Software tutorial/Transformation of data in a linear model

<rst> <rst-options: 'toc' = False/> <rst-options: 'reset-figures' = False/> This is shown by example for a few different types of transformations:

.. list-table::

  :widths: 15 10 30
  :header-rows: 1

  * - Description
    - Desired model
    - Formula function in R
  * - Fit only an intercept
    - :math:`y = b_0`
    - ``lm(y ~ 1)``  *the character is a "one"*
  * - Standard, univariate model
    - :math:`y = b_0 + b_1 x`
    - ``lm(y ~ x)``
  * - Force intercept to zero (check the degrees of freedom!)
    - :math:`y = b_1 x`
    - ``lm(y ~ x + 0)``
  * - Transformation of an :math:`x`  
    - :math:`y = b_0 + b_1\sqrt{x}`
    - ``lm(y ~ sqrt(x))``
  * - Transformation of :math:`y`
    - :math:`\log(y) = b_0 + b_1 x`
    - ``lm(log(y) ~ x)``
  * - Transformation of :math:`y`
    - :math:`100/y= b_0 + b_1 x`
    - ``lm(100/y ~ x)``
  * - Transformation of :math:`x`: **+, -, /, and ^ do not work on the right hand side!**
    - :math:`y= b_0 + \dfrac{b_1}{x}`
    - ``lm(y ~ 1/x)`` *will work, but is not doing what you expect*. It is fitting a model :math:`y = b_0`!! Be careful.
  * - Most transformations of :math:`x` must be wrapped in an AsIs ``I()`` operation:
    - :math:`y= b_0 + \dfrac{b_1}{x}`
    - ``lm(y ~ I(1/x))``  *will work as expected*
  * - Another use of the AsIs ``I()`` operation
    - :math:`y= b_0 + b_1 x^2`
    - ``lm(y ~ I(x^2))``
  * - Another use of the AsIs ``I()`` operation
    - :math:`y= b_0 + b_1 (x - \bar{x})`
    - ``lm(y ~ I(x - mean(x)))``

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