Extrapolation (conversion) between distributions?

Question

Perhaps this represents a trivial question but, if I have two subsets determined by an independent variable of a sample population, which are statistically distinct with respect to a common numerical response variable and wanted to extrapolate (convert) between the two, a simple way might be to add (or subtract) the difference between the respective medians (the populations are not normally distributed) to or from that of the subset one was wanting to convert. However, this would convey the variance of the starting subset to the converted set, which would not be a true conversion, or would it?

To better explain this, my example evaluates the valve replacement size accommodated for replacement valves in two random subsets (those receiving either valve type A or valve type B)  of patients otherwise controlled for confounding covariates. If I wanted to extrapolate which valve type B sizes would be suitable in the subgroup receiving valve type A, I am assuming that the variance in the extrapolated group should be the same as for the reference group, but can I simply use the difference in medians to calculate the conversion?

I've read a bit about inverse transform sampling and wondered if that may provide a more accurate transformation across the range of sizes. Is there a simple approach to this that I could follow using JMP?

34South · Answer

Thanks, I did not expect such a detailed response, although the scripting is a little above my pay grade. I'm very proficient at scripting in something like FileMaker, but my skills are still rudimentary in JMP.  Nevertheless, I did get it to run.

I did have the thought to stretch the one set into the other, but hadn't yet attempted it, but your idea to match quantiles helps a lot and makes a lot of sense.

Again, many thanks!

ih · Answer

I cannot speak as to whether this is statistically valid or accepted, but perhaps you could determine the quantile of the value in the first distribution and then find the value correlating with that quantile in the new distribution.

For example:

Names default to here(1);

//Origin value is in normal distribution
OriginalValue = 10;
OriginalMean = 8;
OriginalSD = 2;
OriginalQuantile = Normal Distribution( OriginalValue, OriginalMean, OriginalSD );

//Projected value will be in Gamma distribution
ProjectedAlpha = 4;
ProjectedScale = 3;
ProjectedThreshold = .2;
(ProjectedValue = Gamma Quantile( OriginalQuantile, ProjectedAlpha, ProjectedScale, ProjectedThreshold ) );

//Show results
Graph Builder(
	Size( 779, 296 ),
	Show Control Panel( 0 ),
	Show Legend( 0 ),
	Variables( X( :Column 1 ), Y( :Column 1 ) ),
	Elements( Points( X, Y, Legend( 3 ) ), Smoother( X, Y, Legend( 4 ) ) ),
	SendToReport(
		Dispatch(
			{},
			"Column 1",
			ScaleBox,
			{Format( "Best", 12 ), Min( 0 ), Max( 35 ),
			Inc( 2 ), Minor Ticks( 1 ), 
			Add Ref Line( OriginalValue, "Solid", "gray", "Original\!n" || format(OriginalValue, Fixed Dec, 2), 2 ),
			Add Ref Line( ProjectedValue, "Solid", "magenta", "Projected\!n" || format(ProjectedValue, Fixed Dec, 2), 2 )}
		),
		Dispatch(
			{}, "Column 1", ScaleBox( 2 ),
			{Format( "Best", 12 ), Min( 0 ), Max( .225 ), Inc( 0.5 ),
			Minor Ticks( 1 )}
		),
		Dispatch(
			{},
			"graph title",
			TextEditBox,
			{Set Text( "Distribution Projection" )}
		),
		Dispatch( {}, "X title", TextEditBox, {Set Text( "" )} ),
		Dispatch( {}, "Y title", TextEditBox, {Set Text( "" )} ),
		Dispatch(
			{},
			"Graph Builder",
			FrameBox,
			{Add Graphics Script(
				2,
				Description( "" ),
				Pen Color( "gray" );
				Y Function( Normal Density( x, OriginalMean, OriginalSD ), x );
			), Add Graphics Script(
				3,
				Description( "" ),
				Pen Color( "magenta" );
				Y Function( Gamma Density( x, ProjectedAlpha, ProjectedScale, ProjectedThreshold ), x );
			)}
		)
	)
);