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<p/> <p>Let <inline-formula> <graphic file="1029-242X-2009-832686-i1.gif"/></inline-formula> denote the open unit disk in the complex plane and let <inline-formula> <graphic file="1029-242X-2009-832686-i2.gif"/></inline-formula> denote the normalized area measure on <inline-formula> <graphic file="1029-242X-2009-832686-i3.gif"/></inline-formula>. For <inline-formula> <graphic file="1029-242X-2009-832686-i4.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2009-832686-i5.gif"/></inline-formula> a twice differentiable, nonconstant, nondecreasing, nonnegative, and convex function on <inline-formula> <graphic file="1029-242X-2009-832686-i6.gif"/></inline-formula>, the Bergman-Orlicz space <inline-formula> <graphic file="1029-242X-2009-832686-i7.gif"/></inline-formula> is defined as follows <inline-formula> <graphic file="1029-242X-2009-832686-i8.gif"/></inline-formula> Let <inline-formula> <graphic file="1029-242X-2009-832686-i9.gif"/></inline-formula> be an analytic self-map of <inline-formula> <graphic file="1029-242X-2009-832686-i10.gif"/></inline-formula>. The composition operator <inline-formula> <graphic file="1029-242X-2009-832686-i11.gif"/></inline-formula> induced by <inline-formula> <graphic file="1029-242X-2009-832686-i12.gif"/></inline-formula> is defined by <inline-formula> <graphic file="1029-242X-2009-832686-i13.gif"/></inline-formula> for <inline-formula> <graphic file="1029-242X-2009-832686-i14.gif"/></inline-formula> analytic in <inline-formula> <graphic file="1029-242X-2009-832686-i15.gif"/></inline-formula>. We prove that the composition operator <inline-formula> <graphic file="1029-242X-2009-832686-i16.gif"/></inline-formula> is compact on <inline-formula> <graphic file="1029-242X-2009-832686-i17.gif"/></inline-formula> if and only if <inline-formula> <graphic file="1029-242X-2009-832686-i18.gif"/></inline-formula> is compact on <inline-formula> <graphic file="1029-242X-2009-832686-i19.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2009-832686-i20.gif"/></inline-formula> has closed range on <inline-formula> <graphic file="1029-242X-2009-832686-i21.gif"/></inline-formula> if and only if <inline-formula> <graphic file="1029-242X-2009-832686-i22.gif"/></inline-formula> has closed range on <inline-formula> <graphic file="1029-242X-2009-832686-i23.gif"/></inline-formula>.</p>