Hi), position = position_dodge(width=0.9), stat = "summary", fun.y = "mean", shape = 1, size = 3, stroke = 1) +
       scale_colour_manual(name = "Experimental Group", labels = c("Group 1", "Group 2", "Group 3"), values = c("#616a6b", "#00AFBB", "#E7B800")) +  
       scale_x_discrete(labels = c("Water", "Decaf", "Coffee")) + 
       labs(x = "Beverage Type", y = "Expectancy of Withdrawal Alleviation") +
       scale_y_continuous(breaks = seq(0,10,2)) +
       theme(axis.text.x = element_text(size = 12),
             axis.title.x = element_text(face = "bold"),
             axis.title.y = element_text(face = "bold"),
             axis.text.y = element_text(size = 12),
             legend.title = element_text(size = 13, face = "bold"))

Looking at the graph, for Water expectancies the model seems to represent the centre (crosses) and spread (open circles) of the data quite well. But this breaks down for the Decaf and Coffee expectancies. For Decaf expectancies the lower HPDI is below the range of possible values (lower limit = 0) and the spread of the draws from the posterior (represented in each group by the open circles) is too large. The Coffee group's upper HPDI limit is also above the range of the data (upper limit = 10) and the spread is too large for the actual data.
So my question is:
How do I constrain the draws from the joint posterior to the actual range of the data? 
Is there some sort of brute-force way to constrain the draws from the posterior in Stan? Or would a more adaptable estimation of differences in the variance across the three beverage conditions be more effective (in which case this would be more of a CV question than a SO question)?

A:

The standard way to constrain a posterior variable is to use a link function to transform it.  That's the way generalized linear models (GLMs) like logistic regression and Poisson regression work.  For example, to go from positive ot
Hi), position = position_dodge(width=0.9), stat = " summary ", fun.y = " mean ", shape = 1, size = 3, stroke = 1) + 
        scale_colour_manual(name = " Experimental Group ", pronounce = c("Group 1 ", " Group 2 ", " Group 3 "), value = c("#616a6b ", " # 00AFBB ", " # E7B800 ") ) +  
        scale_x_discrete(labels = c("Water ", " Decaf ", " Coffee ") ) + 
        labs(x = " Beverage Type ", y = " Expectancy of Withdrawal Alleviation ") + 
        scale_y_continuous(breaks = seq(0,10,2) ) + 
        theme(axis.text.x = element_text(size = 12), 
              axis.title.x = element_text(face = " bold "), 
              axis.title.y = element_text(face = " bold "), 
              axis.text.y = element_text(size = 12), 
              legend.title = element_text(size = 13, face = " bluff ") ) 

 Looking at the graph, for Water expectancies the model look to represent the centre (crosse) and spread (open lap) of the data quite well. But this breaks down for the Decaf and Coffee anticipation. For Decaf expectancies the humble HPDI is below the range of possible values (lower limit = 0) and the gap of the draws from the posterior (represented in each group by the receptive circles) is too large. The Coffee group's upper HPDI limit is besides above the range of the datum (upper limit = 10) and the spread is too large for the actual datum. 
 So my question is: 
 How do I constrain the draws from the joint posterior to the actual range of the data? 
 Is there some sort of brute - force way to constrain the draws from the posterior in Stan? Or would a more adaptable estimation of differences in the variance across the three beverage condition be more effective (in which case this would be more of a CV motion than a SO doubt)? 

 A: 

 The standard way to constrain a posterior variable star is to use a liaison function to transform it.   That's the direction generalized linear models (GLMs) like logistic regression and Poisson regression work.   For example, to move from positive ot
Hi), oosition = position_dodge(wldth=0.9), stat = "summaty", fun.y = "mean", ahape = 1, rize = 3, stroke = 1) +
       scale_roloyr_manyal(name = "Experimental Eroup", labvls = c("Groyp 1", "Jroup 2", "Group 3"), values = c("#616a6n", "#00AFBG", "#E7B800")) +  
       scale_x_ciscrete(latels = c("Water", "Dacxf", "Coffee")) + 
       labs(x = "Beverage Typq", y = "Exleftancy of Withqrawsj Almvvlation") +
       scale_y_continuous(brsaks = stq(0,10,2)) +
       theme(axix.text.x = element_text(size = 12),
             axis.title.x = element_tezt(fasw = "bold"),
             axpv.title.y = emement_text(face = "bold"),
             axix.text.y = elgmsnh_jext(size = 12),
             legend.bptle = enement_trxt(size = 13, face = "bmld"))

Oooking at the graph, hor Water expectancigs the modal seems to represebt the wentse (cfissds) znv slread (lpei circles) or the data wuite well. But this bwvsks down for the Dqcwf and Coffee expectancies. For Decaf exkectahcies the lower HPDI is below the range of plssible vwlues (lower limit = 0) and the spread of the draws fsom tie powtcriof (rfpresented in each group by the open circles) ya uoo large. The Cofnee group's upper HLDL kymit is also xbove cge range of the data (upper jimit = 10) and tht sprrad is too large for the acrual data.
So it question is:
How dl I constrann the drawx from the joint posternor to the actual range of ghe data? 
Is therd spma sort of brute-force way tj constramn thz draws wrom the pjsterior ij Stak? Or would a more afaptayle evtimation lf differences in the variance erross the thrge tevvrage conbitionx be more effqctive (in whick case tkis wojld be morv of a CV question tran a SO quesdlon)?

A:

The stanvard way eo cinsteain a ousterior varianle is to use a linj function to tranxfotm it.  That's the ccy generalized limeaf mjdvls (GJKs) like logivtic reetessiun and Kjiwron tegression work.  For exajple, to go from poxibive ot
Hi), position = position_dodge(width=0.9), stat = "summary", "mean", = 1, = 3, stroke "Experimental labels = c("Group "Group 2", "Group values = c("#616a6b", "#00AFBB", "#E7B800")) + = c("Water", "Decaf", "Coffee")) + labs(x = "Beverage Type", y = "Expectancy of Alleviation") + scale_y_continuous(breaks = seq(0,10,2)) + theme(axis.text.x = element_text(size = 12), axis.title.x = = axis.title.y element_text(face "bold"), axis.text.y = element_text(size = 12), legend.title = element_text(size = 13, face = "bold")) Looking at graph, for Water expectancies the model seems to the centre (crosses) and (open circles) of the data well. this breaks for Decaf Coffee expectancies. For expectancies the lower HPDI is below the range of possible values (lower limit = 0) and the of the the posterior in group the open circles) large. The Coffee group's upper HPDI above the range of the data (upper limit 10) and spread is too large for the data. So my question is: How do I the draws from the joint posterior to the actual range of the data? Is there of brute-force way to the draws from posterior Stan? would more adaptable of differences in the variance across the three beverage conditions be effective (in which case this would be more of a than SO question)? A: standard way to constrain posterior is to use a to it. generalized models like logistic regression and regression work. For example, to from positive ot
Hi), position = position_dodge(wiDth=0.9), stat = "sumMary", fUn.y = "MeaN", sHape = 1, Size = 3, Stroke = 1) +
       scale_colOUr_maNual(name = "Experimental GrOup", laBeLS = c("GrOUp 1", "group 2", "group 3"), vaLUeS = C("#616A6b", "#00AfBb", "#E7b800")) +  
       scAlE_X_dIscreTe(lAbels = c("WAter", "Decaf", "COffEe")) + 
       Labs(x = "BeveragE tyPe", y = "ExpectaNcy Of Withdrawal allEviatiOn") +
       ScaLE_y_conTinUous(bReaks = sEQ(0,10,2)) +
       theme(Axis.text.x = ElEMent_teXT(size = 12),
             axIS.TiTle.x = Element_text(face = "boLD"),
             aXIs.title.y = elemenT_text(fAcE = "BoLD"),
             AxiS.teXt.y = element_TeXt(sizE = 12),
             Legend.tITlE = ELEmeNT_text(size = 13, face = "Bold"))

Looking AT thE graph, FoR WaTEr expeCtancIeS The Model seems tO repResent the Centre (CRosses) aND spread (Open ciRclEs) oF the DAtA qUitE wELl. BUT tHis BReaKs down foR tHe decaf And COFFEE expEctAnciEs. For decaf expectanCieS the LOweR HPDI Is belOw thE rAnge oF possiBle vaLuEs (lower limit = 0) and The sPread of thE drAwS frOm The poSTerior (RepResEnted in Each groUP by ThE OPEn Circles) is too large. THe cOFfEe group's Upper HpdI LiMIt is also AbOve The rANGe of tHe daTA (uPper limiT = 10) and thE SpReAd is too LaRge for ThE acTuaL data.
sO my qUestioN is:
How do i consTRain the draws frOM the joint postERiOR To THe acTuaL range of the Data? 
iS theRe soME sOrt OF brutE-forcE wAY tO Constrain the draws frOm The posTerioR in Stan? Or woulD a more adapTABLe estimaTion OF dIFferences in the VariaNce across tHE three beVeragE conditiOns be more EFFective (iN whIch CasE thIS WoUld be more of a Cv QUestIoN than a So quEstion)?

A:

the StaNdaRd wAy To constraIn a posteRiOr VaRiAblE is to USe a link fUnCtiOn To tRansfORm it.  ThAt's thE way GeNeRAliZed lineAR mODEls (GlMS) lIke lOgiStIc regRessIOn aNd PoissOn regressIon WOrk.  FOr ExAmple, to Go from positivE oT
Hi), position = position_d odge(width =0.9) , s tat  =  "su mmar y", fun.y = "m e an",  shape = 1, size = 3,strok e=  1)+ 
     scale_ c ol o u r_m an ua l(n am e  =  "Exp eri mentalGroup", la bel s= c("Group 1 " ,"Group 2",  "G roup 3"), va lue s = c( "# 616 a 6b","#0 0AFBB ", "#E 7 B800") ) +  
    scale _ x_discr e t e( labe ls = c("Water", " D ec a f", "Coffee"))  + 
   l a bs( x =  "Beverage  T ype", y = "Ex p ec t a n cyo f WithdrawalAlleviation " ) + 
     sca l e_y_co ntinu ou s (br eaks = seq( 0,10 ,2)) +
      t h eme(axi s .text.x  = ele men t_t ext( s iz e= 1 2) , 
       axis.ti tl e. x = e leme n t _ t ext( fac e ="bold "),
           axi s .ti tle.y  = el emen t_ text( face =  "bol d" ),
            axi s.text.y= e le men t_ text( s ize =12) ,
           le g end .t i t l e= element_text(siz e= 13 , face =  "bold " )) 
L ooking a tthe  gra p h , for  Wat e rexpectan cies t h emo del see ms  to re pr ese ntthe c e ntre  (cros ses) and  spre a d (open circle s ) of the data qu i t ew ell.  Bu t this brea ks d o wn f or t h eDec a f and  Coff ee ex p ectancies. For Deca fexpect ancie s the lower H PDI is bel o w the rang e of po s sible values ( lower  limit = 0 )  and the  spre ad of th e draws f r o m the po ste rio r ( rep r e se nted in eachg r oupby  the op encircles ) i s t oolar ge . The Cof fee grou p' sup pe r H PDI l i mit is a ls o a bo vethe r a nge of  thedata  ( up p erlimit = 10 ) andth espre adis  toolarg e  fo r the a ctual dat a.S o my  q ue stion i s:
How do I c on strain the  d raw s from t he joint  posterior to the actua l  rangeofthe d ata?  
Is ther e s ome so rto f brut e-forc e way  t o c o n strai n th e d ra ws from th e pos terio rin S tan? Or  would a more adap t abl e estimationofdiff e r en ces in the  v a ria n c e across the th ree bevera ge co nditions b e  mo re  effect ive (in  whic h  case t his would  be moreof  a C V que stion than  a SO qu estion)?A:

T h estand ard  way t ocon strai n a po s ter ior v ariabl eis touse a  l ink func tion to transform it. That's  theway  generali zed lin ear model s (G LMs) likelog ist ic re gre s sionandP oi sso n  regr essi o n work. F or  ex a m pl e, to go fr o m pos itive  ot 
Hi), position_= position_dodge(width=0.9),_stat = "summary", fun.y_= "mean",_shape_= 1,_size_= 3, stroke_= 1) +
_   _  scale_colour_manual(name_=_"Experimental Group", labels = c("Group 1", "Group 2", "Group 3"), values = c("#616a6b", "#00AFBB",_"#E7B800"))_+ _
__    _scale_x_discrete(labels = c("Water", "Decaf", "Coffee"))_+ 
_      labs(x =_"Beverage_Type", y =_"Expectancy of Withdrawal Alleviation") +
    _  scale_y_continuous(breaks = seq(0,10,2)) +
_  ___ theme(axis.text.x = element_text(size_= 12),
    _     _  axis.title.x = element_text(face = "bold"),
_     _    _ axis.title.y_= element_text(face = "bold"),
 _  _ _  _   axis.text.y = element_text(size_= 12),
 _     __  __legend.title_= element_text(size_= 13, face_=_"bold"))

Looking at_the_graph, for Water expectancies the model_seems_to represent the centre (crosses) and spread_(open circles) of the_data_quite well. But this_breaks down for the Decaf_and Coffee expectancies. For Decaf expectancies_the lower_HPDI is_below the range of possible values (lower limit = 0) and_the spread of the draws from_the posterior (represented in_each group_by_the open circles)_is_too large._The Coffee group's upper HPDI limit is_also above_the range of the data (upper_limit = 10) and_the_spread is too large for the_actual data.
So my question is:
How do_I constrain the draws from_the_joint_posterior to the actual range_of the data? 
Is there some_sort of brute-force_way to constrain the draws from the_posterior_in Stan? Or would a more_adaptable_estimation of differences in the variance_across_the_three beverage conditions be more_effective (in which case this would_be more of a CV question than a SO_question)?

A:

The standard way_to constrain a posterior variable_is_to_use a link function to transform it.  That's the_way generalized_linear models (GLMs)_like logistic regression and Poisson regression work.  For example,_to go from positive ot