# Converting relative effect to absolute effect in log model

I have the following model;

log(daily sales) = intercept + B1*(event dummy) + error

My response variable(daily sales) is basically a daily time series and 'event dummy' is an indicator variable whose value is 1 when a certain event is ON and 0 when OFF.

I have the value of B1 now; say, 0.3. This means that the mean daily sales is multiplied by exp(0.3) as the event dummy gets bigger by 1 unit.

But I have to convert this multiplicative effect into some 'absolute effect' in daily sales. Which is correct among the following suggestions? Or are there any other solution that is more appropriate?

1. (exp(0.3) - 1) * mean daily sales when event dummy is 0
2. ( (exp(0.3) - 1) / exp(0.3) ) * mean daily sales when event dummy is 1

[explanation for the 1st approach] (exp(0.3) - 1) is the growth rate corresponding to 1 unit increase in event dummy. But this 'growth' means the 'growth compared to the mean daily sales when the event is OFF', so the formula came out.

[explanation for the 2nd approach] Likewise, (exp(0.3) - 1) is the growth rate corresponding to 1 unit increase in event dummy. Then, 'mean daily sales when the event is ON' is actually the mean daily sales AFTER the multiplicative effect has taken place. So we have to re-construct the mean daily sales IF the effect wasn't there, and examine the difference between the two. So the formula came out.

## 1 Answer

Let $$Y=$$ daily sale, $$X = 1$$ when a certain event is ON and =0 when OFF.

Then the model is: $$\log(Y) = \beta_0 +\beta_1X +\epsilon$$ where $$\beta_0$$ is intercept and $$\beta_1$$ is the effect of event ON (B1 in your question), $$\epsilon \sim N(0,\sigma^2)$$.

Based on this model, we have: $$E(\log(Y|X=0)) = \beta_0 ==> E(Y|X=0) \approx e^{\beta_0}\qquad \quad \text{ (event OFF)}$$ $$E(\log(Y|X=1)) = \beta_0 + \beta_1 ==>E(Y|X=1) \approx e^{\beta_0 + \beta_1} \qquad \text{ (event ON)}$$

If sample size is large, such that the variances of $$\beta$$s' estimate are small, the approximation is reliable. Otherwise you need to adjust by using the variance.

• Thank you for your answer! So the absolute effect per day is the difference between those two expectations. – th-ahn Nov 10 '18 at 11:09
• The log transformation is used. So the difference $E(Y|X=1)-E(Y|X=0)$ depends on $E(Y|X=0)$. The best way to describe the effect of ON and OFF is the ratio of the means. $E(Y|X=1)/E(Y|X=0) = e^{\beta_1}$. It means given mean sales when OFF is $Z$, then the means sale will be $Ze^{\beta_1}$ when ON. – user158565 Nov 10 '18 at 15:44