Adkins accepts he is favourite for sack first
Peter Odemwingie scored twice to lift West Brom up to fifth place as they inflicted Saints' eighth defeat in 10 league games since their promotion from the Championship.
"Someone is always going to be the favourite to be sacked," Adkins told reporters. "If you're bottom of the league, you should be the favourite probably.
"I'm very positive though. I know I'm a better manager today than when I joined Southampton."
Southampton have collected four out of a possible 30 points, have lost all five away games, conceded 28 goals, are five points behind 17th-placed Aston Villa and are set for a long battle against an immediate return to the second tier.
"If I don't believe we can do it, then the players aren't going to believe we can do it," said Adkins, whose team have not played all that badly but have not had the breaks.
Adkins, 47, took over at Southampton two years ago and has led the side to two successive promotions.
They host Swansea City on Saturday before travelling to Loftus Road to face fellow strugglers Queens Park Rangers on November 17 ahead of matches against Newcastle United and Norwich City.