Day 2 · Interactive

Link functions: what a coefficient really means

A GLM runs a straight line on the link scale, then bends it back to the response scale. Drag the coefficients and watch how the same slope means an additive, multiplicative, or odds effect depending on the link.

Show the plotted values as a table

Things to try

  1. Identity is a straight line. With the identity link the response is the linear predictor: a 1-unit change in x always adds β₁, everywhere.
  2. Log link is multiplicative. Switch to Log. Now a 1-unit change in x multiplies the mean by e^β₁. Notice the curve bends and can never go negative, which is perfect for counts.
  3. Logit link is S-shaped. Switch to Logit. The response is a probability trapped between 0 and 1. The same β₁ moves the probability a lot in the middle and barely at all near 0 or 1. That is why we talk about odds ratios, not probability differences.
  4. The "divide by 4" rule. On the logit, drag x to where the curve is steepest (p ≈ 0.5) and compare the local probability change to β₁/4.
  5. Compounding. Set Δx to 3 on the log link. The multiplier over three units is (e^β₁)³ = e^{3β₁}, not 3·e^β₁. Additive on the link scale, multiplicative on the response scale.
  6. Two pictures of one model. The lower panel is always a straight line, whatever the link. The upper panel is that same line pushed through the inverse link. A GLM never fits a curve; it fits a line and bends it back.

For 16 worked biological examples, see the log/logit intuition guide. ← Back to the Day 2 lab