Relaxing the equal variance assumption
Back when we first discussed Measurement & Sensitivity, we mentioned that the noise distribution and the signal-plus-noise distribution are assumed to have equal variance. It turns out that this assumption is often violated in real world scenarios where we would like to apply SDT. For example, studies of recognition memory consistently find evidence of unequal variance (Glanzer, Kim, Hilford, & Adams, 1999; Ratcliff, Sheu, & Gronlund, 1992). This is a problem for equal variance SDT because it can’t account for these observed patterns of performance.
Fortunately, SDT can be generalized to account for distributions with unequal variance. First, we will quantify the relative variance for the signal-plus-noise distribution as compared to the noise distribution as σ. When the variance, σ, is equal to one then the distributions have equal variance. When it is smaller than one, the signal-plus-noise distribution has a smaller variance, and when it is larger than one, the signal-plus-noise distribution has a larger variance.
You can explore this here:
Our new parameter Variance, σ, has been added to the model diagram to indicate it’s relationship to the width of the Signal-plus-Noise Distribution. You can now drag that distribution up and down to adjust the Variance. Note that when you pull it up, this makes the distribution taller but thinner and thus σ is smaller, and when you drag it down, this makes the distribution shorter but wider and thus σ is larger.
As in previous model explorations, the table of outcomes, ROC space, and the model diagram are linked so they all update simultaneously.
You will notice that this example was set up to show zROC space by default. Observe what happens to the iso-sensitivity curve as you adjust the variance… It changes slope! Granted, you can still see that it changes, somehow, in non-transformed ROC space, but zROC space turns that complex shape-shift into a simple change in slope. Indeed, in the literature, unequal variance is usually determined by testing if the slope of an empirically derived zROC curve is not equal to one (Glanzer et al., 1999; Ratcliff et al., 1992).
Calculating unequal variance d′ and c
As suggested by the diagram of the SDT model, we still have a formal mathematical relationship between performance and model parameters, it’s just a bit more complicated now, since it has to account for the inequality of variance.
Here is the equation for sensitivity, taking variance into account:
And here is the equation for bias, taking variance into account:
Iso-contours with unequal variance
You can also get a sense of the effects of unequal variance on the relationship of sensitivity and bias to hit rate and false alarm rate by looking at how the iso-sensitivity and iso-bias contours change as the variance is manipulated:
In this example, you can only change the Variance in the SDT model, to make it easier to focus on its effects on the iso-contours.
Calculating hit rate and false alarm rate with unequal variance
To make model predictions with unequal variance SDT, we will need to calculate hit rate, taking variance into account:
And we will need to calculate false alarm rate, taking variance into account:
Spurious correlation when unequal variance data analyzed with equal variance model
As a final note, let’s observe what can happen if our data is produced by a mechanism that follows unequal variance SDT, but we erroneously analyze it using the equal variance model.
Imagine we have a number of conditions and for each condition we calculate sensitivity and bias using equal variance SDT. According to our calculations, d′ and c are both different for each condition, and curiously, as shown in the example below, they are correlated with each other. We might be tempted to consider what additional process may be acting in our participants to cause this linkage of sensitivity and bias. But, in fact, this seeming relationship is spurious. When unequal variance is taken into account, all of the conditions have the same sensitivity and only the bias is varying between conditions.
In ROC space, the single shared iso-sensitivity curve and each individual iso-bias curve show the “true” values based on the unequal variance model shown in the SDT model diagram. On the other hand, the iso-sensitivity contours are from the equal variance model. The spurious values for sensitivity from the equal variance model are different for each point, whereas the correct values all fall on the same curve.
If nothing else, this last example demonstrates the importance of understanding and being aware of the assumptions of a model, so that you can keep your eye out for characteristic patterns suggestive of failures of those assumptions.