Stat 112: Lecture 14 Notes

• Finish Chapter 6: – Checking and Remedying Constant Variance

Assumption (Section 6.5)– Checking and Remedying Normality

Assumption (Seciton 6.6) – Outliers and Influential Points (Section 6.7)

• Homework 4 is due next Thursday. • Please let me know of any ideas you want

to discuss for the final project.

Assumptions of Multiple Linear Regression Model

1. Linearity:

2. Constant variance: The standard deviation of Y for the subpopulation of units with is the same for all subpopulations.

3. Normality: The distribution of Y for the subpopulation of units with is normally distributed for all subpopulations.

4. The observations are independent.

1 0 1 1( | , , )K K KE Y X X X X

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Checking Constant Variance Assumption

• Residual plot versus explanatory variables should exhibit constant variance.

• Residual plot versus predicted values should exhibit constant variance (this plot is often most useful for detecting nonconstant variance)

Heteroscedasticity• When the requirement of a constant variance is violated

we have a condition of heteroscedasticity.• Diagnose heteroscedasticity by plotting the residual

against the predicted y.

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How much traffic would a building generate?

• The goal is to predict how much traffic will be generated by a proposed new building of 150,000 occupied sq ft. (Data is from the MidAtlantic States City Planning Manual.)

• The data tells how many automobile trips per day were made in the AM to office buildings of different sizes.

• The variables are x = “Occupied Sq Ft of floor space in the building (in 1000 sq ft)” and

Y = “number of automobile trips arriving at the building per day in the morning”.

Bivariate Fit of AM Trips By Sq Ft (1000)

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AM Trips = -4.55 + 1.515 Sq Ft (1000) Summary of Fit RSquare 0.857071 Observations (or Sum Wgts) 61 Analysis of Variance Source DF Sum of Squares Mean Square F Ratio Model 1 3810173.9 3810174 353.7926 Error 59 635401.2 10770 Prob > F C. Total 60 4445575.1 <.0001

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The heteroscedasticity shows here.

• A brief list of transformations» y’ = y1/2 (for y > 0)

• Use when the spread of residuals increases with

» y’ = log y (for y > 0)• Use when the spread of residuals increases with • Use when the distribution of the residuals is skewed to

the right.

» y’ = y2

• Use when the spread of residuals is decreasing with , • Use when the error distribution is left skewed

Reducing Nonconstant Variance/Nonnormality by Transformations

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Bivariate Fit of Log AM Trips By Occup. Sq. Ft. (1000)

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To try to fix heteroscedasticity we transform Y to Log(Y)

This fixes heteroskedasticity

BUT it creates a nonlinear pattern.

Bivariate Fit of Log AM Trips By Log(OccupSqFt)

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Linear Fit Log AM Trips = 0.6393 + 0.8033 Log(OccupSqFt)

Summary of Fit RSquare 0.827

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To fix nonlinearity we now transform x to Log(x), without changing the Y axis anymore.

The resulting pattern is both satisfactorily homoscedastic AND linear.

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Log AM Trips Predicted

Often we will plot residuals versus predicted.

For simple regression the two residual plots are equivalent

Checking Normality

• If the disturbances are normally distributed, about 68% of the standardized residuals should be between –1 and +1, 95% should be between –2 and +2 and 99% should be between –3 and +3.

• Standardized residual for observation i=

• Graphical methods for checking normality: Histogram of (standardized) residuals, normal quantile plot of (standardized) residuals.

Normal Quantile Plots

• Normal quantile (probability) plot: Scatterplot involving ordered residuals (values) with the x-axis giving the expected value of the kth ordered residual on the standard normal scale (residual / ) and the y-axis giving the actual residual.

• JMP implementation: Save residuals, then click Analyze, Distribution, red triangle next to Residuals and Normal Quantile Plot.

• If the residuals follow approximately a normal distribution, they should fall approximately on a straight line.

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1. Residual plots for the original data:

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Normal Quantile Plot

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Traffic and office space ex. Cont.

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Log AM Trips Predicted

Residuals from the final Log – Log fit

Importance of Normality and Corrections for Normality

• For point estimation/confidence intervals/tests of coefficients and confidence intervals for mean response, normality of residuals is only important for small samples because of Central Limit Theorem. Guideline: Do not need to worry about normality if there are 30 observations plus 10 additional observations for each additional explanatory variable in multiple regression beyond the first one.

• For prediction intervals, normality is critical for all sample sizes.

• Corrections for normality: transformations of y variable (see earlier slide)

Order of Correction of Violations of Assumptions in Multiple Regression

• First, focus on correcting a violation of the linearity assumption.

• Then, focus on correction violations of constant variance after the linearity assumption is satifised. If constant variance is achieved, make sure that linearity still holds approximately.

• Then, focus on correctiong violations of normality. If normality is achieved, make sure that linearity and constant variance still approximately hold.

Outliers and Influential Observations in Simple Regression• Outlier: Any really unusual observation.• Outlier in the X direction (called high leverage point):

Has the potential to influence the regression line.• Outlier in the direction of the scatterplot: An

observation that deviates from the overall pattern of relationship between Y and X. Typically has a residual that is large in absolute value.

• Influential observation: Point that if it is removed would markedly change the statistical analysis. For simple linear regression, points that are outliers in the x direction are often influential.

Outliers in Direction of Scatterplot• Residual • Standardized Residual:

• Under multiple regression model, about 5% of the points should have standardized residuals greater in absolute value than 2, 1% of the points should have standardized residuals greater in absolute value than 3. Any point with standardized residual greater in absolute value than 3 should be examined.

• To compute standardized residuals in JMP, right click in a new column, click Formula and create a formula with the residual divided by the RMSE.

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Housing Prices and Crime Rates• A community in the Philadelphia area is interested in

how crime rates are associated with property values. If low crime rates increase property values, the community might be able to cover the costs of increased police protection by gains in tax revenues from higher property values.

• The town council looked at a recent issue of Philadelphia Magazine (April 1996) and found data for itself and 109 other communities in Pennsylvania near Philadelphia. Data is in philacrimerate.JMP. House price = Average house price for sales during most recent year, Crime Rate=Rate of crimes per 1000 population.

Bivariate Fit of HousePrice By CrimeRate

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Center City Philadelphia is a high leverage point. Gladwyne and Haverford are outliers in the direction of the scatterplot (their house price is considerably higher than one would expect given their crime rate).

Which points are influential?B iv a r i a t e F i t o f H o u s e P r ic e B y C r im e R a t e

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A l l o b s e r v a t i o n s L i n e a r F i t H o u s e P r ic e = 1 7 6 6 2 9 .4 1 - 5 7 6 . 9 0 8 1 3 C r im e R a te

W i t h o u t C e n t e r C i t y P h i l a d e l p h i a L i n e a r F i t H o u s e P r ic e = 2 2 5 2 3 3 .5 5 - 2 2 8 8 . 6 8 9 4 C r im e R a te

W i t h o u t G l a d w y n e L i n e a r F i t H o u s e P r ic e = 1 7 3 1 1 6 . 4 3 - 5 6 7 . 7 4 5 0 8 C r im e R a te

Center City Philadelphia is influential; Gladwyne is not. In general, points that have high leverage are more likely to be influential.

Excluding Observations from Analysis in JMP

• To exclude an observation from the regression analysis in JMP, go to the row of the observation, click Rows and then click Exclude/Unexclude. A red circle with a diagonal line through it should appear next to the observation.

• To put the observation back into the analysis, go to the row of the observation, click Rows and then click Exclude/Unexclude. The red circle should no longer appear next to the observation.

Formal measures of leverage and influence

• Leverage: “Hat values” (JMP calls them hats)• Influence: Cook’s Distance (JMP calls them Cook’s D

Influence).• To obtain them in JMP, click Analyze, Fit Model, put Y

variable in Y and X variable in Model Effects box. Click Run Model box. After model is fit, click red triangle next to Response. Click Save Columns and then Click Hats for Leverages and Click Cook’s D Influences for Cook’s Distances.

• To sort observations in terms of Cook’s Distance or Leverage, click Tables, Sort and then put variable you want to sort by in By box.

Distributions Cook's D Influence HousePrice

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Center City Philadelphia has both influence (Cook’s Distance much Greater than 1 and high leverage (hat value > 3*2/99=0.06). No otherobservations have high influence or high leverage.

Rules of Thumb for High Leverage and High Influence

• High Leverage Any observation with a leverage (hat value) > (3 * # of coefficients in regression model)/n has high leverage, where

# of coefficients in regression model = 2 for simple linear regression.

n=number of observations. • High Influence: Any observation with a Cook’s

Distance greater than 1 indicates a high influence.

What to Do About Suspected Influential Observations?

See flowchart attached to end of slides

Does removing the observation change the

substantive conclusions?• If not, can say something like “Observation x

has high influence relative to all other observations but we tried refitting the regression without Observation x and our main conclusions didn’t change.”

• If removing the observation does change substantive conclusions, is there any reason to believe the observation belongs to a population other than the one under investigation?– If yes, omit the observation and proceed.– If no, does the observation have high leverage

(outlier in explanatory variable).• If yes, omit the observation and proceed. Report that

conclusions only apply to a limited range of the explanatory variable.

• If no, not much can be said. More data (or clarification of the influential observation) are needed to resolve the questions.

General Principles for Dealing with Influential Observations

• General principle: Delete observations from the analysis sparingly – only when there is good cause (observation does not belong to population being investigated or is a point with high leverage). If you do delete observations from the analysis, you should state clearly which observations were deleted and why.