Solved – Interpreting intercept for the log model in linear regression in R for small predictor

linear modelrregression coefficients

I have a dataset. Assume that y is the dependent variable and x is the independent variable. My goals for this analysis is mainly on the following hypothesis:

  1. Expecting x=0 to imply y=0
  2. Expecting a significant relationship between x and y

To achieve this, I am trying to determine the best transformation of x and y to fit the best linear model in R. So, the final model I got is $\sqrt y$ against ln(x). When I fit the model in R, I obtain the following for the coefficients:

  Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.319615   0.028743   11.12 2.93e-10 ***
x           0.150139   0.009959   15.08 9.76e-13 ***
---

Questions:

  1. I am trying to interpret the Intercept term. Since the p-value is much less than 5% significance level, can I say that the intercept is significantly different from 0? However, this model is undefined for x=0, hence I'm not sure if this interpretation is valid. I was thinking of will it be OK if I were to refit the linear model for smaller x. < Solved >

  2. To address the above question, the problem as seen from this model is that I can't test for hypothesis 1. Would be very thankful if anyone could provide some help.

Best Answer

The intercept term does not refer to when x=0, since your x is actually ln(x). Instead, the intercept refers to when ln(x)=0, which occurs when the old x=1. At that point (in the new space), $\hat y$ (i.e., $\widehat{\sqrt{y}}$) differs significantly from 0.

It may help you to read this excellent CV thread: Interpretation of log transformed predictor.

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