Brown Sticks???
Are they flexible? What I mean is can you bend them a little, or do they snap like dry twigs?
If they do, then they are dead parts.
Most mini-and full sized roses I have seen have a good deal of green twigs on them. Rose Trees, stemming from one solid branch, have a slightly brown 'trunk' only because it is thick.
I just ran outside to look at my roses: Here is my 'report':
This years growth is green - all green.
Last years growth (about 1/2 in diameter) is mostly green shot with brown. The heavy trunk of my rose tree which is over an inch in diameter is browner, but still has green in it.
I looked at all my roses, the bush variety and the climbing 'old fashion' variety - there are no purely brown twigs that are alive.
I tell you all of these so you can better judge if we are talking about dead wood or green (live) wood on your rose.
I assume that since there are no leaves on these stems that these are already dead. Prune all the way back to the first notch (where leaves would grow) of the stem where it is still green, leaving the notch.
While you are trimming off dead branches I suggest you seep a cup of tea, preferably Lipton's normal tea - put two tea bags in the cup, I like strong tea.
I would suggest that you look under the leaves, do they look normal? Are they splotchy with yellow spots? Brown spots? Are there any critters nipping at the roses nose? If yes, treat accordingly. Splotchy colour may be disease. check out
http://ia.essortment.com/rosesdiseases_rksn.htm for more information about common rose diseases. If you run a picture search of Rose Diseases you may find pictures where you can compare.
Assuming that the rose is healthy, not infested, no disease, maybe it is time to think about a bigger pot.
If you can, pick up the pot and look at it's holes, do you see roots? If your pot doesn't have holes, then you must replant into one that does. Roses like moist, but drained soil.
If you see roots, time to move into a larger pot.
I have found when I propagate roses that having the soil moist, and going up to a pot which is 1 to 2 inches wider(around the sides of the starting pot) and about an inch deeper works best. I DO NOT break up the roots, I pull the root ball out whole, laying it on a layer of soil in the new post, and filling around, lightly tamping the soil down.
By the Time you are finished pruning, that tea should be cool enough to drink, Now, pour it into the rose's pot. (Not hot, but warm is ok)
I give my indoor and potted plants a cup a tea once a month - they love it.
Cheers
David
I have mini roses that are in a pot right now. The plant is alive, because, there are a few green leaves. There was also a rose, but, roses don't last long, you know. 
One half has the "brown sticks", and the other half has leaves and stuff. You said to prune all the way back to the first notch, where it is green. Well...I don't see any green on these "sticks"... In fact, 2 of the little "sticks" came right out..like they were never even "attached"...The others seem to be more "attached", though.
