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3.2 Modelling Linear Associations

Identifying Explanatory and Response Variables

  • It is important to correctly select the explanatory and response variables when using regression, or the relationship will be incorrect.
  • The explanatory variable is the variable which is used to explain or predict the response variable.
  • In a conventional x-y dataset, the x variable is the explanatory variable and y is the response variable.

Fitting Least Squares Models

  • Start by identifying the explanatory and response variables.
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3.1 Least Squares Linear Regression

The Idea behind Least Squares Regression

  • In order to conveniently estimate the expected values of one variable based on another, we often create a mathematical model which fits, as closely as possible, the data we have collected. In Further Maths, we will only deal with linear regression, where we try to come up with a straight line that fits our data.
  • In least squares regression, we try to find that “best fit” by finding a line that minimises the value of the sum of squared residuals (i.e. we take the difference between each datapoint and the line, then square each and add them all together).
  • The resulting line is of the form


where y and x are the response and explanatory variables, respectively, and a and b are constants which must be determined.

  • Least squares linear regression is only appropriate if:
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