Consider random variables \mathbf X= (X_1, X_2) and Y. Assume that the regression function \mu (x_1, x_2) = \mathbb E[Y \; |\; X = (x_1,x_2)] for a pair (X,Y) happens to be \mu (x) = (ax_1 + bx_2)^3. Which of the following is an appropriate choice for a link function g? In other words, for which g is it true that g(\mu (x)) can be written as a linear function, x^ T \beta for some \beta?

g(\mu ) = \log (\mu )

g(\mu ) = e^{\mu }

g(\mu ) = \mu ^3

g(\mu ) = \sqrt[3]{\mu }
unanswered
If instead \mu (x) = 2^{ax_1}, which of the following are appropriate choices for the link function g? Choose all that apply.

g(\mu ) = \log _2(\mu )

g(\mu ) = \ln (\mu )

g(\mu ) = e^{\mu }

g(\mu ) = \mu ^3

g(\mu ) = \sqrt[3]{\mu }

1 answer

For the first question, we want to find a link function g such that g(μ(x)) can be written as a linear function, x^Tβ.

Plugging μ(x) = (ax_1 + bx_2)^3 into each of the given choices for g:

g(μ) = log(μ) does not result in a linear function.

g(μ) = e^μ does not result in a linear function.

g(μ) = μ^3 does result in a linear function, as μ^3 = (ax_1 + bx_2)^3 = (a^3)x_1^3 + 3(a^2b)x_1^2x_2 + 3(ab^2)x_1x_2^2 + (b^3)x_2^3, which is a linear function of x = (x_1, x_2).

g(μ) = ∛(μ) does not result in a linear function.

Therefore, the appropriate choice for a link function g is g(μ) = μ^3.

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For the second question, we want to find appropriate choices for the link function g given μ(x) = 2^(ax_1).

Plugging μ(x) = 2^(ax_1) into each of the given choices for g:

g(μ) = log_2(μ) does result in a linear function, as log_2(μ) = log_2(2^(ax_1)) = ax_1.

g(μ) = ln(μ) does not result in a linear function.

g(μ) = e^μ does not result in a linear function.

g(μ) = μ^3 does not result in a linear function.

g(μ) = ∛(μ) does not result in a linear function.

Therefore, the appropriate choices for a link function g are g(μ) = log_2(μ) and g(μ) = ax_1.