class: center, middle

# Modeling

.large[Kelly McConville | Math 141 | Week 4 | Fall 2020]

---

## Announcements/Reminders

* Project Assignment 1 is due on Friday October 2nd (end of day) on Gradescope.

* After your lab this week, you should get started on Lab 4.
    + No coding, just narrative.

* Quizzes: We have decided not to include quizzes as a form of assessment for the course.

---

## Week 4 Topics

* Finish up a couple more Data Wrangling examples

* Data collection

* **Modeling**

---

# Goals for Today

* Introduce statistical modeling

* Begin with the simple linear regression model

* Measuring correlation

---

## Statistical Models

### Two Main Motivations

You can often tell the modeling motivation from the research question.  We will look at studies that ask the following questions:

> "Can I use remotely sensed data to predict forest types in Alaska?"

&rarr; Motivation: Predict new observations

> "Do left-handed people live shorter lives than right-handed people?"

&rarr; Motivation: Describe relationships

We will focus mainly on descriptive modeling in this course.  If you want to learn more about predictive modeling, take Math 243: Statistical Learning!

---

## Form of the Model

$$
y = f(x) + \epsilon
$$

**Goal:**

&rarr; Determine a reasonable form for `$f()$`. (Ex: Line, curve, ...)

&rarr; Estimate `$f()$` with `$\hat{f}()$` using the data.

&rarr; Generate predicted values: `$\hat{y} = \hat{f}(x)$`.

---

### Simple Linear Regression Model

Consider this model when:

* Response variable `$(y)$`: quantitative

* Explanatory variable `$(x)$`: quantitative
    + Have only ONE explanatory variable.

--
    
* AND, `$f()$` can be approximated by a line.

---

### Example: Is there a linear relationship between tree diameter and tree height for the trees at the Woodstock Community Center?

---

### Example: Trees at the Woodstock CC

```r
library(pdxTrees)
woodstock_cc <- get_pdxTrees_parks(park = 
 "Woodstock Community Center")

ggplot(data = woodstock_cc, mapping = aes(x = DBH, 
                                          y = Tree_Height)) +
  geom_point()
```

Linear trend? Direction of trend?

---

### Example: Trees at the Woodstock CC

```r
library(pdxTrees)
woodstock_cc <- get_pdxTrees_parks(park = 
 "Woodstock Community Center")

ggplot(data = woodstock_cc, mapping = aes(x = DBH, 
                                          y = Tree_Height)) +
  geom_point() +
  stat_smooth(method = "lm", se = FALSE)
```

Linear trend? Direction of trend?

---

### Example: Trees at the Woodstock CC

```r
library(pdxTrees)
woodstock_cc <- get_pdxTrees_parks(park = 
 "Woodstock Community Center")

ggplot(data = woodstock_cc, mapping = aes(x = DBH, 
                                          y = Tree_Height)) +
  geom_point() +
  stat_smooth(method = "lm", se = FALSE)
```

**A simple linear regression model would be suitable for these data.**

* But first, let's describe more plots!

---

class: center, middle, inverse

# But before that: It's time for Trend Stretches!

---

**Need a summary statistics that quantifies the strength and relationship of the linear trend!**

---

## (Sample) Correlation Coefficient

* Measures the strength and direction of linear relationship between two quantitative variables

* Symbol: `$r$`

* Always between -1 and 1

* Sign indicates the direction of the relationship

* Magnitude indicates the strength of the linear relationship

```r
woodstock_cc %>%
  summarize(cor_ht_dbh = cor(Tree_Height, DBH))
```

```
## # A tibble: 1 x 1
## cor_ht_dbh
## <dbl>
## 1 0.828
```

---

Any guesses on the correlations for A, B, C, or D?

```r
dat %>%
  summarize(A = cor(x, y1), B = cor(x, y2),
            C = cor(x, y3), D = cor(x, y4))
```

```
## # A tibble: 1 x 4
## A B C D
## <dbl> <dbl> <dbl> <dbl>
## 1 0.695 -0.217 -0.815 -0.113
```

---

## New Example

```r
# Correlation coefficients
dat2 %>%
  group_by(dataset) %>%
  summarize(cor = cor(x, y))
```

```
## # A tibble: 13 x 2
## dataset cor
## <chr> <dbl>
## 1 away -0.0641
## 2 bullseye -0.0686
## 3 circle -0.0683
## 4 dino -0.0645
## 5 dots -0.0603
## 6 h_lines -0.0617
## 7 high_lines -0.0685
## 8 slant_down -0.0690
## 9 slant_up -0.0686
## 10 star -0.0630
## 11 v_lines -0.0694
## 12 wide_lines -0.0666
## 13 x_shape -0.0656
```

* Conclude that `$x$` and `$y$` have the same relationship across these different datasets because the correlation is the same?

---

###  Always graph the data when exploring relationships!

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### Simple Linear Regression

Let's return to the Example: Trees at the Woodstock CC

* A line is a reasonable model form.

* Where should the line be?
    + Slope? Intercept?

---

### Simple Linear Regression

Let's return to the Example: Trees at the Woodstock CC

* A line is a reasonable model form.

* Where should the line be?
    + Slope? Intercept?

---

### Simple Linear Regression

Let's return to the Example: Trees at the Woodstock CC

* A line is a reasonable model form.

* Where should the line be?
    + Slope? Intercept?

---

### Simple Linear Regression

Let's return to the Example: Trees at the Woodstock CC

* A line is a reasonable model form.

* Where should the line be?
    + Slope? Intercept?