Q: Is there is a way to have a home psychological evaluation done for my elderly mother who is depressed and anxious but refuses to go to the doctor about it?
A: Depressed, anxious people are often reluctant to leave their "comfort zones," as they're already feeling overwhelmed. The resulting social withdrawal is actually among the symptoms that treatment is intended to address.
Encouraging the person to get out and do things tends to be more effective than reinforcing their desire to "stay put'" For this reason (along with some complex issues related to liability), most psychologists would opt not to do in-home evaluation or treatment unless the person is truly (physically) bed-ridden.
Your best bet might be to encourage your elderly mother to get a routine medical checkup, stressing the importance of prevention and good health to normalize the experience. Assessment of mood and behavior is a routine part of the standard checkup—and you can certainly offer your observations about her anxiety and depression prior to the meeting, to ensure that those issues are addressed thoroughly. The doctor may be able to persuade your mother to get out of the house, to exercise, to take medication, and perhaps to see a therapist.