Introduction to Linear Regression
Relationships Between Variables
- In a statistical model, we generally have one variable that is the output and one or more variables that are the inputs.
- Response variable
- a.k.a. \(y\), dependent
- The quantity you want to understand
- In this class – always numerical
- Explanatory variable
- a.k.a. \(x\), independent, explanatory, predictor
- Something you think might be related to the response
- Either numerical or categorical
Visualizing Linear Regression
- The scatterplot has been called the most “generally useful invention in the history of statistical graphics.”
- It is a simple two-dimensional plot in which the two coordinates of each dot represent the values of two variables measured on a single observation.
Characterizing Relationships
Form (e.g. linear, quadratic, non-linear)
Direction (e.g. positive, negative)
Strength (how much scatter/noise?)
Unusual observations (do points not fit the overall pattern?)
Data for Today
The ncbirths dataset is a random sample of 1,000 cases taken from a larger dataset collected in North Carolina in 2004.
Each case describes the birth of a single child born in North Carolina, along with various characteristics of the child (e.g. birth weight, length of gestation, etc.), the child’s mother (e.g. age, weight gained during pregnancy, smoking habits, etc.) and the child’s father (e.g. age).
Your Turn!
How would your characterize this relationship?
- form
- direction
- strength
- unusual observations
Cleaning & Filtering the Data
It seems like pregnancies with a gestation less than 28 weeks have a non-linear relationship with a baby’s birth weight, so we will filter these observations out of our dataset.
Change in scope of inference
Removing these observations narrows the population of births we are able to make inferences onto! In this case, what population could we infer our findings onto?
Summarizing a Linear Relationship
Correlation:
strength and direction of a linear relationship between two quantitative variables
- Correlation coefficient between -1 and 1
- Sign of the correlations shows direction
- Magnitude of the correlation shows strength
Anscombe Correlations
Four datasets, very different graphical presentations
- same mean and standard deviation in both \(x\) and \(y\)
- same correlation
- same regression line
For which of these relationships is correlation a reasonable summary measure?
Calculating Correlation in R
What if I ran get_correlation(births_post28, weight ~ weeks) instead? Would I get the same value?
Linear regression:
we assume the the relationship between our response variable (\(y\)) and explanatory variable (\(x\)) can be modeled with a linear function, plus some random noise
\(response = intercept + slope \cdot explanatory + noise\)
Writing the Regression Equation
Population Model
\(y = \beta_0 + \beta_1 \cdot x + \epsilon\)
\(y\) = response
\(\beta_0\) = population intercept
\(\beta_1\) = population slope
\(\epsilon\) = errors / residuals
Sample Model
\(\widehat{y} = b_0 + b_1 \cdot x\)
\(b_0\) = sample intercept
\(b_1\) = sample slope
Why does this equation have a hat on \(y\)?
Linear Regression with One Numerical Explanatory Variable
Step 1: Fit a linear regression
Step 2: Obtain coefficient table
| term | estimate | std_error | statistic | p_value | lower_ci | upper_ci |
|---|---|---|---|---|---|---|
| intercept | -5.003 | 0.582 | -8.603 | 0 | -6.144 | -3.862 |
| weeks | 0.316 | 0.015 | 21.010 | 0 | 0.287 | 0.346 |
get_regression_table()
This function lives in the moderndive package, so we will need to load in this package (e.g., library(moderndive) if we want to use the get_regression_table() function.
Our focus (for now…)
Estimated regression equation
\[\widehat{y} = b_0 + b_1 \cdot x\]
Write out the estimated regression equation!
How do you interpret the intercept value of -5.003?
How do you interpret the slope value of 0.316?
Obtaining Residuals
\(\widehat{weight} = -5.003+0.316 \cdot weeks\)
What would the residual be for a pregnancy that lasted 39 weeks and whose baby weighed 7.63 pounds?
Linear Regression with One Categorical Explanatory Variable
Step 1: Finding distinct levels
Step 2: Fit a linear regression
Step 3: Obtain coefficient table
| term | estimate | std_error | statistic | p_value | lower_ci | upper_ci |
|---|---|---|---|---|---|---|
| intercept | 7.246 | 0.046 | 158.369 | 0.000 | 7.157 | 7.336 |
| habit: smoker | -0.418 | 0.128 | -3.270 | 0.001 | -0.668 | -0.167 |
🤔
Step 4: Write Estimated Regression Equation
| term | estimate | std_error | statistic | p_value | lower_ci | upper_ci |
|---|---|---|---|---|---|---|
| intercept | 7.246 | 0.046 | 158.369 | 0.000 | 7.157 | 7.336 |
| habit: smoker | -0.418 | 0.128 | -3.270 | 0.001 | -0.668 | -0.167 |
\[\widehat{weight} = 7.23 - 0.4 \cdot Smoker\]
But what does \(Smoker\) represent???
Categorical Explanatory Variables
\(x\) is a categorical variable with levels:
"nonsmoker""smoker"
We need to convert to:
- a “baseline” group
- “offsets” / adjustments to the baseline
Choosing a Baseline Group
| term | estimate | std_error | statistic | p_value | lower_ci | upper_ci |
|---|---|---|---|---|---|---|
| intercept | 7.246 | 0.046 | 158.369 | 0.000 | 7.157 | 7.336 |
| habit: smoker | -0.418 | 0.128 | -3.270 | 0.001 | -0.668 | -0.167 |
Based on the regression table, what habit group was chosen to be the baseline group?
Rewriting in Terms of Indicator Variables
\[\widehat{weight} = 7.23 - 0.4 \cdot 1_{smoker}(x)\]
where
\(1_{smoker}(x) = 1\) if the mother was a "smoker"
\(1_{smoker}(x) = 0\) if the mother was a "nonsmoker"
Obtaining Group Means
\[\widehat{weight} = 7.23 - 0.4 \cdot 1_{Smoker}(x)\]
Given the equation, what is the estimated mean birth weight for nonsmoking mothers?
For smoking mothers?
Causal Inference
We just concluded that babies born to a "smoker" weigh, on average, 0.4 pounds less than babies born to a "nonsmoker".
Can we conclude that smoking caused these babies to weigh less? Why or why not?
Midterm Project Step 1
Project Proposal
Choose a dataset
Choose one numerical response variable
Choose one numerical explanatory variable
Choose a second explanatory variable, it can be either numerical or categorical
Checking values of your numerical variable(s)
Your numerical variable cannot have a small number of values (e.g., 2 or 3). You can use the distinct() function to determine the unique values of your variable. For example, by running distinct(hbr_maples, year) I would discover that year only has two values (2003 and 2004), meaning year is not eligible to be a numerical response or explanatory variable. It could, however, be a categorical explanatory variable!
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