University of Texs MD Anderson Cncer Center Deprtment of Biosttistics Inequlity Clcultor, Version 3.0 November 5, 013 User s Guide 0. Overview The purpose of the softwre is to clculte the probbility tht one rndom vrible is greter thn nother. The two rndom vribles re ssumed to follow the sme stndrd distribution fmily, with different prmeter vlues. The distributions currently supported re: bet, gmm, inverse gmm, norml, log norml nd Weibull. The softwre my be downloded from the web site: http://biosttistics.mdnderson.org/softwredownlod/ This softwre occsionlly sends usge sttistics nd crsh reports to our biosttistics softwre support tem to improve your experience using it. License: This progrm is distributed t no cost to the user. However, redistribution of this progrm is not permitted. Rther thn giving this progrm to someone else, plese hve them obtin their own copy directly from M. D. Anderson Cncer Center. This llows us to keep record of who is using the softwre nd llows us to notify ll users when progrm enhncements become vilble. NO WARRANTY We provide bsolutely no wrrnty of ny kind, either expressed or implied, including but not limited to the implied wrrnties of merchntbility nd fitness for prticulr purpose. The entire risk s to the qulity nd performnce of the progrm is with the user. Should this progrm prove defective, the user ssumes the cost of ll necessry servicing, repir, or correction. In no event shll The University of Texs or ny of its component institutions, including M. D. Anderson Cncer Center, be lible for dmges, including ny lost profits, lost monies, or other specil, incidentl or consequentil dmges rising out of the use of or inbility to use the progrm (including but not limited to loss of dt or its nlysis being rendered inccurte or losses sustined by third prties).
1. Description For two generl continuous rndom vribles nd Y, the probbility tht > Y is given by P( > Y ) = f ( t) F ( t dt Y ) where f is the PDF (probbility density function) of nd F Y is the CDF (cumultive distribution function) of Y. The progrm computes this probbility for the cse of nd Y following the sme stndrd distribution fmily. In the exmple of Figure 1, we let ~ Norml(0, 1) nd Y ~ Norml(1, 1) nd click on Clculte to obtin the result Prob( > Y) = 0.3975, long with the complementry vlue Prob(Y > ) = 0.7605. Figure 1:
. Shift Prmeter For the cse of nonzero delt shift prmeter, the probbility inequlity is given by P( > Y + delt ) = f ( t) F ( t delt dt. Y ) In the exmple of Figure, we let ~ bet(6, 6), Y ~ bet(3, 3) nd delt = 0.1, giving the result 0.3448. If delt = 0 then the result would be 0.5 due to symmetry. Figure :
3. Clcultions When the rndom vribles nd Y follow one of the specil distributions, there re more efficient methods for computing P(>Y) thn directly pplying the integrtion formul. For exmple, if nd Y follow exponentil distributions (gmm with shpe 1) with mens μ nd μ Y, then µ P( > Y ) = µ + µ Y If nd Y follow the norml distribution with prmeters (μ, σ ) nd (μ Y, σ Y ) respectively, then ( ) µ µ Y P( > Y ) = Φ 1/ σ + σ Y where Φ is the cumultive distribution function of the stndrd norml. More informtion on these clcultions cn be found in the pper Numericl computtion of stochstic inequlity probbilities by John Cook. For nonzero delt nd the norml distribution fmily, Y+delt ~ Norml(μ Y + delt, σ Y ) so tht ( ) µ µ Y delt P( > Y + delt) = Φ 1/ σ + σ Y The other probbility distribution fmilies re not closed under trnsltion (i.e., Y+delt cnnot be prmeterized s stndrd distribution when delt is nonzero), so the probbility inequlity clcultions do use numericl integrtion.
4. Prmeteriztions The softwre generlly follows the conventions (with some exception) in the book Sttisticl Distributions by Merrn Evns, Nichols Hstings, nd Brin Pecock. 4.1 Bet The bet distribution with prmeters nd b nd hs PDF Γ( + b) 1 b 1 x (1 x) Γ( ) Γ( b) with men + b nd vrince b ( + b) ( + b + 1) 4. Gmm The gmm distribution with shpe prmeter nd scle prmeter b hs men b, vrince b nd PDF 1 1 x / b x e Γ( ) b 4.3 Inverse Gmm The inverse gmm distribution with shpe prmeter nd scle prmeter b hs PDF b b / x e + 1 x ( ) Γ If > 1 then the men is b 1 If > then the vrince is b ( 1) ( ) If is distributed s gmm distribution with prmeters (, b) then 1/ is distributed s n inverse gmm with prmeters (, 1/b). Note tht the b in our prmeteriztion of the inverse gmm corresponds to 1/b in nother populr convention.
4.4 Norml The norml distribution prmeterized by its men m nd vrince s s 1 1 ( x m) s e π hs PDF 4.5 Log Norml The log norml distribution is prmeterized by m nd s. If is log norml with these prmeters, log is N(m,s ). Note tht m nd s re not the men nd stndrd devition of but rther of log. The PDF is given by 1 (log( x) m) exp xs π s with men 1 exp m + s nd vrince exp(m + s )(exp( s 4.6 Weibull ) 1) The Weibull distribution hs shpe prmeter nd scle prmeter b. It hs PDF 1 x exp( ( x / b) ) b with men b Γ (( + 1) / ) nd vrince b Γ (( + ) / ) Γ(( + 1) / ) ( )
5. Miscellneous 5.1 Grphic Options The user my customize the PDF plot by right-clicking on the window to bring up menu with the item Grphic Options (Figure 5). Figure 5: Choosing this item will bring up the dilog box in Figure 5b. Any chnge to one of these dilog fields will be reflected immeditely in the grph pne. Clicking the Accept button will sve chnges to the grph, while clicking Cncel will discrd ll chnges.
Figure 5b: The Copy menu item in the popup menu of Figure 5 will copy the grph imge to the clipbord where it cn be psted into nother document. The Sve menu item will bring up "Sve As..." dilog box for selecting the nme nd loction of the file to be sved. The qulity of the grph is dependent on the size of the grph when it is copied or sved. Therefore, it is suggested tht the window be mximized before copying or sving the grph. This will produce clerer imge when enlrged. 5. Error Indictor Ech numericl input field is equipped with n error indictor. When the symbol is displyed, its djcent field contins n invlid entry. By moving the mouse cursor over this symbol, tip will be displyed s shown in Figure 5c. Figure 5c: