IDS 702: Module 2.4

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# IDS 702: Module 2.4
## Model assessment and validation - binned residuals and roc curves
### Dr. Olanrewaju Michael Akande

---

## Model assessment and validation

There are various types of residuals when working with generalized linear models (GLMs). For logistic regression in particular, we have

- .hlight[Response residuals]
  .block[
.small[
`$$e_i = y_i - \hat{\pi}_i.$$`
]
]

- .hlight[Pearson residuals]
  .block[
.small[
`$$e_i^P = \dfrac{y_i - \hat{\pi}_i}{\sqrt{\hat{\pi}_i(1-\hat{\pi}_i)}},$$`
]
]

which are obtained by "normalizing" the response residuals by the estimated Bernoulli standard deviation.

- .hlight[Deviance residuals]
  .block[
.small[
`$$e_i^D = \textrm{sign}(y_i - \hat{\pi}_i) \times 2\left(y_i \textrm{log}\dfrac{1}{\hat{\pi}_i} + (1-y_i) \textrm{log}\dfrac{1}{1-\hat{\pi}_i} \right),$$`
]
]
 
 which are the default in R when using the .hlight[residuals()] function. We will talk a bit more about deviance later, but deviance residuals represent the contributions of individual samples to the deviance.

---
## Model assessment and validation

- Deviance residuals are usually the most appropriate for residual plots, when working with GLMs.

- However, unlike what we had for linear regression, just looking at the residuals does not work well here.
  + They are always positive when `$Y=1$` and always negative when `$Y=0$`.
  
--

+ Also, constant variance is not an assumption of logistic regression.
  
     <div class="question">
    Why is that the case?
    </div>  
    Think about the properties of the Bernoulli distribution when we write `$y_i | x_i \sim \textrm{Bernoulli}(\pi_i)$`
     
--

+ We also do not have normality of residuals to work with either.

---
## Model assessment and validation

- What we can do is check to see if the function of predictors is well specified using .hlight[binned residuals].

- We can assess the overall fit of our model using .hlight[deviance] and .hlight[change in deviance].

- We can also see how well our model predicts (model validation) using
 + Confusion matrix

+ ROC curves

---
## Binned residuals

- Compute raw (response) residuals for fitted logistic regression.

- Order observations by values of predicted probabilities (or predictor values) from the fitted regression.

- Using ordered data, form `$g$` bins of (approximately) equal size. Default: `$g = \sqrt{n}$`.

- Compute average residual in each bin.

- Plot average residual versus average predicted probability (or average predictor value) for each bin.

- Use the .hlight[arm] package in R.

---
## NBA analysis

Recall the NBA data

```r
nba <- read.csv("data/nba_games_stats_reduced.csv",header=T)
nba <- nba[nba$Team=="SAS",]
colnames(nba)[3] <- "Opp"
nba$win <- rep(0,nrow(nba))
nba$win[nba$WINorLOSS=="W"] <- 1
nba$win <- as.factor(nba$win)
nba$Opp_cent <- nba$Opp - mean(nba$Opp)
nbareg <- glm(win~Opp_cent,family=binomial(link=logit),data=nba)
```

---
## NBA analysis

```r
plot(nbareg,which=1)
```

The residuals are the deviance residuals, while the predicted values are on the linear (logit) scale, that is, `$\beta_0 + \beta_1 x_i$`.

Look to see which cases have large absolute values for cases that don't fit well, but not too useful otherwise.

---
## NBA analysis

Plot binned raw residuals versus predicted probabilities (.hlight[arm] package).

```r
binnedplot(fitted(nbareg),residuals(nbareg,"resp"),xlab="Pred. probabilities",col.int="red4",
           ylab="Avg. residuals",main="Binned residual plot",col.pts="navy")
```

Look for "randomness" with almost all points within the red lines.

---
## NBA analysis

- Useful as a "one-stop shopping" plot; especially with many predictors and you want an initial look at model adequacy.

- What we have is mostly good, although model seems to struggle for fitted values over 0.95 or so.

- The red lines represent `$\pm 2$` SE bands, which we would expect to contain about 95% of the observations.

- Too few points here to draw any conclusions!

- You usually want many more data points before these plots start being useful.

---
## NBA analysis

Plot binned raw residuals versus individual predictors.

```r
binnedplot(nba$Opp,residuals(nbareg,"resp"),xlab="Opponent's points (centered)",
           col.int="red4",ylab="Avg. residuals",main="Binned residual plot",col.pts="navy")
```

---
## NBA analysis

- Mostly good, although model seems to struggle for low values of opponent's points.

- Also, too many points (16.7%) outside the bands.

- However, still too few points here for any conclusive takeaways.

- We also know some important predictors are missing by construction...

---
## Deviance

- To assess overall model fit, we can also look at .hlight[deviance].

- Deviance measures how well the model fits the data, when compared to the .hlight[saturated model], that is, an abstract model that fits the sample perfectly.

- Precisely, deviance is defined as the difference of likelihoods between the fitted model and the saturated model:
.block[
.small[
`$$D = - 2 \left[ \text{ Log Likelihood}(\text{Fitted Model}) - \text{ Log Likelihood}(\text{Saturated Model}) \right].$$`
]
]

- However, this "abstract saturated model" will have likelihood equal to one, so that deviance is simply
.block[
.small[
`$$D = - 2\text{ Log Likelihood}(\text{Fitted Model}) = - 2 \sum_{i=1}^n \left[y_i \textrm{log}(\hat{\pi}_{1i}) + (1-y_i) \textrm{log}(1-\hat{\pi}_{1i})\right].$$`
]
]

- Note that .hlight[deviance is always larger or equal than zero], and will only be zero if the fit is "perfect".

- Overall, deviance is a measure of error, so that, .hlight[lower values of deviance means better fit to the data].

---
## Deviance

- Like the metrics used under MLR, it is also often useful to use deviance for a model in relation to another model. We will revisit this soon.

- For now, a model we can use for this comparison is the .hlight[null model], that is, the model with only the intercept.

- Intuitively, this gives us a sense of how much the model improves from the "worst model", by the addition of  the predictors.

- The deviance of the null model, denoted `$D_0$`, is thus referred to as the .hlight[null deviance].

- To get a general sense of how much better the fitted model is to the null model, compare `$D$` to `$D_0$`, usually through the difference `$D_0 - D$`.

- The "larger" this .hlight[change in deviance] `$D_0 - D$` is, the more confident we are that the predictors we have included improve model fit.

- In large samples, `$D_0 - D$` has approximately a chi-squared distribution with degrees of freedom equal to the difference in the number of predictors between the two models.

---
## NBA analysis

For the NBA data for example, we see what looks like a meaningful difference in the two deviance scores.

```r
summary(nbareg)
```

```
## 
## Call:
## glm(formula = win ~ Opp_cent, family = binomial(link = logit), 
##     data = nba)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.2760  -0.7073   0.4454   0.7902   1.9593  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)
## (Intercept)  1.13387    0.15145   7.487 7.06e-14
## Opp_cent    -0.12567    0.01655  -7.594 3.11e-14
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 400.05  on 327  degrees of freedom
## Residual deviance: 313.42  on 326  degrees of freedom
## AIC: 317.42
## 
## Number of Fisher Scoring iterations: 5
```

---
## NBA analysis

- We can formalize this by doing a chi-squared test on the null model vs our fitted model. That is,
  
  ```r
  nbareg_null <- glm(win~1,family=binomial(link=logit),data=nba)
  anova(nbareg_null,nbareg,test= "Chisq")
  ```
  
  ```
  ## Analysis of Deviance Table
  ## 
  ## Model 1: win ~ 1
  ## Model 2: win ~ Opp_cent
  ##   Resid. Df Resid. Dev Df Deviance  Pr(>Chi)
  ## 1       327     400.05                      
  ## 2       326     313.42  1    86.63 < 2.2e-16
  ```

- The low p-value then confirms our previous statement.

- We will revisit this again when we look at logistic regression with multiple predictors.

- We will be able to use deviance for model comparison and selection by looking at the change in deviance `$D_{M_1} - D_{M_2}$`, for two models `$M_1$` and `$M_2$`, where `$M_1$` is nested within `$M_2$`.

---
## Confusion matrix

- We can use the estimated probabilities from our fitted model to predict outcomes, and then compare those to the observed values.

- For example, we could decide to predict `$Y=1$` when the predicted probability exceeds `$0.5$` and predict `$Y=0$` otherwise.

- We then can determine how many cases we classify correctly and incorrectly.

- Resulting `$2 \times 2$` table is called the .hlight[confusion matrix].

- When mis-classification rates are high, model may not be an especially good fit to the data.

---
## Confusion matrix

<table>
  <tr>
    <th> </th>
    <th> </th>
    <th colspan="2">Observed</th>
  </tr>
  <tr>
    <th colspan="2"></th>
    <td style="text-align:center">Y=1</td>
    <td style="text-align:center">Y=0</td>
  </tr>
  <tr>
    <th rowspan="2">Predicted</th>
    <td height="50px">Y=1</td>
    <td style="text-align:center">TP (True Positives)</td>
    <td style="text-align:center">FP (False Positives)</td>
  </tr>
  <tr>
    <td height="50px">Y=0</td>
    <td style="text-align:center">FN (False Negatives)</td>
    <td style="text-align:center">TN (True Negatives)</td>
  </tr>
</table>

- .hlight[True positive rate (TPR)] = `$\dfrac{TP}{TP+FN}$` (also known as .hlight[sensitivity])

- .hlight[False negative rate (FNR)] = `$\dfrac{FN}{TP+FN}$`

- .hlight[True negative rate (TNR)] = `$\dfrac{TN}{FP+TN}$` (also known as .hlight[specificity])

- .hlight[False positive rate (FPR)] = `$\dfrac{FP}{FP+TN}$` (1 - .hlight[specificity])

---
## ROC Curves

- .block[We want high values of sensitivity and low values of (1 - specificity)!]

- The .hlight[receiver operating characteristic (ROC)] curve plots 
  + Sensitivity on Y axis
  + 1 - specificity on X axis

- Evaluated at lots of different values (beyond 0.5) for the threshold.

- Good fitting logistic regression curves toward the upper left corner, with .hlight[area under the curve (AUC)] near one.

- Make ROC curves in R using the pROC package.

- By the way, we also often define .hlight[accuracy] as `$\dfrac{TP + TN}{TP+FN+FP+TN}$`. This estimates how well the model predicts correctly overall.

---
## NBA analysis

Let's look at the confusion matrix for the NBA data. Load the .hlight[arm], .hlight[e1071], .hlight[caret], and .hlight[pROC] packages.

```r
Conf_mat <- confusionMatrix(as.factor(ifelse(fitted(nbareg) >= 0.5, "W","L")),
                            nba$WINorLOSS,positive = "W")
Conf_mat$table
```

```
##           Reference
## Prediction   L   W
##          L  44  19
##          W  54 211
```

```r
Conf_mat$overall["Accuracy"];
```

```
## Accuracy 
## 0.777439
```

```r
Conf_mat$byClass[c("Sensitivity","Specificity")]
```

```
## Sensitivity Specificity 
##   0.9173913   0.4489796
```

.hlight[confusionMatrix] produces a lot of output. Print the .hlight[Conf_mat] object to see all of them.

---
## NBA analysis

```r
invisible(roc(nba$win,fitted(nbareg),plot=T,print.thres=c(0.3,0.5,0.7),legacy.axes=T,
              print.auc =T,col="red3"))
```

---
## NBA analysis

```r
invisible(roc(nba$win,fitted(nbareg),plot=T,print.thres="best",legacy.axes=T,
              print.auc =T,col="red3"))
```

---

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# What's next?

### Move on to the readings for the next module!