72
7 Regression Models and Hypothesis Testing
Here we introduce the matrix J that maps y onto x to explicitely show that the fit
parameters x depend linearly on the measurements y. We denote the matrix by the
symbol J to remind us that it has the same function as a Jacobian in a coordinate
transformation. In our case the transformation is linear, and consequently the Jacobian
has constant matrix elements.
The error bars and the covariance matrix of the fit parameters x j can be calculated by standard error propagation techniques from the covariance matrix of the
measurements y i , which we is given by C i j (y) = =y i y j . The analysis is based
on the realization that the covariance matrix C i j (y) is defined through the second
moments of the deviations y i . On its diagonal C i j (y) contains the squared error
bars of the individual measurements σ
2
i and the off-diagonal elements carry information about correlations among the measurements. If, on the other hand, the measurements are uncorrelated, as we assume them to be, all off-diagonal elements are
zero. In our example C i j (y) is simply the square of the inverse of the matrix ,
namely C(y) =
2
−1 , which has σ
2
1 , . . . , σ
2
n on its diagonal. From (7.5) follows
that small deviations of the measurements y i lead to small deviations of the model
parameters x = J y. This observation allows us to calculate the covariance matrix
C kl (x) = =x k x l . A little algebra then leads to
C(x) = J C(y)J
t
,
(7.6)
which describes how covariance matrices transfrom under a change of variables
given by (7.5). Furthermore, inserting the definition of J from (7.5) in the previous
equation, we find after some additional algebra
C(x) = (A
t
2 A)
−1
,
(7.7)
which allows us to find the covariance matrix C(x) of the fit parameters x j by simple
matrix operations from the error bars of the initial measurements that are buried in
and the system matrix A that initially defined the problem in (7.2).
In order to illustrate the methodology, let us consider a few examples of regression
models taken from several disciplines.
7.2 Examples
Here we illustrate how we can use regression methods to extract information from
data provided in tabular form, either to obtain quantitative numbers or to test some
model.
Our first example addresses the question whether education actually pays off. Do
extra years of education increase the earnings and if so, at which rate. Moreover,
can we find a quantitative description of how well education protects from unemployment? We base our analysis on information for 2018 from the US Bureau of
7 Regression Models and Hypothesis Testing
Here we introduce the matrix J that maps y onto x to explicitely show that the fit
parameters x depend linearly on the measurements y. We denote the matrix by the
symbol J to remind us that it has the same function as a Jacobian in a coordinate
transformation. In our case the transformation is linear, and consequently the Jacobian
has constant matrix elements.
The error bars and the covariance matrix of the fit parameters x j can be calculated by standard error propagation techniques from the covariance matrix of the
measurements y i , which we is given by C i j (y) = =y i y j . The analysis is based
on the realization that the covariance matrix C i j (y) is defined through the second
moments of the deviations y i . On its diagonal C i j (y) contains the squared error
bars of the individual measurements σ
2
i and the off-diagonal elements carry information about correlations among the measurements. If, on the other hand, the measurements are uncorrelated, as we assume them to be, all off-diagonal elements are
zero. In our example C i j (y) is simply the square of the inverse of the matrix ,
namely C(y) =
2
−1 , which has σ
2
1 , . . . , σ
2
n on its diagonal. From (7.5) follows
that small deviations of the measurements y i lead to small deviations of the model
parameters x = J y. This observation allows us to calculate the covariance matrix
C kl (x) = =x k x l . A little algebra then leads to
C(x) = J C(y)J
t
,
(7.6)
which describes how covariance matrices transfrom under a change of variables
given by (7.5). Furthermore, inserting the definition of J from (7.5) in the previous
equation, we find after some additional algebra
C(x) = (A
t
2 A)
−1
,
(7.7)
which allows us to find the covariance matrix C(x) of the fit parameters x j by simple
matrix operations from the error bars of the initial measurements that are buried in
and the system matrix A that initially defined the problem in (7.2).
In order to illustrate the methodology, let us consider a few examples of regression
models taken from several disciplines.
7.2 Examples
Here we illustrate how we can use regression methods to extract information from
data provided in tabular form, either to obtain quantitative numbers or to test some
model.
Our first example addresses the question whether education actually pays off. Do
extra years of education increase the earnings and if so, at which rate. Moreover,
can we find a quantitative description of how well education protects from unemployment? We base our analysis on information for 2018 from the US Bureau of