Confounding and Variable Transformations

# Confounding and Variable Transformations

---

<div class="my-footer">
<span>
by Dr. Lucy D'Agostino McGowan
</span>
</div>

---

## Adjusting for confounders

* What is the relationship between average SAT scores and average teacher salaries?

.small[
<div id="htmlwidget-bb5ee367d3dda39d3eca" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-bb5ee367d3dda39d3eca">{"x":{"filter":"none","data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50"],["Alabama","Alaska","Arizona","Arkansas","California","Colorado","Connecticut","Delaware","Florida","Georgia","Hawaii","Idaho","Illinois","Indiana","Iowa","Kansas","Kentucky","Louisiana","Maine","Maryland","Massachusetts","Michigan","Minnesota","Mississippi","Missouri","Montana","Nebraska","Nevada","New Hampshire","New Jersey","New Mexico","New York","North Carolina","North Dakota","Ohio","Oklahoma","Oregon","Pennsylvania","Rhode Island","South Carolina","South Dakota","Tennessee","Texas","Utah","Vermont","Virginia","Washington","West Virginia","Wisconsin","Wyoming"],[4.405,8.963,4.778,4.459,4.992,5.443,8.817,7.03,5.718,5.193,6.078,4.21,6.136,5.826,5.483,5.817,5.217,4.761,6.428,7.245,7.287,6.994,6,4.08,5.383,5.692,5.935,5.16,5.859,9.774,4.586,9.623,5.077,4.775,6.162,4.845,6.436,7.109,7.469,4.797,4.775,4.388,5.222,3.656,6.75,5.327,5.906,6.107,6.93,6.16],[17.2,17.6,19.3,17.1,24,18.4,14.4,16.6,19.1,16.3,17.9,19.1,17.3,17.5,15.8,15.1,17,16.8,13.8,17,14.8,20.1,17.5,17.5,15.5,16.3,14.5,18.7,15.6,13.8,17.2,15.2,16.2,15.3,16.6,15.5,19.9,17.1,14.7,16.4,14.4,18.6,15.7,24.3,13.8,14.6,20.2,14.8,15.9,14.9],[31.144,47.951,32.175,28.934,41.078,34.571,50.045,39.076,32.588,32.291,38.518,29.783,39.431,36.785,31.511,34.652,32.257,26.461,31.972,40.661,40.795,41.895,35.948,26.818,31.189,28.785,30.922,34.836,34.72,46.087,28.493,47.612,30.793,26.327,36.802,28.172,38.555,44.51,40.729,30.279,25.994,32.477,31.223,29.082,35.406,33.987,36.151,31.944,37.746,31.285],[8,47,27,6,45,29,81,68,48,65,57,15,13,58,5,9,11,9,68,64,80,11,9,4,9,21,9,30,70,70,11,74,60,5,23,9,51,70,70,58,5,12,47,4,68,65,48,17,9,10],[491,445,448,482,417,462,431,429,420,406,407,468,488,415,516,503,477,486,427,430,430,484,506,496,495,473,494,434,444,420,485,419,411,515,460,491,448,419,425,401,505,497,419,513,429,428,443,448,501,476],[538,489,496,523,485,518,477,468,469,448,482,511,560,467,583,557,522,535,469,479,477,549,579,540,550,536,556,483,491,478,530,473,454,592,515,536,499,461,463,443,563,543,474,563,472,468,494,484,572,525],[1029,934,944,1005,902,980,908,897,889,854,889,979,1048,882,1099,1060,999,1021,896,909,907,1033,1085,1036,1045,1009,1050,917,935,898,1015,892,865,1107,975,1027,947,880,888,844,1068,1040,893,1076,901,896,937,932,1073,1001]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>state<\/th>\n      <th>expend<\/th>\n      <th>ratio<\/th>\n      <th>salary<\/th>\n      <th>frac<\/th>\n      <th>verbal<\/th>\n      <th>math<\/th>\n      <th>sat<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"pageLength":5,"columnDefs":[{"className":"dt-right","targets":[2,3,4,5,6,7,8]},{"orderable":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>
]

--
* Are we doing inference or prediction?

---

## Adjusting for confounders

* I fit a linear model for `\(\hat{sat} = \hat\beta_0 + \hat\beta_1 salary\)`

```
## # A tibble: 2 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)  1159.       57.7      20.1  5.13e-25
## 2 salary         -5.54      1.63     -3.39 1.39e- 3
```

* How do we interpret this result?

---

## Adjusting for confounders

* There is a **third variable**, the fraction of students that took the SAT in that state. It is grouped as "Low", "Medium", and, "High".

```
## # A tibble: 4 x 5
##   term          estimate std.error statistic  p.value
##   <chr>            <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)     852.      38.9       21.9  5.56e-26
## 2 salary            1.09     0.988      1.10 2.76e- 1
## 3 frac_groupLOW   150.      12.8       11.7  2.09e-15
## 4 frac_groupMED    38.6     14.1        2.75 8.59e- 3
```

* What is the referent category?
--

* How do you interpret the `\(\hat{\beta}\)` for `frac_groupLOW`?
--

* How do you interpret the `\(\hat{\beta}\)` for `salary` now?

---

## `\(\hat\beta\)` interpretation in multiple linear regression

The coefficient for `\(x\)` is `\(\hat\beta\)` (95% CI: `\(LB_\hat\beta, UB_\hat\beta\)`). A one-unit increase in `\(x\)` yields an expected increase in y of `\(\hat\beta\)`, **holding all other variables constant**.

---

## `\(\hat\beta\)` interpretation in multiple linear regression

The coefficient for average salary is 1.09 (95% CI: -0.90, 3.08).  A one-unit increase in average salary yields an expected increase in average SAT score of 1.09, **holding the fraction of students that took the SAT constant**.

---

## Adjusting for confounders

![](09-variable-transformations_files/figure-html/unnamed-chunk-5-1.png)

---

## Adjusting for confoundrs

![](09-variable-transformations_files/figure-html/unnamed-chunk-6-1.png)

* What is this called? Where the direction reverses?
--

* Notice here the lines are **parallel** so holding the group constant, this is the effect we see.
--

* 😱 what if the lines aren't parallel?
---

## Interactions

* Data looking at the growth rate for kids

.small[
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]

---

## Interactions

![](09-variable-transformations_files/figure-html/unnamed-chunk-8-1.png)

* Will `\(\hat\beta_{age}\)` be positive or negative?

---

## Interactions

* Let's look at this relationship split by `sex` (blue: Girl, black: Boy)

![](09-variable-transformations_files/figure-html/unnamed-chunk-9-1.png)

---

## Interactions

* Let's look at this relationship split by `sex` (blue: Girl, black: Boy)

![](09-variable-transformations_files/figure-html/unnamed-chunk-10-1.png)

* 😱 the lines cross! That means there is an **interaction**, that is the slopes differ based on the group

---

## Interactions

* Let's look at this relationship split by `sex` (blue: Girl, black: Boy)

![](09-variable-transformations_files/figure-html/unnamed-chunk-11-1.png)

* What is the equation for this relationship?

---

## Interactions

`\(Weight = \beta_0 + \beta_1 Age + \beta_2 Girl + \beta_3 Age \times Girl + \epsilon\)`

```r
lm(Weight ~ Age + Sex + Age * Sex, data = Kids198)
```

```
## 
## Call:
## lm(formula = Weight ~ Age + Sex + Age * Sex, data = Kids198)
## 
## Coefficients:
## (Intercept)          Age          Sex      Age:Sex  
##    -33.6925       0.9087      31.8506      -0.2812
```
]

* What does this model become for **boys** (When `Sex = 0`)

--
  * `\(Weight = \beta_0 + \beta_1 Age + \epsilon\)`

--
* What does this model become for **girls** (When `Sex = 1`)

--
  * `\(Weight = \beta_0 + \beta_1 Age + \beta_2 1 + \beta_3 Age \times 1 + \epsilon\)`

--
  * `\(Weight = (\beta_0 + \beta_2) + (\beta_1 + \beta_3) Age  + \epsilon\)`

--
* How do you interpret `\(\hat\beta_0\)` now?

---

## Interactions

```r
lm(Weight ~ Age + Sex + Age * Sex, data = Kids198)
```

* What does this model become for **boys** (When `Sex = 0`)
  * `\(Weight = \beta_0 + \beta_1 Age + \epsilon\)`
* What does this model become for **girls** (When `Sex = 1`)
  * `\(Weight = \beta_0 + \beta_1 Age + \beta_2 1 + \beta_3 Age \times 1 + \epsilon\)`
  * `\(Weight = (\beta_0 + \beta_2) + (\beta_1 + \beta_3) Age  + \epsilon\)`
* How do you interpret `\(\hat\beta_{2}\)` now?

--
  * The difference in intercepts between boys and girls

---

## Interactions

```r
lm(Weight ~ Age + Sex + Age * Sex, data = Kids198)
```

--
  * How much the slope changes as we move from the regression line for boys to that for girls

---

## Interactions

`\(Weight = \beta_0 + \beta_1 Age + \beta_2 Girl + \beta_3 Age \times Girl + \epsilon\)`

* Hypothesis testing: What if you want to test whether the slope is different between groups?
* Is the growth rate different for boys and girls?
* What is `\(H_0\)`?

--
  * `\(H_0: \beta_3 = 0\)`

--
* What is `\(H_A\)`?

--
  * `\(H_A:\beta_3 \neq 0\)`
  
---

## Interactions

```r
lm(Weight ~ Age + Sex + Age * Sex, data = Kids198) %>%
  tidy(conf.int = TRUE)
```

```
## # A tibble: 4 x 7
##   term        estimate std.error statistic  p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept)  -33.7     10.0        -3.37 9.17e- 4  -53.4     -14.0  
## 2 Age            0.909    0.0611     14.9  6.47e-34    0.788     1.03 
## 3 Sex           31.9     13.2         2.41 1.71e- 2    5.73     58.0  
*## 4 Age:Sex       -0.281    0.0816     -3.44 7.00e- 4   -0.442    -0.120
```
]

* What is the result of our hypothesis test?

---

## `\(\hat\beta\)` interpretation for interactions between `\(x\)` and a binary indicator `\(I\)`

The coefficient for the interaction between `\(x\)` and `\(I\)` is `\(\hat\beta\)` (95% CI: `\(LB_\hat\beta, UB_\hat\beta\)`). This means that the effect of `\(x\)` on `\(y\)` differs by `\(\hat\beta\)` when `\(I = 1\)` compared to `\(I = 0\)` **holding all other variables constant***.

--
* You must include this line if there are **additional variables in your model**.

---

## `\(\hat\beta\)` interpretation for interactions between `\(x\)` and a binary indicator `\(I\)`

The coefficient for the interaction between `Age` and `Sex` is -0.28 (95% CI: -0.44, -0.12). This means that the effect of `Age` on `Weight` lower by 0.28 among girls compared to boys.

---

## Non-linear relationships

* Sometimes the relationships between the outcome `\(y\)` and `\(x\)` variables are _nonlinear_. 
* We can use _polynomials_ to address this!
* Returning to the **Diamonds** data, let's say we are interested in predicting Total Price from the Carats.

--
  * Is this an example of inference or prediction?

---

## Non-linear relationships

![](09-variable-transformations_files/figure-html/unnamed-chunk-16-1.png)

---

## Non-linear relationships

```r
lm(TotalPrice ~ Carat, data = Diamonds)
```

![](09-variable-transformations_files/figure-html/unnamed-chunk-18-1.png)

---

## Non-linear relationships

```r
lm(TotalPrice ~ Carat + I(Carat^2), data = Diamonds)
```

![](09-variable-transformations_files/figure-html/unnamed-chunk-20-1.png)

* What is the equation for this relationship?

---

## Interpreting `\(\hat\beta\)`s in the presence of polynomials

`\(Total Price = \beta_0 + \beta_1 Carat + \beta_2 Carat^2 + \epsilon\)`

* What is the interpretation of `\(\hat\beta_1\)`?

--
* Typically, in multiple linear regression, the interpretation of `\(\hat\beta_i\)` is:
a one-unit change in `\(x\)` yields an expected change in `\(y\)` of `\(\hat\beta_i\)` **holding all other variables constant**.

--
  * What does it mean to see a change in `Caret` holding `Carat` `\(^2\)` constant?

--
* When you have a polynomial term, you need to **specify the values you are changing between**, since the change is no longer constant across all values of `\(x\)`.

---

## Interpreting `\(\hat\beta\)` in the presence of polynomials
.small[

```r
lm(TotalPrice ~ Carat + I(Carat^2), data = Diamonds) %>%
  tidy()
```

```
## # A tibble: 3 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    -523.      466.     -1.12 2.63e- 1
## 2 Carat          2386.      753.      3.17 1.66e- 3
## 3 I(Carat^2)     4498.      263.     17.1  5.09e-48
```
]

What is the expected change in `TotalPrice` for a one-unit change in `Carat`, changing from 0.8 to 1.8?

```r
(-522.7 + 2386 * 1.8 + 4498.2 * 1.8^2) - 
  (-522.7 + 2386 * 0.8 + 4498.2 * 0.8^2)
```

```
## [1] 14081.32
```
]
]

```r
2386 * (1.8 - 0.8) + 
  4498.2 * (1.8^2 - 0.8^2)
```

```
## [1] 14081.32
```
]
]

---

## Interpreting `\(\hat\beta\)` in the presence of polynomials
.small[

```r
lm(TotalPrice ~ Carat + I(Carat^2), data = Diamonds) %>%
  tidy()
```

What is the expected change in `TotalPrice` for a one-unit change in `Carat`, changing from 1.8 to 2.8?

```r
2386 * (2.8 - 1.8) + 4498.2 * (2.8^2 - 1.8^2)
```

```
## [1] 23077.72
```
]

--
* Can we talk about `\(\hat\beta_1\)` and `\(\hat\beta_2\)` in the context of a one-unit change in `Carat`?

---
## Interpreting `\(\hat\beta\)` in the presence of polynomials

* `\(\hat\beta\)` coefficients that are transformations of the same `\(x\)` variable **must** be interpreted together

--
* You must first choose to values of `\(x\)` to change between, and then report the change.

--
* A sensible choice for the two `\(x\)` values can be the 25th% quantile and the 75th% quantile.

---

## General `\(\hat\beta\)` interpretation with quadratic terms

The linear term in the model for `\(x\)` has a coefficient of `\(\hat\beta_1\)` (95% CI: `\((LB_{\hat\beta_1}, UB_{\hat\beta_1})\)`). The quadratic term in the model for `\(x\)` has a coefficient of `\(\hat\beta_2\)` (95% CI: `\((LB_{\hat\beta_2}, UB_{\hat\beta_2})\)`). A change in `\(x\)` from `\(a\)` to `\(b\)` yields an expected change in `\(y\)` of `\(\hat\beta_1 (b - a) + \hat\beta_2 (b^2 - a^2)\)` **holding all other variables constant***.

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* You must include this line if there are **additional variables in your model**.

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## Specific `\(\hat\beta\)` interpretation for `\(y = \beta_0 + \beta_1 Carat + \beta_2 Carat^2 + \epsilon\)` model

The linear term in the model for `Carat` has a coefficient of 2386 (95% CI: `\((906, 3866)\)`). The quadratic term in the model for `Carat` has a coefficient of `\(4498\)` (95% CI: `\((3981, 5016)\)`). A change in `Carat` from `\(0.7\)` to `\(1.24\)` yields an expected change in `TotalPrice` of `\(6000.5\)`.

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* Why didn't I say **holding all other variables constant?**

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## Take aways

* The interpretation of `\(\hat\beta\)` in multiple linear regression
  * A one-unit change in `\(x\)` yields an expected change in `\(y\)` of `\(\hat\beta\)` **holding all other included variables constant**

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* If the _slope differs_ between groups (the lines cross in a scatterplot), an **interaction** is present

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* You can include polynomial terms to address **non-linear** relationships

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  * The coefficients for a polynomial must be interpreted together
  
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## <i class="fas fa-laptop"></i> `Diamonds`

- Go to RStudio Cloud and open `Diamonds`
- Fit the model  `\(TotalPrice = \beta_0  + \beta_1Carat + \beta_2 Carat^2 + \beta_3 Color+\epsilon\)`
- Find the 0.25 quantile and 0.75 quantile of `Carat`
- What is the interpretation of `\(\hat\beta_1\)`, `\(\hat\beta_2\)`, and `\(\hat\beta_3\)`?