How about the proposal in Parallel Plot: synchronized axes ?
use same "Y" position to identify the same "value".
This option is already available for columns with modeling type "continuous" (not nominal).
One can adjust the line width to get a better looking graph.
Unfortunately, there is no possibility to use the line width to represent the fraction / population.
Color can be used to identify another type of information:
inital treatment:

number of different treatments:

dt = New Table( "treatment",
Add Rows( 9 ),
New Column( "patient",
Character,
Set Values(
{"V1_0", "V1_1", "V1_2", "V1_3", "V1_4", "V1_5", "V1_6", "V1_7", "V1_8"}
)
),
New Column( "treatment 1",
//"Nominal",
Set Values( [1, 1, 1, 2, 2, 2, 3, 3, 3] )
),
New Column( "treatment 2",
//"Nominal",
Set Values( [1, 2, 3, 1, 2, 3, 1, 2, 3] )
),
New Column( "treatment 3",
//"Nominal",
Set Values( [1, 1, 2, 1, 2, 3, 3, 3, 3] )
)
);
//this functionality is only available for scripters.
//via the GUI, a user can create the equation manually
dt << New Formula Column(
Operation( Category( "Character" ), "Concatenate with Underscore" ),
Columns( :treatment 1, :treatment 2, :treatment 3 )
) << Run Formulas;
Graph Builder(
Size( 568, 240 ),
Show Control Panel( 0 ),
Summary Statistic( "Median" ),
Graph Spacing( 4 ),
Variables(
X( :treatment 1 ),
X( :treatment 2, Position( 1 ) ),
X( :treatment 3, Position( 1 ) ),
Color( :"Concatenate[treatment 1,treatment 2,treatment 3]"n )
),
Elements(
Parallel( X( 1 ), X( 2 ), X( 3 ) )
)
);