The following table summarizes how to interpret a linear regression model with logarithmic transformations:
| Transformation | Model | Interpretation |
|---|---|---|
| No transformations | Y = β0 + β1 X | A 1 unit increase in X is associated with an average change of β1 units in Y. |
| Log-transformed predictor | Y = β0 + β1 | A 1% increase in X is associated with an average change of β1/100 units in Y |
| Log-transformed outcome | log(Y) = β0 + β1 X | A 1 unit increase in X is associated with an average change of 100×β1% in Y |
| Log-log model | log(Y) = β0 + β1 | A 1% increase in X is associated with an average change of β1% in Y |
Next, we will explain where each of these interpretations comes from.
1. For a linear regression model without transformations
Y = β0 + β1X
Interpretation
A 1 unit increase in X is associated with an average increase of β1 units in Y.
(If you are interested, I wrote a separate article on how to interpret linear regression coefficients when X is binary, categorical, or numerical)
2. For a log transformed predictor
Y = β0 + β1 log(X)
Interpretation:
A 1% increase in X is associated with an average change of β1/100 units in Y.
Explanation
Interpreting the coefficient of log(X) by saying that a 1 unit increase in log(X) is associated with a 1 unit increase in Y is not very helpful. After all, what does it mean to increase log(X) by 1?
Instead, we are going to talk about increasing X by 1%.
Increasing X by 1% means that X becomes 1.01X.
So the question becomes: How does Y change when X becomes 1.01X?
Let’s call:
- ΔY: the change in Y
- Ynew: the value of Y after increasing X by 1%
- Yold: the value of Y before increasing X
\(ΔY = Y_{new}-Y_{old}\)
\(ΔY = β_0 + β_1 \log(1.01X)-(β_0 + β_1 \log(X))\)
\(= β_0 + β_1 \log(1.01X)-β_0-β_1 \log(X)\)
\(= β_1 [\log(1.01X)-\log(X)]\)
And since log(a) – log(b) = log(a/b):
\(ΔY = β_1 log(1.01) = β_1 × 0.01 = β_1/100\)
This is why we said that, when X increases by 1%, Y changes by β1/100.
3. For a log transformed outcome
log(Y) = β0 + β1 X
Interpretation
A 1 unit increase in X is associated with an average change of 100×β1% in Y.
Explanation
We want to know how Y changes when we increase X by 1 unit, i.e. when X becomes (X + 1).
Let’s call:
- Ynew: the value of Y after increasing X by 1 unit
- Yold: the value of Y before increasing X
\(\log(Y_{new})-\log(Y_{old}) = β_0 + β_1 (X+1) – (β_0 + β_1 X)\)
\(\log(Y_{new})-\log(Y_{old}) = β_1\)
And since log(a) – log(b) = log(a/b), then:
\(\log(\frac{Y_{new}}{Y_{old}}) = β_1\)
So, \(\frac{Y_{new}}{Y_{old}} = e^{β_1}\)
Now let’s transform \(\frac{Y_{new}}{Y_{old}}\) to a percent change in Y, by subtracting 1 and multiplying by 100.
So we will do this for both sides of the equation:
\((\frac{Y_{new}}{Y_{old}}-1) × 100 = (e^{β_1}-1) × 100\)
\((\frac{Y_{new}-Y_{old}}{Y_{old}}) × 100 = (e^{β_1}-1) × 100\)
So, when X increases by 1 unit, the percent change in Y will be \((e^{β_1}-1) × 100\).
And when β1 < 0.1: \((e^{β_1}-1) × 100 ≈ 100 × β_1\).
The interpretation becomes:
When X increases by 1 unit, Y changes by 100×β1%.
4. For a log-log model
log(Y) = β0 + β1 log(X)
Interpretation
A 1% increase in X is associated with an average change of β1% in Y.
Explanation
A 1% increase in X means that X becomes 1.01X.
Let’s call:
- Ynew: the value of Y after increasing X by 1%
- Yold: the value of Y before increasing X
\(\log(Y_{new})-\log(Y_{old}) = β_0 + β_1 \log(1.01X) – (β_0 + β_1 \log(X))\)
\(\log(Y_{new})-\log(Y_{old}) = β_0 + β_1 \log(1.01X) – β_0 – β_1 \log(X)\)
And since log(a) – log(b) = log(a/b), then:
\(\log(\frac{Y_{new}}{Y_{old}}) = β_1 \log(1.01)\)
Exponentiating both sides we get:
\(\frac{Y_{new}}{Y_{old}} = e^{β_1 \log(1.01)}\)
\(\frac{Y_{new}}{Y_{old}} = 1.01^{β_1}\)
Now let’s transform \(\frac{Y_{new}}{Y_{old}}\) to a percent change, by subtracting 1 and multiplying by 100.
So we will do this for both sides of the equation:
\((\frac{Y_{new}}{Y_{old}}-1) × 100 = (1.01^{β_1} – 1) × 100\)
\((\frac{Y_{new}-Y_{old}}{Y_{old}}) × 100 = (1.01^{β_1} – 1) × 100\)
So, when X increases by 1%, the percent change in Y will be \((e^{β_1}-1) × 100\).
And when β1 < 1: \((1.01^{β_1} – 1) × 100 ≈ β_1\).
The interpretation becomes:
When X increases by 1%, Y changes by β1%.
References
- Vittinghoff E, Glidden DV, Shiboski SC, McCulloch CE. Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models. 2nd ed. 2012 edition. Springer; 2011.
- Yang J. Interpreting Regression Coefficients for Log-Transformed Variables. https://cscu.cornell.edu/wp-content/uploads/83_logv.pdf