald's test) P(LR-test)`
##`mfgrp: ref.=Uninfected                                     0.0822`    
##   `< 10                1.693 (1.083,2.647)  0.021`                    
##   `10-49               1.456 (0.878,2.414)  0.1457`                   
##   `50+                 2.049 (0.955,4.394)  0.0655`                   
##                                                                     
##Log-likelihood = -586.86507
##No. of observations = 4205
##AIC value = 1181.73014
```

---
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# Logistic Regression 2: models with more than one variable

---
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## Number of deaths, by visual impairment and age group

```r
mortality %>% 
  group_by(agegrp, died, vimp) %>% 
  summarise(n = n())
```

```
## # A tibble: 16 x 4
## # Groups:   agegrp, died [8]
##       agegrp  died                  vimp     n
##    <dbl+lbl> <dbl>             <dbl+lbl> <int>
##  1 0 [15-34]     0 0 [Normal]             2301
##  2 0 [15-34]     0 1 [Visually impaired]    22
##  3 0 [15-34]     1 0 [Normal]               29
##  4 0 [15-34]     1 1 [Visually impaired]     2
##  5 1 [35-54]     0 0 [Normal]             1271
##  6 1 [35-54]     0 1 [Visually impaired]   124
##  7 1 [35-54]     1 0 [Normal]               38
##  8 1 [35-54]     1 1 [Visually impaired]    10
##  9 2 [55-64]     0 0 [Normal]              212
## 10 2 [55-64]     0 1 [Visually impaired]    69
## 11 2 [55-64]     1 0 [Normal]               15
## 12 2 [55-64]     1 1 [Visually impaired]    11
## 13 3 [65+]       0 0 [Normal]               90
## 14 3 [65+]       0 1 [Visually impaired]    72
## 15 3 [65+]       1 0 [Normal]               15
## 16 3 [65+]       1 1 [Visually impaired]    17
```

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## Number of deaths, by visual impairment and age group

<table class="table table-striped table-bordered" style="margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="border-bottom:hidden" colspan="1"></th>
<th style="border-bottom:hidden; padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="8"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">died (0 = no, 1 = yes)</div></th>
</tr>
<tr>
<th style="border-bottom:hidden" colspan="1"></th>
<th style="border-bottom:hidden; padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">agegrp 15-34</div></th>
<th style="border-bottom:hidden; padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">agegrp 35-54</div></th>
<th style="border-bottom:hidden; padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">agegrp 55-64</div></th>
<th style="border-bottom:hidden; padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">agegrp 65+</div></th>
</tr>
  <tr>
   <th style="text-align:right;"> vimp </th>
   <th style="text-align:right;"> 0 </th>
   <th style="text-align:right;"> 1 </th>
   <th style="text-align:right;"> 0 </th>
   <th style="text-align:right;"> 1 </th>
   <th style="text-align:right;"> 0 </th>
   <th style="text-align:right;"> 1 </th>
   <th style="text-align:right;"> 0 </th>
   <th style="text-align:right;"> 1 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 2301 </td>
   <td style="text-align:right;"> 29 </td>
   <td style="text-align:right;"> 1271 </td>
   <td style="text-align:right;"> 38 </td>
   <td style="text-align:right;"> 212 </td>
   <td style="text-align:right;"> 15 </td>
   <td style="text-align:right;"> 90 </td>
   <td style="text-align:right;"> 15 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 22 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 124 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 69 </td>
   <td style="text-align:right;"> 11 </td>
   <td style="text-align:right;"> 72 </td>
   <td style="text-align:right;"> 17 </td>
  </tr>
</tbody>
</table>

```
##  --------------------------- 
##            ------Died------- 
##  Visually         0       1  
##  impaired                    
##  --------------------------- 
##  0          3874.00   97.00  
##               97.56    2.44  
##                              
##  1           287.00   40.00  
##               87.77   12.23  
##  ---------------------------
```

---
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### The odds of visual impairment increase with age.

- agegrp = 13-34: `$$(22 + 2) / (29 + 2301) = 0.010$$`  
- agegrp = 35-54: `$$(124 + 10) / (1271 + 38) = 0.102$$`
- agegrp = 55-64: `$$(69 + 11) / (212 + 15) = 0.352$$`
- agegrp = 65+:   `$$(72 + 17) / (90 + 15) = 0.847$$`

---
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### The odds of death also increase with age.

- agegrp = 13-34: `$$(29 + 2) / (2301 + 22) = 0.013$$`
- agegrp = 35-54: `$$(38 + 10) / (1271 + 124) = 0.034$$`
- agegrp = 54-64: `$$(15 + 11) / (212 + 69) = 0.093$$`
- agegrp = 65+: `$$(15 + 17) / (90 + 72) = 0.198$$`

---
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### Exercise 3:

- Fill in the **odds of death odds ratios** of death in the visually impaired and unimpaired individuals, by **age group**.

|  agegrp   | vimp = 0 | vimp = 1 | odds ratio |
| :------- | :------: | :------: | :--------: |
| 0 (15-34) | 0.01260  |          |            |
| 1 (35-54) | 0.02990  |          |            |
| 2 (55-64) | 0.07075  |          |            |
|  3 (65+)  | 0.16667  |          |            |

---
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### Exercise 3: answers

- Fill in the **odds of death odds ratios** of death in the visually impaired and unimpaired individuals, by **age group**.

|  agegrp   | vimp = 0 | vimp = 1 | odds ratio |
| :------- | :------: | :------: | :--------: |
| 0 (15-34) | 0.01260  |  0.0909  |   7.214    |
| 1 (35-54) | 0.02990  | 0.08065  |   2.6973   |
| 2 (55-64) | 0.07075  | 0.15942  |   2.2533   |
|  3 (65+)  | 0.16667  | 0.23611  |   1.4166   |

---
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## Adjusting for confouding using logistic regression

```r
vimp_logm2 <- glm(died ~ vimp + as.factor(agegrp), 
                  data = mortality, 
                  family = binomial(link = "logit"))
library(epiDisplay)
logistic.display(vimp_logm2, decimal = 3)
```

```
##  
##                          OR lower95ci upper95ci     Pr(>|Z|)
## vimp               2.202266  1.408816  3.442592 5.327933e-04
## as.factor(agegrp)1 2.354987  1.484023  3.737116 2.774844e-04
## as.factor(agegrp)2 5.415484  3.082732  9.513468 4.199207e-09
## as.factor(agegrp)3 9.901309  5.541161 17.692306 9.839612e-15
```

---
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## What do we conclude from this output?

- after controlling for age there is strong evidence of an association between visual impairment and odds of death.

- On average, being visually impaired is associated with approximately a doubling in the odds of death. Comparing the adjusted OR (2.20) with the crude odds ratio that we obtained in the previous session (5.57) suggests **important confounding by age**, such that a failure to take age into account in the analysis leads to **an overestimate of the average effect of visual impairment**.

---
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### Look at the output from another format

```r
vimp_logm2 <- glm(died ~ vimp + as.factor(agegrp), 
                  data = mortality, 
                  family = binomial(link = "logit"))
summary(vimp_logm2, decimal = 3)
```

```
## 
## Call:
## glm(formula = died ~ vimp + as.factor(agegrp), family = binomial(link = "logit"), 
##     data = mortality)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -0.7109  -0.2473  -0.1619  -0.1619   2.9468  
## 
## Coefficients:
##                    Estimate Std. Error z value Pr(>|z|)    
## `(Intercept)         -4.3286     0.1809 -23.931  < 2e-16 ***`
## `vimp                 0.7895     0.2279   3.464 0.000533 ***`
## `as.factor(agegrp)1   0.8565     0.2356   3.635 0.000277 ***`
## `as.factor(agegrp)2   1.6893     0.2875   5.876 4.20e-09 ***`
## `as.factor(agegrp)3   2.2927     0.2962   7.741 9.84e-15 ***`
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1213.8  on 4297  degrees of freedom
## Residual deviance: 1091.7  on 4293  degrees of freedom
## AIC: 1101.7
```

???
how do we explain the output on log scale?

---
#### we can write down the formula for this model as:

`$$\log{\mathbf{odds}_{\mathbf{death}}} = \log{\mathbf{odds}_{\mathbf{baseline}}} + \log{\mathbf{OR}_{Vimp}} + \log{\mathbf{OR}_{agegrp}}$$`

| Component of model | Parameter name in output |  Estimate  | Interpretation                       |
|--------------------|--------------------------|:----------:|--------------------------------------|
| Baseline           | (intercept)              |   -4.3286  | log-Odds in baseline group           |
| visual impairment  | Normal (baseline, =0)    | (0, fixed) | log(1) = 0                           |
|                    | Visually impaired (=1)   |   0.7895   | log-Odds ratio, vimp=1 vs vimp=0     |
| Age group          | 15-34 (base, =0)         | (0, fixed) | log(1) = 0                           |
|                    | 35-54 (=1)               |   0.8565   | log-Odds ratio, agegrp 1 vs agegrp 0 |
|                    | 55-64 (=2)               |   1.6893   | log-Odds ratio, agegrp 2 vs agegrp 0 |
|                    | 65+ (=3)                 |   2.2927   | log-Odds ratio, agegrp 3 vs agegrp 0 |

---
class: middle
### Look at the observed odds and odds ratios again

- we make no assumption about how the effects of visual impairment and age group combine.

- However, in the logistic regression model that we have just fitted, **we made the assumption that the effect of visual impairment (i.e. the odds ratio) is the same in each age group**: i.e. we assumed that the effect of visual impairment (measured as an odds ratio) is not modified (changed) by/does not vary with age group: i.e. that age group is not an effect modifier for/does not interact with visual impairment.

---
class: middle, center

# The end

## slide address: https://wangcc.me/logistic-reg-CSS/