### Residual Analysis

Residual Analysis
Chapter 8
Model Assumptions
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Independence (response variables yi are
independent)- this is a design issue
Normality (response variables are normally
distributed)
Homoscedasticity (the response variables
have the same variance)
Best way to check assumptions: check the
assumptions on the random errors
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They are independent
They are normally distributed
They have a constant variance σ2 for all settings of the
independent variables (Homoscedasticity)
They have a zero mean.
If these assumptions are satisfied, we may use the normal density
as the working approximation for the random component. So,
the residuals are distributed as:
εi ~ N(0,σ2)
Plotting Residuals
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To check for homoscedasticity (constant variance):
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Produce a scatterplot of the standardized residuals against
the fitted values.
Produce a scatterplot of the standardized residuals against
each of the independent variables.
If assumptions are satisfied, residuals should vary
randomly around zero and the spread of the residuals
should be about the same throughout the plot (no
systematic patterns.)
Homoscedasticity is probably violated if…
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The residuals seem to increase or decrease in
average magnitude with the fitted values, it is
an indication that the variance of the residuals
is not constant.
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The points in the plot lie on a curve around
zero, rather than fluctuating randomly.
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A few points in the plot lie a long way from
the rest of the points.
Heteroscedasticity
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Not fatal to an analysis; the analysis is
weakened, not invalidated.
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Detected with scatterplots and rectified through
transformation.
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http://www.pfc.cfs.nrcan.gc.ca/profiles/wulder/mvstats/transform_e.html
http://www.ruf.rice.edu/~lane/stat_sim/transformations/index.html
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Cautions with transformations
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Difficulty of interpretation of transformed
variables.
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The scale of the data influences the utility of
transformations.
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If a scale is arbitrary, a transformation can be more
effective
If a scale is meaningful, the difficulty of interpretation
increases
Normality
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The random errors can be regarded as a random
sample from a N(0,σ2) distribution, so we can check
this assumption by checking whether the residuals
might have come from a normal distribution.
We should look at the standardized residuals
Options for looking at distribution:
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Histogram, Stem and leaf plot, Normal plot of residuals
http://statmaster.sdu.dk/courses/st111/module04/index.html
Histogram or Stem and Leaf Plot
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Does distribution of residuals approximate a
normal distribution?
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Regression is robust with respect to
nonnormal errors (inferences typically still
valid unless the errors come from a highly
skewed distribution)
Normal Plot of Residuals
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A normal probability plot is found by plotting the
residuals of the observed sample against the
corresponding residuals of a standard normal
distribution N (0,1)
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If the plot shows a straight line, it is reasonable to
assume that the observed sample comes from a normal
distribution.
If the points deviate a lot from a straight line, there is
evidence against the assumption that the random errors
are an independent sample from a normal distribution.
http://www.skymark.com/resources/tools/normal_test_plot.asp
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2
:Variance of the random error ε
If this value equals zero
 all random errors equal 0
 Prediction equation (ŷ) will be equal to mean value E(y)
If this value is large
 Large (absolute values) of ε
 Larger deviations between ŷ and the mean value E(y).
2
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The larger the value of , the greater the error in estimating
the model parameters and the error in predicting a value of
y for specific values of x.
Variance is estimated with
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s
2
(MSE)
The units of the estimated variance are
squared units of the dependent variable y.
For a more meaningful measure of variability,
we use s or Root MSE.
The interval ±2s provides a rough estimation
with which the model will predict future
values of y for given values of x.
Why do residual analysis?
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Following any modeling procedure, it is a
good idea to assess the validity of your model.
Residuals and diagnostic statistics allow you
to identify patterns that are either poorly fit by
the model, have a strong influence upon the
estimated parameters, or which have a high
leverage. It is helpful to interpret these
diagnostics jointly to understand any potential
problems with the model.
Outliers
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What:
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Why:
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An observation with a residual that is larger than 3s or a
standardized residual larger than 3 (absolute value)
A data entry or recording error, Skewness of the
distribution, Chance, Unassignable causes
Then what?
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Eliminate? Correct? Analyze them?
How much influence do they have? How do I know?
 Minitab Storage: Diagnostics (Leverages, Cook’s
Distance, DFFITS)
Leverages- p. 403
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Identifies observations with unusual or outlying xvalues.
Leverages fall between 0 and 1. A value greater than
2(p/n) is large enough to suggest you should
examine the corresponding observation.
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p: number of predictors (including constant)
n: number of observations
Minitab identifies observations with leverage over
3(p/n) with an X in the table of unusual observations
Cook’s Distance (D)- p. 405
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An overall measure of the combined impact of each
observation on the fitted values. Calculated using
leverage values and standardized residuals.
Considers whether an observation is unusual with
respect to both x- and y-values.
Observations with large D values may be outliers.
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Compare D to F-distribution with (p, n-p) degrees of
freedom. Determine corresponding percentile.
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Less than 20%- little influence
Greater than 50%- major influence
DFFITS- p. 408
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The difference between the predicted value when all
observations are included and when the ith observation is
deleted.
Combines leverage and Studentized residual (deleted t
residuals) into one overall measure of how unusual an
observation is
Represent roughly the number of standard deviations that the
fitted value changes when each case is removed from the data
set. Observations with DFFITS values greater than 2 times
the square root of (p/n) are considered large and should be
examined.
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p is number of predictors (including the constant)
n is the number of observations
Summary of what to look for
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Look for values that stand out in diagnostic measures
Rules of thumb
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Leverages (HI): values greater than 2(p/n)
Cook’s Distance: values greater than 50% of comparable F
(p, n-p) distribution
p
2
DFFITS: values greater than
n
Let’s do an example
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Returning to the Naval Base data set
Finally, are the residuals correlated?
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Durbin-Watson d statistic (p. 415)
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Range: 0 ≤ d ≤ 4
Uncorrelated: d is close to 2
Positively correlated: d is closer to zero
Negatively correlated: d is closer to 4.