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When we say "Linear regression" do we think of the model being linear or the hypothesis function being linear?

When we think only of linear regression do we think of the function being linear?

When we say "Logistic regression is also a linear regression." do we think of the model being linear?

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Linear regression assumes that there is approximately a linear relationship between a predictor variable $X$ and a quantitative response $Y$. Mathematically, we can write this linear relationship as $Y ≈ β_0 + β_1X$.

In many situations, the response variable $Y$ is instead qualitative (e.g. categorical). Predicting a qualitative response for an can be referred to as classifying. Logistic regression is one of classifiers. Logistic regression models the probability that $Y$ belongs to a particular category. Mathematically, we can write logistic relationship as $P(X) ≈ β_0 + β_1X$.

In both cases $X$ is liner variable.

See for details An Introduction to Statistical Learning with Applications in R by G. James, D. Witten, T. Hastie and R. Tibshirani

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  • $\begingroup$ Minor quibble: in logistic regression we model a transformation $g(P(Y = 1 \vert X))$ as a linear function of $X$, where $g : (0,1) \rightarrow \mathbb R$ is our link function. $\endgroup$
    – jld
    Sep 2, 2016 at 14:43
  • $\begingroup$ Linear regression does not assume that $Y$ and $X$ are linearly related. Linear regression models that the response variable is linear with respect to the parameters i.e. $\beta$. $Y = \beta_0 + \beta_1 {x_1}^2 + \beta_2 \log({x_2})$ is a valid equation for a linear regression model. $\endgroup$ Oct 12, 2016 at 6:43

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