MLR Interpreting Effect Sizes

class: center, middle, inverse, title-slide

.title[
# MLR Interpreting Effect Sizes
]
.subtitle[
## Stat 21
]
.author[
### Suzanne Thornton
]
.institute[
### Swarthmore College
]

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## MLR Interpretation
### Example

Suppose we are interested in predicting the amount of time required by a driver to service soft drink vending machines in particular area. This service activity includes stocking the machine with beverage products and minor maintenance or house-keeping. 
 
We suspect that the two most important variables affecting the delivery time `$(Y)$` are the number of cases of product being stocked `$(x_1)$` and the distance the driver must walk to the machine `$(x_2)$`.

---
## MLR Interpretation
### Example

**Case 1:** All predictors are numerical.

`$$Y \mid (x_1, x_2) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \epsilon$$`

**Q1:** What is the interpretation of `$\beta_3$`?

--
**Q2:** What if we transform the response?

--
**Q3:**What if we transform a predictor?

--
The *interpretation* of all `$\beta$` coefficients involves assuming the values of the other predictor terms are not changing, or, that they are held at the same, constant value/level.

---
## MLR Interpretation
### Interaction Effects

Another relationship we may want to capture would allow for an interaction between two or more of the predictor variables.

**Examples**

- As automobiles age, the annual cost-per-mile-driven of keeping them in working order increases, i.e., the effect of mileage on maintenance cost depends on the age of a car.

- As workers become more experienced, their level of education becomes less of a factor in determining job performance, i.e., the effect of years-of-education on productivity depends on a worker’s experience.

- Advertising an upcoming sporting event has a more beneficial impact if the visiting team is one of the league leaders, i.e., the effect of advertising on ticket sales depends on the visitor’s league standing.

- Purchasers of condominiums in a high-rise will pay a premium to be on a lower floor if the condominium has a beach view, and will pay extra to be on an upper floor if the unit has an inland view, i.e., the effect of “floor number” on the market value of a condominium depends on the view from the unit.

.footnote[Source: https://www.kellogg.northwestern.edu/faculty/weber/emp/_session_2/interactions.htm]

---
## MLR Interpretation
### Interaction Effects

Another relationship we may want to capture would allow for an interaction between two or more of the predictor variables.

**Note** For two predictors to have an interacting effect on a response is not the same thing as one predictor variable being dependent on another predictor. E.g. If cars are selected at random (and not, say, just selected from a sample say of city employees only) then age and annual mileage will vary independently in the cars being studied, but the interaction between age and mileage would still be present.

.footnote[Source: https://www.kellogg.northwestern.edu/faculty/weber/emp/_session_2/interactions.htm]

---
## MLR Interpretation
### Interaction Effects

**Case 1:** All predictors are numerical.

`$$Y \mid (x_1, x_2, x_3) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_1 x_2 + \epsilon$$` 
This type of model suggests that the effect of one of the predictors on `$Y$` depends on the size of another predictor!

**Q:** What is the average effect on `$Y$` if `$x_1$` increases by one unit?

--
The answer depends on the value of `$x_2$`!

To see this, plug in `$x_1=0$` and simplify the regression model and then plug in `$x_1=1$` and simplify the regression model. Then look at the difference in these two equations.

---
## MLR Interpretation
### Interaction Effects

`$$Y \mid (x_1, x_2, x_3) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \epsilon$$`

**Case 2:** Some predictors are categorical and some are numerical.

Suppose `$x_1$` and `$x_2$` are indicator variables for a categorical predictor with three levels (A, B, C). Let `$x_1 = \begin{cases}1, \text{if Level A} \\ 0, \text{otherwise} \end{cases}$` and `$x_2 = \begin{cases}1, \text{if Level B} \\ 0, \text{otherwise} \end{cases}$`.

A model with an interaction between these two predictor variables (the numerical `$x_3$` and the categorical predictor with three levels) looks like

`$$Y \mid (x_1, x_2, x_3) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_1 x_3 + \beta_5 x_2 x_3 + \epsilon.$$`

This type of model suggests that the effect of `$x_3$` on `$Y$` depends on the level of the categorical variable.

--
Now the interpretation of holding other predictor terms at a constant value/level isn't necessarily possible!

---
## MLR Interpretation
### Interaction Effects

**Case 2:** Some predictors are categorical and some are numerical.

A model with an interaction between these two predictor variables (the numerical `$x_3$` and the categorical predictor with three levels) looks like

`$$Y \mid (x_1, x_2, x_3) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_1 x_3 + \beta_5 x_2 x_3 + \epsilon.$$`

**Q:** What is the average effect on `$Y$` if `$x_3$` increases by one unit?

--
The answer depends on the values of `$x_1$` and `$x_2$`!

To see this, plug in `$x_3=0$` and simplify the regression model and then plug in `$x_3=1$` and simplify the regression model. Then look at the difference in these two equations.

---
## MLR Interpretation
### Interaction Effects

**Case 2:** Some predictors are categorical and some are numerical.

A model with an interaction between these two predictor variables (the numerical `$x_3$` and the categorical predictor with three levels) looks like

`$$Y \mid (x_1, x_2, x_3) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_1 x_3 + \beta_5 x_2 x_3 + \epsilon.$$`

**Q:** What is the average effect on `$Y$` changing from Level C to Level A?

--
The answer depends on the value of `$x_3$`!

Again, to see this, write out the regression model for an individual from Level C and then write out the regression model for an individual from Level A. Then look at the difference in these two equations.

---
## MLR Interpretation
### Group worksheet

```r
library("tidyverse")
mileage_dat <- read_table("~/GitHub/Stat21/Data/mileage.txt", 
                       skip=2, 
                       col_names = c("car", "mpg", "displacement", "weight", "transmission_type"), 
                       cols(transmission_type = col_factor()))  %>% na.omit
## Note that transmission_type is already a factor! 
head(mileage_dat)
```

```
## # A tibble: 6 × 5
##   car          mpg displacement weight transmission_type
##   <chr>      <dbl>        <dbl>  <dbl> <fct>            
## 1 Apollo      18.9          350   3910 A                
## 2 Omega       17            350   2860 A                
## 3 Nova        20            250   3510 A                
## 4 Monarch     18.2          351   3890 A                
## 5 Duster      20.1          225   3365 M                
## 6 JensonConv  11.2          440   4215 A
```

---
## MLR Interpretation
### Group worksheet

**Model 1:** Fit a main effects model with transmission type (A as the reference level) and weight as the predictors and mileage as the response:

`$$\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 * \text{weight} + \hat{\beta}_2 * \text{transmission_typeM},$$`

where `$weight=$` the weight of the vehicle and 
`$$\text{transmission_typeM} = \begin{cases}1, \quad \text{if transmission type is manual}\\ 0, \quad \text{otherwise} \end{cases}.$$`

```r
mod1 <- lm(mpg ~ weight + transmission_type, mileage_dat)
summary(mod1)$coefficients[,1]
```

```
##        (Intercept)             weight transmission_typeM 
##       34.205671014       -0.004226654        3.715761765
```

---
## MLR Interpretation
### Group worksheet

**Model 2:** Fit a main effects model with transmission type (M as the reference level) and weight as the predictors and mileage as the response:

`$$\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 * \text{weight} + \hat{\beta}_2 * \text{transmission_typeA},$$`
  where `$weight=$` the weight of the vehicle and 
`$$\text{transmission_typeA} = \begin{cases}1, \quad \text{if transmission type is automatic}\\ 0, \quad \text{otherwise} \end{cases}.$$`

```r
mod2 <- lm(mpg ~ weight + relevel(transmission_type, "M"), mileage_dat)
summary(mod2)$coefficients[,1]
```

```
##                      (Intercept)                           weight 
##                     37.921432779                     -0.004226654 
## relevel(transmission_type, "M")A 
##                     -3.715761765
```

---
## MLR Interpretation
### Group worksheet

**Model 3:** Fit an interaction effects model with transmission type (A as the reference level) and weight as the predictors and mileage as the response:

`$$\hat{y} = \hat{\beta}_0 + (\hat{\beta}_1 * \text{weight}) + (\hat{\beta}_2 * \text{transmission_typeM}) + (\hat{\beta}_4 * \text{weight} * \text{transmission_typeM}),$$`
  where 
`$$\text{transmission_typeM} = \begin{cases}1, \quad \text{if transmission type is manual}\\ 0, \quad \text{otherwise} \end{cases}.$$`

```r
mod3 <- lm(mpg ~ weight + transmission_type + weight:transmission_type, mileage_dat)
summary(mod3)$coefficients[,1]
```

```
##               (Intercept)                    weight        transmission_typeM 
##              29.453069625              -0.003036693              28.655350372 
## weight:transmission_typeM 
##              -0.009480683
```

---
## MLR Interpretation
### Group worksheet

**Model 3:** Fit an interaction effects model with transmission type (A as the reference level) and weight as the predictors and mileage as the response:

`$$\hat{y} = \hat{\beta}_0 + (\hat{\beta}_1 * \text{weight}) + (\hat{\beta}_2 * \text{transmission_typeM}) + (\hat{\beta}_4 * \text{weight} * \text{transmission_typeM}),$$`
`$$\text{transmission_typeM} = \begin{cases}1, \quad \text{if transmission type is manual}\\ 0, \quad \text{otherwise} \end{cases}.$$`

```r
mod3 <- lm(mpg ~ weight + transmission_type + weight:transmission_type, mileage_dat)
summary(mod3)$coefficients[,1]
```