```{r include = FALSE}
# OPTIONAL: Set a more color blind friendly palette 
palette("Okabe-Ito")
scale_colour_discrete <- function(...) scale_colour_manual(values = palette())
scale_fill_discrete   <- function(...) scale_fill_manual(values = palette())

knitr::opts_chunk$set(
  collapse = TRUE, 
  warning = FALSE,
  message = FALSE,
  fig.height = 2.75, 
  fig.width = 4.25,
  fig.env='figure',
  fig.pos = 'h',
  fig.align = 'center')
```






# IMPORTANT REMINDERS

- The "Notes" sections of this document for your reference. Leave any code in 
those sections as is.
- When working on the exercises, copy-paste any starter code that I give you into a 
new code chunk, below, and edit from there.









# Notes: k-fold Cross-Validation {-}


**CONTEXT**

- **world = supervised learning**       
    We want to build a model some output variable $y$ by some predictors $x$.

- **task = regression**       
    $y$ is quantitative

- **algorithm = linear regression model**       
    We'll assume that the relationship between $y$ and $x$ can be represented by
    
    $$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_p x_p + \varepsilon$$




**GOAL: model evaluation**        

We want more **honest** metrics of prediction quality that 

(1) assess how well our model predicts **new outcomes**; and 
(2) help prevent **overfitting**.
    


**Why is overfitting so bad?**

Not only can overfitting produce misleading models, it can have serious societal impacts. Examples:

- [A former Amazon algorithm](https://www.reuters.com/article/us-amazon-com-jobs-automation-insight/amazon-scraps-secret-ai-recruiting-tool-that-showed-bias-against-women-idUSKCN1MK08G) built to help sift through resumes was _overfit_ to its current employees in leadership positions (who weren't representative of the general population or candidate pool).

- Facial recognition algorithms are often _overfit_ to the people who build them (who are not broadly representative of society). As one example, this has led to [disproportionate bias in policing](https://www.nytimes.com/2019/07/08/us/detroit-facial-recognition-cameras.html). For more on this topic, you might check out [Coded Bias](https://www.youtube.com/watch?v=jZl55PsfZJQ), a documentary by Shalini Kantayya which features MIT Media Lab researcher Joy Buolamwini.

- Polygenic risk scores (PRSs), which aim to predict a person's risk of developing a particular disease/trait 
based on their genetics, are often _overfit_ to the data on which they are built (which, historically, 
has exclusively---or at least primarily---included individuals of European ancestry). 
As a result, PRS predictions tend to be more accurate in European populations and new
research suggests that their [continued use in clinical settings could exacerbate health disparities](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6563838/). 



We can use **k-fold cross-validation** to estimate the typical error in our model predictions for *new* data:

- Divide the data into $k$ folds (or groups) of approximately equal size.    
- Repeat the following procedures for each fold $j = 1,2,...,k$:    
    - Remove fold $j$ from the data set.    
    - Fit a model using the data in the other $k-1$ folds (training).    
    - Use this model to predict the responses for the $n_j$ cases in fold $j$: $\hat{y}_1, ..., \hat{y}_{n_j}$.    
    - Calculate the MAE for fold $j$ (testing): $\text{MAE}_j = \frac{1}{n_j}\sum_{i=1}^{n_j} |y_i - \hat{y}_i|$.
- Combine this information into one measure of model quality:
    $$\text{CV}_{(k)} = \frac{1}{k} \sum_{j=1}^k \text{MAE}_j$$



![](https://kegrinde.github.io/stat253_coursenotes/images/crossval.png)












# Small Group Discussion: Algorithms and Tuning {-}


**Definitions:**

- **algorithm** = a step-by-step procedure for solving a problem (Merriam-Webster)

- **tuning parameter** = a *parameter* or *quantity* upon which an algorithm depends, that must be *selected* or *tuned* to "optimize" the algorithm

![](https://c1.wallpaperflare.com/preview/461/820/840/music-low-electric-bass-strings.jpg){width=250px}^[https://www.wallpaperflare.com/grayscale-photography-of-guitar-headstock-music-low-electric-bass-wallpaper-zzbyn]
![](https://p1.pxfuel.com/preview/870/881/120/mixer-music-audio-studio-sound-studio-sound-mixer.jpg){width=250px}


**Prompts:**

1. **Algorithms**

a. Why is $k$-fold cross-validation an *algorithm*?



b. What is the *tuning parameter* of this algorithm and what values can this take?






2. **Tuning the k-fold Cross-Validation algorithm**

Let's explore k-fold cross-validation with some personal experience. Our class has a representative sample of cards from a non-traditional population (no "face cards", not equal numbers, etc). We want to use these to predict whether a new card will be odd or even (a classification task).

a. Based on *all* of our cards, do we predict the next card will be odd or even?


b. You've been split into 2 groups. Use 2-fold cross-validation to estimate the possible error of using *our* sample of cards to predict whether a *new* card will be odd or even. How's this different than *validation*?


c. Repeat for 3-fold cross-validation. Why might this be better than 2-fold cross-validation?



d. Repeat for LOOCV, i.e. n-fold cross-validation where n is the number of students in this room. Why might this be worse than 3-fold cross-validation?



e. What value of k do you think practitioners typically use?



3. **R Code Preview**

We've been doing a 2-step process to build **linear regression models** using the **tidymodels** package:

```{r eval = FALSE}
# STEP 1: model specification
lm_spec <- linear_reg() %>%
  set_mode("regression") %>% 
  set_engine("lm")
  
# STEP 2: model estimation
my_model <- lm_spec %>% 
  fit(
    y ~ x1 + x2,
    data = sample_data
  )
```


For k-fold cross-validation, we can *tweak* STEP 2.

Discuss the code and why we need to set the seed.

```{r eval = FALSE}
# k-fold cross-validation
set.seed(___)
my_model_cv <- lm_spec %>% 
  fit_resamples(
    y ~ x1 + x2, 
    resamples = vfold_cv(sample_data, v = ___), 
    metrics = metric_set(mae, rsq)
  )
```









# Notes: R code {-}


Suppose we wish to build and evaluate a linear regression model of `y` vs `x1` and `x2` using our `sample_data`. 


**First, load the appropriate packages**

```{r eval = FALSE}
# Load packages
library(tidyverse)
library(tidymodels)
```




**Obtain k-fold cross-validated estimates of MAE and $R^2$**

(Review example 3 above for discussion of these steps.)


```{r eval = FALSE}
# model specification
lm_spec <- linear_reg() %>%
  set_mode("regression") %>% 
  set_engine("lm")

# k-fold cross-validation
# For "v", put your number of folds k
set.seed(___)
model_cv <- lm_spec %>% 
  fit_resamples(
    y ~ x1 + x2,
    resamples = vfold_cv(sample_data, v = ___), 
    metrics = metric_set(mae, rsq)
)
```





**Obtain the cross-validated metrics**

```{r eval = FALSE}
model_cv %>% 
  collect_metrics()
```






**Details: get the MAE and R-squared for each test fold**

```{r eval = FALSE}
# MAE for each test fold: Model 1
model_cv %>% 
  unnest(.metrics)
```








# Exercises

```{r message = FALSE, warning = FALSE}
# Load packages and data
library(tidyverse)
library(tidymodels)
humans <- read.csv("https://kegrinde.github.io/stat253_coursenotes/data/bodyfat50.csv") %>% 
  filter(ankle < 30) %>% 
  rename(body_fat = fatSiri)
```




## EXERCISE 1: Review -- In-sample metrics

Use the `humans` data to build two separate models of `height`:

```{r eval = FALSE}
# STEP 1: model specification
lm_spec <- ___() %>% 
  set_mode(___) %>% 
  set_engine(___)
```

```{r}

```


```{r eval = FALSE}
# STEP 2: model estimation
model_1 <- ___ %>% 
  ___(height ~ hip + weight + thigh + knee + ankle, data = humans)
model_2 <- ___ %>% 
  ___(height ~ chest * age * weight * body_fat + abdomen + hip + thigh + knee + ankle + biceps + forearm + wrist, data = humans)
```

```{r}

```





Calculate the **in-sample** R-squared for both models:

```{r eval = FALSE}
# IN-SAMPLE R^2 for model_1 = ???
model_1 %>% 
  ___()

# IN-SAMPLE R^2 for model_2 = ???
model_2 %>% 
  ___()
```

```{r}

```




Calculate the **in-sample** MAE for both models:

```{r eval = FALSE}
# IN-SAMPLE MAE for model_1 = ???
model_1 %>% 
  ___(new_data = ___) %>% 
  mae(truth = ___, estimate = ___)

# IN-SAMPLE MAE for model_2 = ???
model_2 %>% 
  ___(new_data = ___) %>% 
  mae(truth = ___, estimate = ___)
```

```{r}

```





\
\


## EXERCISE 2: In-sample model comparison

Which model seems "better" by the in-sample metrics you calculated above? Any concerns about either of these models?










\
\


## EXERCISE 3: 10-fold CV

Complete the code to run 10-fold cross-validation for our two models.

`model_1`: `height ~ hip + weight + thigh + knee + ankle`       
`model_2`: `height ~ chest * age * weight * body_fat + abdomen + hip + thigh + knee + ankle + biceps + forearm + wrist`

```{r eval = FALSE}
# 10-fold cross-validation for model_1
set.seed(253)
model_1_cv <- ___ %>% 
  ___(
    ___,
    ___ = vfold_cv(___, v = ___), 
    ___ = metric_set(mae, rsq)
  )
```

```{r}

```


```{r eval = FALSE}
# 10-fold cross-validation for model_2
set.seed(253)
model_2_cv <- ___ %>% 
  ___(
    ___,
    ___ = vfold_cv(___, v = ___), 
    ___ = metric_set(mae, rsq)
  )
```

```{r}

```
    


\
\



## EXERCISE 4: Calculating the CV MAE

a. Use `collect_metrics()` to obtain the cross-validated MAE and $R^2$ for both models.

```{r eval = FALSE}
# HINT
___ %>% 
  collect_metrics()
```
    
b. Interpret the cross-validated MAE *and* $R^2$ for `model_1`.    
    







\
\


## EXERCISE 5: Details -- fold-by-fold results

`collect_metrics()` gave the final CV MAE, or the average MAE across all 10 test folds. `unnest(.metrics)` provides the MAE from *each* test fold.

a. Obtain the fold-by-fold results for the `model_1` cross-validation procedure using `unnest(.metrics)`.       

```{r eval = FALSE}
# HINT
___ %>% 
  unnest(.metrics)
```

```{r}

```



b. Which fold had the worst average prediction error and what was it?



c. Recall that `collect_metrics()` reported a final CV MAE of 1.87 for `model_1`. Confirm this calculation by wrangling the fold-by-fold results from part a.



   
    
    
    


\
\





## EXERCISE 6: Comparing models

The table below summarizes the in-sample and 10-fold CV MAE for both models.    
    
Model        IN-SAMPLE MAE  10-fold CV MAE
----------- -------------- ---------------
`model_1`             1.55            1.87
`model_2`             0.64            2.47


a. Based on the in-sample MAE alone, which model appears better?    


b. Based on the CV MAE alone, which model appears better?    


c. Based on all of these results, which model would you pick?


d. Do the in-sample and CV MAE suggest that `model_1` is overfit to our `humans` sample data? What about `model_2`?





\
\


## EXERCISE 7: LOOCV

a. Reconsider `model_1`. Instead of estimating its prediction accuracy using the 10-fold CV MAE, use the LOOCV MAE. THINK: How many people are in our `humans` sample?



b. How does the LOOCV MAE compare to the 10-fold CV MAE of 1.87? NOTE: These are just two different *approaches* to estimating the same thing: the typical prediction error when applying our model to new data. Thus we should expect them to be similar.    



c. Explain why we technically don't *need* to `set.seed()` for the LOOCV algorithm.





\
\



## EXERCISE 8: Data drill

a. Calculate the average height of people under 40 years old vs people 40+ years old.

```{r}

```

b. Plot height vs age among our subjects that are 30+ years old.

```{r}

```

c. Fix this code:       

```{r eval = FALSE}
model_3<-lm_spec%>%fit(height~age,data=humans)
model_3%>%tidy()
```

```{r}

```





## EXERCISE 9: Reflection (Part 1)

The "regular" exercises are over but class is not done! Your group should agree to either work on HW2 OR the remaining reflection questions. 
    
This is the end of Unit 1 on "Regression: Model Evaluation"! Let's reflect on the technical content of this unit:

- What was the main motivation / goal behind this unit?

- What are the four main questions that were important to this unit?

- For each of the following tools, describe how they work and what questions they help us address:        
    - R-squared
    - residual plots
    - MAE
    - in-sample MAE
    - validation
    - cross-validation

- In your own words, define the following: overfitting, algorithm, tuning parameter.

- Review the new tidymodels syntax from this unit. Identify key themes and patterns.





\
\


## EXERCISE 10: Reflection (Part 2)

The reflection above addresses your **learning goals** in this course. Consider the other components that have helped you worked toward this learning throughout Unit 1.
    
a. With respect to **collaboration**, reflect upon your strengths and what you might change in the next unit:  
    
    - How actively did you contribute to group discussions?
    - How actively did you include all other group members in discussion?
    - In what ways did you (or did you not) help create a space where others feel comfortable making mistakes & sharing their ideas?

b. With respect to **engagement**, reflect upon your strengths and what you might change the next unit:
    
    - Did you regularly attend, be on time for, & stay for the full class sessions?
    - Have you not missed more than 3 in-person class sessions?
    - Were you actively present during class (eg: not on your phone, not working on other courses, etc)?
    - Did you stay updated on Slack?
    - When you had questions, did you ask them on Slack or in OH?

c. With respect to **preparation**, how many of checkpoints 1--3 did you complete and pass?

d. With respect to **exploration**, did you complete and pass HW1? Are you on track to complete and pass HW2?
    
  
    



\
\


## Done!

- Knit/render your notes.
- Check the solutions in the online manual.
- Check out the wrap-up steps online
- Work on your homework!



    