Suppose we have a regression equation of the form (see Multiple Regression Basic Concepts):
y = b0 + b1x1 + b2x2 + b3x3
Let’s suppose that y, x2, and x3 are continuous variables where y = the annual salary of a person in a company, x2 = number of years the person has worked in the company, and x3 = the number of years of education. Let’s also suppose that x1 is a categorical variable where x1 = 0 for male and x2 = 1 for female. We now describe how to interpret the regression coefficients bi.
Intercept
b0 = the value of y when x1 = x2 = x3 = 0. Thus, b0 = the average salary of a male (x1 = 0) with no education who was just hired by the company. Perhaps no one meets these criteria, but in any case, this coefficient is still useful in making predictions.
Categorical variable coefficient
b1 = the average increase in the value of y when x1 = 1 over that when x1 = 0. Thus, b1 + b0 is the average salary for the average salary of a female (x1 = 1) with no education who was just hired by the company. If b1 > 0 then on average females’ salaries are higher than males’. If b1 < 0 then on average males’ salaries are higher than females’.
Continuous variable coefficient
b2 = the increase in y for each increase of 1 unit in x2 when the other variables (x1 and x3) are held constant. Thus, on average, for each additional year of employment at the company, the salary of the person increases by b2, assuming no change in education (x3 is held constant). This is true for males as well as females (since x1 is also held constant).
The situation is similar for coefficient b3. Suppose b3 = 500, then if the average male’s salary after 10 years in the company is 60,000, the average male’s salary after 11 years is 60,500, and the average salary after 9 years is 59,500.
Note that if x2 and x3 are correlated (which is usually the situation), then the situation becomes more complicated. E.g increasing either of these by one changes the value of the other, and so you won’t be able to keep it constant on average.
Log transformation
Suppose now that your regression model is
ln y = b0 + b1x1 + b2x2 + b3x3
Thus, on average, for each additional year of employment at the company, the log of a person’s salary increases by b2, assuming no change in education or gender (x1 and x3 are held constant). This is equivalent to saying that on average each increase of e units of employment results in one extra unit of salary.
Reference
Grace-Martin, K. (2021) Interpreting regression coefficients. The analysis factor
https://www.theanalysisfactor.com/interpreting-regression-coefficients/