In signal detection theory, one typically uses "signal" and "no signal" responses to analyze the data (that is, the analysis is based on a discrete choice for each trial, generating the four possible outcomes "hit", "miss", "false alarm" and "correct rejection").
Is there any standard way to also incorporate response time into the signal detection analysis?
I'd suggest checking out the Linear Ballistic Accumulator (Donkin et al., 2011) model for a scenario like this. While LBA can be used to model any number of alternatives in a speeded choice task, to model signal detection you'd want to model just two accumulators, one for the "signal" response and one for the "no signal" response. With this scenario, response bias will manifest as a difference between the accumulators in terms of their distance from starting point to criterion, while discriminability will manifest as the mean drift rate (which will probably be the same across the two accumulators).
References
Donkin, C., Brown, S., & Heathcote, A. (2011). Drawing conclusions from choice response time models: A tutorial using the linear Ballistic accumulator. Journal of Mathematical Psychology, 55, 140-151.
In addition to Mike's answer, see the Ratcliff diffusion model E.g.:
Ratcliff, R., & Rouder, J. N. (1998). Modeling response times for two–choice decisions. Psychological Science, 9, 347–356.
Ratcliff, R., & Tuerlinckx, F. (2002). Estimating parameters of the diffusion model: Approaches to dealing with contaminant reaction times and parameter variability. Psychonomic Bulletin & Review, 9, 438–481.
You should probably also check out: Pleskac & Busemeyer (2010). Two-stage dynamic signal detection: A theory of choice, decision time, and confidence. Psychological Review.
Also, I believe Busemeyer has a dynamic signal detection theory paper but I don't know that it has been published. The Pleskac & Busemeyer paper probably draws on this unpublished manuscript.
I believe you may be looking for Fuzzy SDT which allows for the incorporation of response time into SDT, among other things:
Hancock, P.A., Masalonis, A.J., & Parasuraman, R. (2000) "On the theory of fuzzy signal detection: theoretical and practical considerations" Theor. Issues In Ergon. Sci. 1(3):207-230 pdf
As per self-regulatory theory people can have two types of regulatory focus: promotion and prevention.
Promotion focus is an eagerness-strategy where if we draw an analogy with SDT, one is more eager to detect signals even though there may be a few false alarms. One is ok with errors of commission. This type of motivation will reflect in seeded responses and is in direct opposition to accuracy that is modeled by the other preventive focus.
In preventive focus, you are more vigilant and on-guard , more focused on avoiding failure or in terms of SDT, ready to miss the signal but not miss out a correct rejection. The more cautious approach will lead to less fast repose but more accurate response bias.
Thus, the way I think speed or response time is to be incorporated in SDT is by pitting it against accuracy and it measuring and indicating the propensity to detect signal even at the cost of false alarms.The more the response time is less for a subject the more eagerness or promotion related focus he has - of detecting signal or achieving success.
Of course this is lay mans integration of response time/ speed with SDT; if you are looking for more mathematical models , perhaps the other answers will work better.
