No results by using different 'Distance Coefficients' is because layer_data has not 'setDistanceCoefficient' method. Desired method is in idw_interpolator.
So, by using same code in your previous question about QgsIDWInterpolator, I had to add only one code line:
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idw_interpolator.setDistanceCoefficient(10)
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for producing desired result in following code:
layer = iface.activeLayer()
layer_data = QgsInterpolator.LayerData()
layer_data.source = layer
layer_data.zCoordInterpolation = False
layer_data.interpolationAttribute = 1 # the 2nd field in the layer contains the data to interpolate
layer_data.mInputType = 1
idw_interpolator = QgsIDWInterpolator([layer_data])
idw_interpolator.setDistanceCoefficient(10) #added line
rasOut = r"/home/zeito/pyqgis_data/BiomassNOx_Emissions_tpa2.tif"
export_path = rasOut # e.g. r"C:\Temp\AnalysisResults\BiomassNOx_Emissions_tpa.tif"
rect = layer.extent()
res = 10
ncols = int( ( rect.xMaximum() - rect.xMinimum() ) / res )
nrows = int( (rect.yMaximum() - rect.yMinimum() ) / res)
output = QgsGridFileWriter(idw_interpolator,export_path,rect,ncols,nrows)
output.writeFile()
for producing visible changes in interpolated raster layer.
After running above code with same point layer in your previous question, I got result observed in following image (different of my answer in your previous question):
So, you should modify your code as follows:
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idw_interpolator = QgsIDWInterpolator([layer_data])
idw_interpolator.setDistanceCoefficient(10)
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for obtaining an equivalent result.