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\begin_body

\begin_layout Part*
Chapter 11.
 Linear Statistical Models 
\end_layout

\begin_layout Chapter*
11.1 The Method of Least Squares 
\end_layout

\begin_layout Standard
When each observation from an experiment is a pair of numbers, it is often
 important to try to predict one of the numbers from the other.
 Least squares is a method for constructing a predictor of one of the variables
 from the other by making use of sample of observed pairs.
\end_layout

\begin_layout Section*
Fitting a Straight Line.
\end_layout

\begin_layout Section*
The Least-Squares Line
\end_layout

\begin_layout Standard
One method of constructing a straight line to fit observed values is called
 the method of least squares, which chooses the line to minimize the sum
 of the squares of the vertical deviations of all the points from the line.
 We shall now study the method of least squares in more detail.
\end_layout

\begin_layout Theorem*
11.1.1.
 Least Squares.
 Let 
\begin_inset Formula $(x_{1},y_{1}),...,(x_{n},y_{n})$
\end_inset

 be a set of n points.
 The straight line that minimizes the sum of the squares of the vertical
 deviations of all the points from the line has the following slope and
 intercept:
\end_layout

\begin_layout Theorem*
\begin_inset Formula 
\[
\hat{\beta}_{1}=\frac{\sum_{i=1}^{n}(y_{i}-\bar{y})(x_{i}-\bar{x})}{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}\text{ (11.1.1)}
\]

\end_inset


\end_layout

\begin_layout Theorem*
\begin_inset Formula 
\[
\hat{\beta}_{0}=\bar{y}-\hat{\beta}_{1}\bar{x}
\]

\end_inset


\end_layout

\begin_layout Theorem*
where 
\begin_inset Formula $\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}$
\end_inset

 and 
\begin_inset Formula $\bar{y}=\frac{1}{n}\sum_{i=1}^{n}y_{i}$
\end_inset

.
\end_layout

\begin_layout Theorem*
So 
\begin_inset Formula $y=\beta_{0}+\beta_{1}x$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Definition*
11.1.1 Least-Squares Line.
 Let 
\begin_inset Formula $\hat{\beta}_{0}$
\end_inset

 and 
\begin_inset Formula $\hat{\beta}_{1}$
\end_inset

 be as defined in (11.1.1).
 The line defined by the equation 
\begin_inset Formula $y=\hat{\beta}_{0}+\hat{\beta}_{1}x$
\end_inset

 is called the least-squares line.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Section*
Fitting a Polynomial by the Method of Least Squares
\end_layout

\begin_layout Standard
Suppose now that instead of simply fitting a straight line to 
\begin_inset Formula $n$
\end_inset

 plotted points, we wish to fit polynomial of degree 
\begin_inset Formula $k$
\end_inset

 (
\begin_inset Formula $k\ge2)$
\end_inset

.
 Such a polynomial will have the following form: 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
y=\beta_{0}+\beta_{1}x+\beta_{2}x^{2}+...+\beta_{k}x^{k}\text{ (11.1.6)}
\]

\end_inset


\end_layout

\begin_layout Standard
As before, these equations in Eq.
 (11.1.8) are called the normal equations.
 If the normal equations have a unique solution, that solution provides
 the minimum value for 
\begin_inset Formula $Q$
\end_inset

 (in Eq (11.1.17)).
 A necessary and sufficient condition for unique solution is that the determinan
t of the 
\begin_inset Formula $(k+1)(k+1)$
\end_inset

 matrix formed by the coefficients of 
\begin_inset Formula $\beta_{0},...,\beta_{k}$
\end_inset

 in Eq.
 (11.1.8) is not zero.
\end_layout

\begin_layout Section*
Fitting a Linear Function of Several Variables.
\end_layout

\begin_layout Standard
Suppose that for each patient 
\begin_inset Formula $i$
\end_inset

 
\begin_inset Formula $(i=1,...,n)$
\end_inset

 we measure her reaction 
\begin_inset Formula $y_{i}$
\end_inset

 to drug 
\begin_inset Formula $B$
\end_inset

, her reaction 
\begin_inset Formula $x_{i1}$
\end_inset

 to drug 
\begin_inset Formula $A$
\end_inset

, and also her values 
\begin_inset Formula $x_{i2},...,x_{ik}$
\end_inset

 for the other variables.
 Suppose also that in order to fit these observed values for 
\begin_inset Formula $n$
\end_inset

 patients, we wish to consider a linear function having the form 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
y=\beta+\beta_{1}x_{1}+....+\beta_{n}x_{n}(11.1.11)
\]

\end_inset


\end_layout

\begin_layout Standard
It should be noted that the problem of fitting a polynomial of degree 
\begin_inset Formula $k$
\end_inset

 involving only one variable, as specified by Eq.
 (11.1.6), can be regarded as a special case of the problem of fitting a linear
 function involving serveral variables, as specified by Eq.
 (11.1.11).
 To make Eq.(11.1.11) applicable to the problem of fitting a polynomial having
 the form given in Eq.
 (11.1.16), we define the 
\begin_inset Formula $k$
\end_inset

 variables 
\begin_inset Formula $x_{1},...,x_{k}$
\end_inset

 simply as 
\begin_inset Formula $x_{1}=x,x_{2}=x^{2},...,x_{k}=x^{k}$
\end_inset

.
 
\end_layout

\begin_layout Section*
Summary 
\end_layout

\begin_layout Standard
The method of least squares allows the calculation of a predictor for one
 variable (
\begin_inset Formula $y)$
\end_inset

 based on one or more other variables (
\begin_inset Formula $x_{1},...,x_{k}$
\end_inset

) of the form 
\begin_inset Formula $\beta_{0}+\beta_{1}x_{1}+....+\beta_{k}x_{k}$
\end_inset

.
 The coefficients 
\begin_inset Formula $\beta_{0},...,\beta_{k}$
\end_inset

 are chosen so that the sum of squared difference between observed values
 of 
\begin_inset Formula $y$
\end_inset

 and observed values of 
\begin_inset Formula $\beta+\beta_{1}x_{1}+....+\beta_{k}x_{k}$
\end_inset

 is as small as possible.
 Algebraic formaluas for the coefficients are given for the case 
\begin_inset Formula $k=1$
\end_inset

, but most statistic computer software will calculate the coefficients more
 easily.
\end_layout

\end_body
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