Linear Models (1)

class: center, middle, inverse, title-slide

# Linear Models (1)
### Yue Jiang
### Duke University

---

### A quick survey

https://duke.qualtrics.com/jfe/form/SV_cBmbywxmqDYyvhI

---

### Stop-question-frisk

.small[Photo - Adapted from Kena Betancur, Agence France-Presse/Getty Images]

- Police protocol intended to reduce crime by stopping, questioning, and 
searching civilians
- Instituted in early 2000s, peaking in 2011 with almost 700,000 stops
- Fraught with controversy - many assert unfair targeting of Black and Hispanic
citizens

---

### Stop-question-frisk

- Gelman, Fagan, and Kiss (JASA, 2007) found evidence of racial disparities,
even after adjusting for potential confounders
- A 2013 class-action lawsuit determined SQF was being used unconstitutionally; 
number of events fell sharply, with only around 10,000 stops per year from 
2016 onwards

---

### Writing a model

```r
# Note: you'll have to do the data cleaning yourself...
# Keep it clear and reproducible!

sqf <- read_csv("dat/sqf_cleaned.csv")
```

```r
table(sqf$build_clean)
```

```
## 
##   HEA   MED OTHER   THN 
##  3085 12443  1354 16310
```

```r
table(sqf$any_weapon)
```

```
## 
##     0     1 
## 30096  3096
```

---

### Writing a model

```r
summary(sqf$stop_duration_min)
```

```
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    0.00    5.00    8.00   11.75   14.00  999.00
```

```r
summary(sqf$suspect_age)
```

```
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    0.00   19.00   25.00   28.48   35.00   99.00
```

---

### Writing a model

.question[
Let's ignore model assumption issues and diagnostics for now. Suppose you wanted
to create a linear model that related the stop duration of an SQF event to the
build of the suspect, whether they had a weapon on their person, and their age.
How would you express such a model mathematically?
]

---

### What could go wrong?

.question[
What issue(s) do you see with the model below? (Again, ignore issues of whether
the model itself is appropriate; assume we just want a linear model relating
the outcome of stop duration to the three predictors)
]

`\begin{align*}
duration = build + weapon + age
\end{align*}`

---

### What could go wrong?

.question[
How about here?
]

`\begin{align*}
duration = \beta_1 build + \beta_2 weapon + \beta_3 age
\end{align*}`

---

### What could go wrong?

.question[
Or here?
]

`\begin{align*}
duration &= \beta_1 I(build = HEA) + \beta_2 I(build = MED) + \\
&\mathrel{\phantom{=}} \beta_3 I(build = OTHER) + \beta_4 weapon + \\
&\mathrel{\phantom{=}} \beta_5 age
\end{align*}`

---

### What could go wrong?

.question[
Are we finished?
]

`\begin{align*}
duration_i &= \beta_0 + \beta_1 I(build_i = HEA) + \beta_2 I(build_i = MED) + \\
&\mathrel{\phantom{=}} \beta_3 I(build_i = OTHER) + \beta_4 weapon_i + \\ 
&\mathrel{\phantom{=}} \beta_5 age_i
\end{align*}`

---

### What could go wrong?

.question[
What are the parameters in the model written below? How can we write it 
succintly in matrix notation?
]

where

`\begin{align*}
\epsilon_i \stackrel{i.i.d.}{\sim} N(0, \sigma^2)
\end{align*}`

---

### Another model

.question[
Say instead of duration being the outcome variable, suppose we're interested in
the log-odds of having a "long" SQF event (an event lasting over half an hour). 
If we fit a logistic regression model for this question, what would our model 
look like?
]

```r
sqf %>% 
  mutate(long_sqf = ifelse(stop_duration_min > 30, 1, 0)) %>% 
  count(long_sqf)
```

```
##   long_sqf     n
## 1        0 31731
## 2        1  1461
```

---

### A Bayesian model

.question[
Let's return to our original OLS model. Suppose we wanted to fit a Bayesian
linear regression model. How would you formulate it? What priors on the 
parameters might make sense?
]

---

### References

[1] Stop, Question, and Frisk Data, 2003 - 2019. Accessed on NYPD website.

[2] Gelman A, Fagan J, Kiss A (2012). "Stop-and-frisk policy in the context of 
claims of racial bias." Journal of the American Statistical Association. 
102(479): 813 - 823.

[3] Floyd, et al. v. City of New York, et al. United States District Court for
the Southern District of New York. 959 F. Supp. 2d 540 (2013).

[4] Mummolo, J (2018). "Modern Police Tactics, Police-Citizen Interactions, and 
the Prospects for Reform". The Journal of Politics. 80: 1–15.