# Interpret Log Transformations in Linear Regression

The following table summarizes how to interpret a linear regression model with logarithmic transformations:

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?

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