Interpretations for SLR models
Mathematical Justification
SLR
For the typical SLR model,
\[ \widehat{y}(x^*) = \beta_0 + \beta_1 \cdot x^* \]
So then,
\[\begin{align*} \widehat{y}(x^* + 1) &= \beta_0 + \beta_1 \cdot (x^* + 1) \\ &= \beta_0 + \beta_1 \cdot x^* + \beta_1 \\ &= \widehat{y}(x^*) + \beta_1 \end{align*}\]
So a one unit increase in \(x\) results in a \(\beta_1\) unit increase in \(y\).
log-transformed explanatory variable
If the explanatory variable is log-transfored, then:
\[ \widehat{y}(x^*) = \beta_0 + \beta_1 \cdot \log{x^*} \]
So consider multiplying the explanatory variable by \(c\),
\[\begin{align*} \widehat{y}(cx^* ) &= \beta_0 + \beta_1 \cdot \log{(cx^*)} \\ &= \beta_0 + \beta_1 \left( \log{c} + \log{x^*} \right) \\ &= \beta_0 + \beta_1 \cdot \log{c} + \beta_1 \cdot \log{x^*} \\ &= \widehat{y}(x^*) + \beta_1 \cdot \log{c} \end{align*}\]
So a \(c\)-fold increase in \(x\) results in a \(\beta_1 \cdot \log{c}\) increase in \(y\).
Note that trying to add 1 in this case isn’t going to get you anywhere, because you’ll end up with \(\log{(x^* + 1)}\) and we can’t separate that using basic log properties!
log-transformed response variable
If the response variable is log-transfored, then:
\[ \widehat{\log{y(x^*)}} = \beta_0 + \beta_1 \cdot x^* \]
So then consider adding 1 to the explanatory variable. Then:
\[\begin{align*} \widehat{\log{y(x^* + 1)}} &= \beta_0 + \beta_1 \cdot (x^* + 1) \\ &= \beta_0 + \beta_1 \cdot x^* + \beta_1 \\ &= \widehat{\log{y(x^*)}} + \beta_1 \\ \exp{ \left( \widehat{\log{y(x^* + 1)}} \right) } &= \exp{ \left( \widehat{\log{y(x^*)}} + \beta_1 \right) } \\ \widehat{y(x^* + 1)} &= \exp{ \left( \widehat{\log{y(x^*)}} \right) } \cdot \exp{\beta_1} \\ &= \widehat{y(x^*)} \cdot e^{\beta_1} \end{align*}\]
So a one unit increase in \(x\) results in a \(e^{\beta_1}\) multiplicative change in the median value of \(y\).
Double log-models
Now combine the previous two approaches.
\[ \widehat{\log{y(x^*)}} = \beta_0 + \beta_1 \cdot \log{x^*} \]
And then,
\[\begin{align*} \widehat{\log{y(cx^* )}} &= \beta_0 + \beta_1 \cdot \log{cx^*} \\ &= \beta_0 + \beta_1 \left( \log{c} + \log{x^*} \right) \\ &= \beta_0 + \beta_1 \cdot \log{c} + \beta_1 \cdot \log{x^*} \\ &= \widehat{\log{y(x^*)}} + \beta_1 \cdot \log{c} \\ \exp{ \left( \widehat{\log{y(cx^* )}} \right) } &= \exp{ \left( \widehat{\log{y(x^* )}} + \beta_1 \cdot \log{c} \right) } \\ \widehat{y(cx^* )} &= \exp{ \left( \widehat{\log{y(x^*)}} \right) } \cdot \exp{ \left( \beta_1 \cdot \log{c} \right) } \\ &= \widehat{y(x^*)} \cdot c^{\beta_1} \end{align*}\]
So a \(c\)-fold increase in the explanatory variable results in a \(c^{\beta_1}\)-fold increase in the median values of the response variable.