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

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.

• What I meant was click the ASK QUESTION at the top, & ask a totally new question, not edit this one. It will also help to say what x & y are (eg, blood pressure, stock prices, etc), why you need a model of them, & why the model should show y=0 when x=0. – gung - Reinstate Monica Sep 19 '14 at 2:30
• If you can say -- what does $y$ represent? – Glen_b Sep 19 '14 at 9:17
• @Glen_b y is a ratio of two numbers, each is a count on the number of objects. – user106113 Sep 19 '14 at 21:33
• Question continues in a new question thread. stats.stackexchange.com/questions/116106/… – user106113 Sep 19 '14 at 21:47
• The denominator of $y$ can occasionally be smaller than the numerator? It seems to behave like there's an upper boundary near 1 (as you'd see with a count divided by a total count - like "proportion of people with brown eyes"), but at least one of the observations exceeds 1. Trying to understand why it's nearly limited to 1 but not quite. – Glen_b Sep 19 '14 at 23:19

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.