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NAG[g02cac] NAG[nag_simple_linear_regression] - Simple linear regression with or without a constant term, data may be weighted
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Calling Sequence
g02cac(mean, x, y, a, b, a_serr, b_serr, rsq, rss, df, 'n'=n, 'wt'=wt, 'fail'=fail)
nag_simple_linear_regression(. . .)
Parameters
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mean - String;
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On entry: indicates whether nag_simple_linear_regression (g02cac) is to include a constant term in the regression.
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 The regression constant  is included.
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Constraint:   "Nag_AboutMean" or "Nag_AboutZero". .
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x - Vector(1..n, datatype=float[8]);
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Constraint: all the values of  must not be identical. .
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y - Vector(1..n, datatype=float[8]);
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Constraint: all the values of  must not be identical. .
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a - assignable;
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Note: On exit the variable a will have a value of type float.
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On exit: if  then a is the regression constant  , otherwise a is set to zero.
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b - assignable;
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Note: On exit the variable b will have a value of type float.
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On exit: the regression coefficient  .
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a_serr - assignable;
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Note: On exit the variable a_serr will have a value of type float.
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On exit: the standard error of the regression constant  .
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b_serr - assignable;
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Note: On exit the variable b_serr will have a value of type float.
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On exit: the standard error of the regression coefficient  .
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rsq - assignable;
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Note: On exit the variable rsq will have a value of type float.
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On exit: the coefficient of determination,  .
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rss - assignable;
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Note: On exit the variable rss will have a value of type float.
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On exit: the sum of squares of the residuals about the regression.
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df - assignable;
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Note: On exit the variable df will have a value of type float.
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On exit: the degrees of freedom associated with the residual sum of squares.
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'n'=n - integer; (optional)
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Default value: the first dimension of the arrays x, y, wt.
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On entry: the number of observations,  .
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if  ,  ;
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if  ,.
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'wt'=wt - Vector(1..n, datatype=float[8]); (optional)
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Constraint: ![wt = NULL*wt[i] and NULL*wt[i] >= 0.](/support/helpjp/helpview.aspx?si=10001/file08412/math210.png) , for  . .
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'fail'=fail - table; (optional)
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The NAG error argument, see the documentation for NagError.
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Description
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Purpose
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nag_simple_linear_regression (g02cac) performs a simple linear regression with or without a constant term. The data is optionally weighted.
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Description
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nag_simple_linear_regression (g02cac) fits a straight line model of the form,
where  is the expected value of the variable  , to the data points
![(x[1],,,y[1]),(x[2],,,y[2]),..,(x[n],,,y[n])](/support/helpjp/helpview.aspx?si=10001/file08412/math249.png)
such that
![y[i]=a+(bx)[i]+e[i],i=1,2,..,n(n,>,2)](/support/helpjp/helpview.aspx?si=10001/file08412/math253.png)
where the ![e[i]](/support/helpjp/helpview.aspx?si=10001/file08412/math256.png) values are independent random errors. The  th data point may have an associated weight ![w[i]](/support/helpjp/helpview.aspx?si=10001/file08412/math260.png) , these may be used either in the situation when var ![epsilon[i] = sigma^2/w[i]](/support/helpjp/helpview.aspx?si=10001/file08412/math262.png) or if observations have to be removed from the regression by having zero weight or have been observed with frequency ![w[i]](/support/helpjp/helpview.aspx?si=10001/file08412/math264.png) .
The regression coefficient,  , and the regression constant,  are estimated by minimizing
![sum(w[i]*e[i]^2, i = 1 .. n)](/support/helpjp/helpview.aspx?si=10001/file08412/math273.png)
if the weights option is not selected then ![w[i] = 1.0](/support/helpjp/helpview.aspx?si=10001/file08412/math276.png) .
The following statistics are computed:
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the estimate of regression constant  ,
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the estimate of regression coefficient (y[i],-,y))/(∑w[i](x[i],-,x)^2)](/support/helpjp/helpview.aspx?si=10001/file08412/math286.png) ,
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the residual sum of squares [i])^2](/support/helpjp/helpview.aspx?si=10001/file08412/math290.png) ,
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where the weighted means  and  are
![x=(∑w[i]x[i])/(∑w[i]) y=(∑w[i]y[i])/(∑w[i])](/support/helpjp/helpview.aspx?si=10001/file08412/math299.png)
The number of degrees of freedom associated with  is
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![df=∑w[i]-2](/support/helpjp/helpview.aspx?si=10001/file08412/math306.png) where 
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![df=∑w[i]-1](/support/helpjp/helpview.aspx?si=10001/file08412/math311.png) where 
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Note: the weights should be scaled to give the correct degrees of freedom in the case var ![epsilon[i] = sigma^2/w[i]](/support/helpjp/helpview.aspx?si=10001/file08412/math316.png) .
The  value or coefficient of determination
[i],-,(y)[i])^2)/(∑w[i](y[i],-,y)^2)=(∑w[i](y[i],-,y)^2-rss)/(∑w[i](y[i],-,y)^2)](/support/helpjp/helpview.aspx?si=10001/file08412/math323.png)
This measures the proportion of the total variation about the mean  that can be explained by the regression.
The standard error for the regression constant 
![a_serr=sqrt((rss)/(df)(1/(∑w[i]),+,((x)^2)/(∑w[i](x[i],-,x)^2)))=sqrt((rss)/(df)1/(∑w[i])(∑w[i]x[i]^2)/(∑w[i](x[i],-,x)^2))](/support/helpjp/helpview.aspx?si=10001/file08412/math333.png)
The standard error for the regression coefficient 
^2))](/support/helpjp/helpview.aspx?si=10001/file08412/math339.png)
Similar formulae can be derived for the case when the line goes through the origin, that is  .
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Error Indicators and Warnings
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"NE_BAD_PARAM"
On entry, argument mean had an illegal value.
"NE_INT_ARG_LT"
On entry, n must not be less than 1:  if  .
"NE_NEG_WEIGHT"
On entry, at least one of the weights is negative.
"NE_SW_LOW"
On entry, the sum of elements of wt must be greater than 1.0 if  or greater than 2.0 if  .
"NE_WT_LOW"
On entry, wt must contain at least 1 positive element if  or at least 2 positive elements if  .
"NE_X_OR_Y_IDEN"
On entry, all elements of x and/or y are equal.
"NE_ZERO_DOF_RESID"
On entry, the degrees of freedom for the residual are zero, i.e., the designated number of arguments  the effective number of observations.
"NW_RSS_EQ_ZERO"
Residual sum of squares is zero, i.e., a perfect fit was obtained.
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Accuracy
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The computations are believed to be stable.
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Further Comments
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The time taken by the function depends on  . The function uses a two-pass algorithm.
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Examples
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mean := "Nag_AboutMean":
n := 8:
x := Vector([1, 0, 4, 7.5, 2.5, 0, 10, 5], datatype=float[8]):
y := Vector([20, 15.5, 28.3, 45, 24.5, 10, 99, 31.2], datatype=float[8]):
wt := Vector([1, 1, 1, 1, 1, 1, 1, 1], datatype=float[8]):
NAG:-g02cac(mean, x, y, a, b, a_serr, b_serr, rsq, rss, df, 'n' = n, 'wt' = wt):
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