Unfortunately, the prior respondent logicality is simply inaccurate here. Of course positive reinforcement works! I used it all the time to raise my kids and they are of course, exceptional stellar adults! Withhold reinforcement and behavior tends to stop i.e., stop paying experts on this website for answering questions and most would surely leave to do something else with their time. (In fact, most I know who have left have done so because too many customers don't hit the Accept button and reinforce their effort with a payment following their effort)I Anyway, I constantly use positive reinforcement in my everyday work---last week I decided to increase the general level of student commentary in one of my undergraduate classes because the same students were answering all of the questions. Answer? Use positive reinforcement. I simply announced that I would provide one exam point of extra credit of reward (reinforcement) each week to students who offered to answer questions. Five minutes later? The number of raised hands to answer questions increased 80% and remains high to this day. And of course it works with dogs--EVERY competent dog training program uses the same principles of food reward, clickers conditioned or paired with reward, petting, verbal cues--initially -tied to reward. I work alongside the world's leading expert on training of parrots and pet birds and she uses nothing but reinforcement. So of course this stuff works. (Check out the American Humane Society for instance:
I think you have figured out that there may be a link to your younger child being present in the home now and your daughter's refusal to go to bed. You didn't provide enough detail for me to comment about the approaches you are using so far. But obviously, the sticker system is not in fact, reinforcing to her. For example, there is almost surely a problem with the proportion of stickers she has to earn for each 'prize', and the prize may be of no value to her. So I suspect that nothing you are doing is in fact, reinforcing the actions of going to bed, as you think they are. I deal with this stuff with my grandson who has a chaotic homelife and when he is visitng for a full week, we find it helps to get him on a structured schedule to get him to bed on time---if we don't, he is miserable the next day because he simply stays up until 1 a.m. or so, and we have to get to bed because we do have to work the next day!. We cannot stay up until he decides he wants to go to bed. He is of course, allowed to lay in bed and stay away as long as he wants but has to stay in bed. I'm wondering what role your daughter plays in caretaking your younger son. For example, a good approach to toilet training as you know, is to have the child who is learning to go to the bathroom help teach a drinking-wetting-doll to go to the toilet, and receive verbal praise and a 'treat' initially, to reward the behavior. There are training toilets that provide a rewarding, interesting sound when water/urine hits the bottom of the toilet that helps here as well. But the point is that it might help to have your daughter participate and actually guide the bedtime ritual of her younger brother---helping him get ready for bed, helping give him his bedtime story (she can read to him with you), and she would then receive lots of compliments and praise (rewards) for doing so. You are reinforcing through active modeling behavior the 'going to bed' sequence of behaviors, which increases the likelihood she will engage in the same behavior. It may be that your daughter is resisting going to bed if she has the belief that her brother gets to stay up, or if he fusses about going to bed after she is in bed, he gets extra, reinforcing attention for this because he gets to stay up a bit longer.
After she participates in the brother-going-to-bed routine and gets hugs with him as he prepares for bedtime, she gets her pjs on---followed by lots of praise and hugs. She gets into bed in preparation for her storytime. The story is read to her only when she is in bed. She is then given instructions that you will check up on her and sing a verse of song if she can lie quietly in bed and not fuss, for say, 3 minutes. So you do check back on her and compliment her for her success in staying quietly in bed, every few minutes. After several nights, you do what is called you increase the time intervals between visits, shaping her behavior. I'm outlining principles of behavior here, not necessarily the details of it. Next morning, you can show her a success chart with stars on it and she can receive an additional reward for having a successful, prior evening.
Now, I don't know how you handled the tantruming behavior but I would advise ignoring it. Let her tear up her room but of course, the next day, she has to work alongside you doing most of the tidying up after school before she gets to play or watch TV or video games or whatever she does for fun. (Play is allowed to only follow the work, reinforcing it). If you pay a lot of attention to her immediately after she tantrums, you are reinforcing her for carrying out the tantrum. What you can do is simply ignore the episode, wait for a period of 5 mintues of peace and calm and then go to her room, as her if she is ready to get prepared to go to bed, stay with her while she goes through the ritual (brushing teeth, PJs, climbing into bed, preparing for her story or bedtime song). It is the CHAINING or sequencing of these behaviors, with use of positive reinforcement at each step of the chain that gradually builds a behavioral 'habit'---one cue leads to the next. I will pause her and solicit your feedback. The principles of behavior are certainly correct but getting the steps, sequences, proper reinforcement just right is the problem here.