# How to interpret a regression coefficient for the reciprocal of an independent variable?

Does anyone know how to interpret a coefficient when the variable in the model is the reciprocal of the original variable? I have an inverse equation, where $\text{time} = \beta_0 + \beta_1(1/\text{horsepower})$, where $\text{time}$ is how long it takes to accelerate a car. I need to hypothesize the sign of the variable $\text{horsepower}$. But to do that, I need to understand what it means.

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I assume you want to know how long it takes to accelerate to a given speed? – gung Dec 25 '12 at 14:32

The interpretation of a beta is the same whether the variable is in its original form or a reciprocal. Specifically, holding all else equal, a one unit change in the variable (in whatever form it has been entered into the model), will correspond to $\beta_1$ units change in the response.

What you need to understand is the meaning of $1/\text{horsepower}$. I'm not sure what the substantive meaning of this is for the potential acceleration of a car, but to the extent that more horsepower enables a car to accelerate faster, more horsepower means a smaller reciprocal horsepower, so the sign of $\beta_1$ would be negative.

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When fitting a nonlinear relationship it is often better to create a plot showing the predicted relationship and interpret the plot than to try to interpret the slope directly (in this case it is not too hard, but the plot will probably still be more meaningful).

Choose a sequence of horsepower values within the meaningful range of the variable (and the range of the actual data you have), then find the predicted times for each based on the fitted model. Now plot those values and interpret the plot, what happens with time as horsepower increases?

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