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.. note:: A particularly useful tutorial for the theory of least squares are Chapters 5, 9 and 10 of the book `Introductory Statistics with R <https://www.google.ca/search?q=Introductory+Statistics+with+R+by+Dalgaard>`_ by Dalgaard. You might be able access the PDF version from your company or university's subscription.


The ``lm(...)`` function is the primary tool to build a linear model in R.  The input for this function must be a formula object (type ``help(formula)`` for further info).  In the example below the formula is ``y ~ x``.  This says: "calculate for me the linear model that relates :math:`x` to :math:`y`"; or alternatively and equivalently: "build the linear model where :math:`y` is regressed on :math:`x`".

.. code-block:: s

	x <- c(1, 2, 3, 4, 5)
	y <- c(2, 3, 4, 4, 5)
	model <- lm(y~x)

The output from ``lm`` is a linear model *object*, also called an ``lm`` object.  In R you can get a description of most objects when using the ``summary(...)`` command.

.. code-block:: s

	summary(model)
	Call:
	lm(formula = y ~ x)

	Residuals:
	        1         2         3         4         5 
	-2.00e-01  1.00e-01  4.00e-01 -3.00e-01  2.29e-16 

	Coefficients:
	            Estimate Std. Error t value Pr(>|t|)   
	(Intercept)   1.5000     0.3317   4.523  0.02022 * 
	x             0.7000     0.1000   7.000  0.00599 **
	---
	Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

	Residual standard error: 0.3162 on 3 degrees of freedom
	Multiple R-squared: 0.9423,	Adjusted R-squared: 0.9231 
	F-statistic:    49 on 1 and 3 DF,  p-value: 0.005986

This output gives you the intercept and slope for the equation :math:`y = b_0 + b_1 x` and in this case it is :math:`y = 1.5 + 0.7x`.  The residual standard error, :math:`S_E = 0.3162` and :math:`R^2 = 0.9423`.  
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