Week 07 - Regression

# Week 07 - Regression
## Model Fit, LM for Randomised Experiments, Multiple Predictors, Merging Data Sets in R
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### Danilo Freire
### 8th March 2019

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# Today's Agenda

* How to assess model fit with `$R^2$`

* Analyse randomised experiments with `lm()`

* Merge data sets with `merge(x, y, by = )`
]
---

# Review

* A few things to remember:

* .orange[Regression equation]: `$Y = \alpha + \beta X + \epsilon$`
  - `$\alpha$` is the intercept (value of `$Y$` when `$X = 0$`)
  - `$\beta$` is the slope (increase/decrease in `$Y$` when `$X$` varies by one unit)
  - `$\epsilon$` is the error term (difference from points to fitted line; mean zero)
]
---

# Linear Models

```r
bivariate <- read.csv("https://raw.githubusercontent.com/pols1600/pols1600.github.io/master/datasets/prediction/bivariate_data.csv")
bivariate <- subset(bivariate, Year == 2010)
model.cm <- lm(Child.Mortality ~ log(GDP), data = bivariate)
model.cm
```

```
## 
## Call:
## lm(formula = Child.Mortality ~ log(GDP), data = bivariate)
## 
## Coefficients:
## (Intercept)     log(GDP)  
##      276.58       -26.13
```
* `$Y = 276.58 - 26.13X + \epsilon$`
]
---

# R-Squared

* `$R^2$`, _coefficient of determination_

`$$R^2 = 1 - \frac{\textsf{SSR}}{\textsf{Total sum of squares (TSS)}}  \ = \ 1 - \frac{\sum_{i=1}^n \hat\epsilon_i^2}{\sum_{i=1}^n (Y_i - \overline{Y})^2}$$`
* `$R^2$` is also defined as the _explained variance_ in Y

* Goes from 0 to 1

* We can use `summary(model)` to see it
]
---

# Example

```r
summary(model1)
```

```
## 
## Call:
## lm(formula = Child.Mortality ~ log(GDP), data = bivariate)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -46.733 -15.699  -4.320   6.974 133.772 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  276.582     15.534   17.80   <2e-16 ***
## log(GDP)     -26.127      1.691  -15.45   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 26.78 on 168 degrees of freedom
##   (21 observations deleted due to missingness)
*## Multiple R-squared:  0.587,	Adjusted R-squared:  0.5845 
## F-statistic: 238.7 on 1 and 168 DF,  p-value: < 2.2e-16
```
]
---

# Caveats

* `$R^2$` .orange[does not] indicate whether:
  - _the independent variable causes the dependent variable_
  - the model is correctly specified 
  - the model suffers from ommitted variable bias
  - there are enough data points to make a valid conclusion
]
---

# Linear Models and RCTs

.font150[
* .orange[When the data come from a randomised experiment,] model parameters have a causal interpretation

* Treatment status as the independent variable (0 or 1)
  - 0 indicates control group
  - 1 indicates treatment group
]
--
.font150[
* `$Y = \alpha + \beta \times treatment + \epsilon$`
]
--
.font150[
* What is the interpretation of `$\alpha$` here?

* What is the interpretation of `$\beta$` here?
]
---

# Women as Policy Makers

* Observational studies: compare policies adopted by female politicians with those adopted by male politicians

* Randomised natural experiment:
  - one third of village council heads reserved for women
  - assigned at the level of Gram Panchayat (village council) since mid-1990s
  - each GP has multiple villages

* Hypothesis: female politicians represent the interests of female voters

* Female voters complain about drinking water while male voters complain about irrigation
]
---

# Data

| Name         | Description                                                                                                                |
| :----------- | :------------------------------------------------------------------------------------------------------------------------- |
| `GP`         | An identifier for the Gram Panchayat (village council)                                                                     |
| `village`    | identifier for each village                                                                                                |
| `reserved`   | binary variable indicating whether the GP was reserved for women leaders or not                                            |
| `female`     | binary variable indicating whether the GP had a female leader or not                                                       |
| `irrigation` | variable measuring the number of new or repaired irrigation facilities in the village since the reserve policy started     |
| `water`      | variable measuring the number of new or repaired drinking-water facilities in the village since the reserve policy started |

```r
women <- read.csv("https://raw.githubusercontent.com/pols1600/pols1600.github.io/master/datasets/prediction/women.csv")
names(women)
```

```
## [1] "GP"         "village"    "reserved"   "female"     "irrigation"
## [6] "water"
```
---

# Models

```r
tapply(women$female, women$reserved, mean)
```

```
##          0          1 
## 0.07476636 1.00000000
```

* Does it change the policy outcomes?

```r
# drinking-water facilities
tapply(women$water, women$reserved, mean)
```

```
##        0        1 
## 14.73832 23.99074
```

```r
# irrigation facilities
tapply(women$irrigation, women$reserved, mean)
```

```
##        0        1 
## 3.387850 3.018519
```
]
---

# Slope Coefficient = Difference-in-Means

```r
tapply(women$water, women$reserved, mean)
```

```
##        0        1 
*## 14.73832 23.99074
```

```r
mean(women$water[women$reserved == 1]) - mean(women$water[women$reserved == 0])
```

```
*## [1] 9.252423
```

```r
lm(water ~ reserved, data = women)
```

```
## 
## Call:
## lm(formula = water ~ reserved, data = women)
## 
## Coefficients:
## (Intercept)     reserved  
*##      14.738        9.252
```
]
---

# Resume Experiment Revisited

```r
resume <- read.csv("https://raw.githubusercontent.com/pols1600/pols1600.github.io/master/datasets/causality/resume.csv")
model1 <- lm(call ~ race, data = resume)
model1
```

```
## 
## Call:
## lm(formula = call ~ race, data = resume)
## 
## Coefficients:
## (Intercept)    racewhite  
##     0.06448      0.03203
```

```r
tapply(resume$call, resume$race, mean)
```

```
##      black      white 
## 0.06447639 0.09650924
```

```r
# baseline (black) + the effect of having a white-sounding name
0.06448 + 0.03203
```

```
## [1] 0.09651
```
]
---

# Call Rates and Gender

```r
model2 <- lm(call ~ sex, data = resume)
model2
```

```
## 
## Call:
## lm(formula = call ~ sex, data = resume)
## 
## Coefficients:
## (Intercept)      sexmale  
##    0.082488    -0.008645
```

```r
tapply(resume$call, resume$sex, mean)
```

```
##     female       male 
## 0.08248799 0.07384342
```
]
---

# Race + Gender

```r
model3 <- lm(call ~ race + sex, data = resume)
model3
```

```
## 
## Call:
## lm(formula = call ~ race + sex, data = resume)
## 
## Coefficients:
## (Intercept)    racewhite      sexmale  
##    0.066534     0.032130    -0.009128
```

```r
# Call rates for a white male
 0.066534 + 0.032130 - 0.009128
```

```
## [1] 0.089536
```
* Regression Equation: `$Y = 0.066 + 0.032*white - 0.009*male + \epsilon$`
]
---

# Interpreting Multiple Predictors

* Let's interpret the coefficient for _white_ in `$Y = 0.066 + 0.032 \times white - 0.009 \times male + \epsilon$`

* We say: "_all else equal, having a white-sounding name increases the change of getting a job call in about 3%. Since candidates with black-sounding names a job call rate of about 6%, candidates with white-sounding names are 50% more likely to get a call_"
]

---

# Adjusted R-Squared

.font140[
* When we have more than one independent variable, we need to modify the `$R^2$` formula to account for those additional variables

* `$R^2$` measures the overall impact of _all_ variables at once, but some might just add noise to the model

* Every predictor added to a model increases `$R^2$` and never decreases it

* Adjusted `$R^2$` compensates for the addition of variables and only increases _if the new term enhances the model_

* It is usually lower than regular `$R^2$` but not much
]
---

# Adjusted R-Squared

```r
summary(model.cm)
```

---

# Merging Data Sets in R

* If the data refer to the same units/indiviuals, .orange[it is essential that both data sets have the same identification variable]

* Use `merge(x, y, by =)` function
]

```r
pres2008 <- read.csv("https://raw.githubusercontent.com/pols1600/pols1600.github.io/master/datasets/prediction/pres08.csv")
names(pres2008)
```

```
## [1] "state.name" "state"      "Obama"      "McCain"     "EV"
```

```r
intrade08 <- read.csv("https://raw.githubusercontent.com/pols1600/pols1600.github.io/master/datasets/prediction/intrade08.csv")
names(intrade08)
```

```
## [1] "X"         "day"       "statename" "PriceD"    "VolumeD"   "PriceR"   
## [7] "VolumeR"   "state"
```

```r
*pres.merged <- merge(pres2008, intrade08, by = "state")
```
---

# Merging Data Sets in R

```r
names(pres.merged)
```

```
##  [1] "state"      "state.name" "Obama"      "McCain"     "EV"        
##  [6] "X"          "day"        "statename"  "PriceD"     "VolumeD"   
## [11] "PriceR"     "VolumeR"
```

```r
head(pres.merged)
```

```
##   state state.name Obama McCain EV     X        day statename PriceD
## 1    AK     Alaska    38     59  3 24073 2008-02-28    Alaska    7.5
## 2    AK     Alaska    38     59  3 25195 2008-03-21    Alaska    7.5
## 3    AK     Alaska    38     59  3  8773 2007-05-04    Alaska   10.0
## 4    AK     Alaska    38     59  3 25603 2008-03-29    Alaska    8.0
## 5    AK     Alaska    38     59  3 25246 2008-03-22    Alaska    7.5
## 6    AK     Alaska    38     59  3 23665 2008-02-20    Alaska    7.5
##   VolumeD PriceR VolumeR
## 1       0   92.5       0
## 2       0   92.5       0
## 3       0   90.0       0
## 4       0   92.0       0
## 5       0   92.5       0
## 6       0   92.5       0
```
]
---

# Questions?

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# Homework

# See you!