Preheat oven to 350 degrees F (175 degrees C). Sift flour, baking powder, soda, salt and allspice together and set aside.
In a large bowl, cream the butter and sugar until light and fluffy. Add eggs and egg white and beat well. Add the mashed banana, vanilla and milk. Add the flour mixture and mix until well blended.
Divide batter into two 9 inch pans. Bake at 350 degrees F (175 degrees C) for 25 to 30 minutes, or until a toothpick inserted into the center of the cake comes out clean. Cool completely and frost with Cream Cheese Frosting.