I'm working with a team that is pressing hard to do a conjoint analysis on what is essentially a 2x2 factorial design. I get the feeling, though, that this is just their favourite hammer and not necessarily a good idea, so I'd like to push back in an informed way. What assumptions does a conjoint analysis make about the data? When is it not a good idea? What would it make of a factorial design?
Conjoint analysis is not an analysis method per se, but rather a family of choice-based methods for collecting preference data. These methods include (among others) best-worst scaling (MaxDiff), full profile rating, binary choice or discrete choice experiments, graded pairs and constant sum paired comparisons.
They are typically based on random utility theory, which holds that latent utility ($U$) is a combination of a systematic component ($v$) and a stochastic component ($ \epsilon $): $$ U=v+\epsilon $$ The systematic component, $v$, is assumed to be a function of the attributes levels presented in the conjoint choice tasks, i.e. $ v=f(x_1,...,x_n)$. $\epsilon$ is not observable and makes the analysis probabilistic from the perspective of the analyst. The assumed distribution of $\epsilon$ determines the model used to analyze the choice data. For example, in a binary choice analysis, assuming a normal distribution leads to a probit model, while a EV1 distribution leads to a logit. The coefficients from the choice model represent the part-worth utility associated with a 1-unit change in each attribute level.
If you want more information on CA, you might be interested in this book... Applied Choice Analysis. For more detail on discrete choice specifically, the second edition of Kenneth Train's fantastic book is available online for free... Discrete Choice Methods with Simulation
Obviously the validity of these estimates depends on the quality of the experimental design, but the theory of experimental design in another matter altogether. If you're interested in this, you should visit http://support.sas.com/resources/papers/tnote/tnote_marketresearch.html.
And to add to DarkPrivateer's excellent answer, almost all conjoint studies are based on factorial designs.
In your case I take it that your 2*2 factorial design creates options something like:
Red Nike Red Adidas Blue Nike Blue Adidas
The respondents' evaluation task (choice, ranking, Best-worst) gives a score for each option. Depending on the task, and underlying assumptions of the task & choice processes & error term, and whether data are treated at the individual level or aggregated to create a single model for a whole group, the scores a simply regressed against the components (attributes) of the options (here colour and brand) to infer the value of each attribute level.
As with Regression, ANOVA, etc., the technique is robust to moderate violations of the underlying statistical assumptions.
If your problem is a 2*2 factorial as I have presented, then the final data-gathering process may be the same whether you call it conjoint analysis or not. Conjoint Analysis more often is used when there are many attributes, each with several attribute levels. This usually calls for fractional factorial designs, or some sort of dynamically changing evaluation set if you want to get fancy.
If your team is advocating using a branded conjoint analysis technique such as Sawtooth then they're just using a sledge hammer to crack a wallnut. Or they don't understand what they're doing, but they do know how to press buttons on a black box.