Poisson Regression - Basic concepts

class: left, middle, inverse, title-slide

.title[
# Poisson Regression - Basic concepts
]
.author[
### Mabel Carabali
]
.institute[
### EBOH, McGill University
]
.date[
### updated: 2024-11-27
]

---

class: middle

---

class: middle
**Expected competencies:**
- Knowledge about linear, logistic and binomial regressions.
- Knowledge about count data and Poisson distribution.

### Objectives
- Revise Poisson distribution and model assumptions
- Provide tools to estimate Risk Ratios, Risk Difference and Rate Ratios using Poisson regressions

---
class:middle
### Poisson distribution
The Poisson distribution gives the probability mass function (discrete) of all possible numbers of new cases, from 0 to `$\infty$`.     
<br>   
If `$Y_i$`= # of new cases with `$\lambda$` = expected number of cases in a given time period and `$$Y_i \sim  Poisson (\lambda)$$`

then the probability mass function that `$Y=k$`, is: 
`$$P(Y=k) = \dfrac{\lambda^{k}e^{-\lambda}}{k!}$$`

for `$k = 0, 1, 2,$` ... and `$λ > 0$`, where `$λ$` is both the mean and the variance of `$K$`; where 
`$λ$` = expected number of cases in a given time period.

---
class:middle
### Poisson distribution
This can be also presented as:
`$$y_i ∼Poisson(e^{X_iβ})$$`

where the linear predictor `$X_i β$` is the logarithm of the expected value of measurement `$y_i$`, or `$ln(\lambda)$`.

- In count-data regressions, each unit `$i$` corresponds to a setting (typically a spatial location or a time interval) in which `$y_i$` events are observed.

--
<br>

- Under the Poisson model, `$sd( y_i ) = \sqrt E(y_i)$`; thus if the model accurately describes the data, we also have a sense of how much variation we would expect from the fitted curve.

**.red[For a Poisson random variable, the  variance = mean =]** **E(Y) = `$\lambda$`**

---
class:middle
### Poisson distribution
<img src="L21_EPIB704_Poisson_Regression_files/figure-html/unnamed-chunk-2-1.png" width="60%" style="display: block; margin: auto;" />
--
**Checking distribution**
.pull-left[

```r
mean(dat$art)
```

```
## [1] 1.692896
```
]
.pull-right[

```r
sd(dat$art); var(dat$art)
```

```
## [1] 1.926069
```

```
## [1] 3.709742
```
]

---
class:middle
### Poisson considerations

- As `$λ$` increases Poisson distribution approximates to a normal distribution
- When `$n$` is large and `$p$` is very small, Poisson distribution approximates the binomial distribution.

.pull-left[
**Example:** Consider binomial distribution with probability = 0.01. What is the probability of 6 occurrences in 1200 tries?

```r
# binomial
dbinom(6,1200,0.01)
```

```
## [1] 0.02516172
```

```r
# poisson
dpois(6,1200*0.01) # lambda = n*p 
```

```
## [1] 0.02548128
```
]

.pull-right[
**To approximate the true (binomial) results:**

To approximate the mean
- `$µ = E(X) = λ = np$`

To approximate the standard deviation.
- `$σ = sqrt(λ)= sqrt(np)$`
]

---
class:middle
### Poisson considerations

---
class:middle
## Poisson Model Assumptions

**Poisson Response:** The response variable is a count per unit of time or space.

**Independence:** The observations must be independent of one another.

**Mean = Variance:** Conditional means equal the conditional variances.

**Linearity:** The log of the mean rate, log(λ), must be a linear function of x.
- the mean values of  `$Y$` at each level of  `$X$`, `$λ_{Y|X}$`, fall on a curve, not a line, although the logs of the means should follow a line.

---
class:middle
## Poisson Model

- A generalized linear model with a log link and a Poisson distribution,
- Used to model count variables 
- A natural model for rates in data sets with person time (e.g: Mortality).

.blue[
- Under predicts zero counts and over predicts the first set of counts
- Assumes same rate of outcome among the individual observations but incorporating “observed heterogeneity” relax this assumption.
]

---
class:middle
### Illustration using the articles and PhD years after graduation
.pull-left[
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      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="<span class='gt_from_md'><strong>N = 915</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height: 0;"><sup>1</sup></span>"><span class='gt_from_md'><strong>N = 915</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height: 0;"><sup>1</sup></span></th>
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<td headers="stat_0" class="gt_row gt_center"><br /></td></tr>
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<td headers="stat_0" class="gt_row gt_center">1.69; 1.00 (0.00, 2.00)</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">fem</td>
<td headers="stat_0" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Men</td>
<td headers="stat_0" class="gt_row gt_center">494 / 915 (54%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Women</td>
<td headers="stat_0" class="gt_row gt_center">421 / 915 (46%)</td></tr>
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<td headers="stat_0" class="gt_row gt_center">606 / 915 (66%)</td></tr>
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<td headers="stat_0" class="gt_row gt_center">309 / 915 (34%)</td></tr>
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<td headers="stat_0" class="gt_row gt_center"><br /></td></tr>
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]
.pull-right[
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<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="<span class='gt_from_md'><strong>Characteristic</strong></span>"><span class='gt_from_md'><strong>Characteristic</strong></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="<span class='gt_from_md'><strong>N = 915</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height: 0;"><sup>1</sup></span>"><span class='gt_from_md'><strong>N = 915</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height: 0;"><sup>1</sup></span></th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">profage</td>
<td headers="stat_0" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Mean; Median (Q1, Q3)</td>
<td headers="stat_0" class="gt_row gt_center">33.2; 33.0 (31.0, 36.0)</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">kid5</td>
<td headers="stat_0" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    0</td>
<td headers="stat_0" class="gt_row gt_center">599 / 915 (65%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    1</td>
<td headers="stat_0" class="gt_row gt_center">195 / 915 (21%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    2</td>
<td headers="stat_0" class="gt_row gt_center">105 / 915 (11%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    3</td>
<td headers="stat_0" class="gt_row gt_center">16 / 915 (1.7%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="font-weight: bold;">ment</td>
<td headers="stat_0" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Mean; Median (Q1, Q3)</td>
<td headers="stat_0" class="gt_row gt_center">9; 6 (3, 12)</td></tr>
  </tbody>
  
  <tfoot class="gt_footnotes">
    <tr>
      <td class="gt_footnote" colspan="2"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height: 0;"><sup>1</sup></span> <span class='gt_from_md'>n / N (%)</span></td>
    </tr>
  </tfoot>
</table>
</div>

]
---
class:middle
### Illustration using the articles and PhD years after graduation

```r
pois0 <- glm(art ~ 1, family= "poisson", data = dat)
```
--
** Exponentiated coefficient**

```
##             exp(Est.) 2.5% 97.5% z val. p
## (Intercept)      1.69 1.61  1.78  20.72 0
```

--
**Summary**

```
## 
## Call:
## glm(formula = art ~ 1, family = "poisson", data = dat)
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  0.52644    0.02541   20.72   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for poisson family taken to be 1)
## 
##     Null deviance: 1817.4  on 914  degrees of freedom
## Residual deviance: 1817.4  on 914  degrees of freedom
## AIC: 3487.1
## 
## Number of Fisher Scoring iterations: 5
```

---
class:middle
**Illustration using the articles and PhD years after graduation**
<table class="table table-striped table-hover table-condensed table-responsive" style="width: auto !important; margin-left: auto; margin-right: auto;">
<tbody>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Observations </td>
   <td style="text-align:right;"> 915 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Dependent variable </td>
   <td style="text-align:right;"> art </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Type </td>
   <td style="text-align:right;"> Generalized linear model </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Family </td>
   <td style="text-align:right;"> poisson </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Link </td>
   <td style="text-align:right;"> log </td>
  </tr>
</tbody>
</table> <table class="table table-striped table-hover table-condensed table-responsive" style="width: auto !important; margin-left: auto; margin-right: auto;">
<tbody>
  <tr>
   <td style="text-align:left;font-weight: bold;"> 𝛘²(0) </td>
   <td style="text-align:right;"> -0.00 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Pseudo-R² (Cragg-Uhler) </td>
   <td style="text-align:right;"> 0.00 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Pseudo-R² (McFadden) </td>
   <td style="text-align:right;"> 0.00 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> AIC </td>
   <td style="text-align:right;"> 3487.15 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> BIC </td>
   <td style="text-align:right;"> 3491.97 </td>
  </tr>
</tbody>
</table> <table class="table table-striped table-hover table-condensed table-responsive" style="width: auto !important; margin-left: auto; margin-right: auto;border-bottom: 0;">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> exp(Est.) </th>
   <th style="text-align:right;"> 2.5% </th>
   <th style="text-align:right;"> 97.5% </th>
   <th style="text-align:right;"> z val. </th>
   <th style="text-align:right;"> p </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;font-weight: bold;"> (Intercept) </td>
   <td style="text-align:right;"> 1.69 </td>
   <td style="text-align:right;"> 1.61 </td>
   <td style="text-align:right;"> 1.78 </td>
   <td style="text-align:right;"> 20.72 </td>
   <td style="text-align:right;"> 0.00 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> Standard errors: MLE</td></tr></tfoot>
</table>

---
class:middle
### Illustration using the articles and PhD years after graduation

---
class:middle
### Illustration using the articles and PhD years after graduation
If the follow-up time is not specified, the software assumes that person time is one for everyone, and Poisson regression actually produces the RISK ratio:

.pull-left[

```r
pois1 <- glm(art ~ fem + mar, 
*            poisson(link="log"),
             data = dat) 
round(jtools::summ(pois1, confint= T, 
                   exp = T)$"coeftable", 2) ; #summary(pois1)
```

```
##             exp(Est.) 2.5% 97.5% z val.    p
## (Intercept)      1.89 1.77  2.03  18.19 0.00
## femWomen         0.79 0.71  0.87  -4.49 0.00
## marSingle        0.97 0.87  1.09  -0.50 0.62
```
]

.pull-right[
<img src="images/poisPRM.gif" width="120%" style="display: block; margin: auto;" />
]

---
class:middle
### Illustration using the articles and PhD years after graduation
To get the RATE ratio instead of RISK ratio, we need to tell the software where to find the follow-up time:

```r
*dat$logprofage <- log(dat$profage)
#summary(dat$profage)
*pois2a <- glm(art ~ fem + mar + offset(logprofage),
             poisson(link="log"), data = dat)
```

```
##             exp(Est.) 2.5% 97.5% z val.   p
## (Intercept)      0.06 0.05  0.06 -81.49 0.0
## femWomen         0.79 0.72  0.88  -4.30 0.0
## marSingle        0.97 0.87  1.08  -0.52 0.6
```

**.purple[The exponentiated coefficient give then the Rate Ratio.]**

---
class:middle
### Illustration using the articles and PhD years after graduation

<img src="images/artpred_T.gif" width="60%" style="display: block; margin: auto;" />
--
**comparing "goodness-of-fit" using AIC (Akaike Information Criteria)**

```
##         Empty     Risk     Rate
## [1,] 3487.147 3467.868 3470.165
```

---
class:middle
### Illustration using the articles and PhD years after graduation

```r
*pois2b <- glm(art ~ fem + mar + offset(profage),
             poisson(link="log"), data = dat)
round(jtools::summ(pois2b, confint= T, exp = T)$"coeftable", 2)
```

```
##             exp(Est.) 2.5% 97.5%  z val.    p
## (Intercept)      0.00 0.00  0.00 -997.79 0.00
## femWomen         1.04 0.94  1.15    0.76 0.45
## marSingle        0.71 0.64  0.79   -6.15 0.00
```

.red[Using the variable age without the log-transformation changes the estimates and the likelihood is affected as evidenced by the AIC]

**comparing "goodness-of-fit" using AIC (Akaike Information Criteria)**

```
##         Empty     Risk     Rate Rate_nolog
## [1,] 3487.147 3467.868 3470.165   14060.01
```
---
class:middle
### Illustration using the articles and PhD years after graduation
<img src="images/artpred_T1.png" width="60%" style="display: block; margin: auto;" />
- Under predicts zero counts and over predicts the first set of counts
- Assumes same rate of outcome among the individual observations but incorporating “observed heterogeneity” relax this assumption.

---
class:middle
### Poisson Models for Risk Ratios and Risk Differences
Recall our example: **Wells in Bangladesh**

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#qeayjdrzqo div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after {
  height: 0px !important;
}
</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="<span class='gt_from_md'><strong>Characteristic</strong></span>"><span class='gt_from_md'><strong>Characteristic</strong></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="<span class='gt_from_md'><strong>N = 3,020</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height: 0;"><sup>1</sup></span>"><span class='gt_from_md'><strong>N = 3,020</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height: 0;"><sup>1</sup></span></th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="label" class="gt_row gt_left">switch</td>
<td headers="stat_0" class="gt_row gt_center">1,737 (58%)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">arsenic</td>
<td headers="stat_0" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Median (Q1, Q3)</td>
<td headers="stat_0" class="gt_row gt_center">1.30 (0.82, 2.20)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Min, Max</td>
<td headers="stat_0" class="gt_row gt_center">0.51, 9.65</td></tr>
    <tr><td headers="label" class="gt_row gt_left">dist100</td>
<td headers="stat_0" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Median (Q1, Q3)</td>
<td headers="stat_0" class="gt_row gt_center">0.37 (0.21, 0.64)</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Min, Max</td>
<td headers="stat_0" class="gt_row gt_center">0.00, 3.40</td></tr>
    <tr><td headers="label" class="gt_row gt_left">assoc</td>
<td headers="stat_0" class="gt_row gt_center">1,277 (42%)</td></tr>
  </tbody>
  
  <tfoot class="gt_footnotes">
    <tr>
      <td class="gt_footnote" colspan="2"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height: 0;"><sup>1</sup></span> <span class='gt_from_md'>n (%)</span></td>
    </tr>
  </tfoot>
</table>
</div>
]

--
.pull-right[
<img src="L21_EPIB704_Poisson_Regression_files/figure-html/unnamed-chunk-25-1.png" width="150%" height="150%" style="display: block; margin: auto;" />
]

---
class:middle
### Recall our example: **Wells in Bangladesh**

**.black[Model "switching" to a safer well (outcome) as a function of whether the owner belongs to a community association and As. level.]**
<br>

|Model                       |	Frequentist Estimates|
|:---------------------------|:-------------------:|
|Linear Model `$^1$`     |    RD= -0.036; RR= N/A       |
|Logit Model (Predicted probs.) `$^2$`| RD= -0.033; RR= 0.939|
|AME (prediction `StdGLM`) `$^2$` | RD= -0.031; RR= 0.948 |
|GLM: Log-Binomial `$^2$` |RD= N/A; RR= 0.928|

<br>

`$^1$` Single predictor;  `$^2$` Two predictors
 
---
class:middle
### Poisson Model and Risk Ratios
–our example- **Wells in Bangladesh**

```r
mod.glm.p.rr <- glm(switch ~ assoc + arsenic, data = wells, 
*                   family=poisson(link="log"))
```
--

```
#> 
#> Call:
#> glm(formula = switch ~ assoc + arsenic, family = poisson(link = "log"), 
#>     data = wells)
#> 
#> Coefficients:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  -0.7489     0.0473  -15.83  < 2e-16 ***
#> assoc        -0.0559     0.0488   -1.14     0.25    
#> arsenic       0.1258     0.0192    6.55  5.6e-11 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for poisson family taken to be 1)
#> 
#>     Null deviance: 1921.5  on 3019  degrees of freedom
#> Residual deviance: 1880.2  on 3017  degrees of freedom
#> AIC: 5360
#> 
#> Number of Fisher Scoring iterations: 5
```

---
class:middle
Poisson Model and RRs –our example- **Wells in Bangladesh** 
<table class="table table-striped table-hover table-condensed table-responsive" style="width: auto !important; margin-left: auto; margin-right: auto;">
<tbody>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Observations </td>
   <td style="text-align:right;"> 3020 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Dependent variable </td>
   <td style="text-align:right;"> switch </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Type </td>
   <td style="text-align:right;"> Generalized linear model </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Family </td>
   <td style="text-align:right;"> poisson </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Link </td>
   <td style="text-align:right;"> log </td>
  </tr>
</tbody>
</table> <table class="table table-striped table-hover table-condensed table-responsive" style="width: auto !important; margin-left: auto; margin-right: auto;">
<tbody>
  <tr>
   <td style="text-align:left;font-weight: bold;"> 𝛘²(2) </td>
   <td style="text-align:right;"> 41.25 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Pseudo-R² (Cragg-Uhler) </td>
   <td style="text-align:right;"> 0.02 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> Pseudo-R² (McFadden) </td>
   <td style="text-align:right;"> 0.01 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> AIC </td>
   <td style="text-align:right;"> 5360.21 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> BIC </td>
   <td style="text-align:right;"> 5378.25 </td>
  </tr>
</tbody>
</table> <table class="table table-striped table-hover table-condensed table-responsive" style="width: auto !important; margin-left: auto; margin-right: auto;border-bottom: 0;">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> exp(Est.) </th>
   <th style="text-align:right;"> 2.5% </th>
   <th style="text-align:right;"> 97.5% </th>
   <th style="text-align:right;"> z val. </th>
   <th style="text-align:right;"> p </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;font-weight: bold;"> (Intercept) </td>
   <td style="text-align:right;"> 0.47 </td>
   <td style="text-align:right;"> 0.43 </td>
   <td style="text-align:right;"> 0.52 </td>
   <td style="text-align:right;"> -15.83 </td>
   <td style="text-align:right;"> 0.00 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> assoc </td>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:right;"> 0.86 </td>
   <td style="text-align:right;"> 1.04 </td>
   <td style="text-align:right;"> -1.14 </td>
   <td style="text-align:right;"> 0.25 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> arsenic </td>
   <td style="text-align:right;"> 1.13 </td>
   <td style="text-align:right;"> 1.09 </td>
   <td style="text-align:right;"> 1.18 </td>
   <td style="text-align:right;"> 6.55 </td>
   <td style="text-align:right;"> 0.00 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> Standard errors: MLE</td></tr></tfoot>
</table>

---
class:middle
### Poisson Model and Risk Ratios, our example- Wells in Bangladesh

```
#>             exp(Est.)  2.5% 97.5% z val.        p
#> (Intercept)     0.473 0.431 0.519 -15.83 1.82e-56
#> assoc           0.946 0.859 1.041  -1.14 2.52e-01
#> arsenic         1.134 1.092 1.178   6.55 5.61e-11
```

<br>
- Poisson models provides RISK ratios with biased confidence intervals.
- The model assumes a Poisson distribution when in fact the outcomes are binary.  
- To overcome issues with the variance, we could use a variance estimator that is robust to the misspecification of the model.

---
class:middle
### Poisson Model and RRs (Robust Variance)
–our example- **Wells in Bangladesh**

```r
jtools::summ(mod.glm.p.rr, confint=T, exp=T, 
*            robust= "HC0")$"coeftable"
```

```
#>             exp(Est.)  2.5% 97.5% z val.         p
#> (Intercept)     0.473 0.445 0.502 -24.48 2.63e-132
#> assoc           0.946 0.889 1.006  -1.77  7.74e-02
#> arsenic         1.134 1.110 1.159  11.50  1.33e-30
```

**Quasi-Poisson Risk Ratio**

```r
mod.glm.qp.rr <- glm(switch ~ assoc + arsenic, data = wells, 
*                    family= quasipoisson(link="log"))
jtools::summ(mod.glm.qp.rr, confint=T, exp=T)$"coeftable"
```

```
#>             exp(Est.)  2.5% 97.5% t val.         p
#> (Intercept)     0.473 0.445 0.502 -24.31 2.19e-119
#> assoc           0.946 0.888 1.006  -1.76  7.90e-02
#> arsenic         1.134 1.107 1.162  10.06  1.87e-23
```
.small[*Note: Pseudo-R2 for quasibinomial/quasipoisson families is calculated by refitting the fitted and null models as
binomial/poisson.]

---
class:middle
### Poisson Model and Risk Differences
–our example- **Wells in Bangladesh**

```r
mod.glm.p.rd <- glm(switch ~ assoc + arsenic, data = wells, 
*                   family=poisson(link="identity"))
```

```
#> 
#> Call:
#> glm(formula = switch ~ assoc + arsenic, family = poisson(link = "identity"), 
#>     data = wells)
#> 
#> Coefficients:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)   0.4388     0.0278   15.80  < 2e-16 ***
#> assoc        -0.0268     0.0275   -0.97     0.33    
#> arsenic       0.0891     0.0138    6.47  9.8e-11 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for poisson family taken to be 1)
#> 
#>     Null deviance: 1921.5  on 3019  degrees of freedom
#> Residual deviance: 1876.2  on 3017  degrees of freedom
#> AIC: 5356
#> 
#> Number of Fisher Scoring iterations: 5
```

---
class:middle
### Poisson Model and Risk Differences

```r
jtools::summ(mod.glm.p.rd, confint=T)$"coeftable"
```

```
#>                Est.    2.5%  97.5% z val.        p
#> (Intercept)  0.4388  0.3844 0.4932 15.800 3.09e-56
#> assoc       -0.0268 -0.0806 0.0271 -0.974 3.30e-01
#> arsenic      0.0891  0.0621 0.1161  6.470 9.83e-11
```

**Quasi-Poisson Risk Difference (Robust Variance)**

```r
mod.glm.qp.rd1 <- glm(switch ~ assoc + arsenic, data = wells, 
*                     family= quasipoisson(link="identity")) #
jtools::summ(mod.glm.qp.rd1, confint=T)$"coeftable"
```

```
#>                Est.    2.5%   97.5% t val.         p
#> (Intercept)  0.4388  0.4033 0.47429  24.23 1.19e-118
#> assoc       -0.0268 -0.0619 0.00837  -1.49  1.36e-01
#> arsenic      0.0891  0.0715 0.10674   9.92  7.54e-23
```

---
class:middle
**Why and When do we use Poisson for Risk Ratios and Risk Differences?**

**.purple[Poisson regression]**
- Used for rare outcomes 
- When the follow-up has different lengths

**.black[Logistic regression:]**
Used with same length of follow-up time

**.blue[Poisson approximation to Binomial]**
- Cohort studies where all patients have equal follow-up times. 
- Poisson regression can be used similarly as logistic regression, with a time-at-risk value specified as 1 for each subject. 
- If the model adequately fits the data, this provides a correct estimate of the AdjRR

<br>

.small[Chen et al.: Comparison of robustness to outliers between robust poisson models and log-binomial models when estimating relative risks for common binary outcomes: a simulation study. BMC Medical Research Methodology 2014 14:82.
]

---
class:middle
## Comparing Results
**.black[Model "switching" to a safer well (outcome) as a function of whether the owner belongs to a community association and As. level.]**
<br>

`$^1$` Single predictor;  `$^2$` Two predictors

---
class:middle
**Why and When do we use Poisson for Risk Ratios and Risk Differences?**

- For common outcomes, Poisson regression is likely to compute a confidence interval(s) that is conservative, suggesting less precision than is true.

- Poisson regression produces wider confidence intervals (compared with a log-binomial model and stratified analysis) because Poisson errors are overestimates of binomial errors when the outcome is common

- Poisson errors approximately equal binomial errors when the outcome is rare

<br>

.small[
Guangyong Zou. A Modified Poisson Regression Approach to Prospective Studies with Binary Data, AJE, Vol 159: 7 (2004).702–706 (https://doi.org/10.1093/aje/kwh090).

Chen et al. Comparison of robustness to outliers between robust poisson models and log-binomial models when estimating relative risks for common binary outcomes: a simulation study. BMC Med Res Method 2014 14:82.
]

---
class:middle
### Robust Poisson models
- The Poisson regression uses a logarithm as the natural link function under the generalized linear model framework.

- The robust Poisson regression model uses the classical sandwich estimator under the generalized estimation equation (GEE) framework to correct the inflated variance (also known as over-dispersion) in the standard Poisson regression.

- The technique is known as modified Poisson regression or pseudo-likelihood estimation.

- This correction can be achieved by using the `robust` option form `jtools` in `R` or the `quasipoisson` from `glm`, also using the REPEATED statement in SAS Proc GENMOD; the ROBUST option in STATA’s Poisson procedure.

---
class:middle
### Robust Poisson models vs. Log-Binomial

- In presence of linear confounders, the two models yield comparable relative biases.

- With the non-linear confounders, the robust Poisson model outperform the log-binomial model. Larger differences with rare outcomes.

- The robust Poisson models are more robust to outliers compared to the log-binomial models when estimating relative risks or risk ratios for common binary outcomes.

.small[Chen et al. Comparison of robustness to outliers between robust poisson models and log-binomial models when estimating relative risks for common binary outcomes: a simulation study. BMC Medical Research Methodology 2014 14:82.
]

---
class:middle
### Log-Binomial vs Robust Poisson?
- Of the two methods, the log-binomial method is generally preferred due to the fact that the MLEs estimated by the log-binomial models are more efficient compared to the pseudo-likelihood estimators used by the robust Poisson models.

- Spiegelman and Hertzmark recommend using the log-binomial models over the robust Poisson models when convergence is not an issue.

- It appears that the gain in efficiency is beneficial to log-binomial models only for samples of small sizes.

- Due to the concern of lack of efficiency for the robust Poisson models for small samples, log-binomial may still be the choice when the sample size is small.

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### Other Count Models

- Negative Binomial (Overdispersed Poisson Models)

- Zero Inflated Models

- Zero Truncated Models

.pull-right[
### .red[… Out of the scope for this session ]
]

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###  QUESTIONS?

## COMMENTS?

# RECOMMENDATIONS?

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## Distributions
<img src="images/distributions.png" width="80%" style="display: block; margin: auto;" />

Vittinghoff E., Glidden D.V., Shiboski S.C., McCulloch C.E. (2012) Generalized Linear Models. In: Regression Methods in Biostatistics. Statistics for Biology and Health. Springer, Boston, MA

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### Log-Binomial vs Robust Poisson?
- Software utilizes iterative weighted least squares (IWLS*) approach or variations of IWLS to find MLEs for generalized linear models. 
- For log-binomial models, the weights used by the IWLS approach contain the term 1/(1-p), where `$p = exp(XTβ)$` with a range from 0 to 1. 
- The MLE of a log-binomial model is likely to be too sensitive to outliers because a very large p has a large influence on the weights.
- MLE and pseudolikelihood estimators are deteriorated in presence of outliers. 
- The level of deterioration differed when the relationships between the confounder and the outcome was not in a simple form

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### Maximum likelihood estimate of lambda 
`$$MLE = \ell(\lambda; y_1, \dots, y_n) = \sum_{i = 1}^n \log f(y_i; \lambda) = \sum_{i = 1}^n \log \frac{\lambda^{y_1}exp^{-\lambda}}{y_1!}$$`

Take first derivative (notation `$\prime = \frac{d(.)}{d\lambda}$`)

`$\ell^\prime(\lambda; y_1, \dots, y_n) =  \sum_{i = 1}^n \log \frac{\lambda^{y_1}exp^{-\lambda}}{y_1!}^\prime$`
                                      `$= \sum_{i = 1}^n \left\{ -\lambda + y_i \cdot \log(\lambda) - \log(y_i!) \right\}^\prime$`
                                      `$= \sum_{i = 1}^n \left\{ -1 + y_i \cdot \frac{1}{\lambda} \right\}$`
                                      `$= -n + \frac{1}{\lambda} \sum_{i = 1}^n y_i$`    
                                      
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### Maximum likelihood estimate of lambda                                       
The first-order condition for maximizing the log-likelihood sets its derivative to zero

`$$\ell^\prime(\lambda; y_1, \dots, y_n)  = 0$$`
`$$-n + \frac{1}{\lambda} \sum_{i = 1}^n y_i = 0$$`
`$$n \cdot \lambda  = \sum_{i = 1}^n y_i$$`
`$$\lambda = \frac{1}{n} \sum_{i = 1}^n y_i = \bar y$$`
  
Thus, the maximum likelihood estimator is simply the empirical mean `$\hat \lambda = \bar y$`

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## Poisson distribution (PD)
If instead are given the average rate, *r*, at which events occur then `$\lambda = rt$`, and
`$$P(k\, events\, in\, interval\, t) = \dfrac{rt^{k}e^{-rt}}{k!}$$`
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## Poisson distribution (PD)
PD a special case of the binomial, with trials n -> `$\infty$` and P(success in any trial) -> 0       
If p is small, binomial P(k successes) `$\approx$` poisson P(k with `$\lambda$` = np)     
<br>
**Example:** Consider binomial distribution with probability = 0.01. What is the probability of 8 occurences in 1000 tries?

```r
# binomial
dbinom(8,1000,0.01)
```

```
#> [1] 0.113
```

```r
# poisson
dpois(8,1000*.01) # lambda = n*p 
```

```
#> [1] 0.113
```

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### Offset vs Exposure  
In most applications of count-data regression, there is a baseline or exposure, some value such as the  number of person years that the counts occurred over

We can  model `$y_i$` as the number of cases in a process with rate `$\theta_i$` and exposure `$\mu_i$`  
`$$y_i \sim negative \, binomial (\mu_i \theta_i, \phi)$$`

where, as before, `$\theta_i = e^{X_i \beta}$` and includes Poisson regression as the special case of  φ → ∞

The logarithm of the exposure, `$log(\mu_i)$`, is called the offset in GLM terminology.

The regression coefficients `$\beta$` now reflect the associations between the predictors and `$\theta_i$` (for example, the rate of deaths)

Putting the logarithm of the exposure into the model as an offset, is equivalent to  including it as a regression predictor, but with its coefficient fixed to the value 1.  Another option is to include it as a predictor and let its coefficient be estimated from the data.

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## AIC	=	Akaike information criterion

`$AIC = - 2(log-likelihood) + kn_{par}$`, where `$k=2$`, equivalently

`$AIC = 2k-2 \mathrm {\ln}(\hat {L})$`

`$k$` =	number of estimated parameters in the model
`$\hat {L}$`	=	maximum value of the likelihood function for the model

Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models.

Hence, AIC provides a means for model selection

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## BIC	=	Bayesian Information Criterion

`$BIC = - 2(log-likelihood) + kn_{par}$`,

where `$k= log(n)$`,

and `$n$` is the number of observations