How long is this piece of string?
What exactly is the make/model of the analyser? Have you any circuit information?
What is the normal running current of the device? Is 160mA a large or small amount relatively speaking?
Does the pushbutton switch the supply directly or does it just supply a control signal to an electronic switch?
Are there any components such as capacitors which are directly across the power rails regardless of the ON/OFF state of the unit?