The Toffees have won just once in nine games, against Leicester last time out at Goodison Park, in all competitions.
A defeat against Crystal Palace would put Saturday's opponents, currently 14th, just a point behind Everton - who go into the weekend just seven points off the bottom three.
Allardyce thought the threat of relegation had been extinguished after an impressive start to his career at the club following the sacking of Ronald Koeman.
However, a slump in form has them looking over their shoulders again.
'It's won three, drawn two, lost one since I've been here so our home form is very good,' said Allardyce.
'It's our best way of staying in the top half and, first of all, avoiding the threat of relegation.
'Our home form has to be continued, starting with Palace on Saturday, along the lines of how it was against Leicester.
'When you are looking at us in the light of away results (one win in 13 matches) it is not so good but our home form is pretty good at this moment in time.
'There is only Manchester United who have beaten us at Goodison in the last six games and that is a positive for us to look at.'
Allardyce admits his side have been boosted by the news Palace winger Wilfried Zaha has been ruled out for a month with a knee injury.
The meeting pits two former England managers against each other and the Toffees boss has been impressed with the results Roy Hodgson has achieved, having taken over with the Eagles in deep trouble.
'It's a great turnaround for Roy and the lads. It was always in those players and Roy has managed to bring it out of them,' Allardyce added.
'It is basically the same players I had (last season) and they have some really good strengths in the side.
'Yes we have to be concerned about Crystal Palace's strength but I think Wilf may have a problem and may not be playing so that will be an advantage to us.
'They have been very good and very resilient (starting) from a difficult position.
'But at Goodison it is about us, about us playing to our best and trying to win the game.'