Regression Diagnostics Introduction to Regression 1 Why do we need - - PowerPoint PPT Presentation

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Regression Diagnostics Introduction to Regression 1 Why do we need - - PowerPoint PPT Presentation

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Regression Diagnostics Introduction to Regression 1 Why do we need to do all this? Theory based on assumptions Focuses on the residuals and fitted values Validate the model Gives us clues how to change the model Is it

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Regression Diagnostics

Introduction to Regression

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Why do we need to do all this?

  • Theory based on assumptions
  • Focuses on the residuals and fitted values
  • Validate the model
  • Gives us clues how to change the model
  • Is it appropriate?
  • Lots of statistical tests based on certain

assumptions

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What shall we look at?

  • Calculate residuals for each case
  • Observed value – Predicted value
  • Standardise them by dividing by their SD

(approx.)

  • Different types of standardised residuals
  • We need to do a series of plots
  • Remember the model
  • Constant variance

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Standardised Residuals

  • Should be

– Size??? – Independent – Normally distributed – Constant – Unrelated to the fitted values – Unrelated to the independent variables

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Plots to do compute

  • Normal probability plot of residuals
  • Look for large standardised residuals
  • Check values
  • Plot residuals vs fitted/ predicted values
  • Plot residuals vs each independent variable
  • Plot residuals against time (if that is

appropriate)

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The regression equation is sqrtrooms = 0.200 + 1.90 sqrtcrews

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Leverage

  • Measure the distance from the x-values to the

mean of the x-values

  • May influence results
  • p-predictors
  • High values for leverage > 2 ∗ (𝑞+1)

𝑜

  • Be careful here

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Outliers and Bad leverage points

  • Examine them and see if they are different
  • Flag a problem with model
  • Consider fitting another model

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Cooks distance

  • Measures the influence of an observation on

the set of regression coefficients . Influential

  • bservations can be leverage points, outliers,
  • r both.
  • Look for gaps
  • Function of leverage and standardised

residuals

  • Suggested cutoffs are 4/(n-2).
  • What happens when you omit points

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Makey up data

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SRESID Leverage Cooks 1.54 0.26 0.42

  • 4.35

0.26 3.35

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Results

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Including all points

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And more

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Without x=20, and y=10 point

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Without point x=20, y=95

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DFITS

  • Measures the influence of each observation
  • n the fitted values
  • Roughly the number of standard deviations

that the fitted value changes when each

  • bservation is removed from the data set and

the model is refit.

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What model to fit?

  • Suppose we start with
  • Salaries = α+β1Experience+ε – linear model
  • And look at residuals vs fitted values

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So what model should we fit?

  • Salaries = α+β1Experience+ε – linear model
  • Should create a new variable
  • Experience*Experience and added it to model
  • Salaries = α+β1Experience+β2Exper*Exper+ ε
  • Polynomial model

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Oregon Housing

  • Description
  • 76 single-family homes in Eugene, Oregon

during 2005

  • Estate agents have their methods of

determining price

  • Seller wanted a method of determining asking

price

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Variables

  • Price (thousands of $)
  • Floor size (thousands of sq ft)
  • Age of house
  • Number of bedrooms
  • Number of bathrooms
  • Garage size
  • School area
  • Lot size (1:11)- interesting variable too.
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Coding of Lot size

Category Lot Size 1 0-3k 2 3-5k 3 5-7k 4 7-10k 5 10-15k 6 15-20k 7 20K-1acre 8 1-3ac 9 3-5ac 10 5-10ac 11 10-20ac

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0-3k = 0-3,000 sq ft 1 acre = 43,560 sq ft

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Model

  • Going to focus on three variables Price, Size

and Age

  • Age is coded as (Year – 70)/10
  • Going to fit two models
  • Price = α+β1*Size+ β2*Age+ε
  • Price = α+β1*Size+ β2*Age+ β3*Age*Age+ ε
  • First we draw some graphs

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First model

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Residuals vs Age

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Second model

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Article

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Modeling Home Prices Using Realtor Data Iain Pardoe Lundquist College of Business, University of Oregon Journal of Statistics Education Volume 16, Number 2 (2008), www.amstat.org/publications/jse/v16n2/datasets.pardoe.html

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Conclusion

  • Be sure to run diagnostics
  • Examine the plots
  • Check funny points
  • Try out some changes

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Added variable plots

  • Added-variable plots enable us to visually

assess the effect of each predictor, having adjusted for the effects of the other predictors.

  • Y and two predictor variables X and Z
  • Regress Y on X – calculate residuals – Set 1
  • Regress Z on X – calculate residuals – Set 2
  • Plot Set 1 residuals vs Set 2 residuals

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And more…

  • Residuals from Y and X = part of Y not

predicted by X

  • Residuals from Z and X = part of Z not

predicted by X

  • Added-variable plot for predictor variable Z

shows that part of Y that is not predicted by X against that part of Z that is not predicted by X

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Another dataset

  • Price = the price (in $US) of dinner (including one

drink and a tip)

  • Food = customer rating of the food (out of 30)
  • Décor = customer rating of the decor (out of 30)
  • Service = customer rating of the service (out of

30)

  • East = 1 (0) if the restaurant is east (west) of Fifth

Avenue

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Added Variable plots

  • For Food variable
  • Price vs Décor, service and East – calculate

residuals

  • Food vs Décor, service and East- calculate

residuals

  • Plot residuals against each other

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