Threshold and Bias: Drawing a Line in the Sand
Making a decision
On the previous page, we explored the first step in signal detection according to SDT: taking a measurement. Now it is time to consider the second step: making a decision. Our model will use the evidence to decide whether it is going to respond ‘present’ or ‘absent’.
According to SDT, a threshold is set, such that on each trial, the evidence is either greater than or less than that threshold. A simple rule is then used to decide on a response: if the evidence is below the threshold respond ‘absent’, and if the evidence is above the threshold respond ‘present’.
Here, it is the model that is responding to the stimulus on each trial based on the evidence and the threshold, instead of you, our intrepid participant.
The box representing the measurement for each trial indicates whether it was a ‘Present’ response or an ‘Absent’ response. The vertical bar represents the threshold.
A higher or lower threshold
In the example above, the threshold was set at the neutral point, where the evidence is equally suggestive of signal and noise. But SDT doesn’t require this. When the threshold is moved to one side or the other, we say the model is exhibiting bias.
The threshold can be set higher, so that stronger evidence must be measured for the model to respond ‘present’. In this case, the model will respond ‘present’ for a narrow range of values, and we say that the model has a conservative bias:
Or the threshold can be set lower, so that even with weak evidence the model will respond ‘present’. In this case, the model will respond ‘present’ for a wider range of values, and we say that the model has a liberal bias:
Note that the use of conservative and liberal here refer to their non-political meanings of “strict or cautious” versus “lenient or tolerant”.
Parameterizing our threshold with bias, c
The location of our threshold determines the response bias of our model, defined as c. The bias, c, lies along a continuum from negative infinity to infinity. Zero indicates a neutral bias. Positive numbers indicate a conservative bias. The larger the positive bias, the more evidence necessary before a ‘present’ response is given. Negative numbers indicate a liberal bias. The larger the negative bias, the less evidence necessary to respond ‘present’.
Explore how the bias determines how little or how much evidence is necessary to respond ‘present’ versus ‘absent’:
The distance from the origin, or neutral point, to the threshold is explicitly labeled with c. This is a live graph, so you can drag the threshold line or its handle to adjust its position. If there are evidence measurements for individual trials, they will change between ‘Present’ responses and ‘Absent’ responses based on the location of the threshold, allowing you to see the effect of Bias on behavior.
If you are wondering why bias is represented with the symbol c, perhaps it is because the threshold is often referred to as a criterion in the work on SDT. However, this reason was not explicitly stated in the original paper introducing c (Ingham, 1970).
Note that when c is zero, the threshold is precisely where the two distributions intersect. In other words, the model will respond ‘present’ whenever it is more likely that the signal is indeed present, and it will respond ‘absent’ whenever it is more likely that the signal is indeed absent. This is not coincidental, and hints at the strong link between SDT and decision making based on maximum likelihood and the likelihood ratio (Creelman, 2015; Peterson et al., 1954). In fact, the early work on SDT usually expressed the bias as β, in terms of the likelihood ratio of signal and noise (Peterson et al., 1954; Tanner & Swets, 1954). It was only later that c was introduced and suggested as a better measure of response bias for most purposes (Ingham, 1970; Macmillan & Creelman, 1990; Stanislaw & Todorov, 1999).