Is this equation:

$$\log{(y)} = a + bx$$

semi-log or log-linear mode (or it is the same thing)?

I have two models: linear (1) and semi-log (2). The values of $R^{2}$, adjusted $R^{2}$, and Standard Error are:

  • Linear: $R^{2} = 0.6780,~\mathrm{adj.}~R^{2} = 0.6513,~~\mathrm{SE}=94.101$
  • Semi-log: $R^{2} = 0.5803,~\mathrm{adj.}~R^{2} = 0.5455,~~\mathrm{SE}=0.5493$

How to interpret this values especially from the second model?


This is an answer to the first part of the question regarding the description of the model: $$\log{(y)} = a + bx.......(1)$$ It is important to distinguish: i) whether a model is linear in the sense of the Classical Linear Regression Model (CLRM), and ii) whether a model has linear functional form. Model (1) is linear in the first sense because it is linear in the parameters $a$ and $b$, and this is not affected by the log of $y$. Similarly, models (2), (3) and (4) below are all linear in the CLRM sense: $$y = a + bx.......(2)$$ $$y = a + b*log(x).......(3)$$ $$log(y) = a + b*log(x).......(4)$$ However, of the above models only Model (2) has linear functional form. Models (1) and (3) could both be said to have semi-log functional form, although it is better I suggest to be more precise and indicate which variable is logged by describing (1) as semi-log (dependent) and (3) as semi-log (independent). The functional form of Model (4) is sometimes described as log-linear and sometimes as double log.

  • 1
    $\begingroup$ @Navi Welcome to the site. I am appreciative of your rapid acceptance of my answer - 3 minutes may be some sort of record!. What most participants do however is allow time both for feedback on answers, via votes and comments, and for any further answers, and then judge which answer is best. In this case, moreover, my answer only addressed the first part of your question. So please feel free to accept another answer, and unaccept mine, if you have second thoughts. $\endgroup$ – Adam Bailey Jul 17 '14 at 22:14

I have never heard the term "semi-log regression" in 20 years. It may be in use in some substantive areas. Log linear analysis is something else - it is used when you have multiple categorical variables.

Both of your models are linear regressions. It's just that the second uses the log of y rather than y. $R^2$ has the same meaning as usual - it is the proportion of variance in y explained by the model. Adjusted $R^2$ is one way of penalizing for complexity. Since your model has only one independent variable, it is very close to the unadjusted $R^2$

EDIT answer to comment For SE of the regression see this article. It is entirely reasonable that this changes a lot when you change the scale of the dependent variable.

  • $\begingroup$ @ Peter Flom Thanks for you answer. How to interpret SE values i.e. such a great difference between this two? Can SE be 94.101 when R2 = 0.6780 is much lower? $\endgroup$ – quirik Jul 17 '14 at 9:41
  • $\begingroup$ What is the SE value? What is it the standard error of? $\endgroup$ – Peter Flom - Reinstate Monica Jul 17 '14 at 15:00
  • $\begingroup$ @Navi to clarify Peter's last comment: in your equation I can off the top of my head think of four standard errors: (1) the SE of $\hat{a}$, (2) the SE of $\hat{b}$ the SE of $\hat{\log(y)}$ (i.e. the predicted value of the regression line), and the SE of $\tilde{\log(y)}$ (i.e. of the predicted value of $\log(y)$ for a given value of $x$. Which SE is your question about? $\endgroup$ – Alexis Jul 17 '14 at 16:39
  • $\begingroup$ @Peter Flom Standard error of regression $\endgroup$ – quirik Jul 17 '14 at 16:43
  • $\begingroup$ OK, I will edit my answer. $\endgroup$ – Peter Flom - Reinstate Monica Jul 17 '14 at 17:37

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