  This paper aims at studying the Iwasawa $\lambda$-invariant of the
$p$-primary Selmer group. We study the growth behaviour of $p$-primary Selmer
groups in $p$-power degree extensions over non-cyclotomic
$\mathbb{Z}_p$-extensions of a number field. We prove a generalization of
Kida's formula in such a case. Unlike the cyclotomic $\mathbb{Z}_p$-extension,
where all primes are finitely decomposed; in the $\mathbb{Z}_p$-extensions we
consider, primes may be infinitely decomposed. In the second part of the paper,
we study the relationship for Iwasawa invariants with respect to congruences,
obtaining refinements of the results of R. Greenberg-V. Vatsal and K. Kidwell.
As an application, we provide an algorithm for constructing elliptic curves
with large anticyclotomic $\lambda$-invariant. Our results are illustrated by
explicit computation.
