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 }
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.
To determine an appropriate choice for the link function g, we need to find a form of g such that g(μ(x)) can be expressed as a linear function, x^Tβ, for some β.
Given that μ(x) = (ax_1 + bx_2)^3, we can analyze the options:
1. g(μ) = log(μ): This is not an appropriate choice because log(μ) is not a linear function of μ.
2. g(μ) = e^μ: This is not an appropriate choice because e^μ is not a linear function of μ.
3. g(μ) = μ^3: This is an appropriate choice because (μ(x))^3 can be written as a linear function of x. Specifically, we have μ^3 = (ax_1 + bx_2)^3 = (x_1a + x_2b)^3 = x^Tβ, where β = [a, b].
4. g(μ) = ∛(μ): This is not an appropriate choice because ∛(μ) is not a linear function of μ.
Therefore, the appropriate choice for the link function g is g(μ) = μ^3.
Let's move on to the second part of the question.
Given that μ(x) = 2^(ax_1), we can analyze the options:
1. g(μ) = log_2(μ): This is an appropriate choice because log_2(μ) is a linear function of μ. Specifically, we have log_2(μ) = log(μ) / log(2) = (ax_1) / log(2) = x^Tβ, where β = [a / log(2), 0].
2. g(μ) = ln(μ): This is not an appropriate choice because ln(μ) is not a linear function of μ.
3. g(μ) = e^μ: This is not an appropriate choice because e^μ is not a linear function of μ.
4. g(μ) = μ^3: This is not an appropriate choice because μ^3 is not a linear function of μ.
5. g(μ) = ∛(μ): This is not an appropriate choice because ∛(μ) is not a linear function of μ.
Therefore, the appropriate choice(s) for the link function g are g(μ) = log_2(μ).
I hope this clarifies the appropriate choices for the link function g in both scenarios!