Imagine two people reviewing the same evidence. One rarely chooses the highest confidence rating. The other uses it freely. Their answers differ, but that alone does not tell us who distinguishes signal from noise more accurately.

Signal detection theory separates two ideas: how much the evidence distributions overlap, and where a person places a decision criterion. Under an equal-variance Gaussian model, discriminability is:

[d’=.]

Moving the criterion changes the answers without changing this separation. See this introduction to signal detection theory.

A small example

Take noise with mean 0 and signal with mean 1.5. Give both distributions a standard deviation of 1. The resulting (d’) is 1.5.

Call an observation “signal” when it exceeds a threshold. At a threshold of 0.5, the illustrative hit rate is about 84%, and the false-alarm rate is about 31%. Move the threshold to 1.0: the rates fall to about 69% and 16%.

These are calculated probabilities from the example model, not observations from a study. The stricter rule changes both rates. It does not change (d’).

From two answers to six

A six-point response scale needs five ordered thresholds. Moving these thresholds changes how much probability falls into each category.

This is why I find the picture useful: it separates a change in the response rule from a change in the underlying distributions.

Try it

Move a threshold in the interactive figure. Watch the category probabilities change while the distributions stay fixed. Reset the figure to compare with the starting point.

The demonstration assumes normal distributions with equal variance. Real data may require a different model. Observed ratings alone do not establish why two groups differ.