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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 ***
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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.

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  • $\begingroup$ 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. $\endgroup$ – gung Sep 19 '14 at 2:30
  • $\begingroup$ If you can say -- what does $y$ represent? $\endgroup$ – Glen_b Sep 19 '14 at 9:17
  • $\begingroup$ @Glen_b y is a ratio of two numbers, each is a count on the number of objects. $\endgroup$ – user106113 Sep 19 '14 at 21:33
  • $\begingroup$ Question continues in a new question thread. stats.stackexchange.com/questions/116106/… $\endgroup$ – user106113 Sep 19 '14 at 21:47
  • $\begingroup$ 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. $\endgroup$ – Glen_b Sep 19 '14 at 23:19
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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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  • $\begingroup$ Thanks for the clarification, I totally missed it. If this is the case, how should I test whether at x=0, y is not significantly different from 0? $\endgroup$ – user106113 Sep 19 '14 at 0:11
  • $\begingroup$ Is the point of your analysis to determine if y=0 when x=0? If so, using transformations (eg ln & sqrt) may not be the way to go. Is x=0 within, or close to, the range of x's you have in your dataset? $\endgroup$ – gung Sep 19 '14 at 0:13
  • $\begingroup$ It is one of the reality check of the model, that when x=0, y should be close to zero. This transformation seems to be the only one that satisfy the model assumptions on residuals. And for the second question, my x ranges from 0.2 to 200. $\endgroup$ – user106113 Sep 19 '14 at 0:17
  • $\begingroup$ What's wrong with the residuals? Are they non-normal? Is the variance not constant? Is there a curvilinear relationship that isn't being picked up? Bear in mind that when x=0, ln(x)=-infty. $\endgroup$ – gung Sep 19 '14 at 0:22
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    $\begingroup$ Why not try regressing y on x & x^2? Non-normal residuals usually aren't that big a deal. How far from normal are they & how much data do you have? $\endgroup$ – gung Sep 19 '14 at 0:58

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