3.5 Residual Plots and Residual Analysis

Residual Plots

  • A residual is the name given to the difference in response variable value between a datapoint and the value predicted at the corresponding explanatory variable value by the fitted model.
  • Data points above the fitted line will have a positive residual value and those below the line will have a negative residual value.
  • A residual plot is similar to a scatterplot with each residual value placed at the same value along the x-axis as the corresponding datapoint, and along the y-axis at its residual value.
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3.4 Coefficient of Determination and Measures of Strength

Coefficient of Determination; r^2

  • The coefficient of determination gives a quantitative way of determining how much of the variation of the response variable is explained by variation in the explanatory variable.
  • It is represented by a lower-case r with a 2 superscript and can be calculated by squaring the correlation coefficient:

r^{2}=\left(\frac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{(n-1) s_{x} s_{y}}\right)^{2}

  • When calculating the coefficient of determination, you will get a decimal answer, however, when interpreting the value, you should convert it into a percentage (multiply by 100).
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3.3 Using the Formula for a Fitted Line


  • After fitting a model to a dataset (through linear regression), we can use that model to estimate values we don’t have data points for.
  • When estimating values that lie within the range of available raw data points, we refer to it as interpolating.
  • Interpolation is considered accurate if the fit has high strength and sufficient data points were used.

Example: if a linear fit is creating using data points ranging in value from 1 to 10, estimating the value of the response variable when the explanatory variable has a value of 2 would be considered interpolation.


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