Posted: November 17th, 2015

Regression

Regression

1. Consider the simple regression model yi = 5 + ß1xi + ?i with E(?i) = 0, var(?) = s
2
, and
cov(?i
;?j ) = 0. Answer the following questions:
(a) Prove that the least squares estimate of ß1 is ߈
1 =
?
i?
(yi-5)xi
i
x
2
i
(b) Find var(߈
1)
(c) Find E(yi – ߈
1xi)
(d) Find cov(yi – yˆi
, yj – yˆj )
1
2. Consider the simple regression model yi = ß0 + ß1xi + ?i
, ?is are Gaussian with E(?i) = 0,
var(?) = s
2
, and cov(?i
;?j ) = 0
(a) Recall that ˆyi =
?n
j=1 wijyj
, where wij =
1
n + ?
(xi-x¯)(xj-x¯)
n
k=1(xk-x¯)
2 i = 1, · · · , n.
i. Show that ˆyi = ¯y if xi = ¯x, in other words the regression line goes through the
point (¯x, y¯)
ii. Using the expression above (in 2 (a)) to derive var(ˆyi)
iii. Show that var(ˆyi) = var(¯y)
2
3. For the following residuals plots discuss the possible violations of regression model assumptions
and suggest remedial measures
-0.5 0.0 0.5 1.0 1.5 2.0
-10 -5 0 5 10
Fitted values
Residuals
Residuals vs Fitted
9
51 46
-2 -1 0 1 2
-4 -2 0 2 4
Theoretical Quantiles
Standardized residuals
Normal Q-Q
9
51 46
-0.5 0.0 0.5 1.0 1.5 2.0
0.0 0.5 1.0 1.5 2.0
Fitted values
Standardized residuals
Scale-Location
9
51 46
0.00 0.02 0.04 0.06 0.08
-4 -2 0 2 4
Leverage
Standardized residuals
Cook’s distance 1
0.5
0.5
Residuals vs Leverage
51 46
9
0 2 4 6
-1 0 1 2
Fitted values
Residuals
Residuals vs Fitted
77 25
70
-2 -1 0 1 2
-1 0 1 2 3
Theoretical Quantiles
Standardized residuals
Normal Q-Q
2577
70
0 2 4 6
0.0 0.5 1.0 1.5
Fitted values
Standardized residuals
Scale-Location
77 25
70
0.00 0.01 0.02 0.03 0.04 0.05
-2 -1 0 1 2 3
Leverage
Standardized residuals
Cook’s distance
Residuals vs Leverage
77 25
39
3
4. True or False.
(a) Statistically significant correlation or evidence of a statistically significant effect always
implies a causal relationship
(b) Consideration must always be given to the size of the data set as this is related to the
power of the analysis to detect differences in a given size
(c) When working with a categorical covariate, the reference category has to be chosen.
There are two considerations to be made in selecting a reference category, the ease of
interpretation and the number of data points in the category.
(d) To allow a fair comparisons between the different model fits, it is important that the
models are being fitted to the same data set
(e) A principal components analysis is done on the explanatory variables that identify
vectors (i.e., the linear combinations of variables) that account, successively, for the
smallest variation in the observations of the explanatory variables
(f) The principal components analysis is done in complete disregard of observed variability
in the response.
(g) The main limitation of principal components regression lies in the difficulties of interpretation
of the principal components
(h) Aside from designing manipulative experiments to break correlations among explanatory
variables, no technique exists that allows researchers to infer the different functional
relationships between the response and explanatory variables
(i) In addition to fundamental shortcomings with regard to finding the best model, step wise
procedures are known to suffer from a multiple-testing problem
(j) Significance tests based on step wise procedures lead to decreased Type I error rates

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