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204.1.4 How good is my Regression Line

Mathematics behind a good fit.

Link to the previous post : https://statinfer.com/204-1-3-practice-regression-line-fitting/

In this post we will understand the mathematics behind a good regression line.

How good is my regression line?

  • Take an (x,y) point from data.
  • Imagine that we submitted x in the regression line, we got a prediction as ypred
  • If the regression line is a good fit then the we expect ypred=y or (y-ypred) =0
  • At every point of x, if we repeat the same, then we will get multiple error values (y-ypred) values
  • Some of them might be positive, some of them may be negative, so we can take the square of all such errors
SSE=(yy^)2
  • For a good model we need SSE to be zero or near to zero
  • Standalone SSE will not make any sense, For example SSE= 100, is very less when y is varying in terms of 1000’s. Same value is is very high when y is varying in terms of decimals.
  • We have to consider variance of y while calculating the regression line accuracy
  • Error Sum of squares (SSE- Sum of Squares of error)
    SSE=(yy^)2
  • Total Variance in Y (SST- Sum of Squares of Total)
    SST=(yy¯)2
    SST=(yy^+y^y¯)2
    SST=(yy^+y^y¯)2
    SST=(yy^)2+(y^y¯)2
    SST=SSE+(y^y¯)2
    SST=SSE+SSR
  • So, total variance in Y is divided into two parts,
    • Variance that can’t be explained by x (error)
    • Variance that can be explained by x, using regression

Explained and Unexplained Variation

  • Total variance in Y is divided into two  parts,
    • Variance that can be explained by x, using regression
    • Variance that can’t be explained by x
      SST=SSE+SSR
      TotalsumofSquares=SumofSquaresError+SumofSquaresRegression
      SST=(yy¯)2SSE=(yy^)2SSR=(y^y¯)2

In next session we will figure out Rsquared which a statistical measure of closeness of datapoints to the fitted regression line.

The next post is about R squared in python.

Link to the next post : https://statinfer.com/204-1-5-r-squared-in-python/

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