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FM Least Squares Regression

3.6 Least Squares Regression for Transformed Data

Note: if you cannot remember how to interpret least squares regression lines, revise notes for 3.3 Using the Formula for a Fitted Line.

Guideline to Analyse Least Squares Linear Regression Relationships for Transformed Data

  • Analysing a least squares linear fit for transformed data is similar to the process for non-transformed data, however you must keep in mind the association is not between the explanatory variable and response variable, it is between the transformed variable and the non-transformed variable (which will be either the explanatory or response variable).
  • When interpreting the meaning of the coefficient of determination, it gives an indication of what percentage of variation in the transformed variable is explained by variation of the non-transformed variable, or visa-versa (e.g. for an explanatory variable squared transformation, the coefficient indicates what percentage of variation in y can be explained by variation in x^2).
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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

y=a+bx

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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