STAT1008 Study Guide - Final Guide: Dependent And Independent Variables, Scatter Plot
2.6 Two Quantitative Variables: Linear Regression
The Regression Line
• The regression line provides a model of a linear association between two variables, and we
can use the regression line on a scatterplot to give a predicted variable of the response
variable, based on a given value of the explanatory variable.
• The estimated regression line is .
•
• Unlike correlation, for linear regression it does matter which is the explanatory and which is
the response variable.
• Note: The regression line to predict y from x is not the same as the regression line to predict
x from y.
Predicted and Actual Values
• The observed response value, y, is the response value observed for a particular data point.
• The predicted response variable, , is the response variable that would be predicted for a
given x value, based on a model.
• The best fitting line is what makes the predicted value closest to the actual values.
Residuals
• The residual for each data point = observed - predicted = y –
• The residual is also the vertical distance from each point to the line.
• Points above the line will have positive residuals and points below will have negative.
Least Squares Line
• The least squares line is the line which minimises the sum of squared residuals.
• Minimise
i
• Rely on technology for finding the least square line.
• "least squares line" = "regression line"
• If we add up all the residuals from the regression line, the sum will always be zero.
Slope and Intercept
• For the estimated regression line is → a is the intercept and b is the slope.
• Slope: increase in predicted y for every unit increase in x.
• Intercept: predicted y value when x = 0.
Regression Cautions
1. Do not use the regression equation or line to predict outside the range of x values available
in your data.
o If none of the x values are anywhere near 0, then the intercept is meaningless.
o It is helpful to think about units when interpreting a regression equation.
o
2. Although the regression line can be calculated for any set of paired quantitative variables, it
is only appropriate to use a regression line when there is a linear trend in the data.
3. Outliers have a strong influence on the regression line.
o Data points for which the explanatory value is an outlier are often called influential
points.
4. Higher values of x may lead to higher (or lower) predicted values of y, but this does not
mean that changing x will cause y to increase or decrease.
o Causation can only be determined randomly (which is rarely the case for a
continuous explanatory variable).
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