My only new the ropes by 19mos. but refused to try, even tho' all her friends were trained. Just like yours, she'd refuse at home. Her doc was brilliant. With strong-willed ones, you have to act cool as a cucumber, even tho' you're frustrated out of your gourd. When at home, encourage her when she goes through the motions, even tho' no results. Say: "no big deal, at least next time, if you have to go, you know what to do." Act poker-faced and unemotional when "changing" her. [The idea is she gets NO attention over this.] Make sure her pull-ups are NOT the cute, soft ones, but the boring cheap plain ones. Tell her you can't afford the fancy ones. When shopping, make sure you "accidentally" walk by the cute cotton underwear dept. Let her browse if she wants. Mention that she can get the cool ones when she's ready. [My girl found the character panties she couldn't live w/out.] It's amazing how quickly this worked for us. The hardest part is covering up your own frustration. It is ESSENTIAL that you act like all this is no big deal. It will work wonders in your relationship all the way through. Strong will is good....in this world, it is a strength for the child. A definite challenge for the parents, but a blessing in disguise.
Good luck, and stay "light",