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

Transformation | Model | Interpretation |
---|---|---|

No transformations | `Y = β` | A 1 unit increase in X is associated with an average change of β._{1} units in Y |

Log-transformed predictor | `Y = β` | A 1% increase in X is associated with an average change of β_{1}/100 units in Y |

Log-transformed outcome | `log(Y) = β` | A 1 unit increase in X is associated with an average change of 100×β_{1}% in Y |

Log-log model | `log(Y) = β` | A 1% increase in X is associated with an average change of β_{1}% in Y |

Next, we will explain where each of these interpretations comes from.

## 1. For a linear regression model without transformations

`Y = β`_{0} + β_{1}X

### Interpretation

A 1 unit increase in X is associated with an average increase of β_{1}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 = β`_{0} + β_{1} `log(`

X`)`

### Interpretation:

A 1% increase in X is associated with an average change of β_{1}/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**Y**: the value of Y after increasing X by 1%_{new}**Y**: the value of Y before increasing X_{old}

\(ΔY = Y_{new}-Y_{old}\)

\(ΔY = β_0 + β_1 \log(1.01X)-(β_0 + β_1 \log(X))\)

\(= β_0 + β_1 \log(1.01X)-β_0-β_1 \log(X)\)

\(= β_1 [\log(1.01X)-\log(X)]\)

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 β_{1}/100.

## 3. For a log transformed outcome

`log(Y) = β`_{0} + β_{1} X

### Interpretation

A 1 unit increase in X is associated with an average change of 100×β_{1}% 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:

**Y**: the value of Y after increasing X by 1 unit_{new}**Y**: the value of Y before increasing X_{old}

\(\log(Y_{new})-\log(Y_{old}) = β_0 + β_1 (X+1) – (β_0 + β_1 X)\)

\(\log(Y_{new})-\log(Y_{old}) = β_1\)

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 β_{1} < 0.1: \((e^{β_1}-1) × 100 ≈ 100 × β_1\).

The interpretation becomes:

When X increases by 1 unit, Y changes by 100×β_{1}%.

## 4. For a log-log model

`log(Y) = β`_{0} + β_{1} `log(`

X`)`

### Interpretation

A 1% increase in X is associated with an average change of β_{1}% in Y.

### Explanation

A 1% increase in X means that X becomes 1.01X.

Let’s call:

**Y**: the value of Y after increasing X by 1%_{new}**Y**: the value of Y before increasing X_{old}

\(\log(Y_{new})-\log(Y_{old}) = β_0 + β_1 \log(1.01X) – (β_0 + β_1 \log(X))\)

\(\log(Y_{new})-\log(Y_{old}) = β_0 + β_1 \log(1.01X) – β_0 – β_1 \log(X)\)

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 β_{1} < 1: \((1.01^{β_1} – 1) × 100 ≈ β_1\).

The interpretation becomes:

When X increases by 1%, Y changes by β_{1}%.

## 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