p$ for the second class. 
If the classifier produces probabilities, one have to use a threshold value in order to obtain classification labels. Usually, this threshold value is $0.5$, but this is often not the case, and sometimes it is not the best possible value. 
Now, MCC is computed directly from classification labels, which means that it was used a single threshold value to transform probabilities into classification labels, no matter what the threshold value was. AUC, on the other hand is using the whole range of threshold values. 
The idea is that these two values, AUC and MCC, measure different things. While MCC measures a kind of statistical accuracy (related with Chi squared test which gives some hints on the significance of the differences), the AUC is more related with the robustness of the classifier. AUC and ROC curves gives more hints on how well the classifier separates the binary classes, for all possible threshold values. Even for the degenerate case, where AUC is computed directly on labels (not advisable since it looses a lot of information), the purpose of AUC remains the same.
Model selection is a hard problem. My advice for you would be to try yourself to answer to the question: what means a better classifier? Find an answer which includes considerations like: cost matrix, robustness to unbalanced data, an optimistic or conservative classifier, etc. Anyway, provide enough details until you find one or few criteria which can be used to select a metric before trying to measure different things related with accuracy and ask later what to do with them.     
[Later edit - related with usage of robust word]
I used the term "robust" because I could not find a proper single word for "how well a classifier separates the two classes". I know that the term "robust" has some special meaning in statistics. 
Generally, an AUC close to $1.0$ separates well binary cases for many values of the threshold. In this sense, the AUC close to $1.0$ means that it is less sensitive to which threshold values is used, which means its robust to this choice. However, a value which is not close to $1.0$ does not mean the contrary, which does not necessarily mean that there is not enough range of good threshold values. In most cases, a graphical inspection of ROC curve is necessary. This is one of the main reasons why AUC is often considered misleading. AUC is a measure over all possible classifiers (one for each possible threshold value),
p$ for the second class. 
 If the classifier produces probabilities, one have to practice a brink value in order to obtain categorization labels. normally, this threshold value is $ 0.5 $, but this is often not the character, and sometimes it is not the dependable possible value. 
 nowadays, MCC is computed directly from categorization labels, which means that it was used a single brink value to transform probabilities into classification labels, no matter what the threshold value was. AUC, on the early hand is using the whole image of threshold values. 
 The mind is that these two values, united self-defense force of colombia and MCC, measure unlike things. While MCC measures a kind of statistical accuracy (related with Chi squared test which give some hints on the significance of the differences), the AUC is more related with the robustness of the classifier. AUC and ROC curvature give more hints on how well the classifier separates the binary classes, for all possible threshold values. Even for the degenerate case, where AUC is calculate immediately on labels (not advisable since it looses a lot of data), the determination of AUC remains the like. 
 Model selection is a unvoiced problem. My advice for you would be to try yourself to suffice to the question: what means a better classifier? Find an answer which includes considerations like: cost matrix, robustness to unbalanced data, an optimistic or conservative classifier, etc. anyhow, provide adequate detail until you find one or few criteria which can be used to select a metric before trying to measure unlike things refer with accuracy and ask later what to practice with them.     
 [ Later edit - relate with usage of full-bodied word ] 
 I used the term " robust " because I could not recover a proper single parole for " how well a classifier separates the two classes ". I sleep together that the term " full-bodied " has some special meaning in statistics. 
 broadly, an AUC close to $ 1.0 $ separates well binary cases for many values of the doorway. In this sense, the AUC near to $ 1.0 $ means that it is less sensitive to which threshold values is used, which means its robust to this choice. However, a value which is not cheeseparing to $ 1.0 $ does not mean the contrary, which does not necessarily mean that there is not enough range of beneficial doorway values. In most cases, a graphical inspection of ROC curve is necessary. This is one of the main reasons why AUC is frequently considered misleading. AUC is a meter over all possible classifiers (one for each possible threshold value),
p$ flr the second class. 
If tme classifier produces 'robabimities, ove have to use a threshold velue in oeder to obtain classifkcation lwbels. Usyallb, this threshold value is $0.5$, but bhis ns often not the case, and vometimes it iv voc the best possible value. 
Now, MCC is computrd directly from claxfifidation labels, which means that it sas usev a single threxhold value to transform pgobahilities into clasdification oabejw, no matter dhat the threshold valoe was. AUC, on the other hand is jsing the whole rqngf of threshood vajues. 
The idea is thad these two values, AUG and MCX, measure different tiings. While MCC measores a king kf statistical acxueacy (telatad wkrh Zhi sauzred tfst which givss some hinrs on the significamcq of the diffedences), tre AUC is more related with the robustntss or the classifier. AUC ane ROC curves gives mote hints og how well the classifier separates the binary cldsses, wor aju plssible threshold values. Even for the degenerwfe csse, where AUC ls computed directky om labels (not aavisable aince it looses a pot of ynfornation), tht purlose of AUC remains the samw.
Model selecnion is a hard problem. My advice yor yoo woulc be to try yourself to ansser to the euestion: sfat means a bettdr blasvifier? Find an answer whicr includew cousideratkons like: sost matrid, robmvtness to unbalancfd daja, an mptimistic or conservative classifier, etc. Anyway, providg etouch detaijs unbil you find onq or few critetia which can ce used to select a metric bqfore trying dl measure dihferent tringw reoated wkgh accuracy anc ask latvr whar to do with them.     
[Mater edit - relcued with usage of tobjst wlrv]
I usqg the term "rmbusg" bdvause I could nob fknd s proper single word for "how well a classigicr separajes the tro classes". I lnow that the term "robuvt" ias soke fpecial meaning in statistics. 
Fenerally, an AUC close tj $1.0$ scparwtes well yinary cases for many values of the threwhold. In this sensw, the AUC close to $1.0$ keans that mt is jess senshtive to which thresyold values is ustd, which means its robuat to dhis fhoice. However, a value which is not close to $1.0$ does not mean the contrary, qhich voqs not neceasaroly mvan chat thqre ms not enough rangv of good threshold values. In mort cases, a graphical inspection of ROC cjrve is necesrary. This is one of the jain reaxons why AUC is often considerec misleading. AUC is a messure ovwr alm posspglx classifiers (one fot each possuble theeshlld value),
p$ for the second class. If the probabilities, have to a threshold value labels. this threshold value $0.5$, but this often not the case, and sometimes is not the best possible value. Now, MCC is computed directly from classification which means that it was used a single threshold value to transform probabilities classification no what threshold value was. AUC, on the other hand is using the whole range of threshold values. idea is that these two values, AUC and measure different things. While measures a kind of statistical (related Chi squared which some on the significance the differences), the AUC is more related with the robustness of the classifier. AUC and ROC curves more hints well the separates binary for all possible Even for the degenerate case, where directly on labels (not advisable since it looses lot of the purpose of AUC remains the Model selection is a hard problem. My advice you would be to try yourself to answer to the question: what means a better an answer which includes like: cost matrix, to data, optimistic conservative classifier, Anyway, provide enough details until you find one or few criteria can be used to select a metric before trying to things with accuracy and later what to do them. edit - related with robust I "robust" I not find a proper word for "how well a separates the two classes". "robust" has some special meaning in statistics. Generally, AUC close to $1.0$ separates well binary for many values of the threshold. In this sense, the AUC close $1.0$ means is less sensitive to which threshold values is which means its robust this choice. However, a value which is not close $1.0$ not mean contrary, which does necessarily mean that is not enough good threshold In cases, of ROC curve is necessary. This one the main reasons why AUC often AUC is a measure over all possible classifiers (one for each possible threshold value),
p$ for the second class. 
If the clAssifier prOduceS prObaBiLitiEs, onE have to use a thrESholD value in order to obtain cLassiFiCAtioN LaBels. USually, tHIs THResHoLd ValUe IS $0.5$, bUt thiS is Often noT the case, anD soMeTimes it is not THe Best possibLe vAlue. 
Now, MCC is ComPuted dIrEctLY from ClaSsifiCation LAbels, wHich means ThAT it was USed a sinGLE tHresHold value to transfORm PRobabilities inTo clasSiFIcATIon LabEls, no matteR wHat thE ThreshoLD vALUE waS. aUC, on the other Hand is using THe wHole raNgE of THreshoLd valUeS. 
the Idea is that tHese Two values, aUC and mcC, measuRE differEnt thiNgs. whiLe MCc MeAsUreS a KInd OF sTatIStiCal accurAcY (rElateD witH cHI SquaRed Test Which Gives some hintS on The sIGniFicanCe of tHe diFfErencEs), the AuC is mOrE related with the RobuStness of tHe cLaSsiFiEr. AUC ANd ROC cUrvEs gIves morE hints oN How WeLL THe Classifier separateS tHE BiNary clasSes, for ALl PoSSible thrEsHolD valUES. Even For tHE dEgeneratE case, wHErE AuC is comPuTed dirEcTly On lAbels (NOt adVisablE since it LooseS A lot of informatIOn), the purpose oF aUc REmAIns tHe sAme.
Model selEctiON is a Hard PRoBleM. my advIce foR yOU wOUld be to try yourself tO aNswer tO the qUestion: what meAns a better CLASsifier? FInd aN AnSWer which includEs conSiderationS Like: cost MatriX, robustnEss to unbaLANced data, An oPtiMisTic OR CoNservative claSSIfieR, eTc. AnywaY, prOvide enOugH deTaiLs uNtIl you find One or few CrItErIa WhiCh can BE used to sElEct A mEtrIc befORe tryiNg to mEasuRe DiFFerEnt thinGS rELAted WiTh AccuRacY aNd ask LateR WhaT to do wiTh them.     
[LatEr eDIt - reLaTeD with usAge of robust woRd]
i used the teRm "RobUst" becAUSe I could Not find a proper single worD For "how wEll A clasSifiEr separatEs tHe two cLasSEs". I knoW that tHe terM "rObuST" Has soME SpEciAl Meaning in sTATisTics. 
GEnEralLy, an AUC Close to $1.0$ separates weLL biNary cases for mAny ValuES Of The THrEShoLd. iN thIS Sense, the AUC closE to $1.0$ means thAt IT iS less sensiTIve To Which thReshold ValueS Is used, wHich means Its robust To This CHOicE. However, a vAlue whicH is not cloSE to $1.0$ doES nOt meaN thE contrArY, whIch doEs not nECesSarilY mean tHaT there Is not EnOugh rangE of good threshold values. IN most cAses, a GraPhical insPecTIon Of ROC curvE is nEcessary. ThIs iS onE of thE maIN reasOns wHY AuC iS Often ConsIDered mislEAdIng. auc iS a measure ovER ALl pOssibLe cLAssifiErs (oNe for each possible THreshold value),
p$ for the second class. If the cla ssifi erpro du cesprob abilities, one have  to use a threshold va lue i no rder to  obta in clas s if i c ati on  l abe ls .  U suall y,this th reshold va lue  i s $0.5$, but th is is ofte n n ot the case,  an d some ti mes it is  no t the  bestp ossibl e value.
N o w, MCC is comp u t ed  dir ectly from classi f ic a tion labels, w hich m ea n st h atitwas used a  s ingle thresho l dv a l uet o transform p robabilitie s  in to cla ss ifi c ationlabel s, nomatter what  the  threshol d valu e  was. A U C, on t he oth erhan d is us in g t he who l eran g e o f thresh ol dvalue s. T h e idea  is  tha t the se two values , A UC a n d M CC, m easur e di ff erent  thing s. Wh il e MCC measuresa ki nd of sta tis ti cal  a ccura c y (rel ate d w ith Chi  square d  te st w h ic h gives some hints  o n th e signif icance of  t h e differ en ces ), t h e  AUCis m o re  related  witht he  r obustne ss  of th ecla ssi fier. AUCand RO C curves  give s  more hints on how well thec la s s if i er s epa rates the b inar y  cla sses ,  f ora ll po ssibl et hr e shold values. Evenfo r thedegen erate case, w here AUC i s c omputeddire c tl y  on labels (no t adv isable sin c e it loo ses a  lot ofinformati o n ), the p urp ose  of  AU C re mains the sam e . 
Mod el  select ion  is a h ard  pr obl em.  M y advicefor youwo ul dbe  to  tryy ourselfto  an sw erto th e  quest ion:what  m ea n s a  better cl a s sifi er ?Find  an  a nswer  whi c h i ncludes  consider ati o ns l ik e:  cost m atrix, robust ne ss to unba la nce d data , an optim istic or conservative c l assifie r,etc.Anyw ay, provi deenough  de t ails u ntil y ou fi nd  on e or fe w cr ite ri a which ca n beusedto  sel ect a m etric before tryin g  to  measure diff ere nt t h i ng s r e la t edwi t h a c c uracy and ask l ater whatto do  with them .    
[Later  edit -  rela t ed with  usage of  robust w or d]
I u sed  the term"robust"  becauseI  coul d  n ot fi nda prop er  si ngleword f o r " how w ell acl assifi er se pa rates th e two classes". I knowthat t he te rm"robust"has som e special  mea ning in st ati sti cs. Gen e rally , an AU C c l ose t o $1 . 0$ separa t es  we l l  b inary cases f o r m any v alu e s of t he t hreshold. In this sense, the AUC  clo s e  to  $1 . 0$ m ea ns that it isles ss e nsitiveto  which thre shold va lu e s isused,whichmeans i t s  r o bust t o th ischoice. H owe ve r , a val ue  w h ich is  not  c lose t o $1.0 $  doe s not mean the con trary , which doe s not  n ecessar i ly m ean that t here is not  enoug h ra nge o f goodth reshol d v al ues. In mo s t cases,a gra phicalin spec tio n of R OC c u r ve is  nec es sar y. This i s on e  o ft hemain  reas on s wh y AUC iso ften con sid e red mis le adi n g . AUCi sa measure ov erall p o s sible clas s ifie r s( one f or eac h poss ible th r esh ol d value ),
p$ for_the second_class. 
If the classifier_produces probabilities,_one_have to_use_a threshold value_in order to_obtain classification labels. Usually,_this threshold value_is_$0.5$, but this is often not the case, and sometimes it is not the_best_possible value._
Now,_MCC_is computed directly from classification_labels, which means that it_was used_a single threshold value to transform probabilities into_classification_labels, no matter_what the threshold value was. AUC, on the other_hand is using the whole range_of threshold values._
The_idea_is that these two_values, AUC and MCC, measure different_things. While MCC measures a kind_of statistical accuracy (related with Chi squared_test which gives some hints on_the significance of the differences),_the AUC_is more related with the_robustness of the_classifier. AUC_and ROC curves_gives more hints on how well_the classifier separates_the binary classes, for all possible_threshold_values. Even for_the_degenerate_case, where_AUC is computed_directly_on labels_(not_advisable since it looses a lot_of_information), the purpose of AUC remains the_same.
Model selection is a_hard_problem. My advice for_you would be to try_yourself to answer to the question:_what means_a better_classifier? Find an answer which includes considerations like: cost matrix, robustness_to unbalanced data, an optimistic or_conservative classifier, etc. Anyway,_provide enough_details_until you find_one_or few_criteria which can be used to select_a metric_before trying to measure different things_related with accuracy and_ask_later what to do with them._    
[Later edit_- related with usage of_robust_word]
I_used the term "robust" because_I could not find a proper_single word for_"how well a classifier separates the two_classes"._I know that the term "robust"_has_some special meaning in statistics. 
Generally,_an_AUC_close to $1.0$ separates well_binary cases for many values of_the threshold. In this sense, the AUC close to_$1.0$ means that_it is less sensitive to_which_threshold_values is used, which means its robust to this choice._However, a_value which is_not close to $1.0$ does not mean the contrary, which_does not necessarily mean that there is_not enough range of good threshold values. In most cases, a_graphical inspection of ROC curve is necessary. This_is one of the main reasons why_AUC is often considered_misleading._AUC is a measure over all possible_classifiers_(one_for_each possible threshold_value),