1. Unit 3 – Linear regression

2. Linear relationships Variables Response Variable Explanatory Variable
When dealing with linear relationships we need 2 variables!
Variables

3. Strength + Direction + Form
There is a …
Strength + Direction + Form
…relationship between x and y.
Linear
Curved
Scattered
Weak
Moderate
Strong
Positive
Negative
Describing a Linear Relationship

4. Finding the line of best fit
𝑦 = 𝐵 𝑜 + 𝐵 1 x
Slope = 𝑆 𝑦 𝑆 𝑥 * r
Y-intercept
X value
Predicted Y
Slope
Equation

5. CAUTION: When using a line of best fit
Extrapolation : using your equation to predict outside the range of the data you used to come up with your equation.
Lurking variables : an underlying variable that is causing the relationship to look different than it is in reality
Outliers : your equation is strongly influenced by outliers
Claiming X causes Y. They just have a linear relationship
Warning!

6. Sentences Interpreting Linear Relationship Output
Slope
As __X___ increases by 1, ___Y___ will be expected to increase/decrease by ___slope_____
Y-intercept
When ___X__ is 0, ___Y___ will be expected to be ___Y-intercept___
R-squared
___ 𝑹 𝟐 (as percent) __ of the variability in ___Y___ can be explained by ___X__
Interpreting Linear Relationship Output

7. What is a residual?
It’s the difference between the actual value and the predicted value.
Residual = 𝑦− 𝑦
Residual

8. What do you want residuals to look like
What do you want residuals to look like? (To be confident your line is a good fit)
Scattered or have a pattern?
All above 0, all below 0, or half and half?
Equal variance around 0, or not equal variance around 0.
Residuals

9. SO…. How do you know if you can do a linear regression
relationship between X and Y is linear (can check by looking at a scatterplot of x and y or the residual plot)
no obvious lurking variables (you can kind of assume this for this course)
simple random sample (was the data taken from a SRS)
constant variance of the residuals (plot of residuals)
residuals vary according to a normal distribution (normal quantile plot of residuals)
Warning

10. Confidence Intervals for the slope
𝑏 1 ± 𝑡 𝛼 2 ;𝑛−2 ∗𝑠𝑒 𝑏 1
standard error se(b1) can be obtained from the JMP output.
𝑏 1 is the predicted value of the slope
Degrees of freedom = n-2
I am ___% confident that the true slope between __X__ and __Y__ is between _____ and ______

11. Hypothesis testing for the slope
𝐻 0 : 𝐵 1 =0
𝐻 𝐴 : 𝐵 1 ≠0
Obtain a t-stat
Obtain a p-value.
Decide to reject Ho.
If you reject Ho, then there is sufficient evidence to suggest there is a linear relationship between ___explanatory___ and ___response___