remedial measures, brown-forsythe test, f testfwood/teaching/w4315/fall2009/...remedial measures,...
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![Page 1: Remedial Measures, Brown-Forsythe test, F testfwood/Teaching/w4315/Fall2009/...Remedial Measures, Brown-Forsythe test, F test Dr. Frank Wood Frank Wood, fwood@stat.columbia.edu Linear](https://reader030.vdocuments.us/reader030/viewer/2022040221/5e3b02c92549f66dbf07cac7/html5/thumbnails/1.jpg)
Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 1
Remedial Measures,
Brown-Forsythe test,
F test
Dr. Frank Wood
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 2
Remedial Measures
• How do we know that the regression function is a good explainer of the observed data?
– Plotting
– Tests
• What if it is not? What can we do about it?
– Transformation of variables (next lecture)
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 3
Graphical Diagnostics for the Predictor Variable
• Dot Plot
– Useful for visualizing distribution of inputs
• Sequence Plot
– Useful for visualizing dependencies between
error terms
• Box Plot
– Useful for visualizing distribution of inputs
• Toluca manufacturing example again: production time vs. lot size.
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 4
Dot Plot
• How many observations per input value?
• Range of inputs?
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 5
Sequence Plot
• If observations are made over time, is there a correlation between input and position in observation sequence?
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 6
Box Plot
• Shows
– Median
– 1st and 3rd quartiles
– Maximum and minimum
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 7
Residuals
• Remember, the definition of residual:
• And the difference between that and the unknown
true error
• In a normal regression model the ǫi’s are assumed
to be iid N(0,σ) random variables. The observed
residuals ei should reflect these properties.
ei = Yi − Yi
ǫi = Yi − E{Yi}
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 8
Remember: residual properties
• Mean
• Variance
ei =
∑ei
n= 0
s2 =
∑(ei−e)2
n−2 = SSE
n−2 =MSE
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 9
Nonindependence of Residuals
• The residuals ei are not independent random variables– The fitted values are based on the same
fitted regression line.• The residuals are subject to two constraints
– Sum of the ei’s equals 0
– Sum of the products Xi ei‘s equals 0
• When the sample size is large in comparison to the number of parameters in the regression model, the dependency effect among the residuals ei can reasonably safely be ignored.
Yi
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 10
Definition: semistudentized residuals
• Like usual, since the standard deviation of ǫiis σ (itself estimated by MSE1/2) a natural form
of standardization to consider is
• This is called a semistudentized residual.
e∗i= ei−0√
MSE
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 11
Departures from Model…
• to be studied by residuals
– Regression function not linear
– Error terms do not have constant variance
– Error terms are not independent
– Model fits all but one or a few outlier observations
– Error terms are not normally distributed
– One or more predictor variables have been
omitted from the model
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 12
Diagnostics for Residuals
• Plot of residuals against predictor variable
• Plot of absolute or squared residuals against predictor variable
• Plot of residuals against fitted values
• Plot of residuals against time or other sequence
• Plots of residuals against omitted predictor variables
• Box plot of residuals
• Normal probability plot of residuals
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 13
Diagnostic Residual Plots
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 14
Scatter and Residual Plot
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 16
Nonconstancy of Error Variance
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 17
Presence of Outliers
• Outliers can strongly effect the fitted values of the regression line.
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 18
Outlier effect on residuals
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 19
Nonindependence of Error Terms
• Sequential observations
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 20
Non-normality of Error Terms
• Distribution plots
• Comparison of Frequencies
• Normal probability plot
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 21
Normal probability plot
• For a N(0,MSE1/2) random variable, a good approximation of the expected value of the kth
smallest observation in a random sample of size n is
• A normal probability plot consists of plotting the expected value of the kth smallest observation against the observed kth smallest observation
√MSE[z(k−.375
n+.25 )]
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 22
Omission of Important Predictor Variables
• Example
– Qualitative variable
• Type of machine
• Partitioning data can
reveal dependence on
omitted variable(s)
• Works for quantitative
variables as well
• Can suggest that
inclusion of other inputs
is important
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 23
Tests Involving Residuals
• Tests for randomness
• Tests for constancy of variance
• Tests for outliers
• Tests for normality
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 24
Correlation Test for Normality
• Calculated the coefficient of correlation between residuals ei
and their expected values under normality
• Tables (B.6 in the book) given critical values for the null hypothesis (normally distributed errors) holding
r =√R2
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 25
Tests for Constancy of Error Variance
• Brown-Forsythe test does not depend on normality of error terms.
– The Brown-Forsythe test is applicable to simple
linear regression when
• The variance of the error terms either increases or
decreases with X
• Sample size is large enough to ignore dependencies
between the residuals
• Basically a t-test for testing whether the means of two normally distributed populations are the same
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 26
Brown-Forsythe Test
• Divide X into X1 (the low values of X) and X2
(the high values of X)
• Let ei1 be the error terms for X1 and vice versa
• Let n = n1 + n2
• The Brown-Forsythe test uses the absolute deviations of the residuals around their group median
d1i = |e1i − e1|
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 27
Brown-Forsythe Test
• The test statistic for comparing the means of the absolute deviations of the residuals around the group medians is
where
s2 =
∑(di1−d1)2+
∑(di2−d2)2
n−2
t∗BF
= d1−d2s
√1
n1+ 1
n2
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 28
Brown-Forsythe Test
• If n1 and n2 are not extremely small
approximately
• From this confidence intervals and tests can be constructed.
t∗BF
∼ t(n− 2)
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 29
F test for lack of fit
• Formal test for determining whether a specific type of regression function adequately fits the data.
• Assumptions (usual) : – Y | X
• iid
• normally distributed
• same variance σ
• Requires: repeat observations at one or more X levels (called replicates)
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 30
Example
• 12 similar branches of a bank offered gifts for setting up money market accounts
• Minimum initial deposits were specified to qualify for the gift
• Value of gift was proportional to the specified minimum deposit
• Interested in: relationship between specified minimum deposit and number of new accounts opened
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 31
F Test Example Data and ANOVA Table
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 33
Data Arranged To Highlight Replicates
• The observed value of the response variable for the ith replicate for the jth level of X is Yij.
• The mean of the Y observations at the level X = Xj is Yj
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 34
Full Model vs. Regression Model
• The full model is
where
– µj are parameters j = 1,…,c
– ǫij are iid N(0,σ)
• Since the error terms have expectation zero
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 35
Full Model
• In the full model there is a different mean (a free parameter) for each Xi
• In the regression model the mean responses are constrained to lie on a line
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 36
Fitting the Full Model
• The estimators of µj are simply
• The error sum of squares for the full model therefore is
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 37
Degrees of Freedom
• Ordinary total sum of squares has n-1 degrees of freedom.
• Each of the j terms is a ordinary total sum of squares
– Each then has nj -1 degrees of freedom
• The number of degrees of freedom of SSPE is the sum of the component degrees of freedom
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 38
General Linear Test
• Remember: the general linear test proposes a reduced model null hypothesis
– this will be our normal regression model
• The full model will be as described (one independent mean for each level of X)
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 39
SSE For Reduced Model
• The SSE for the reduced model is as before
– remember
– and has n-2 degrees of freedom
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 41
F Test Statistic
• From the general linear test approach
a little algebra takes us to the next slide
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 42
F Test Rule
• From the F test we know that large values of F* lead us to reject the null hypothesis
• For this example we have
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 43
Example Conclusion
• If we set the significance level to
• And look up the value of the F inv-cdf
• We can conclude that the null hypothesis should be rejected.
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 44
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Frank Wood, [email protected] Linear Regression Models Lecture 7, Slide 45