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Smutek to uczucie, jak gdyby się tonęło, jak gdyby grzebano cię w ziemi.
Call the variable on the vertical axis Y and the one on
the horizontal axis X. A slope line is a linear relationship between X and Y :
yi = α + βxi + εi, i = 1, . . . , n.
(3.27)
Here, α is the intercept and β is the slope of the line. The errors (or deviations from the
line) are denoted as εi and are assumed to have zero mean and finite variance σ2. The task
of finding (α, β) in (3.27) is referred to as a linear adjustment.
In Section 3.6 we shall derive estimators for α and β more formally, as well as accurately
describe what a “good” estimator is. For now, one may try to find a “good” estimator (α, b
β)
b
via graphical techniques. A very common numerical and statistical technique is to use those
α and b
β that minimize:
b
n
X
(α, b
β) = arg min
(y
b
i − α − βxi)2.
(3.28)
(α,β) i=1
The solutions to this task are the estimators:
sXY
b
β
=
(3.29)
sXX
α = y − b
βx.
(3.30)
b
The variance of b
β is:
σ2
V ar( b
β) =
.
(3.31)
n · sXX
The standard error (SE) of the estimator is the square root of (3.31),
σ
SE( b
β) = {V ar( b
β)}1/2 =
.
(3.32)
(n · sXX)1/2
We can use this formula to test the hypothesis that β=0. In an application the variance
σ2 has to be estimated by an estimator σ2 that will be given below. Under a normality
b
assumption of the errors, the t-test for the hypothesis β = 0 works as follows.
One computes the statistic
b
β
t =
(3.33)
SE( b
β)
and rejects the hypothesis at a 5% significance level if | t |≥ t0.975;n−2, where the 97.5%
quantile of the Student’s tn−2 distribution is clearly the 95% critical value for the two-sided
test. For n ≥ 30, this can be replaced by 1.96, the 97.5% quantile of the normal distribution.
An estimator σ2 of σ2 will be given in the following.
b
3.4
Linear Model for Two Variables
97
pullovers data
200
sales (X2)
150
100
80
90
100
110
120
price (X2)
Figure 3.5.
Regression of sales (X1) on price (X2) of pullovers.
MVAregpull.xpl
EXAMPLE 3.10 Let us apply the linear regression model (3.27) to the “classic blue” pullovers.
The sales manager believes that there is a strong dependence on the number of sales as a
function of price. He computes the regression line as shown in Figure 3.5.
How good is this fit? This can be judged via goodness-of-fit measures. Define
y
α + b
βx
bi = b
i,
(3.34)
as the predicted value of y as a function of x. With y the textile shop manager in the above
b
example can predict sales as a function of prices x. The variation in the response variable
is:
n
X
nsY Y =
(yi − y)2.
(3.35)
i=1
98
3
Moving to Higher Dimensions
The variation explained by the linear regression (3.27) with the predicted values (3.34) is: n
X(ybi − y)2.
(3.36)
i=1
The residual sum of squares, the minimum in (3.28), is given by:
n
X
RSS =
(yi − ybi)2.
(3.37)
i=1
An unbiased estimator σ2 of σ2 is given by RSS/(n − 2).
b
The following relation holds between (3.35)–(3.37):
n
n
n
X
X
X
(yi − y)2 =
(y
(y
y
bi − y)2 +
i − bi)2,
(3.38)
i=1
i=1
i=1
total variation = explained variation + unexplained variation.
The coefficient of determination is r2:
n
P (ybi − y)2
explained variation
r2 = i=1
=
·
(3.39)
n
total variation
P (yi − y)2
i=1
The coefficient of determination increases with the proportion of explained variation by the
linear relation (3.27). In the extreme cases where r2 = 1, all of the variation is explained by
the linear regression (3.27). The other extreme, r2 = 0, is where the empirical covariance is
sXY = 0. The coefficient of determination can be rewritten as
n
P (yi − ybi)2
r2 = 1 − i=1
.
(3.40)
n
P (yi − y)2
i=1
From (3.39), it can be seen that in the linear regression (3.27), r2 = r2
is the square of
XY
the correlation between X and Y .
EXAMPLE 3.11 For the above pullover example, we estimate
α = 210.774
and
b
β = −0.364.
b
The coefficient of determination is
r2 = 0.028.
The textile shop manager concludes that sales are not influenced very much by the price (in
a linear way).
3.4
Linear Model for Two Variables
99
pullover data
sales
90
100
price
Figure 3.6. Regression of sales (X1) on price (X2) of pullovers. The overall
mean is given by the dashed line.
MVAregzoom.xpl
The geometrical representation of formula (3.38) can be graphically evaluated using Fig-
ure 3.6. This plot shows a section of the linear regression of the “sales” on “price” for the
pullovers data. The distance between any point and the overall mean is given by the distance
between the point and the regression line and the distance between the regression line and
the mean. The sums of these two distances represent the total variance (solid blue lines
from the observations to the overall mean), i.e., the explained variance (distance from the
regression curve to the mean) and the unexplained variance (distance from the observation
to the regression line), respectively.
In general the regression of Y on X is different from that of X on Y . We will demonstrate
this using once again the Swiss bank notes data.
EXAMPLE 3.12 The least squares fit of the variables X4 (X) and X5 (Y ) from the genuine
bank notes are calculated. Figure 3.7 shows the fitted line if X5 is approximated by a linear
100
3
Moving to Higher Dimensions
Swiss bank notes
11
10
9