I don't have any clever map sort or grep approaches, but the code could be simplified. For the sequence of if-then statements, one could use

my @institutions = qw|hospital1 hospital2|;
my %seen;
$seen{$_}++ foreach @institutions;

if (!defined $institution) {
    $institution_table{"unaffiliated"} ++;
} elsif (exists $seen{lc $institution}) {
    $institution_table{lc $institution} ++;
} else {
    $institution_table{"other"} ++;
}
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For the loop, here is a slightly more clever method:

my $cat_age = int ($age/5) - 9;
my @age_cats = qw|fortyfive_fifty fifty_fiftyfive fiftyfive_sixty|;

foreach (keys %institution_table) {
 $institution_table{ $_ }{ $age_cats[$cat_age] }++;
}
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One could use a map for the loop, but the foreach is faster.

-Mark