Software tutorial/Transformation of data in a linear model

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This is shown by example for a few different types of transformations:

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