Something has become the No. 1 Toilet Training Problem in America: children who refuse to use the toilet for bowel movements. In almost every case, these kids' parents missed the boat when it came to training, meaning they didn't start until well after their kids' second birthdays.
Now, let's take up the issue of how to toilet train between 18 and 24 months.
Approach toilet training with the same casual, matter-of-fact attitude with which you approached teaching your child to eat with a spoon.
Despite psychobabble to the contrary, neither is fraught with apocalyptic psychological ramifications. When spoon-training your child, you encouraged without being silly, conveyed a clear expectation and tolerated the temporary mess. If you can muster that same attitude in potty training, you're halfway there.
Second, put a potty or two in that area or those areas of the home where your toddler spends most of his time. Keep it in the bathroom and you inadvertently invoke the Out of Sight, Out of Mind Principle. If your home layout requires it, put out two potties. They should be simple, not ones that do silly, superfluous things like play Barney songs.
Third, set aside a week during which you can spend most of your time at home. Your ability to maintain a calm focus during this teaching time will help your toddler remember what you are expecting of him.
Fourth, if you're working with a girl, she should spend her day naked from the waist down. Don't even use diapers for naps. Dress a boy in nothing but the thinnest cotton underwear you can find. The point, in either case, is for the "mess" to travel unimpeded down the child's legs, which they do not like. Just take a deep breath and clean it up.
Fifth, as soon as your child wakes up in the morning, take him or her to the potty. When he/she is seated, walk away, saying, "Call me when you've done something and I'll clean you up." Do not hover or act nervous. If your child refuses to sit, so be it. Make this into a power struggle and you're done for.
Sixth, whether or not your child produces at the first sitting of the day, set a timer to ring every 60 minutes or so (different kids, different intervals). Call it the "potty bell." When it rings, simply announce that it's time to sit on the potty. Remember, don't hang around.
Seventh, clean up messes without drama. Nonetheless, don't tell your child that making a mess is okay. It's not. When he/she has an accident, tell him, "Make sure you sit on the potty next time." Be firm, but don't be angry.
Eighth, respond to successes positively, but not overly enthusiastically. Do not give rewards.
Last, when the process is complete, hire a carpet cleaner. The typical cost of said removal is why I call this method "Naked and $75."