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PSYCHOMETRICA

Functions for Fitting and Plotting Psychometric Data
;[s]
3:0,0;15,1;67,0;68,-1;
2:2,25,18,Times,1,24,0,0,0;1,21,15,Times,1,20,0,0,0;
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©1995,1996 Andrew B. Watson
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Credits & Version
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Copyright 1995 by Andrew B. Watson. All rights reserved. Do not distribute or reproduce without permission. No warranties expressed or implied. This is Version 2.0 , compatible with Mathematica 2.2.2.
;[s]
3:0,0;182,1;193,0;201,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
History
:[font = subsection; inactive; preserveAspect; startGroup]
1.0->2.0
:[font = text; inactive; preserveAspect; endGroup; endGroup]
Changed label location in PsychometricPlot.
Took shift argument out of all psychometric functions, and all examples using such functions.
Several arguments changed to options in PsychometricFit. Added FitLimits option.
PsychometricFit now accepts a list of tables or the output of QuestExperiment.
PsychometricPlot accepts multiple fits.
Added PNames to each psychometric function.
Complete print of parameters in PsychometricPlot.
Underflow messages turned Off and On in psychometric functions.
Term "value" changed to "strength" throughout.
:[font = section; inactive; Cclosed; dontPreserveAspect; startGroup]
Introduction
:[font = text; inactive; dontPreserveAspect]
These functions implement general fitting and plotting of psychometric data. Weibull, Normal and Logistic psychometric functions are also defined. Some background reading on the fitting method is available in:

Watson, A. B. (1979). Probability summation over time. Vision Research 19, 515-522.
:[font = text; inactive; preserveAspect; endGroup]
This Notebook begins with function definitions and some documentation. If you are not concerned with implementation, and want only to use the functions, take a look at the Tutorial.
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Preliminaries
:[font = input; initialization; preserveAspect; endGroup]
*)
Needs["Utilities`FilterOptions`"]
(*
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Psychometric Functions
:[font = text; inactive; preserveAspect]
The following are definitions for a number of commonly used psychometric functions, which describe probability of a positive response (yes, or correct) as a function of some measure of stimulus strength. They are typically cumulative probability distribution functions, ammended by guessing and lapsing factors. All the functions defined here follow a standard form f[x_,p_List] where p is a list of parameters. This standard form allows the functions to be used by fitting routines and by the Quest staircase procedure (see Quest.ma). User-written functions  should follow the same convention to obtain these capabilities.
:[font = subsection; inactive; preserveAspect; startGroup]
LogWeibullPF 
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The Weibull function has proven usefull as a description of the human psychometric function (proportion correct versus signal strength) for visual detection of threshold stimuli. The following are some useful references.
:[font = text; inactive; preserveAspect]
Nachmias, J. (1981). On the psychometric function for contrast detection. Vision Research 21(2), 215-223.

Quick, R. F. (1974). A vector magnitude model of contrast detection. Kybernetik 16, 65-67.

Watson, A. B. (1979). Probability summation over time. Vision Research 19, 515-522.

Weibull, W. (1951). A statistical distribution function of wide applicability. J. appl Mech. 18, 292-297.
:[font = text; inactive; preserveAspect]
As a psychometric function, the Weibull is expressed as a function of a logarithmic (or decibel) range, which we call the LogWeibullPF function. The LogWeibullPF function is actually equivalent to the cumulative density function of the density known alternately as the Log Weibull distribution, the Extreme Value Distribution, the Gumbel distribution, or the Fisher-Tippet distribution.
:[font = info; active; initialization; dontPreserveAspect]
*)
LogWeibullPF::usage = "LogWeibullPF[x_,{threshold_,slope_,guess_,lapse_}]]: This is a Weibull psychometric function, defined on a decibel (dB) abscissa. The stimulus strength in dB is x, the threshold in dB is threshold, the slope is slope, the guessing rate is  guess, the lapsing rate is lapse. The function returns a probability of success.";
(*
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Clear[LogWeibullPF]
:[font = input; initialization; preserveAspect]
*)
LogWeibullPF[x_,{threshold_,slope_,guess_,lapse_}] :=  
 Module[{p}, 
  Off[General::under];
  p=N[guess + (1-guess-lapse) (1-
			Exp[- (dBInverse[x - threshold]) ^ slope])];
	 On[General::under];
	 p]
(*
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This version is retained for backward compatibility.
:[font = input; initialization; preserveAspect]
*)
LogWeibullPF[x_,threshold_,p_List]:= 
   LogWeibullPF[x,Prepend[p,threshold]]
(*
:[font = input; initialization; preserveAspect]
*)
PNames[LogWeibullPF] ^:=
 {"threshold","slope","guess","lapse"}
(*
:[font = input; preserveAspect]
Plot[LogWeibullPF[x,{0.,1.,0,0}],{x,-600000,600000}
,PlotRange->{0,1} ,Frame->True,FrameLabel->{"Strength","Probability"}];
:[font = subsubsection; inactive; preserveAspect; startGroup]
Examples
:[font = input; preserveAspect; startGroup]
LogWeibullPF[Range[-6,6],{0.,4.,0,.01}]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{0.06053494713099659, 0.0942109561444001, 0.1451022417501507, 
 
  0.2199031393582535, 0.3251258513634773, 0.4632386505411203, 
 
  0.6257993532402722, 0.7870800125873235, 0.909696073983521, 
 
  0.971521031684096, 0.988199379192804, 0.989955054069535, 
 
  0.989999870421928}
;[o]
{0.0605349, 0.094211, 0.145102, 0.219903, 0.325126, 0.463239, 0.625799, 
 
  0.78708, 0.909696, 0.971521, 0.988199, 0.989955, 0.99}
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Some illustrative plots
:[font = input; preserveAspect; startGroup]
Plot[LogWeibullPF[x,{0.,4.,0,.01}],{x,-6,6} ,PlotRange->{0,1} ,Frame->True,FrameLabel->{"Strength","Probability"}];
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:[font = subsubsection; inactive; preserveAspect; startGroup]
Comparisons with built-in Weibull and ExtremeValueDistribution
:[font = text; inactive; preserveAspect]
This section considers the option of using the built-in Statistical distributions to define LogWeibullPF. It is not currently used because it is slower.
:[font = input; preserveAspect]
Needs["Statistics`ContinuousDistributions`"]
:[font = name; inactive; preserveAspect]
Alternate Definition
:[font = text; inactive; preserveAspect]
The LogWeibullPF could be defined in terms of the ExtremeValueDistribution, like so:
:[font = input; preserveAspect]
Clear[LogWeibullPF3]
:[font = input; preserveAspect]
LogWeibullPF3[x_,{threshold_,slope_,guess_,lapse_}] :=
		N[guess + (1. - guess - lapse)*
		(1.-	CDF[ExtremeValueDistribution[threshold
		        ,20./(slope Log[10.])],-x])]
:[font = text; inactive; preserveAspect]
We verify that it produces results equivalent to LogWeibullPF, but is about two times slower.
:[font = input; preserveAspect; startGroup]
Print[Timing[tst=Table[Table[LogWeibullPF3[y, {0, 4, 0, 0}],{y,-20,20}],{8}];]];
Plus @@ Flatten[tst]
:[font = print; inactive; preserveAspect]
{0.85 Second, Null}
:[font = output; output; inactive; preserveAspect; endGroup]
174.0258953890841
;[o]
174.026
:[font = input; preserveAspect; startGroup]
Print[Timing[tst=Table[Table[LogWeibullPF[y,{ 0, 4, 0, 0}],{y,-20,20}],{8}];]];
Plus @@ Flatten[tst]
:[font = print; inactive; preserveAspect]
{0.45 Second, Null}
:[font = output; output; inactive; preserveAspect; endGroup]
174.0258953890841
;[o]
174.026
:[font = name; inactive; preserveAspect]
Derivation
:[font = input; preserveAspect; startGroup]
CDF[WeibullDistribution[slope,threshold],x]
:[font = output; output; inactive; preserveAspect; endGroup]
1 - E^(-(x/threshold)^slope)
;[o]
                   slope
     -(x/threshold)
1 - E
:[font = input; preserveAspect; startGroup]
CDF[ExtremeValueDistribution[a,b],y]
:[font = output; output; inactive; preserveAspect; endGroup]
E^(-E^(-((-a + y)/b)))
;[o]
   -((-a + y)/b)
 -E
E
:[font = input; preserveAspect; startGroup]
Exp[-dBInverse[dB[x] - dB[threshold]]^slope]//Simplify
:[font = output; output; inactive; preserveAspect; endGroup]
E^(-(10^((1.*(-1.*Log[alpha] + 1.*Log[x]))/Log[10]))^beta)
;[o]
     (1. (-1. Log[alpha] + 1. Log[x]))/Log[10] beta
 -(10                                         )
E
:[font = input; preserveAspect]
E^(-(E^((-Log[threshold] + Log[x])))^slope)
:[font = input; Cclosed; preserveAspect; startGroup]
E^(-(E^(slope (-Log[threshold] + Log[x]))))
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
E^(-E^(beta*(-Log[alpha] + Log[x])))
;[o]
   beta (-Log[alpha] + Log[x])
 -E
E
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Timing test
:[font = text; inactive; preserveAspect]
PowerMac 9500
:[font = input; preserveAspect]
test = Table[Random[Real,{-4,4}],{10000}];
:[font = input; preserveAspect; startGroup]
Timing[LogWeibullPF[test,{1.,4.,.5,.01}];]
:[font = output; output; inactive; preserveAspect; endGroup]
{4.599999999999909*Second, Null}
;[o]
{4.6 Second, Null}
:[font = input; preserveAspect; startGroup]
(msec / Second/ point) %[[1]]/10
:[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup]
(0.4599999999999909*msec)/point
;[o]
0.46 msec
---------
  point
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
LogisticPF 
:[font = text; inactive; preserveAspect]
The logistic is a widely used psychometric function. Human contrast detection psychometric functions are typically parallel on a log (or decibel) abscissa, so here the parameters x and threshold must be interpreted as logarithmic (typically decibel) units.
:[font = info; active; initialization; dontPreserveAspect]
*)
LogisticPF::usage = "LogisticPF[x_,{threshold_,slope_,guess_,lapse_}]: The LogisticPF psychometric function, defined on a decibel (dB) abscissa. The stimulus strength in dB is x, the threshold in dB is threshold, the slope is slope, the guessing rate is  guess, the lapsing rate is lapse. The function returns a probability of success.";
(*
:[font = input; preserveAspect]
Clear[LogisticPF]
:[font = input; initialization; preserveAspect]
*)
LogisticPF[x_,{threshold_,slope_,guess_,lapse_}]:=  
 N[guess + (1 - guess - lapse) /
   (1 + Exp[ slope (threshold - x)])]
(*
:[font = text; inactive; preserveAspect]
We place all of the parameters except x into a list so that this function may be used in the various other functions that accept a psychometric function name and a list of parameters (see PsychometricFit). 
:[font = input; initialization; preserveAspect]
*)
PNames[LogisticPF] ^:=
 {"threshold","slope","guess","lapse"}
(*
:[font = input; preserveAspect; startGroup]
CDF[LogisticDistribution[threshold,1/slope],x]
:[font = output; output; inactive; preserveAspect; endGroup]
(1 + E^(slope*(threshold - x)*Sign[slope^(-1)]))^(-1)
;[o]
                   1
----------------------------------------
     slope (threshold - x) Sign[1/slope]
1 + E
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Examples and Discussion
:[font = text; inactive; preserveAspect]
Here we plot a number of examples, first varying threshold (threshold) and then slope (slope). The other parameters are appropriate for two-alternative forced-choice psychometric data.
:[font = input; preserveAspect]
Plot[#,{x,-10,10},PlotRange->{0,1}]& /@
{Table[LogisticPF[x,{a,1,.5,.01}],{a,-3,3}]
,Table[LogisticPF[x,{0, 2^b,.5,.01}],{b,-2,2}]};
:[font = text; inactive; preserveAspect]
As a rule of thumb, the slope parameter of the logistic on a decibel abscissa should be about 0.18 times the comparable slope parameter for the LogWeibullPF
:[font = input; preserveAspect; startGroup]
slope = 4.;
factor = .18;
Plot[{ LogWeibullPF[x,{0, slope        ,.5,0}]
      ,LogisticPF[ x,{0, slope factor ,.5,0}]}
  ,{x,-10,10},PlotRange->{0,1}
  ,PlotStyle->{RGBColor[1,0,0],RGBColor[0,1,0]}];
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slope = 4.;
factor = .18;
Plot[{ LogWeibullPF[x,{0, slope        ,.5,0}]
      ,LogisticPF[ x,{0, slope factor ,.5,0}]}
      ,{x,-10,10}
  ,PlotRange->{0,1} ,Frame->True
  ,PlotStyle->{GrayLevel[0],Dashing[{.01}]}
  ,FrameLabel->{"Strength","Probability"}];
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:[font = subsubsection; inactive; preserveAspect; startGroup]
Timing test
:[font = text; inactive; preserveAspect]
PowerMac 9500
:[font = input; preserveAspect]
test = Table[Random[Real,{-4,4}],{10000}];
:[font = input; preserveAspect; startGroup]
Timing[LogisticPF[test,{0.,.2,.5,.01}];]
:[font = output; output; inactive; preserveAspect; endGroup]
{4.216666666666243*Second, Null}
;[o]
{4.21667 Second, Null}
:[font = input; preserveAspect; startGroup]
Timing[LogisticPF[test,{0.,.2,.5,.01}];]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup]
{4.166666666656966*Second, Null}
;[o]
{4.16667 Second, Null}
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
NormalPF
:[font = text; inactive; preserveAspect]
The Normal CDF is a widely used psychometric function. Human contrast detection psychometric functions are typically parallel on a log (or decibel) abscissa, so here the parameters x, mean, and sd should usually be interpreted as logarithmic (typically decibel) units.
:[font = info; active; initialization; dontPreserveAspect]
*)
NormalPF::usage = "NormalPF[x_,{threshold_,slope_,guess_,lapse_}]: The NormalPF psychometric function is the cumulative distribution function of a Normal density with mean threshold and standard deviation 1/slope. The stimulus strength is x. The guessing rate is guess, the lapsing rate is lapse. The function returns a probability of success.";
(*
:[font = input; preserveAspect]
Clear[NormalPF]
:[font = input; initialization; preserveAspect]
*)
NormalPF[x_,{threshold_, slope_, guess_, lapse_}] := guess + 0.5 (1. - guess - lapse) *
 (1. + Check[Erf[0.7071067811865476 * slope * 
 (-threshold + x)],1,General::under])
(*
:[font = input; initialization; preserveAspect]
*)
PNames[NormalPF] ^:=
 {"mean","sd^-1","guess","lapse"}
(*
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Experimenting with underfolw
:[font = input; preserveAspect]
NormalPF[x_,{threshold_, slope_, guess_, lapse_}] := guess + 0.5 (1. - guess - lapse) *
 (1. + Erf[(0.7071067811865476 * slope * 
 (-threshold + x))])
:[font = input; preserveAspect]
Plot[NormalPF[x,{0.,1.,0,0}],{x,-600,600000}
,PlotRange->{0,1} ,Frame->True,FrameLabel->{"Strength","Probability"}];
:[font = input; preserveAspect; endGroup]
NormalPF[x_,{threshold_, slope_, guess_, lapse_}] :=   
 Module[{p}, 
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  p = guess + 0.5 (1. - guess - lapse) *
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 	On[General::under];
	 p]
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Some illustrative plots
:[font = input; preserveAspect; startGroup]
Plot[NormalPF[x,{0.,1.,.5,.01}],{x,-6,6}
,PlotRange->{0,1} ,Frame->True,FrameLabel->{"Strength","Probability"}];
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup]
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MathPictureEnd

:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Timing test
:[font = text; inactive; preserveAspect]
PowerMac 9500
:[font = input; preserveAspect]
test = Table[Random[Real,{-4,4}],{10000}];
:[font = input; preserveAspect; startGroup]
Timing[NormalPF[test,{1.,1.,.5,.01}];]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{3.833333333343034*Second, Null}
;[o]
{3.83333 Second, Null}
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Derivation
:[font = input; Cclosed; preserveAspect; startGroup]
?Erf
:[font = print; inactive; preserveAspect; endGroup]
Erf[z] gives the error function erf(z). Erf[z0, z1] gives the generalized error function erf(z1) -
   erf(z0).
:[font = input; Cclosed; preserveAspect; startGroup]
(1/( s Sqrt[2 Pi])) Integrate[Exp[- (1/(2 s^2)) (x-m)^2],x]
:[font = output; output; inactive; preserveAspect; endGroup]
Erf[(-m + x)/(2^(1/2)*s)]/2
;[o]
     -m + x
Erf[---------]
    Sqrt[2] s
--------------
      2
:[font = input; preserveAspect]
Needs["Statistics`ContinuousDistributions`"]
:[font = input; Cclosed; preserveAspect; startGroup]
CDF[NormalDistribution[mean,sd],x]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup; endGroup]
(1 + Erf[(-mean + x)/(2^(1/2)*sd)])/2
;[o]
        -mean + x
1 + Erf[----------]
        Sqrt[2] sd
-------------------
         2
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Fitting and Plotting Psychometric Data
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
PsychometricFit
:[font = info; active; initialization; preserveAspect]
*)
PsychometricFit::usage = "PsychometricFit[data_]: Fits a set of psychometric data with a psychometric function. The function returns the input data, the function name, and the specified and estimated parameters. The fit is based upon a maximum likelihood method, as described in: Watson, A. B. (1979) Probability summation over time, Vision Research 19, 515-522. data is a list of triples of the form {strength,no,yes}, where strength is a measure of signal strength, no is the number of no or incorrect responses, and yes is the number of yes or correct reponses. The function also accepts a list of such tables, and produces one fit per table. The function also accepts the data structure produced by QuestExperiment from the package Quest.ma. The option PsychometricFunction (default = LogWeibullPF) allows selection of an alternate psychometric function. The psychometric function must adhere to a standard format: function[x_, parameters_List]. The option PsychometricParameters (default = {Automatic, 4., 0.5, .01}) allows selection of initial or final values of psychometric function parameters. The value Automatic in the first position indicates that the initial value will be estimated from the data (the largest strength in the input data). Another option is FitParameters (default = {True}) which indicates whether each parameter should be estimated. Unspecified values default to False. For example, to estimate only the first and third parameters, use FitParameters->{True,False,True}. By convention, the first parameter is threshold and the second parameter is slope. This convention is relevant only in functions such as PsychometricPlot, which accepts directly the output of PsychometricFit. A final option is FitLimits, which specifies the bounds on each parameter (default = {{-$MaxMachineNumber,$MaxMachineNumber}, {0,$MaxMachineNumber}, {0,1}, {0,1}}";
(*
:[font = input; initialization; preserveAspect]
*)
Options[PsychometricFit] = {PsychometricFunction->LogWeibullPF, PsychometricParameters->{Automatic,4.,0.5,0.01}, FitParameters->{True},
FitLimits->{{-$MaxMachineNumber,$MaxMachineNumber},{0,$MaxMachineNumber},{0,1},{0,1}}};
(*
:[font = input; initialization; preserveAspect]
*)
PsychometricFit[data_,opts___Rule] :=
 Module[{fit, function, limits, params, err, args, val, no, yes, total, psy, tmp,dummy, dum1, dum2, dum3, dum4, dum5,dum6 ,dum7, dum8},
 {fit,params,function,limits} =
  {FitParameters, PsychometricParameters, 
    PsychometricFunction,FitLimits} /.
     {opts} /. Options[PsychometricFit];
	{val,no,yes} = Transpose[data];
 dummy =
  Take[{dum1,dum2,dum3,dum4,dum5,dum6,dum7,dum8}
   ,Length[params]];
	If[params[[1]]==Automatic
	    ,params[[1]] = Sort[val][[-1]]];
	If[params[[1]]==0.,params[[1]] = Sort[val][[-2]]];
	args = {err};
 fit = Flatten[Append[fit
    ,Table[False,{Length[params]-Length[fit]}]]];
	Table[ If[ fit[[i]],
	 args=Append[args,{dummy[[i]],params[[i]] {1.,.9},Sequence @@ limits[[i]]}]
	 ,dummy[[i]] = params[[i]]]
	  ,{i,Length[params]}];
	total = no+yes;
	err := (psy = function[#,dummy]& /@ val;
		    Chop[Apply[Plus, NLogP[yes, yes/(total psy)] +
			           NLogP[no, no/(total (1-psy))]]]);
	tmp = Check[FindMinimum @@ args
		 ,{,(#->Null)& /@ dummy},FindMinimum::fmlim];
  {data,function,{dummy /. tmp[[2]], tmp[[1]]}}
 ]
(*
:[font = input; initialization; preserveAspect]
*)
PsychometricFit[data_,opts___Rule] :=
PsychometricFit[SequenceToTable[First[data]],opts] /;
 Depth[data] == 4 &&
 Not[Equal @@ (Last /@ (Dimensions /@ data))]
(*
:[font = input; initialization; preserveAspect; endGroup]
*)
PsychometricFit[data_,opts___Rule] :=
 PsychometricFit[#,opts]& /@ data /;
 Depth[data] == 4 &&
 Equal @@ (Last /@ (Dimensions /@ data))
(*
:[font = subsection; inactive; Cclosed; dontPreserveAspect; startGroup]
PsychometricPlot
:[font = info; active; initialization; dontPreserveAspect]
*)
PsychometricPlot::usage = "PsychometricPlot[fitdata]: Plots a set of psychometric data and a psychometric function, as returned by PsychometricFit. See that function for details.
Also draws approximate Bernoulli confidence intervals around each data point, unless the option PsychErrorBars->False is given. To suppress printing of parameters and error of the fit, use the option PsychLegend->False. Options to ListPlot may also be included.";
(*
:[font = input; initialization; preserveAspect]
*)
Options[PsychometricPlot] = {PsychErrorBars->True, PsychLegend->True};
(*
:[font = input; initialization; preserveAspect]
*)
PsychometricPlot[{data_, function_
 ,{params_,error_}}, opts___Rule ] := Module[
 {val,no,yes,xtop,f1,f2,f3,f4,bars,legend,extra=.5, ulcorner},
 {bars,legend} = {PsychErrorBars,PsychLegend} /.
    {opts} /. Options[PsychometricPlot];
 {val,no,yes} = Transpose[data];
 xtop = Max[val]+1;
 f1 = ListPlot[Transpose[{val,yes/(yes+no)}]
   ,FilterOptions[ListPlot,opts], PlotJoined->False
   ,PlotStyle->{PointSize[.03]}
   ,PlotRange->{{Min[val]-1,xtop},{-.001,1.001}}
   ,Frame->True, FrameLabel->{"Strength","Probability"}
   ,Axes->False,DisplayFunction->Identity];
 f2 = Plot[function[x,params]
   ,{x,Min[val]-extra,Max[val]+extra}
   ,DisplayFunction->Identity];
 f3 = Graphics[{GrayLevel[.5]
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 f4 = Graphics[Text[ StringJoin @@ Flatten[
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 Show[f1, f2, If[bars,f3,{}], If[legend,f4,{}] 
   ,FilterOptions[ListPlot,opts]
   ,DefaultFont->{"Courier",10}
   ,DisplayFunction->$DisplayFunction ]];
(*
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:[font = input; initialization; preserveAspect; endGroup]
*)
PsychometricPlot[data_,opts___Rule] :=
 PsychometricPlot[#,opts]& /@ data /;
  Depth[data] == 5
(*
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Approximate error bars
:[font = info; active; initialization; dontPreserveAspect]
*)
BinomialErrorBars::usage = "BinomialErrorBars[failures,sucesses]:  Compute approximate confidence intervals on the probability p of a  binomial random variable, given numbers of failures and successes.";
(*
:[font = text; inactive; preserveAspect]
These functions are used to compute approximate confidence intervals for a binomial random variable (from Abramowitz & Stegun). An alternative solution using the statistical packages ise described in the next section.
:[font = input; initialization; preserveAspect]
*)
rsign[x_] := Floor[x / (x + 1.) + .6]
(*
:[font = input; initialization; preserveAspect]
*)
tail[no_,yes_] := Exp[Log[.05] / (yes + no)]
(*
:[font = input; initialization; preserveAspect]
*)
aprox1[no_,yes_] := (yes + .5 + 1.92 + 1.96 Sqrt[(yes + .5)*
 (1. - (yes + .5) /(yes + no)) + .96]) / (yes + no + 3.842)
(*
:[font = input; initialization; preserveAspect]
*)
aprox2[no_,yes_] := (yes - .5 + 1.92 - 1.96 Sqrt[(yes - .5)*
 (1. - (yes - .5) / (yes + no)) + .96]) / (yes + no + 3.842)
(*
:[font = input; initialization; preserveAspect]
*)
errbr1[no_,yes_] := rsign[no] * aprox1[no,yes] * rsign[yes] +
 (1. - rsign[no]) + (1. - rsign[yes]) * (1. - tail[no,yes]) 
(*
:[font = input; initialization; preserveAspect]
*)
errbr2[no_,yes_] := rsign[no] * aprox2[no,yes] * rsign[yes] +
 (1. - rsign[no]) * tail[no,yes]
(*
:[font = input; initialization; preserveAspect]
*)
BinomialErrorBars[no_,yes_] := {errbr2[no,yes],errbr1[no,yes]}
(*
:[font = input; preserveAspect; startGroup]
BinomialErrorBars[3,4] 
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{0.2023088714168743, 0.881790440185598}
;[o]
{0.202309, 0.88179}
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Precise error bars
:[font = text; inactive; preserveAspect]
This section describes the construction of Binomial confidence limits by way of probability distributions. It is not currently used, both because it is slower, and because it depends upon a root-finding operation which soemtimes fails to converge.
:[font = input; preserveAspect]
Needs["Statistics`DiscreteDistributions`"]
:[font = text; inactive; preserveAspect]
We first write a likelihood function that describes the likelihood of obtaining a particular set of data {no,yes}, given a Binomial random variable with probability p.
:[font = input; preserveAspect]
lik0[no_,yes_,p_] := PDF[BinomialDistribution[yes+no,p],yes]
:[font = text; inactive; preserveAspect]
We exand this function.
:[font = input; Cclosed; preserveAspect; startGroup]
lik0[no,yes,p]
:[font = output; output; inactive; preserveAspect; endGroup]
If[0 <= yes <= no + yes, If[IntegerQ[yes], 
 
   Binomial[no + yes, yes]*p^yes*(1 - p)^(no + yes - yes), 0], 
 
  0]
;[o]
If[0 <= yes <= no + yes, If[IntegerQ[yes], 
 
                            yes        no + yes - yes
   Binomial[no + yes, yes] p    (1 - p)              , 0], 0]
:[font = text; inactive; preserveAspect]
We remove the conditionals (which are always true), defining a new function.
:[font = input; preserveAspect]
lik1[no_,yes_,p_] :=
  Binomial[no + yes, yes]   p^yes (1 - p)^no
:[font = text; inactive; preserveAspect]
To determine the normalized pdf, we integrate lik1 to find the normalizing factor.
:[font = input; Cclosed; preserveAspect; startGroup]
Integrate[ lik1[no,yes,p] ,{p,0,1}]
:[font = output; output; inactive; preserveAspect; endGroup]
(Binomial[no + yes, yes]*Gamma[1 + no]*Gamma[1 + yes])/
 
  Gamma[2 + no + yes]
;[o]
Binomial[no + yes, yes] Gamma[1 + no] Gamma[1 + yes]
----------------------------------------------------
                Gamma[2 + no + yes]
:[font = input; preserveAspect]
norm[no_,yes_] :=
 (Binomial[no + yes, yes] Gamma[1 + no] Gamma[1 + yes]) /
 Gamma[2 + no + yes]
:[font = text; inactive; preserveAspect]
We create a posterior pdf.
:[font = input; preserveAspect]
pdf[no_,yes_,p_] := lik1[no,yes,p] / norm[no,yes]
:[font = text; inactive; preserveAspect]
We will need the cdf of this density, so we get the indefinite integral.
:[font = input; Cclosed; preserveAspect; startGroup]
Integrate[pdf[no,yes,p],{p,0,x}]
:[font = output; output; inactive; preserveAspect; endGroup]
(x^(1 + yes)*Gamma[2 + no + yes]*
 
    Hypergeometric2F1[-no, 1 + yes, 2 + yes, x])/
 
  ((1 + yes)*Gamma[1 + no]*Gamma[1 + yes])
;[o]
  1 + yes
(x        Gamma[2 + no + yes] 
 
    Hypergeometric2F1[-no, 1 + yes, 2 + yes, x]) / 
 
  ((1 + yes) Gamma[1 + no] Gamma[1 + yes])
:[font = text; inactive; preserveAspect]
Using this result, we define a cumulative distribution.
:[font = input; preserveAspect]
cdf[no_,yes_,x_] := (x^(1 + yes) Gamma[2 + no + yes] *
    Hypergeometric2F1[-no, 1 + yes, 2 + yes, x]) /
  ((1 + yes) Gamma[1 + no] Gamma[1 + yes])
:[font = text; inactive; preserveAspect]
The confidence limit on the binomial probability p is reached when a probaility alpha is reached in the cdf.
:[font = input; preserveAspect]
Clear[BinomialCL]
:[font = input; preserveAspect]
BinomialCL[no_,yes_,alpha_,opts___] := FindRoot[
   cdf[no,yes,x] == alpha
      ,{x,N[yes/(yes+no)]}
      ,opts][[1,2]]
:[font = text; inactive; preserveAspect]
This plot compares the approximate error bars defined in the previous section with the more precise ones defined here. The parameter n is the possible number of yes or no responses. Red points are upper limits, green are lower limits. The color sauration corresponds to percent correct.
:[font = input; Cclosed; preserveAspect; startGroup]
n=20;
Show[Graphics[Table[MapThread[
   {Hue[#1,yes/(no+yes),1],
   Point[{BinomialCL[no,yes,#2],#3[no,yes]}]}&,
	  {{1,.29},{.975,.025},{errbr1,errbr2}}] 
  ,{no,1,n},{yes,1,n}] ]
 ,Frame->True,FrameLabel->{"Precise","Approximate"}
 ,PlotRange->{{0,1},{0,1}} ,Axes->False ,AspectRatio->1];
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.92 .00375 L
s
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p
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.96 0 m
.96 .00375 L
s
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[(Precise)] .5 0 0 2 0 0 -1 Mouter Mrotshowa
p
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p
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[(0)] -0.0125 0 1 0 0 Minner Mrotshowa
p
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0 .2 m
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P
[(0.2)] -0.0125 .2 1 0 0 Minner Mrotshowa
p
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0 .4 m
.00625 .4 L
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P
[(0.4)] -0.0125 .4 1 0 0 Minner Mrotshowa
p
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0 .6 m
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P
[(0.6)] -0.0125 .6 1 0 0 Minner Mrotshowa
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0 .8 m
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P
[(0.8)] -0.0125 .8 1 0 0 Minner Mrotshowa
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[(1)] -0.0125 1 1 0 0 Minner Mrotshowa
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.00375 .12 L
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.00375 .24 L
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0 .32 m
.00375 .32 L
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0 .36 m
.00375 .36 L
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0 .44 m
.00375 .44 L
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0 .48 m
.00375 .48 L
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0 .52 m
.00375 .52 L
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p
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0 .56 m
.00375 .56 L
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0 .64 m
.00375 .64 L
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0 .68 m
.00375 .68 L
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0 .72 m
.00375 .72 L
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0 .76 m
.00375 .76 L
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0 .84 m
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0 .88 m
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p
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0 .92 m
.00375 .92 L
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0 .96 m
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[(Approximate)] -0.0125 .5 1 0 90 -1 0 Mouter Mrotshowa
p
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.44 .99625 m
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.48 .99625 m
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1 .32 L
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1 .36 L
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1 .44 L
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1 .48 L
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1 .52 L
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1 .56 L
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1 .64 L
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1 .68 L
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1 .72 L
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1 .76 L
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1 .84 L
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1 .88 L
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1 .92 L
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1 .96 L
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1 0 m
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0 0 m
1 0 L
1 1 L
0 1 L
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1 .5 .5 r
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1 .333 .333 r
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.507 1 .333 r
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1 .25 .25 r
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.94726 .98661 Mdot
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1 .2 .2 r
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1 .167 .167 r
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.96331 .99108 Mdot
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.383 1 .167 r
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1 .143 .143 r
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.96815 .99236 Mdot
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.366 1 .143 r
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1 .125 .125 r
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.97186 .99331 Mdot
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.353 1 .125 r
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1 .111 .111 r
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.97479 .99405 Mdot
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.342 1 .111 r
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1 .1 .1 r
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.334 1 .1 r
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1 .091 .091 r
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.327 1 .091 r
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1 .083 .083 r
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.322 1 .083 r
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1 .077 .077 r
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1 .071 .071 r
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1 .067 .067 r
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.309 1 .067 r
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1 .063 .063 r
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1 .059 .059 r
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1 .056 .056 r
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1 .053 .053 r
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.297 1 .05 r
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1 .048 .048 r
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.295 1 .048 r
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1 .667 .667 r
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1 .5 .5 r
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.63 1 .5 r
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1 .4 .4 r
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1 .333 .333 r
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1 .286 .286 r
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.471 1 .286 r
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1 .25 .25 r
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.92515 .95533 Mdot
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1 .222 .222 r
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1 .2 .2 r
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1 .182 .182 r
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1 .167 .167 r
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1 .154 .154 r
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1 .143 .143 r
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1 .133 .133 r
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1 .091 .091 r
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1 .75 .75 r
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1 .375 .375 r
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1 .273 .273 r
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1 .25 .25 r
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1 .231 .231 r
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1 .214 .214 r
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1 .136 .136 r
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1 .13 .13 r
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1 .267 .267 r
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.545 1 .385 r
.008 w
.35138 .32272 Mdot
P
P
p
p
1 .357 .357 r
.008 w
.83664 .86014 Mdot
P
p
.524 1 .357 r
.008 w
.3838 .35624 Mdot
P
P
p
p
1 .333 .333 r
.008 w
.84802 .87005 Mdot
P
p
.507 1 .333 r
.008 w
.41338 .38682 Mdot
P
P
p
p
1 .313 .313 r
.008 w
.8579 .87865 Mdot
P
p
.491 1 .313 r
.008 w
.44042 .41479 Mdot
P
P
p
p
1 .294 .294 r
.008 w
.86657 .88617 Mdot
P
p
.478 1 .294 r
.008 w
.4652 .44044 Mdot
P
P
p
p
1 .278 .278 r
.008 w
.87424 .89281 Mdot
P
p
.466 1 .278 r
.008 w
.48797 .46402 Mdot
P
P
p
p
1 .263 .263 r
.008 w
.88107 .89872 Mdot
P
p
.455 1 .263 r
.008 w
.50895 .48575 Mdot
P
P
p
p
1 .25 .25 r
.008 w
.88719 .90401 Mdot
P
p
.445 1 .25 r
.008 w
.52834 .50585 Mdot
P
P
p
p
1 .238 .238 r
.008 w
.89271 .90878 Mdot
P
p
.436 1 .238 r
.008 w
.5463 .52446 Mdot
P
P
p
p
1 .227 .227 r
.008 w
.89771 .91309 Mdot
P
p
.428 1 .227 r
.008 w
.56297 .54176 Mdot
P
P
p
p
1 .217 .217 r
.008 w
.90227 .91701 Mdot
P
p
.421 1 .217 r
.008 w
.57849 .55787 Mdot
P
P
p
p
1 .208 .208 r
.008 w
.90644 .92059 Mdot
P
p
.414 1 .208 r
.008 w
.59296 .5729 Mdot
P
P
p
p
1 .2 .2 r
.008 w
.91026 .92388 Mdot
P
p
.408 1 .2 r
.008 w
.60649 .58696 Mdot
P
P
P
p
p
p
1 .857 .857 r
.008 w
.52651 .57981 Mdot
P
p
.894 1 .857 r
.008 w
.03185 .00746 Mdot
P
P
p
p
1 .75 .75 r
.008 w
.60009 .64414 Mdot
P
p
.815 1 .75 r
.008 w
.07485 .0445 Mdot
P
P
p
p
1 .667 .667 r
.008 w
.65245 .69073 Mdot
P
p
.753 1 .667 r
.008 w
.12155 .09037 Mdot
P
P
p
p
1 .6 .6 r
.008 w
.6921 .72624 Mdot
P
p
.704 1 .6 r
.008 w
.16749 .13688 Mdot
P
P
p
p
1 .545 .545 r
.008 w
.72333 .75429 Mdot
P
p
.664 1 .545 r
.008 w
.21094 .18133 Mdot
P
P
p
p
1 .5 .5 r
.008 w
.74865 .77705 Mdot
P
p
.63 1 .5 r
.008 w
.25135 .22282 Mdot
P
P
p
p
1 .462 .462 r
.008 w
.76964 .79591 Mdot
P
p
.602 1 .462 r
.008 w
.28861 .26116 Mdot
P
P
p
p
1 .429 .429 r
.008 w
.78733 .81179 Mdot
P
p
.577 1 .429 r
.008 w
.32287 .29644 Mdot
P
P
p
p
1 .4 .4 r
.008 w
.80247 .82536 Mdot
P
p
.556 1 .4 r
.008 w
.35435 .32887 Mdot
P
P
p
p
1 .375 .375 r
.008 w
.81556 .8371 Mdot
P
p
.538 1 .375 r
.008 w
.38328 .35869 Mdot
P
P
p
p
1 .353 .353 r
.008 w
.82701 .84735 Mdot
P
p
.521 1 .353 r
.008 w
.40993 .38616 Mdot
P
P
p
p
1 .333 .333 r
.008 w
.83711 .85637 Mdot
P
p
.507 1 .333 r
.008 w
.4345 .41151 Mdot
P
P
p
p
1 .316 .316 r
.008 w
.84609 .86439 Mdot
P
p
.494 1 .316 r
.008 w
.45721 .43494 Mdot
P
P
p
p
1 .3 .3 r
.008 w
.85412 .87156 Mdot
P
p
.482 1 .3 r
.008 w
.47825 .45665 Mdot
P
P
p
p
1 .286 .286 r
.008 w
.86135 .878 Mdot
P
p
.471 1 .286 r
.008 w
.49778 .47682 Mdot
P
P
p
p
1 .273 .273 r
.008 w
.8679 .88382 Mdot
P
p
.462 1 .273 r
.008 w
.51595 .49558 Mdot
P
P
p
p
1 .261 .261 r
.008 w
.87385 .88912 Mdot
P
p
.453 1 .261 r
.008 w
.53289 .51309 Mdot
P
P
p
p
1 .25 .25 r
.008 w
.87928 .89395 Mdot
P
p
.445 1 .25 r
.008 w
.54871 .52944 Mdot
P
P
p
p
1 .24 .24 r
.008 w
.88427 .89837 Mdot
P
p
.438 1 .24 r
.008 w
.56352 .54476 Mdot
P
P
p
p
1 .231 .231 r
.008 w
.88886 .90245 Mdot
P
p
.431 1 .231 r
.008 w
.57742 .55913 Mdot
P
P
P
p
p
p
1 .875 .875 r
.008 w
.4825 .53311 Mdot
P
p
.908 1 .875 r
.008 w
.02814 .00652 Mdot
P
P
p
p
1 .778 .778 r
.008 w
.5561 .598 Mdot
P
p
.836 1 .778 r
.008 w
.06674 .03942 Mdot
P
P
p
p
1 .7 .7 r
.008 w
.60974 .64624 Mdot
P
p
.778 1 .7 r
.008 w
.10926 .0809 Mdot
P
P
p
p
1 .636 .636 r
.008 w
.65112 .68378 Mdot
P
p
.731 1 .636 r
.008 w
.15165 .12361 Mdot
P
P
p
p
1 .583 .583 r
.008 w
.68422 .71393 Mdot
P
p
.692 1 .583 r
.008 w
.19223 .16495 Mdot
P
P
p
p
1 .538 .538 r
.008 w
.71139 .73872 Mdot
P
p
.658 1 .538 r
.008 w
.23036 .20397 Mdot
P
P
p
p
1 .5 .5 r
.008 w
.73414 .7595 Mdot
P
p
.63 1 .5 r
.008 w
.26586 .24039 Mdot
P
P
p
p
1 .467 .467 r
.008 w
.75349 .77717 Mdot
P
p
.605 1 .467 r
.008 w
.29878 .27419 Mdot
P
P
p
p
1 .438 .438 r
.008 w
.77017 .7924 Mdot
P
p
.584 1 .438 r
.008 w
.32925 .3055 Mdot
P
P
p
p
1 .412 .412 r
.008 w
.7847 .80566 Mdot
P
p
.565 1 .412 r
.008 w
.35745 .33449 Mdot
P
P
p
p
1 .389 .389 r
.008 w
.79748 .81732 Mdot
P
p
.548 1 .389 r
.008 w
.38358 .36136 Mdot
P
P
p
p
1 .368 .368 r
.008 w
.80881 .82765 Mdot
P
p
.533 1 .368 r
.008 w
.40781 .38629 Mdot
P
P
p
p
1 .35 .35 r
.008 w
.81893 .83686 Mdot
P
p
.519 1 .35 r
.008 w
.43032 .40945 Mdot
P
P
p
p
1 .333 .333 r
.008 w
.82802 .84514 Mdot
P
p
.507 1 .333 r
.008 w
.45128 .43101 Mdot
P
P
p
p
1 .318 .318 r
.008 w
.83624 .85261 Mdot
P
p
.495 1 .318 r
.008 w
.47081 .45112 Mdot
P
P
p
p
1 .304 .304 r
.008 w
.8437 .85939 Mdot
P
p
.485 1 .304 r
.008 w
.48905 .46991 Mdot
P
P
p
p
1 .292 .292 r
.008 w
.8505 .86557 Mdot
P
p
.476 1 .292 r
.008 w
.50612 .48749 Mdot
P
P
p
p
1 .28 .28 r
.008 w
.85674 .87123 Mdot
P
p
.467 1 .28 r
.008 w
.52213 .50398 Mdot
P
P
p
p
1 .269 .269 r
.008 w
.86247 .87643 Mdot
P
p
.459 1 .269 r
.008 w
.53715 .51946 Mdot
P
P
p
p
1 .259 .259 r
.008 w
.86776 .88123 Mdot
P
p
.452 1 .259 r
.008 w
.55128 .53403 Mdot
P
P
P
p
p
p
1 .889 .889 r
.008 w
.44502 .49321 Mdot
P
p
.918 1 .889 r
.008 w
.02521 .00579 Mdot
P
P
p
p
1 .8 .8 r
.008 w
.51776 .55773 Mdot
P
p
.852 1 .8 r
.008 w
.06022 .03539 Mdot
P
P
p
p
1 .727 .727 r
.008 w
.57186 .60675 Mdot
P
p
.798 1 .727 r
.008 w
.09925 .07323 Mdot
P
P
p
p
1 .667 .667 r
.008 w
.61426 .64556 Mdot
P
p
.753 1 .667 r
.008 w
.13858 .11269 Mdot
P
P
p
p
1 .615 .615 r
.008 w
.64862 .67716 Mdot
P
p
.715 1 .615 r
.008 w
.17661 .1513 Mdot
P
P
p
p
1 .571 .571 r
.008 w
.67713 .70345 Mdot
P
p
.683 1 .571 r
.008 w
.21267 .1881 Mdot
P
P
p
p
1 .533 .533 r
.008 w
.70122 .7257 Mdot
P
p
.655 1 .533 r
.008 w
.24651 .22272 Mdot
P
P
p
p
1 .5 .5 r
.008 w
.72188 .7448 Mdot
P
p
.63 1 .5 r
.008 w
.27812 .2551 Mdot
P
P
p
p
1 .471 .471 r
.008 w
.73981 .76137 Mdot
P
p
.608 1 .471 r
.008 w
.30757 .2853 Mdot
P
P
p
p
1 .444 .444 r
.008 w
.75553 .7759 Mdot
P
p
.589 1 .444 r
.008 w
.335 .31343 Mdot
P
P
p
p
1 .421 .421 r
.008 w
.76942 .78874 Mdot
P
p
.572 1 .421 r
.008 w
.36054 .33964 Mdot
P
P
p
p
1 .4 .4 r
.008 w
.7818 .80018 Mdot
P
p
.556 1 .4 r
.008 w
.38435 .36408 Mdot
P
P
p
p
1 .381 .381 r
.008 w
.79291 .81043 Mdot
P
p
.542 1 .381 r
.008 w
.40658 .38689 Mdot
P
P
p
p
1 .364 .364 r
.008 w
.80292 .81968 Mdot
P
p
.529 1 .364 r
.008 w
.42734 .40822 Mdot
P
P
p
p
1 .348 .348 r
.008 w
.81201 .82806 Mdot
P
p
.517 1 .348 r
.008 w
.44678 .42818 Mdot
P
P
p
p
1 .333 .333 r
.008 w
.82028 .83569 Mdot
P
p
.507 1 .333 r
.008 w
.465 .44689 Mdot
P
P
p
p
1 .32 .32 r
.008 w
.82786 .84267 Mdot
P
p
.497 1 .32 r
.008 w
.4821 .46447 Mdot
P
P
p
p
1 .308 .308 r
.008 w
.83481 .84908 Mdot
P
p
.488 1 .308 r
.008 w
.49819 .48099 Mdot
P
P
p
p
1 .296 .296 r
.008 w
.84122 .85498 Mdot
P
p
.479 1 .296 r
.008 w
.51333 .49656 Mdot
P
P
p
p
1 .286 .286 r
.008 w
.84715 .86044 Mdot
P
p
.471 1 .286 r
.008 w
.52762 .51125 Mdot
P
P
P
p
p
p
1 .9 .9 r
.008 w
.41278 .45876 Mdot
P
p
.926 1 .9 r
.008 w
.02283 .00521 Mdot
P
P
p
p
1 .818 .818 r
.008 w
.48414 .52237 Mdot
P
p
.865 1 .818 r
.008 w
.05486 .0321 Mdot
P
P
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p
1 .75 .75 r
.008 w
.53813 .57157 Mdot
P
p
.815 1 .75 r
.008 w
.09092 .0669 Mdot
P
P
p
p
1 .692 .692 r
.008 w
.58104 .61108 Mdot
P
p
.772 1 .692 r
.008 w
.1276 .10355 Mdot
P
P
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p
1 .643 .643 r
.008 w
.6162 .64365 Mdot
P
p
.736 1 .643 r
.008 w
.16336 .13975 Mdot
P
P
p
p
1 .6 .6 r
.008 w
.64565 .67103 Mdot
P
p
.704 1 .6 r
.008 w
.19753 .17453 Mdot
P
P
p
p
1 .563 .563 r
.008 w
.67075 .6944 Mdot
P
p
.676 1 .563 r
.008 w
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P
P
p
p
1 .529 .529 r
.008 w
.69243 .7146 Mdot
P
p
.652 1 .529 r
.008 w
.26019 .23853 Mdot
P
P
p
p
1 .5 .5 r
.008 w
.71136 .73226 Mdot
P
p
.63 1 .5 r
.008 w
.28864 .26765 Mdot
P
P
p
p
1 .474 .474 r
.008 w
.72804 .74783 Mdot
P
p
.611 1 .474 r
.008 w
.31528 .29492 Mdot
P
P
p
p
1 .45 .45 r
.008 w
.74287 .76166 Mdot
P
p
.593 1 .45 r
.008 w
.34021 .32045 Mdot
P
P
p
p
1 .429 .429 r
.008 w
.75614 .77404 Mdot
P
p
.577 1 .429 r
.008 w
.36355 .34436 Mdot
P
P
p
p
1 .409 .409 r
.008 w
.76809 .78519 Mdot
P
p
.563 1 .409 r
.008 w
.38542 .36677 Mdot
P
P
p
p
1 .391 .391 r
.008 w
.7789 .79528 Mdot
P
p
.55 1 .391 r
.008 w
.40594 .38779 Mdot
P
P
p
p
1 .375 .375 r
.008 w
.78875 .80445 Mdot
P
p
.538 1 .375 r
.008 w
.42521 .40754 Mdot
P
P
p
p
1 .36 .36 r
.008 w
.79774 .81284 Mdot
P
p
.526 1 .36 r
.008 w
.44333 .42612 Mdot
P
P
p
p
1 .346 .346 r
.008 w
.80599 .82053 Mdot
P
p
.516 1 .346 r
.008 w
.46039 .44362 Mdot
P
P
p
p
1 .333 .333 r
.008 w
.81359 .82761 Mdot
P
p
.507 1 .333 r
.008 w
.47648 .46012 Mdot
P
P
p
p
1 .321 .321 r
.008 w
.82062 .83415 Mdot
P
p
.498 1 .321 r
.008 w
.49168 .4757 Mdot
P
P
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p
1 .31 .31 r
.008 w
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P
p
.49 1 .31 r
.008 w
.50604 .49044 Mdot
P
P
P
p
p
p
1 .909 .909 r
.008 w
.3848 .42875 Mdot
P
p
.933 1 .909 r
.008 w
.02086 .00473 Mdot
P
P
p
p
1 .833 .833 r
.008 w
.45447 .49111 Mdot
P
p
.877 1 .833 r
.008 w
.05038 .02937 Mdot
P
P
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p
1 .769 .769 r
.008 w
.50798 .54008 Mdot
P
p
.829 1 .769 r
.008 w
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P
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p
1 .714 .714 r
.008 w
.551 .5799 Mdot
P
p
.789 1 .714 r
.008 w
.11824 .09578 Mdot
P
P
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p
1 .667 .667 r
.008 w
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P
p
.753 1 .667 r
.008 w
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P
P
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p
1 .625 .625 r
.008 w
.61672 .64121 Mdot
P
p
.723 1 .625 r
.008 w
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P
P
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p
1 .588 .588 r
.008 w
.64255 .66541 Mdot
P
p
.695 1 .588 r
.008 w
.2153 .19424 Mdot
P
P
p
p
1 .556 .556 r
.008 w
.665 .68648 Mdot
P
p
.671 1 .556 r
.008 w
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P
P
p
p
1 .526 .526 r
.008 w
.68472 .705 Mdot
P
p
.649 1 .526 r
.008 w
.27196 .25208 Mdot
P
P
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p
1 .5 .5 r
.008 w
.70219 .72141 Mdot
P
p
.63 1 .5 r
.008 w
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P
P
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1 .476 .476 r
.008 w
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P
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.612 1 .476 r
.008 w
.3221 .30335 Mdot
P
P
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1 .455 .455 r
.008 w
.7318 .74925 Mdot
P
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.596 1 .455 r
.008 w
.34495 .32671 Mdot
P
P
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1 .435 .435 r
.008 w
.74447 .76115 Mdot
P
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.582 1 .435 r
.008 w
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P
P
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1 .417 .417 r
.008 w
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P
p
.568 1 .417 r
.008 w
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P
P
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1 .4 .4 r
.008 w
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P
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.556 1 .4 r
.008 w
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P
P
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1 .385 .385 r
.008 w
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P
p
.545 1 .385 r
.008 w
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P
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1 .37 .37 r
.008 w
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P
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.534 1 .37 r
.008 w
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P
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1 .357 .357 r
.008 w
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P
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.524 1 .357 r
.008 w
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P
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1 .345 .345 r
.008 w
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P
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.515 1 .345 r
.008 w
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P
P
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1 .333 .333 r
.008 w
.80773 .82059 Mdot
P
p
.507 1 .333 r
.008 w
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P
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1 .917 .917 r
.008 w
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P
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.008 w
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P
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1 .846 .846 r
.008 w
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.886 1 .846 r
.008 w
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P
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1 .786 .786 r
.008 w
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.841 1 .786 r
.008 w
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P
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1 .733 .733 r
.008 w
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P
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.803 1 .733 r
.008 w
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P
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1 .688 .688 r
.008 w
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P
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.769 1 .688 r
.008 w
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P
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1 .647 .647 r
.008 w
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.739 1 .647 r
.008 w
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P
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1 .611 .611 r
.008 w
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.712 1 .611 r
.008 w
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1 .579 .579 r
.008 w
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.688 1 .579 r
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1 .55 .55 r
.008 w
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.667 1 .55 r
.008 w
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1 .524 .524 r
.008 w
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.648 1 .524 r
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P
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1 .5 .5 r
.008 w
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.63 1 .5 r
.008 w
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1 .478 .478 r
.008 w
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1 .458 .458 r
.008 w
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.599 1 .458 r
.008 w
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1 .44 .44 r
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P
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.586 1 .44 r
.008 w
.36918 .35267 Mdot
P
P
p
p
1 .423 .423 r
.008 w
.7452 .76022 Mdot
P
p
.573 1 .423 r
.008 w
.38798 .37189 Mdot
P
P
p
p
1 .407 .407 r
.008 w
.75538 .76984 Mdot
P
p
.561 1 .407 r
.008 w
.40577 .39006 Mdot
P
P
p
p
1 .393 .393 r
.008 w
.76476 .77871 Mdot
P
p
.551 1 .393 r
.008 w
.4226 .40727 Mdot
P
P
p
p
1 .379 .379 r
.008 w
.77344 .78692 Mdot
P
p
.541 1 .379 r
.008 w
.43856 .42358 Mdot
P
P
p
p
1 .367 .367 r
.008 w
.7815 .79454 Mdot
P
p
.531 1 .367 r
.008 w
.4537 .43905 Mdot
P
P
p
p
1 .355 .355 r
.008 w
.789 .80162 Mdot
P
p
.523 1 .355 r
.008 w
.46807 .45375 Mdot
P
P
P
p
p
p
1 .923 .923 r
.008 w
.33868 .37905 Mdot
P
p
.943 1 .923 r
.008 w
.01779 .004 Mdot
P
P
p
p
1 .857 .857 r
.008 w
.4046 .43843 Mdot
P
p
.894 1 .857 r
.008 w
.04331 .02511 Mdot
P
P
p
p
1 .8 .8 r
.008 w
.45646 .48621 Mdot
P
p
.852 1 .8 r
.008 w
.07266 .05311 Mdot
P
P
p
p
1 .75 .75 r
.008 w
.49899 .52586 Mdot
P
p
.815 1 .75 r
.008 w
.10314 .08329 Mdot
P
P
p
p
1 .706 .706 r
.008 w
.5348 .55947 Mdot
P
p
.782 1 .706 r
.008 w
.13343 .11373 Mdot
P
P
p
p
1 .667 .667 r
.008 w
.5655 .5884 Mdot
P
p
.753 1 .667 r
.008 w
.16289 .14353 Mdot
P
P
p
p
1 .632 .632 r
.008 w
.59219 .61362 Mdot
P
p
.727 1 .632 r
.008 w
.19119 .17227 Mdot
P
P
p
p
1 .6 .6 r
.008 w
.61565 .63584 Mdot
P
p
.704 1 .6 r
.008 w
.2182 .19974 Mdot
P
P
p
p
1 .571 .571 r
.008 w
.63645 .65556 Mdot
P
p
.683 1 .571 r
.008 w
.24386 .22588 Mdot
P
P
p
p
1 .545 .545 r
.008 w
.65505 .67322 Mdot
P
p
.664 1 .545 r
.008 w
.2682 .25068 Mdot
P
P
p
p
1 .522 .522 r
.008 w
.67179 .68911 Mdot
P
p
.646 1 .522 r
.008 w
.29124 .27417 Mdot
P
P
p
p
1 .5 .5 r
.008 w
.68694 .7035 Mdot
P
p
.63 1 .5 r
.008 w
.31306 .29642 Mdot
P
P
p
p
1 .48 .48 r
.008 w
.70073 .7166 Mdot
P
p
.615 1 .48 r
.008 w
.33371 .31749 Mdot
P
P
p
p
1 .462 .462 r
.008 w
.71333 .72858 Mdot
P
p
.602 1 .462 r
.008 w
.35326 .33745 Mdot
P
P
p
p
1 .444 .444 r
.008 w
.72489 .73957 Mdot
P
p
.589 1 .444 r
.008 w
.37179 .35636 Mdot
P
P
p
p
1 .429 .429 r
.008 w
.73554 .74969 Mdot
P
p
.577 1 .429 r
.008 w
.38936 .37429 Mdot
P
P
p
p
1 .414 .414 r
.008 w
.74539 .75905 Mdot
P
p
.566 1 .414 r
.008 w
.40603 .39131 Mdot
P
P
p
p
1 .4 .4 r
.008 w
.75452 .76773 Mdot
P
p
.556 1 .4 r
.008 w
.42187 .40747 Mdot
P
P
p
p
1 .387 .387 r
.008 w
.76302 .7758 Mdot
P
p
.546 1 .387 r
.008 w
.43692 .42284 Mdot
P
P
p
p
1 .375 .375 r
.008 w
.77093 .78333 Mdot
P
p
.538 1 .375 r
.008 w
.45125 .43747 Mdot
P
P
P
p
p
p
1 .929 .929 r
.008 w
.31948 .35826 Mdot
P
p
.947 1 .929 r
.008 w
.01658 .00371 Mdot
P
P
p
p
1 .867 .867 r
.008 w
.38348 .41605 Mdot
P
p
.901 1 .867 r
.008 w
.04047 .02341 Mdot
P
P
p
p
1 .813 .813 r
.008 w
.43432 .46302 Mdot
P
p
.861 1 .813 r
.008 w
.06811 .0497 Mdot
P
P
p
p
1 .765 .765 r
.008 w
.47637 .50233 Mdot
P
p
.826 1 .765 r
.008 w
.09695 .0782 Mdot
P
P
p
p
1 .722 .722 r
.008 w
.51203 .53589 Mdot
P
p
.794 1 .722 r
.008 w
.12576 .10709 Mdot
P
P
p
p
1 .684 .684 r
.008 w
.54279 .56497 Mdot
P
p
.766 1 .684 r
.008 w
.15391 .13552 Mdot
P
P
p
p
1 .65 .65 r
.008 w
.56968 .59046 Mdot
P
p
.741 1 .65 r
.008 w
.18107 .16306 Mdot
P
P
p
p
1 .619 .619 r
.008 w
.59342 .61303 Mdot
P
p
.718 1 .619 r
.008 w
.20709 .18949 Mdot
P
P
p
p
1 .591 .591 r
.008 w
.61458 .63316 Mdot
P
p
.697 1 .591 r
.008 w
.23191 .21473 Mdot
P
P
p
p
1 .565 .565 r
.008 w
.63357 .65124 Mdot
P
p
.678 1 .565 r
.008 w
.25553 .23877 Mdot
P
P
p
p
1 .542 .542 r
.008 w
.65072 .66759 Mdot
P
p
.661 1 .542 r
.008 w
.27797 .26162 Mdot
P
P
p
p
1 .52 .52 r
.008 w
.66629 .68244 Mdot
P
p
.645 1 .52 r
.008 w
.29927 .28333 Mdot
P
P
p
p
1 .5 .5 r
.008 w
.6805 .696 Mdot
P
p
.63 1 .5 r
.008 w
.3195 .30394 Mdot
P
P
p
p
1 .481 .481 r
.008 w
.69353 .70842 Mdot
P
p
.616 1 .481 r
.008 w
.3387 .32351 Mdot
P
P
p
p
1 .464 .464 r
.008 w
.70551 .71986 Mdot
P
p
.604 1 .464 r
.008 w
.35694 .3421 Mdot
P
P
p
p
1 .448 .448 r
.008 w
.71658 .73043 Mdot
P
p
.592 1 .448 r
.008 w
.37427 .35978 Mdot
P
P
p
p
1 .433 .433 r
.008 w
.72684 .74021 Mdot
P
p
.581 1 .433 r
.008 w
.39076 .37659 Mdot
P
P
p
p
1 .419 .419 r
.008 w
.73636 .7493 Mdot
P
p
.57 1 .419 r
.008 w
.40645 .39259 Mdot
P
P
p
p
1 .406 .406 r
.008 w
.74524 .75778 Mdot
P
p
.561 1 .406 r
.008 w
.42139 .40783 Mdot
P
P
p
p
1 .394 .394 r
.008 w
.75353 .76569 Mdot
P
p
.552 1 .394 r
.008 w
.43564 .42236 Mdot
P
P
P
p
p
p
1 .933 .933 r
.008 w
.30232 .33961 Mdot
P
p
.951 1 .933 r
.008 w
.01551 .00346 Mdot
P
P
p
p
1 .875 .875 r
.008 w
.36441 .39582 Mdot
P
p
.908 1 .875 r
.008 w
.03799 .02192 Mdot
P
P
p
p
1 .824 .824 r
.008 w
.41418 .4419 Mdot
P
p
.869 1 .824 r
.008 w
.06409 .0467 Mdot
P
P
p
p
1 .778 .778 r
.008 w
.45565 .48076 Mdot
P
p
.836 1 .778 r
.008 w
.09147 .07369 Mdot
P
P
p
p
1 .737 .737 r
.008 w
.49105 .51416 Mdot
P
p
.805 1 .737 r
.008 w
.11893 .10119 Mdot
P
P
p
p
1 .7 .7 r
.008 w
.52175 .54326 Mdot
P
p
.778 1 .7 r
.008 w
.14588 .12836 Mdot
P
P
p
p
1 .667 .667 r
.008 w
.54872 .56891 Mdot
P
p
.753 1 .667 r
.008 w
.17198 .15478 Mdot
P
P
p
p
1 .636 .636 r
.008 w
.57266 .59171 Mdot
P
p
.731 1 .636 r
.008 w
.19708 .18025 Mdot
P
P
p
p
1 .609 .609 r
.008 w
.59406 .61213 Mdot
P
p
.71 1 .609 r
.008 w
.2211 .20465 Mdot
P
P
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p
1 .583 .583 r
.008 w
.61335 .63055 Mdot
P
p
.692 1 .583 r
.008 w
.24402 .22796 Mdot
P
P
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p
1 .56 .56 r
.008 w
.63082 .64726 Mdot
P
p
.674 1 .56 r
.008 w
.26587 .25019 Mdot
P
P
p
p
1 .538 .538 r
.008 w
.64674 .66249 Mdot
P
p
.658 1 .538 r
.008 w
.28667 .27136 Mdot
P
P
p
p
1 .519 .519 r
.008 w
.6613 .67643 Mdot
P
p
.644 1 .519 r
.008 w
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P
P
p
p
1 .5 .5 r
.008 w
.67469 .68924 Mdot
P
p
.63 1 .5 r
.008 w
.32531 .3107 Mdot
P
P
p
p
1 .483 .483 r
.008 w
.68703 .70106 Mdot
P
p
.617 1 .483 r
.008 w
.34326 .32897 Mdot
P
P
p
p
1 .467 .467 r
.008 w
.69845 .712 Mdot
P
p
.605 1 .467 r
.008 w
.36034 .34637 Mdot
P
P
p
p
1 .452 .452 r
.008 w
.70906 .72216 Mdot
P
p
.594 1 .452 r
.008 w
.37663 .36296 Mdot
P
P
p
p
1 .438 .438 r
.008 w
.71893 .73162 Mdot
P
p
.584 1 .438 r
.008 w
.39215 .37878 Mdot
P
P
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p
1 .424 .424 r
.008 w
.72815 .74044 Mdot
P
p
.574 1 .424 r
.008 w
.40697 .39387 Mdot
P
P
p
p
1 .412 .412 r
.008 w
.73677 .7487 Mdot
P
p
.565 1 .412 r
.008 w
.42112 .40829 Mdot
P
P
P
p
p
p
1 .938 .938 r
.008 w
.28689 .3228 Mdot
P
p
.954 1 .938 r
.008 w
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P
P
p
p
1 .882 .882 r
.008 w
.34712 .37744 Mdot
P
p
.913 1 .882 r
.008 w
.03579 .02062 Mdot
P
P
p
p
1 .833 .833 r
.008 w
.39578 .4226 Mdot
P
p
.877 1 .833 r
.008 w
.06052 .04404 Mdot
P
P
p
p
1 .789 .789 r
.008 w
.43661 .46093 Mdot
P
p
.844 1 .789 r
.008 w
.08657 .06968 Mdot
P
P
p
p
1 .75 .75 r
.008 w
.47166 .49407 Mdot
P
p
.815 1 .75 r
.008 w
.11281 .0959 Mdot
P
P
p
p
1 .714 .714 r
.008 w
.50222 .5231 Mdot
P
p
.789 1 .714 r
.008 w
.13865 .12192 Mdot
P
P
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p
1 .682 .682 r
.008 w
.52919 .5488 Mdot
P
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.765 1 .682 r
.008 w
.16376 .14731 Mdot
P
P
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p
1 .652 .652 r
.008 w
.55322 .57175 Mdot
P
p
.743 1 .652 r
.008 w
.18799 .17187 Mdot
P
P
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p
1 .625 .625 r
.008 w
.57479 .59238 Mdot
P
p
.723 1 .625 r
.008 w
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P
P
p
p
1 .6 .6 r
.008 w
.59429 .61106 Mdot
P
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.704 1 .6 r
.008 w
.23352 .21809 Mdot
P
P
p
p
1 .577 .577 r
.008 w
.61202 .62805 Mdot
P
p
.687 1 .577 r
.008 w
.2548 .23972 Mdot
P
P
p
p
1 .556 .556 r
.008 w
.62821 .64358 Mdot
P
p
.671 1 .556 r
.008 w
.27511 .26037 Mdot
P
P
p
p
1 .536 .536 r
.008 w
.64306 .65783 Mdot
P
p
.656 1 .536 r
.008 w
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P
P
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p
1 .517 .517 r
.008 w
.65674 .67097 Mdot
P
p
.643 1 .517 r
.008 w
.31297 .29888 Mdot
P
P
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p
1 .5 .5 r
.008 w
.66939 .68312 Mdot
P
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.63 1 .5 r
.008 w
.33061 .31682 Mdot
P
P
p
p
1 .484 .484 r
.008 w
.68113 .69439 Mdot
P
p
.618 1 .484 r
.008 w
.34744 .33395 Mdot
P
P
p
p
1 .469 .469 r
.008 w
.69204 .70487 Mdot
P
p
.607 1 .469 r
.008 w
.36351 .35031 Mdot
P
P
p
p
1 .455 .455 r
.008 w
.70221 .71464 Mdot
P
p
.596 1 .455 r
.008 w
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P
P
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p
1 .441 .441 r
.008 w
.71173 .72379 Mdot
P
p
.586 1 .441 r
.008 w
.39353 .38087 Mdot
P
P
p
p
1 .429 .429 r
.008 w
.72065 .73235 Mdot
P
p
.577 1 .429 r
.008 w
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P
P
P
p
p
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1 .941 .941 r
.008 w
.27294 .30757 Mdot
P
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.956 1 .941 r
.008 w
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P
P
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p
1 .889 .889 r
.008 w
.33138 .36068 Mdot
P
p
.918 1 .889 r
.008 w
.03383 .01946 Mdot
P
P
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p
1 .842 .842 r
.008 w
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P
p
.883 1 .842 r
.008 w
.05733 .04167 Mdot
P
P
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p
1 .8 .8 r
.008 w
.41907 .44264 Mdot
P
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.852 1 .8 r
.008 w
.08218 .06608 Mdot
P
P
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p
1 .762 .762 r
.008 w
.4537 .47545 Mdot
P
p
.824 1 .762 r
.008 w
.10729 .09114 Mdot
P
P
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p
1 .727 .727 r
.008 w
.48405 .50434 Mdot
P
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.798 1 .727 r
.008 w
.1321 .1161 Mdot
P
P
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p
1 .696 .696 r
.008 w
.51095 .53002 Mdot
P
p
.775 1 .696 r
.008 w
.1563 .14054 Mdot
P
P
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p
1 .667 .667 r
.008 w
.535 .55304 Mdot
P
p
.753 1 .667 r
.008 w
.17972 .16424 Mdot
P
P
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p
1 .64 .64 r
.008 w
.55667 .57381 Mdot
P
p
.734 1 .64 r
.008 w
.20226 .18709 Mdot
P
P
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p
1 .615 .615 r
.008 w
.57632 .59267 Mdot
P
p
.715 1 .615 r
.008 w
.2239 .20905 Mdot
P
P
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1 .593 .593 r
.008 w
.59423 .60988 Mdot
P
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.699 1 .593 r
.008 w
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P
P
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1 .571 .571 r
.008 w
.61064 .62565 Mdot
P
p
.683 1 .571 r
.008 w
.26446 .25024 Mdot
P
P
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1 .552 .552 r
.008 w
.62573 .64016 Mdot
P
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.668 1 .552 r
.008 w
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P
P
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1 .533 .533 r
.008 w
.63966 .65357 Mdot
P
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.655 1 .533 r
.008 w
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P
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1 .516 .516 r
.008 w
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P
p
.642 1 .516 r
.008 w
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P
P
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1 .5 .5 r
.008 w
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P
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.63 1 .5 r
.008 w
.33544 .3224 Mdot
P
P
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p
1 .485 .485 r
.008 w
.67573 .6883 Mdot
P
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.619 1 .485 r
.008 w
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P
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1 .471 .471 r
.008 w
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P
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.608 1 .471 r
.008 w
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P
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1 .457 .457 r
.008 w
.69595 .70778 Mdot
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.598 1 .457 r
.008 w
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P
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1 .444 .444 r
.008 w
.70513 .71662 Mdot
P
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.589 1 .444 r
.008 w
.39488 .38286 Mdot
P
P
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1 .944 .944 r
.008 w
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P
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.959 1 .944 r
.008 w
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P
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1 .895 .895 r
.008 w
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P
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.922 1 .895 r
.008 w
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P
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1 .85 .85 r
.008 w
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P
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.889 1 .85 r
.008 w
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P
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1 .81 .81 r
.008 w
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.008 w
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P
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1 .773 .773 r
.008 w
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P
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.832 1 .773 r
.008 w
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P
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1 .739 .739 r
.008 w
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.807 1 .739 r
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P
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1 .708 .708 r
.008 w
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.008 w
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P
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1 .68 .68 r
.008 w
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.008 w
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1 .654 .654 r
.008 w
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.744 1 .654 r
.008 w
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P
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1 .63 .63 r
.008 w
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.008 w
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P
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1 .607 .607 r
.008 w
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.008 w
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P
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1 .586 .586 r
.008 w
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.694 1 .586 r
.008 w
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1 .567 .567 r
.008 w
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.679 1 .567 r
.008 w
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1 .548 .548 r
.008 w
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.666 1 .548 r
.008 w
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1 .531 .531 r
.008 w
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.008 w
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1 .515 .515 r
.008 w
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P
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1 .5 .5 r
.008 w
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.63 1 .5 r
.008 w
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1 .486 .486 r
.008 w
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.008 w
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1 .472 .472 r
.008 w
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.609 1 .472 r
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1 .459 .459 r
.008 w
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MathPictureEnd

:[font = input; preserveAspect]
tst = Table[Random[Integer,{1,100}],{10},{2}];
:[font = input; Cclosed; preserveAspect; startGroup]
Timing[res = Apply[Function[{x,y},BinomialCL[x,y,#,AccuracyGoal->1]& /@ {.025,.975}],tst,1];]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{7.550000000002911*Second, Null}
;[o]
{7.55 Second, Null}
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
dB, dBInverse
:[font = text; inactive; dontPreserveAspect]
The decibel (dB) is 1/20 of the log to the base 10. These functions convert to and from dB.
:[font = input; initialization; dontPreserveAspect]
*)
dB[x_] := 20. Log[10.,x]
(*
:[font = input; initialization; preserveAspect]
*)
dBInverse[x_] := 10.^( x .05)
(*
:[font = text; inactive; preserveAspect]
Retained for backward compatibility:
:[font = input; initialization; dontPreserveAspect; endGroup]
*)
dBinv := dBInverse
(*
:[font = subsection; active; Cclosed; dontPreserveAspect; startGroup]
Protected log function.
:[font = text; inactive; preserveAspect]
In the computation of log likelihoods, it is occasionally necessary to make use of the following rules, so that the log function does not blow up.
:[font = input; initialization; preserveAspect]
*)
NLogP[0,0] := 0
(*
:[font = input; initialization; preserveAspect]
*)
NLogP[n_,0] := -Sign[n] Infinity
(*
:[font = input; initialization; dontPreserveAspect]
*)
NLogP[n_,d_] := n Log[d]
(*
:[font = input; initialization; dontPreserveAspect; endGroup; endGroup]
*)
SetAttributes[NLogP,Listable]
(*
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Tutorial
:[font = text; inactive; preserveAspect]
In this section I provide some simple examples of how to use the functions PsychometricFit and PsychometricPlot, and how to create and use a new psychometric function.
:[font = subsection; inactive; preserveAspect; startGroup]
Fitting and plotting one set of data
:[font = text; inactive; preserveAspect]
First we create some fictional data. These are triples of the form {strength, # wrong, # right}.
:[font = input; preserveAspect]
data1={{2,4,3},{4,3,4},{6,10,34},{8,1,18},{10,0,10}};
:[font = text; inactive; preserveAspect]
We fit the data. The function returns the data, the name of the fitted psychometric function, and a list containing a list of function parameters and a measure of the error of the fit.
:[font = input; preserveAspect; startGroup]
fit1 = PsychometricFit[data1]
:[font = output; output; inactive; preserveAspect; endGroup]
{{{2, 4, 3}, {4, 3, 4}, {6, 10, 34}, {8, 1, 18}, {10, 0, 10}}, 
 
  LogWeibullPF, {{6.373449329205624, 4., 0.5, 0.01}, 0.4896021602893594}}
;[o]
{{{2, 4, 3}, {4, 3, 4}, {6, 10, 34}, {8, 1, 18}, {10, 0, 10}}, 
 
  LogWeibullPF, {{6.37345, 4., 0.5, 0.01}, 0.489602}}
:[font = text; inactive; preserveAspect]
This result can be plotted directly using the function PsychometricFit, about which we will say more in a moment.
:[font = input; preserveAspect; startGroup]
PsychometricPlot[fit1];
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 48; pictureWidth = 323; pictureHeight = 199; endGroup]
%!
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:[font = text; inactive; preserveAspect]
By default, only the first parameter (threshold) is estimated. To estimate both the threshold and slope, we use the option FitParameters. Note the reduction in the error.
:[font = input; preserveAspect; startGroup]
PsychometricFit[data1,FitParameters->{True,True}]
:[font = output; output; inactive; preserveAspect; endGroup]
{{{2, 4, 3}, {4, 3, 4}, {6, 10, 34}, {8, 1, 18}, {10, 0, 10}}, 
 
  LogWeibullPF, {{6.43045468104664, 5.301292498036729, 0.5, 0.01}, 
 
   0.2824608081873678}}
;[o]
{{{2, 4, 3}, {4, 3, 4}, {6, 10, 34}, {8, 1, 18}, {10, 0, 10}}, 
 
  LogWeibullPF, {{6.43045, 5.30129, 0.5, 0.01}, 0.282461}}
:[font = text; inactive; preserveAspect]
The LogWeibullPF psychometric function has four parameters (threshold, slope, guess, and lapse). Here we estimate threshold and guess, and we alter the default value for the slope parameter.
:[font = input; preserveAspect; startGroup]
PsychometricFit[data1,FitParameters->{True,False,True}
,PsychometricParameters->{Automatic,3.,0.5,0.01}]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{{{2, 4, 3}, {4, 3, 4}, {6, 10, 34}, {8, 1, 18}, {10, 0, 10}}, 
 
  LogWeibullPF, {{5.058810771960051, 3., 0.1623733096620143, 0.01}, 
 
   0.1710268584971723}}
;[o]
{{{2, 4, 3}, {4, 3, 4}, {6, 10, 34}, {8, 1, 18}, {10, 0, 10}}, 
 
  LogWeibullPF, {{5.05881, 3., 0.162373, 0.01}, 0.171027}}
:[font = subsection; inactive; preserveAspect; startGroup]
Fitting a different psychometric function
:[font = text; inactive; preserveAspect]
Another widely used psychometric function is the LogisticPF. We have defined it in such a way that its parameters are identical in structure to those of the LogWeibull, though slightly different in effect.
:[font = text; inactive; preserveAspect]
To change the type of fitted function, we use the option PsychometricFunction.
;[s]
3:0,0;57,1;77,0;79,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect; startGroup]
PsychometricPlot[PsychometricFit[data1
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  ,FitParameters->{True,True} ]];
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:[font = subsection; inactive; preserveAspect; startGroup]
Fitting and plotting several sets of data
:[font = text; inactive; preserveAspect]
Both PsychometricFit and PsychometricPlot accept multiple data sets, as when an experiment contains several conditions.  In this example each of two conditions contributes one list of triples.
:[font = input; dontPreserveAspect]
data2 = {
  {{3, 3, 4}, {5, 10, 34}, {7, 1, 18} }
 ,{{3, 4, 3}, {5, 16, 34}, {7,5,37}, {9, 1, 18} } };
:[font = input; preserveAspect; startGroup]
PsychometricPlot[PsychometricFit[data2]];
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 48; pictureWidth = 323; pictureHeight = 199; startGroup]
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:[font = subsection; inactive; preserveAspect; startGroup]
Creating and fitting a ramp psychometric function
:[font = text; inactive; preserveAspect]
To show how the user may create a new type of psychometric function, we consider a ramp.
:[font = input; preserveAspect]
ramp[x_,{threshold_, slope_, guess_, lapse_}] := Max[guess,Min[1-lapse,.75 + slope (x-threshold)]]
:[font = text; inactive; preserveAspect]
We plot the function with  particular paramaters.
:[font = input; preserveAspect; startGroup]
Plot[ramp[x,{3,.1,.5,.01}],{x,-10,10}
,PlotRange->{0,1}];
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:[font = text; inactive; preserveAspect]
We also indicate the names of the various parameters. This information is used by plotting functions.
:[font = input; preserveAspect]
PNames[ramp] ^:=
 {"threshold","slope","guess","lapse"}
:[font = text; inactive; preserveAspect]
To fit the data with this function, we use the PsychometricFunction option. We also use the PsychometricParameters option to specify a reasonable starting point for the slope parameter.
:[font = input; preserveAspect; startGroup]
PsychometricPlot[PsychometricFit[data1
 ,PsychometricFunction->ramp
 ,FitParameters->{True,True}
 ,PsychometricParameters->{Automatic,.1,0.5,0.01}]];
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 48; pictureWidth = 323; pictureHeight = 199; endGroup; endGroup]
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:[font = subsection; inactive; preserveAspect; startGroup]
What happens when a fit cannot be found?
:[font = text; inactive; preserveAspect]
If a fit cannot be produced, a structure of the correct shape is returned, but with Nulls in place of all parameters and the error (beeps and messages will also be produced). This sometimes happens when the parameter limits are exceeded, or when the error function has no minimum.
:[font = input; preserveAspect; startGroup]
PsychometricFit[{{2,4,3}}]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{{{2, 4, 3}}, LogWeibullPF, {{Null, Null, Null, Null}, Null}}
;[o]
{{{2, 4, 3}}, LogWeibullPF, {{Null, Null, Null, Null}, Null}}
:[font = subsection; inactive; preserveAspect; startGroup]
Some plotting options
:[font = text; inactive; preserveAspect]
The basic plot includes proportion correct (points), fitted function (curve), binomial confidence limits (grey vertical lines), and printed parameters. The latter two are optional.
:[font = input; preserveAspect; startGroup]
PsychometricPlot[fit1,PsychErrorBars->False];
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 48; pictureWidth = 323; pictureHeight = 199; endGroup]
%!
%%Creator: Mathematica
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