Statistical Rethinking - Lecture 06

Lecture 6, Multivariate models (part 2), from "Statistical Rethinking: A Bayesian Course with R Examples"

Transcript

1. Week 3: Multivariate Models Richard McElreath Statistical Rethinking
2. Goals this week • Multivariate Gaussian models • The good: • Reveal spurious correlation • Uncover masked association • The bad: • Correlated predictors • Overfitting looms large
3. Masked association • Sometimes association between outcome and predictor masked by another variable • Tends to arise when • Another predictor associated with outcome in opposite direction • Both predictors associated with one another • Also just noise
4. Eulemur fulvus 0.49 kcal/g 55% neocortex Homo sapiens 0.71 kcal/g 75% neocortex Cebus apella 0.89 kcal/g 68% neocortex Milk and Brain
5. Masked influence • Primate milk data kcal.per.g -2 0 2 4 0.5 0.7 0.9 -2 0 2 4 log(mass) 0.5 0.7 0.9 55 65 75 55 65 75 neocortex.perc library(rethinking) data(milk) d <- milk pairs(~kcal.per.g+log(mass) +neocortex.perc , data=d)
6. Complete cases • Missing values in primate milk data • Drop cases (species) with missing values • Much later, see how to “impute” missing values, so can use all the data .$"(ʃ.ǭɠ&'Ǐ+ -Ǐ"ǮǮ Ǯ --*- $) *+/$(ǭ+- ʃ +-.ǐ !) ʃ (& Ǭ($)0.'*"'ǐ !'$./ ʃ !'$./ƽǐ / ʃ /ǐ Ǒ $)$/$' 1'0 $) ǘ1(($)ǘ $. )*/ !$)$/ :PV TIPVME HFU UIF FSSPS NFTTBHF EJTQMBZFE BCPWF 8IBU IBT HPOF XSPOH IFSF ćJT QBSUJDVMBS FSSPS NFTTBHF NFBOT UIBU UIF MJLFMJIPPE GVODUJPO EJEOU SFUVSO B WBMJE MJLFMJIPPE GPS FWFO UIF TUBSUJOH QBSBNFUFS WBMVFT *O UIJT DBTF UIF DVMQSJU JT UIF NJTTJOH WBMVFT JO UIF ) **-/ 3Ǐ+ - DPMVNO 5BLF B MPPL JOTJEF UIBU DPMVNO BOE TFF GPS ZPVSTFMG 3 DPEF  ɠ) **-/ 3Ǐ+ - &BDI  JO UIF PVUQVU JT B NJTTJOH WBMVF *G ZPV QBTT B WFDUPS MJLF UIJT UP B MJLFMJIPPE GVODUJPO MJLF )*-( JU EPFTOU LOPX XIBU UP EP "ęFS BMM XIBUT UIF MJLFMJIPPE PG B NJTTJOH WBMVF 8IBUFWFS UIF BOTXFS JU JTOU B OVNCFS BOE TP )*-( SFUVSOT B  6OBCMF UP FWFO HFU TUBSUFE (+ PS SBUIFS *+/$( XIJDI EPFT UIF SFBM XPSL HJWFT VQ BOE CBSLT BCPVU 1(($) OPU CFJOH ĕOJUF ćJT JT FBTZ UP ĕY UIPVHI 8IBU ZPV OFFE UP EP IFSF JT NBOVBMMZ ESPQ BMM UIF DBTFT XJUI NJTTJOH WBMVFT .PSF BVUPNBUFE CMBDLCPY DPNNBOET MJLF '( BOE "'( XJMM ESPQ TVDI DBTFT GPS ZPV #VU UIJT JTOU BMXBZT B HPPE UIJOH JG ZPV BSFOU BXBSF PG JU *O UIF OFYU DIBQUFS ZPVMM TFF POF SFBTPO XIZ 4P JOEVMHF NF GPS OPX *UT XPSUI MFBSOJOH IPX UP EP UIJT ZPVSTFMG 5P NBLF B OFX EBUB GSBNF XJUI POMZ DPNQMFUF DBTFT JO JU KVTU VTF 3 DPEF   ʄǤ ǯ *(+' / Ǐ. .ǭǮ ǐ ǰ ćJT NBLFT B OFX EBUB GSBNF  UIBU DPOTJTUT PG UIF  SPXT GSPN  UIBU IBWF WBMVFT JO BMM DPMVNOT /PX MFUT XPSL XJUI UIF OFX EBUB GSBNF "MM UIBU JT OFX JO UIF DPEF JT VTJOH  JOTUFBE PG  [1] 55.16 NA NA NA NA 64.54 64.54 67.64 NA 68.85 58.85 61.69 [13] 60.32 NA NA 69.97 NA 70.41 NA 73.40 NA 67.53 NA 71.26 [25] 72.60 NA 70.24 76.30 75.49 :PV TIPVME HFU UIF FSSPS NFTTBHF EJTQMBZFE BCPWF 8IBU IBT HPOF XSPOH IFSF ćJT QBSUJDVMBS FSSPS NFTTBHF NFBOT UIBU UIF MJLFMJIPPE GVODUJPO EJEOU SFUVSO B WBMJE MJLFMJIPPE GPS FWFO UIF TUBSUJOH QBSBNFUFS WBMVFT *O UIJT DBTF UIF DVMQSJU JT UIF NJTTJOH WBMVFT JO UIF ) **-/ 3Ǐ+ - DPMVNO 5BLF B MPPL JOTJEF UIBU DPMVNO BOE TFF GPS ZPVSTFMG 3 DPEF  ɠ) **-/ 3Ǐ+ - &BDI  JO UIF PVUQVU JT B NJTTJOH WBMVF *G ZPV QBTT B WFDUPS MJLF UIJT UP B MJLFMJIPPE GVODUJPO MJLF )*-( JU EPFTOU LOPX XIBU UP EP "ęFS BMM XIBUT UIF MJLFMJIPPE PG B NJTTJOH WBMVF 8IBUFWFS UIF BOTXFS JU JTOU B OVNCFS BOE TP )*-( SFUVSOT B  6OBCMF UP FWFO HFU TUBSUFE (+ PS SBUIFS *+/$( XIJDI EPFT UIF SFBM XPSL HJWFT VQ BOE CBSLT BCPVU 1(($) OPU CFJOH ĕOJUF ćJT JT FBTZ UP ĕY UIPVHI 8IBU ZPV OFFE UP EP IFSF JT NBOVBMMZ ESPQ BMM UIF DBTFT XJUI NJTTJOH WBMVFT .PSF BVUPNBUFE CMBDLCPY DPNNBOET MJLF '( BOE "'( XJMM ESPQ TVDI DBTFT GPS ZPV #VU UIJT JTOU BMXBZT B HPPE UIJOH JG ZPV BSFOU BXBSF PG JU *O UIF OFYU DIBQUFS ZPVMM TFF POF SFBTPO XIZ 4P JOEVMHF NF GPS OPX *UT XPSUI MFBSOJOH IPX UP EP UIJT ZPVSTFMG 5P NBLF B OFX EBUB GSBNF XJUI POMZ DPNQMFUF DBTFT JO JU KVTU VTF 3 DPEF   ʄǤ ǯ *(+' / Ǐ. .ǭǮ ǐ ǰ ćJT NBLFT B OFX EBUB GSBNF  UIBU DPOTJTUT PG UIF  SPXT GSPN  UIBU IBWF WBMVFT JO BMM DPMVNOT /PX MFUT XPSL XJUI UIF OFX EBUB GSBNF "MM UIBU JT OFX JO UIF DPEF JT VTJOH  JOTUFBE PG  3 DPEF  (ǀǏǀ ʄǤ (+ǭ '$./ǭ &'Ǐ+ -Ǐ" ʋ )*-(ǭ (0 ǐ .$"( Ǯ ǐ (0 ʋ  ɾ +Ƿ) **-/ 3Ǐ+ - Ǯ ǐ
7. Bivariate models Figure 5.8 kcal.per.g ~ dnorm(mu,sigma), mu <- a + bp*neocortex.perc kcal.per.g ~ dnorm(mu,sigma), mu <- a + bm*log(mass)   .6-5*7"3*"5& -*/&"3 .0%&-4 55 60 65 70 75 0.5 0.6 0.7 0.8 0.9 neocortex.perc kcal.per.g -2 -1 0 1 2 3 4 0.5 0.6 0.7 0.8 0.9 log.mass kcal.per.g 6 0.7 0.8 0.9 kcal.per.g 6 0.7 0.8 0.9 kcal.per.g
8. Multivariate model OE UIBU MPHNBTT JT OFHBUJWFMZ DPSSFMBUFE XJUI LJMPDBMPSJFT ćJT JOĘVFODF EPFT HFS UIBO UIBU PG OFPDPSUFY QFSDFOU BMUIPVHI JO UIF PQQPTJUF EJSFDUJPO *U JT RVJUF O UIPVHI XJUI B XJEF DPOĕEFODF JOUFSWBM UIBU JT DPOTJTUFOU XJUI B XJEF SBOHF PG BOE TUSPOHFS SFMBUJPOTIJQT ćJT SFHSFTTJPO JT TIPXO JO UIF VQQFSSJHIU PG 'ĶĴłĿIJ /PX MFUT TFF XIBU IBQQFOT XIFO XF BEE CPUI QSFEJDUPS WBSJBCMFT BU UIF TBNF UJNF U TTJPO ćJT JT UIF NVMUJWBSJBUF NPEFM JO NBUI GPSN LJ ∼ /PSNBM(µJ, σ) µJ = α + βO OJ + βN MPH(NJ) α ∼ /PSNBM(, ) βO ∼ /PSNBM(, ) βN ∼ /PSNBM(, ) σ ∼ 6OJGPSN(, ) F L JT &'Ǐ+ -Ǐ" O JT ) **-/ 3Ǐ+ - BOE N JT (.. 'JUUJOH UIF KPJOU NPEFM J VE FYQFDU CZ OPX ʄǤ (+ǭ '$./ǭ
9. Multivariate model βO ∼ /PSNBM(, ) βN ∼ /PSNBM(, ) σ ∼ 6OJGPSN(, ) "CPWF L JT &'Ǐ+ -Ǐ" O JT ) **-/ 3Ǐ+ - BOE N JT (.. 'JUUJOH UIF KPJOU NPEFM JT KVTU BT ZPVE FYQFDU CZ OPX 3 DPEF  (ǀǏǂ ʄǤ (+ǭ '$./ǭ &'Ǐ+ -Ǐ" ʋ )*-(ǭ (0 ǐ .$"( Ǯ ǐ (0 ʄǤ  ɾ )Ƿ) **-/ 3Ǐ+ - ɾ (Ƿ'*"Ǐ(.. ǐ  ʋ )*-(ǭ ƻ ǐ Ƽƻƻ Ǯ ǐ ) ʋ )*-(ǭ ƻ ǐ Ƽ Ǯ ǐ ( ʋ )*-(ǭ ƻ ǐ Ƽ Ǯ ǐ .$"( ʋ 0)$!ǭ ƻ ǐ Ƽ Ǯ Ǯ ǐ /ʃ Ǯ +- $.ǭ(ǀǏǂǮ  ) / 1 ƽǏǀɳ DŽǂǏǀɳ  ǤƼǏƻǃ ƻǏƿǂ ǤƽǏƻƻ ǤƻǏƼǂ ) ƻǏƻƾ ƻǏƻƼ ƻǏƻƼ ƻǏƻƿ ( ǤƻǏƼƻ ƻǏƻƽ ǤƻǏƼƿ ǤƻǏƻǀ .$"( ƻǏƼƼ ƻǏƻƽ ƻǏƻǃ ƻǏƼǀ #Z JODPSQPSBUJOH CPUI QSFEJDUPS WBSJBCMFT JO UIF SFHSFTTJPO UIF FTUJNBUFE BTTPDJBUJPO PG CPUI XJUI UIF PVUDPNF IBT JODSFBTFE ćF QPTUFSJPS NFBO GPS UIF BTTPDJBUJPO PG OFPDPSUFY QFSDFOU IBT JODSFBTFE NPSF UIBO GPME BOE JUT  JOUFSWBM JT OPX FOUJSFMZ BCPWF [FSP ćF QPTUFSJPS NFBO GPS MPH CPEZ NBTT JT NPSF TUSPOHMZ OFHBUJWF
10. Figure 5.8 Bivariate Multivariate 55 60 65 70 75 0.5 0.6 0.7 0.8 0.9 neocortex.perc kcal.per.g -2 -1 0 1 2 3 4 0.5 0.6 0.7 0.8 0.9 log.mass kcal.per.g 55 60 65 70 75 0.5 0.6 0.7 0.8 0.9 neocortex.perc kcal.per.g -2 -1 0 1 2 3 4 0.5 0.6 0.7 0.8 0.9 log.mass kcal.per.g 'ĶĴłĿIJ ƍƐ .JML FOFSHZ BOE OFPDPSUFY BNPOH QSJNBUFT *O UIF UPQ UXP QMPUT TJNQMF CJWBSJBUF SFHSFTTJPOT PG LJMPDBMPSJFT QFS HSBN PG NJML PO MFę OFPDPSUFY QFSDFOU BOE SJHIU MPH GFNBMF CPEZ NBTT TIPX XFBL BOE VODFS 55 60 65 70 75 0.5 0.6 0.7 0.8 0.9 neocortex.perc kcal.per.g -2 -1 0 1 2 3 4 0.5 0.6 0.7 0.8 0.9 log.mass kcal.per.g 55 60 65 70 75 0.5 0.6 0.7 0.8 0.9 neocortex.perc kcal.per.g -2 -1 0 1 2 3 4 0.5 0.6 0.7 0.8 0.9 log.mass kcal.per.g 'ĶĴłĿIJ ƍƐ .JML FOFSHZ BOE OFPDPSUFY BNPOH QSJNBUFT *O UIF UPQ UXP QMPUT TJNQMF CJWBSJBUF SFHSFTTJPOT PG LJMPDBMPSJFT QFS HSBN PG NJML PO MFę
11. Figure 5.8   .6-5*7"3*"5& -*/&"3 .0%&-4 -FUT QMPU UIF JOUFSWBMT GPS UIF QSFEJDUFE NFBO LJMPDBMPSJFT GPS UIJT OFX NPEFM )FSFT UIF DPEF GPS UIF SFMBUJPOTIJQ CFUXFFO LJMPDBMPSJFT BOE OFPDPSUFY QFSDFOU ćFTF BSF DPVO UFSGBDUVBM QMPUT TP XFMM VTF UIF NFBO MPH CPEZ NBTT JO UIJT DBMDVMBUJPO TIPXJOH POMZ IPX QSFEJDUFE FOFSHZ WBSJFT BT B GVODUJPO PG OFPDPSUFY QFSDFOU ( )Ǐ'*"Ǐ(.. ʄǤ ( )ǭ '*"ǭɠ(..Ǯ Ǯ )+Ǐ. , ʄǤ ƻǑƼƻƻ +- Ǐ/ ʄǤ '$./ǭ ) **-/ 3Ǐ+ -ʃ)+Ǐ. ,ǐ '*"Ǐ(..ʃ( )Ǐ'*"Ǐ(.. Ǯ (0 ʄǤ '$)&ǭ (ǀǏǂ ǐ /ʃ+- Ǐ/ Ǯ (0Ǐ( ) ʄǤ ++'4ǭ (0 ǐ ƽ ǐ ( ) Ǯ (0Ǐ ʄǤ ++'4ǭ (0 ǐ ƽ ǐ  Ǯ +'*/ǭ &'Ǐ+ -Ǐ" ʋ ) **-/ 3Ǐ+ - ǐ /ʃ ǐ /4+ ʃǙ)Ǚ Ǯ '$) .ǭ )+Ǐ. , ǐ (0Ǐ( ) Ǯ '$) .ǭ )+Ǐ. , ǐ (0Ǐ ǯƼǐǰ ǐ '/4ʃƽ Ǯ '$) .ǭ )+Ǐ. , ǐ (0Ǐ ǯƽǐǰ ǐ '/4ʃƽ Ǯ ćJT QMPU JT EJTQMBZFE JO UIF MPXFSMFę PG 'ĶĴłĿIJ ƍƐ ćF BOBMPHPVT QMPU GPS '*"ǭ(..Ǯ JT TIPXO JO UIF MPXFSSJHIU * MFBWF JU UP UIF SFBEFS UP NPEJGZ UIF DPEF BCPWF UP SFQMJDBUF UIF QMPU JO UIF MPXFSSJHIU 8IZ EJE BEEJOH OFPDPSUFY BOE CPEZ NBTT UP UIF TBNF NPEFM MFBE UP MBSHFS FTUJNBUFE FČFDUT PG CPUI ćJT JT B DPOUFYU JO XIJDI UIFSF BSF UXP WBSJBCMFT DPSSFMBUFE XJUI UIF PVU DPNF CVU POF JT QPTJUJWFMZ DPSSFMBUFE XJUI JU BOE UIF PUIFS JT OFHBUJWFMZ DPSSFMBUFE XJUI JU 55 60 65 70 75 0.5 0.6 neocortex.perc kc 55 60 65 70 75 0.5 0.6 0.7 0.8 0.9 neocortex.perc kcal.per.g 'ĶĴłĿIJ ƍƐ .JML FOFSHZ BOE OFPDPS QMPUT TJNQMF CJWBSJBUF SFHSFTTJPOT PG L OFPDPSUFY QFSDFOU BOE SJHIU MPH GFN UBJO BTTPDJBUJPOT )PXFWFS PO UIF C OFPDPSUFY QFSDFOU BOE MPH CPEZ NBTT WBSJBCMFT #PUI OFPDPSUFY BOE CPEZ N
12. Regression as a wicked oracle • Regression automatically focuses on the most informative cases • Cases that don’t help are automatically ignored • But not a kind oracle: (1) omitted variables (2) non-identifiability
13. Why not just add everything? • Could just add all available predictors to model • Almost always a bad idea • Multicollinearity • Loss of interpretability • Loss of precision • Overfitting
14. Multicollinear legs height leg_left leg_right 1 15.384202 7.115039 7.139183 2 12.176479 5.718942 5.729024 3 9.634356 4.278725 4.275795 4 7.671892 3.158348 3.166970 5 8.592127 3.518352 3.543422 6 7.747036 3.397380 3.384179 7 9.623175 4.601825 4.603800 8 7.735412 3.852066 3.848137 9 12.083202 5.502614 5.521156 10 11.080817 4.847354 4.790418 11 11.631615 5.017371 4.996615 12 6.477359 3.023023 3.036469 13 8.870094 3.708882 3.764201 14 12.703396 6.073339 6.076483 15 11.416840 5.444431 5.441192 16 10.758823 5.286965 5.297677 17 11.464688 5.596979 5.604316 18 9.747457 4.003333 4.012955 19 12.211823 6.092597 6.100131 20 12.671249 6.184386 6.193254
15. Multicollinear legs BOE ' "Ǭ-$"#/ #FGPSF ĕUUJOH UIF NPEFM BOE MPPLJOH BU UIF QPTUFSJPS NFBOT IPXFWFS DPO TJEFS XIBU XF FYQFDU 0O BWFSBHF BO JOEJWJEVBMT MFHT BSF  PG IJT PS IFS IFJHIU JO UIFTF TJNVMBUFE EBUB  4P XF TIPVME FYQFDU UIF CFUB DPFďDJFOU UIBU NFBTVSFT UIF BTTPDJBUJPO PG B MFH XJUI IFJHIU UP FOE VQ BSPVOE UIF BWFSBHF IFJHIU  EJWJEFE CZ  PG UIF BWFSBHF IFJHI   ćJT JT /. ≈ . /PX MFUT TFF XIBU IBQQFOT JOTUFBE 3 DPEF  (ǀǏǃ ʄǤ (+ǭ '$./ǭ # $"#/ ʋ )*-(ǭ (0 ǐ .$"( Ǯ ǐ (0 ʄǤ  ɾ 'Ƿ' "Ǭ' !/ ɾ -Ƿ' "Ǭ-$"#/ ǐ  ʋ )*-(ǭ Ƽƻ ǐ Ƽƻƻ Ǯ ǐ ' ʋ )*-(ǭ ƽ ǐ Ƽƻ Ǯ ǐ - ʋ )*-(ǭ ƽ ǐ Ƽƻ Ǯ ǐ .$"( ʋ 0)$!ǭ ƻ ǐ Ƽƻ Ǯ Ǯ ǐ /ʃ Ǯ +- $.ǭ(ǀǏǃǮ  ) / 1 ƽǏǀɳ DŽǂǏǀɳ  ƻǏǃƼ ƻǏƽǃ ƻǏƽǀ ƼǏƾǂ ' ǤƻǏƻǁ ƼǏDŽƿ ǤƾǏǃǂ ƾǏǂǀ - ƽǏƼƼ ƼǏDŽǁ ǤƼǏǂƽ ǀǏDŽƿ .$"( ƻǏǀǂ ƻǏƻƿ ƻǏƿDŽ ƻǏǁǀ ćPTF QPTUFSJPS NFBOT BOE TUBOEBSE EFWJBUJPOT MPPL DSB[Z ćJT JT B DBTF JO XIJDI B HSBQIJDBM WJFX PG UIF +- $. PVUQVU JT NPSF VTFGVM CFDBVTF JU EJTQMBZT UIF QPTUFSJPS NFBOT BOE  JOUFSWBMT JO B XBZ UIBU BMMPXT VT XJUI B HMBODF UP TFF UIBU TPNFUIJOH IBT HPOF XSPOH IFSF  8)&/ "%%*/( 7"3*"#-&4 )6354 sigma br bl a -4 -2 0 2 4 6 Estimate :PVS OVNCFST BOE +- $. QMPU XJMM OPU MPPL FYBDUMZ UIF TBNF EVF UP TJN #VU UIFZ XJMM TIPX UIF TBNF PEE SFTVMU *G CPUI MFHT IBWF BMNPTU JEFOUJDBM M
16. Multicollinear legs • Q: What is value of learning left/right leg, once we already know right/left leg? • A: Almost nothing, on average.   .6-5*7"3*"5& -*/&"3 .0%&-4 1.8 1.9 2.0 2.1 2.2 2.3 0 1 2 3 4 5 6 sum of bl and br Density 'ĶĴłĿIJ ƍƑ -Fę 1PTUFSJPS EJTUSJCVUJPO PG UIF BTTPDJBUJPO PG FBDI MFH XJUI
17. Multicollinear legs SJBCMFT DPOUBJO BMNPTU FYBDUMZ UIF TBNF JOGPSNBUJPO JG ZPV JOTJTU PO EFM UIFO UIFSF XJMM CF B QSBDUJDBMMZ JOĕOJUF OVNCFS PG DPNCJOBUJPOT VDF UIF TBNF QSFEJDUJPOT G UIJT QIFOPNFOPO JT UIBU ZPV IBWF BQQSPYJNBUFE UIJT MJLFMJIPPE ZJ ∼ /PSNBM(µJ, σ) µJ = α + β YJ + β YJ PNF MJLF IFJHIU JO UIF FYBNQMF BOE Y JT B TJOHMF QSFEJDUPS MJLF UIF MFH )FSF Y JT VTFE UXJDF XIJDI JT B QFSGFDU FYBNQMF PG UIF QSPCMFN DBVTFE OUJDBM MFH MFOHUIT 'SPN UIF DPNQVUFST QFSTQFDUJWF UIJT MJLFMJIPPE ZJ ∼ /PSNBM(µJ, σ) µJ = α + (β + β)YJ PVU PG FBDI UFSN ćF QBSBNFUFST β BOE β DBOOPU CF QVMMFE BQBSU SBUFMZ JOĘVFODF UIF NFBO µ 0OMZ UIFJS TVN β+β JOĘVFODFT µ 4P WBSJBCMF Z JT UIF PVUDPNF MJLF IFJHIU JO UIF FYBNQMF BOE Y JT B TJOHMF UIT JO UIF FYBNQMF )FSF Y JT VTFE UXJDF XIJDI JT B QFSGFDU FYBNQMF P VTJOH UIF BMNPTUJEFOUJDBM MFH MFOHUIT 'SPN UIF DPNQVUFST QFSTQF BMMZ ZJ ∼ /PSNBM(µJ, σ) µJ = α + (β + β)YJ *WF EPOF JT GBDUPS YJ PVU PG FBDI UFSN ćF QBSBNFUFST β BOE β DB BVTF UIFZ OFWFS TFQBSBUFMZ JOĘVFODF UIF NFBO µ 0OMZ UIFJS TVN β+ NFBOT UIF QPTUFSJPS EJTUSJCVUJPO FOET VQ SFQPSUJOH UIF QSBDUJDBMMZ J  BOE β UIBU NBLF UIFJS TVN DMPTF UP UIF BDUVBM BTTPDJBUJPO PG Y XJ   .6-5*7"3*"5& -*/&"3 .0%&-4 1.8 1.9 2.0 2.1 2.2 2.3 0 1 2 3 4 5 6 sum of bl and br Density 'ĶĴłĿIJ ƍƑ -Fę 1PTUFSJPS EJTUSJCVUJPO PG UIF BTTPDJBUJPO PG FBDI MFH XJUI
18. Correlated predictors • Multicollinearity: strong correlations among prediction variables  .6-5*7"3*"5& .0%&-4  kcal.per.g 10 30 50 0.5 0.7 0.9 10 30 50 perc.fat 0.5 0.7 0.9 30 50 70 30 50 70 perc.lactose 'JHVSF  " QBJST QMPU PG UIF UPUBM FOFSHZ QFS DFOU GBU BOE QFSDFOU MBDUPTF WBSJBCMFT GSPN UIF QSJNBUF NJML EBUB 1FSDFOU GBU BOE QFSDFOU MBDUPTF BSF TUSPOHMZ OFHBUJWFMZ DPSSFMBUFE XJUI POF BOPUIFS QSPWJEJOH FT TFOUJBMMZ UIF TBNF JOGPSNB UJPO
19. Correlated predictors • perc.fat or perc.lactose alone: Strong association with kcal.per.g  .6-5*7"3*"5& .0%&-4  kcal.per.g 10 30 50 0.5 0.7 0.9 10 30 50 perc.fat 30 50 70 perc.lactose 'JHVSF  " QBJST QMPU PG UIF UPUBM FOFSHZ QFS DFOU GBU BOE QFSDFOU MBDUPTF WBSJBCMFT GSPN UIF QSJNBUF NJML EBUB 1FSDFOU GBU BOE QFSDFOU MBDUPTF BSF TUSPOHMZ OFHBUJWFMZ DPSSFMBUFE XJUI POF BOPUIFS QSPWJEJOH FT TFOUJBMMZ UIF TBNF JOGPSNB UJPO  .6-5*7"3*"5& .0%&-4  10 30 50 0.5 0.7 0.9 perc.fat 'JHVSF  " QBJST QMPU PG UIF UPUBM FOFSHZ QFS DFOU GBU BOE QFSDFOU MBDUPTF WBSJBCMFT GSPN UIF QSJNBUF NJML EBUB 1FSDFOU GBU BOE   .6-5*7"3*"5& -*/&"3 .0%&-4 +- $.ǭ (ǀǏƼƼ ǐ $"$/.ʃƾ Ǯ  ) / 1 ƽǏǀɳ DŽǂǏǀɳ  ƻǏƾƻƼ ƻǏƻƾǁ ƻǏƽƾƼ ƻǏƾǂƼ ! ƻǏƻƼƻ ƻǏƻƻƼ ƻǏƻƻǃ ƻǏƻƼƽ .$"( ƻǏƻǂƾ ƻǏƻƼƻ ƻǏƻǀƿ ƻǏƻDŽƽ  ) / 1 ƽǏǀɳ DŽǂǏǀɳ  ƼǏƼǁǁ ƻǏƻƿƾ ƼǏƻǃƾ ƼǏƽǀƻ ' ǤƻǏƻƼƼ ƻǏƻƻƼ ǤƻǏƻƼƽ ǤƻǏƻƻDŽ .$"( ƻǏƻǁƽ ƻǏƻƻǃ ƻǏƻƿǁ ƻǏƻǂǃ ćF QPTUFSJPS NFBO GPS ! UIF BTTPDJBUJPO PG QFSDFOU GBU XJUI NJML JOUFSWBM [., .] ćF QPTUFSJPS NFBO JO UIF TFDPOE NPEFM GPS
20. Correlated predictors • Together: Neither matters much ZPV NJHIU XPOEFS XIFUIFS UIFTF BTTPDJBUJPOT SFBMMZ IBWF NVDI JOĘVFODF PO UIF PVUDPNF NJML FOFSHZ ćFZ EP 3FNFNCFS UIBU CPUI QSFEJDUPST BSF QFSDFOUT TP BSF QPUFOUJBMMZ MBSHF OVNCFST ćF BCTPMVUF NBHOJUVEF PG SFHSFTTJPO TMPQFT JT OPU BMXBZT NFBOJOHGVM CFDBVTF UIF JOĘVFODF PO QSFEJDUJPO EFQFOET VQPO UIF QSPEVDU PG UIF QBSBNFUFS BOE UIF EBUB :PV IBWF UP DPNQVUF PS QMPU QSFEJDUJPOT VOMFTT ZPV EFDJEF UP TUBOEBSEJ[F BMM PG ZPVS QSFEJDUPST (JWFO UIF TUSPOH BTTPDJBUJPO PG FBDI QSFEJDUPS XJUI UIF PVUDPNF XF NJHIU DPODMVEF UIBU CPUI WBSJBCMFT BSF SFMJBCMF QSFEJDUPST PG UPUBM FOFSHZ JO NJML BDSPTT TQFDJFT ćF NPSF GBU UIF NPSF LJMPDBMPSJFT JO UIF NJML ćF NPSF MBDUPTF UIF GFXFS LJMPDBMPSJFT JO NJML #VU XBUDI XIBU IBQQFOT XIFO XF QMBDF CPUI QSFEJDUPS WBSJBCMFT JO UIF TBNF SFHSFTTJPO NPEFM 3 DPEF  (ǀǏƼƽ ʄǤ (+ǭ '$./ǭ &'Ǐ+ -Ǐ" ʋ )*-(ǭ (0 ǐ .$"( Ǯ ǐ (0 ʄǤ  ɾ !Ƿ+ -Ǐ!/ ɾ 'Ƿ+ -Ǐ'/*. ǐ  ʋ )*-(ǭ ƻǏǁ ǐ Ƽƻ Ǯ ǐ ! ʋ )*-(ǭ ƻ ǐ Ƽ Ǯ ǐ ' ʋ )*-(ǭ ƻ ǐ Ƽ Ǯ ǐ .$"( ʋ 0)$!ǭ ƻ ǐ Ƽƻ Ǯ Ǯ ǐ /ʃ Ǯ +- $.ǭ (ǀǏƼƽ ǐ $"$/.ʃƾ Ǯ  ) / 1 ƽǏǀɳ DŽǂǏǀɳ  ƼǏƻƻǂ ƻǏƽƻƻ ƻǏǁƼǀ ƼǏƾDŽDŽ ! ƻǏƻƻƽ ƻǏƻƻƽ ǤƻǏƻƻƾ ƻǏƻƻǂ ' ǤƻǏƻƻDŽ ƻǏƻƻƽ ǤƻǏƻƼƾ ǤƻǏƻƻƿ .$"( ƻǏƻǁƼ ƻǏƻƻǃ ƻǏƻƿǀ ƻǏƻǂǂ
21. Categorical variables • Un-ordered categorical distinctions become dummy variables (aka indicator) 130 150 170 0.00 0.04 0.08 height (cm) Density height weight age male 1 151.765 47.82561 63 1 2 139.700 36.48581 63 0 3 136.525 31.86484 65 0 4 156.845 53.04191 41 1 5 145.415 41.27687 51 0 6 163.830 62.99259 35 1 7 149.225 38.24348 32 0 8 168.910 55.47997 27 1 9 147.955 34.86988 19 0 10 165.100 54.48774 54 1 11 154.305 49.89512 47 0 12 151.130 41.22017 66 1
22. Categorical variables • Un-ordered categorical distinctions become dummy variables (aka indicator) 130 150 170 0.00 0.04 0.08 height (cm) Density height weight age male 1 151.765 47.82561 63 1 2 139.700 36.48581 63 0 3 136.525 31.86484 65 0 4 156.845 53.04191 41 1 5 145.415 41.27687 51 0 6 163.830 62.99259 35 1 7 149.225 38.24348 32 0 8 168.910 55.47997 27 1 9 147.955 34.86988 19 0 10 165.100 54.48774 54 1 11 154.305 49.89512 47 0 12 151.130 41.22017 66 1
23. Categorical variables • Dummy variables allow each category to have unique intercept 0/1 variable male mean when mi = 0 change in mean when mi = 1 PO JT GFNBMF *U EPFTOU NBUUFS XIJDI DBUFHPSZ‰iNBMFw PS iGFN CZ UIF  ćF NPEFM XPOU DBSF #VU DPSSFDUMZ JOUFSQSFUJOH UI BOE UIBU ZPV SFNFNCFS TP JUT B HPPE JEFB UP OBNF UIF WBSJBCMF BTTJHOFE UIF  WBMVF FČFDU PG B EVNNZ WBSJBCMF JT UP UVSO B QBSBNFUFS PO GPS UIPTF PSZ 4JNVMUBOFPVTMZ UIF WBSJBCMF UVSOT UIF TBNF QBSBNFUFS PČ BOPUIFS DBUFHPSZ ćF NPEFM UP ĕU JT IJ ∼ /PSNBM(µJ, σ) µJ = α + βN NJ T IFJHIU BOE N JT UIF EVNNZ WBSJBCMF JOEJDBUJOH B NBMF JOEJWJE S βN JT OPX UVSOFE PO BOE JOĘVFODFT QSFEJDUJPO GPS UIPTF DBT 8IFO NJ =  JU IBT OP FČFDU PO QSFEJDUJPO 5P ĕU UIJT NPEFM (+ǭ
24. Mean for males: 135 + 7 = 142 Percentile interval? 3 DPEF  (ǀǏƼƾ ʄǤ (+ǭ '$./ǭ # $"#/ ʋ )*-(ǭ (0 ǐ .$"( Ǯ ǐ (0 ʄǤ  ɾ (Ƿ(' ǐ  ʋ )*-(ǭ Ƽǀƻ ǐ Ƽƻƻ Ǯ ǐ ( ʋ )*-(ǭ ƻ ǐ Ƽƻ Ǯ ǐ .$"( ʋ 0)$!ǭ ƻ ǐ ǀƻ Ǯ Ǯ ǐ /ʃ Ǯ +- $.ǭ(ǀǏƼƾǮ  ) / 1 ƽǏǀɳ DŽǂǏǀɳ  ƼƾƿǏǃƽ ƼǏǀDŽ ƼƾƼǏǂƻ ƼƾǂǏDŽƿ ( ǂǏƽDŽ ƽǏƽǃ ƽǏǃƼ ƼƼǏǂǁ .$"( ƽǂǏƾƼ ƻǏǃƾ ƽǀǏǁDŽ ƽǃǏDŽƾ 5P JOUFSQSFU UIFTF FTUJNBUFT ZPV IBWF UP OPUF UIBU UIF QBSBNFUFS α  JT OPX UIF BWFSBHF IFJHIU BNPOH GFNBMFT 8IZ #FDBVTF XIFO NJ =  JOEJDBUJOH B GFNBMF UIF QSFEJDUFE NFBO IFJHIU JT KVTU µJ = α + βN() = α 4P UIF FTUJNBUF TBZT UIBU UIF FYQFDUFE BWFSBHF GFNBMF IFJHIU JT  DN ćF QBSBNFUFS βN UIFO UFMMT VT UIF BWFSBHF EJČFSFODF CFUXFFO NBMFT BOE GFNBMFT  DN 4P UP DPNQVUF UIF BWFSBHF NBMF IFJHIU ZPV KVTU BEE UIFTF UXP FTUJNBUFT  + . = . ćBUT HPPE FOPVHI GPS UIF QPTUFSJPS NFBO PG BWFSBHF NBMF IFJHIU CVU ZPVMM BMTP OFFE UP DPOTJEFS UIF XJEUI PG UIF QPTUFSJPS EJTUSJCVUJPO "OE CFDBVTF UIF QBSBNFUFST α BOE βN BSF DPSSFMBUFE XJUI POF BOPUIFS ZPV DBOU KVTU BEE UPHFUIFS UIF CPVOEBSJFT JO UIF +- $. PVUQVU BOE HFU DPSSFDU CPVOEBSJFT GPS UIFJS TVN #VU BT VTVBM UIF NPTU BDDFTTJCMF XBZ UP EFSJWF B QFSDFOUJMF JOUFSWBM GPS BWFSBHF NBMF IFJHIU JT KVTU UP TBNQMF GSPN UIF QPTUFSJPS ćFO ZPV DBO KVTU BEE TBNQMFT PG  BOE ( UPHFUIFS UP HFU UIF QPTUFSJPS EJTUSJCVUJPO PG UIFJS TVN )FSFT BMM JU UBLFT 3 DPEF  +*./ ʄǤ 3/-/Ǐ.(+' .ǭ(ǀǏƼƾǮ +- $.ǭ(ǀǏƼƾǮ  ) / 1 ƽǏǀɳ DŽǂǏǀɳ  ƼƾƿǏǃƽ ƼǏǀDŽ ƼƾƼǏǂƻ ƼƾǂǏDŽƿ ( ǂǏƽDŽ ƽǏƽǃ ƽǏǃƼ ƼƼǏǂǁ .$"( ƽǂǏƾƼ ƻǏǃƾ ƽǀǏǁDŽ ƽǃǏDŽƾ 5P JOUFSQSFU UIFTF FTUJNBUFT ZPV IBWF UP OPUF UIBU UIF QBSBNFUFS α  JT OPX UIF BWFSBHF IFJHIU BNPOH GFNBMFT 8IZ #FDBVTF XIFO NJ =  JOEJDBUJOH B GFNBMF UIF QSFEJDUFE NFBO IFJHIU JT KVTU µJ = α + βN() = α 4P UIF FTUJNBUF TBZT UIBU UIF FYQFDUFE BWFSBHF GFNBMF IFJHIU JT  DN ćF QBSBNFUFS βN UIFO UFMMT VT UIF BWFSBHF EJČFSFODF CFUXFFO NBMFT BOE GFNBMFT  DN 4P UP DPNQVUF UIF BWFSBHF NBMF IFJHIU ZPV KVTU BEE UIFTF UXP FTUJNBUFT  + . = . ćBUT HPPE FOPVHI GPS UIF QPTUFSJPS NFBO PG BWFSBHF NBMF IFJHIU CVU ZPVMM BMTP OFFE UP DPOTJEFS UIF XJEUI PG UIF QPTUFSJPS EJTUSJCVUJPO "OE CFDBVTF UIF QBSBNFUFST α BOE βN BSF DPSSFMBUFE XJUI POF BOPUIFS ZPV DBOU KVTU BEE UPHFUIFS UIF CPVOEBSJFT JO UIF +- $. PVUQVU BOE HFU DPSSFDU CPVOEBSJFT GPS UIFJS TVN #VU BT VTVBM UIF NPTU BDDFTTJCMF XBZ UP EFSJWF B QFSDFOUJMF JOUFSWBM GPS BWFSBHF NBMF IFJHIU JT KVTU UP TBNQMF GSPN UIF QPTUFSJPS ćFO ZPV DBO KVTU BEE TBNQMFT PG  BOE ( UPHFUIFS UP HFU UIF QPTUFSJPS EJTUSJCVUJPO PG UIFJS TVN )FSFT BMM JU UBLFT 3 DPEF  +*./ ʄǤ 3/-/Ǐ.(+' .ǭ(ǀǏƼƾǮ (0Ǐ(' ʄǤ +*./ɠ ɾ +*./ɠ(  ǭ(0Ǐ(' Ǯ   .6-5*7"3*"5& -*/&"3 .0%&-4 ƽǏǀɳ DŽǂǏǀɳ ƼƾǃǏǂǀƽƻ ƼƿǀǏƿƼƼƽ 8PSLJOH XJUI TBNQMFT BVUPNBUJDBMMZ IBOEMFT UIF TVCUMF QSPCMFN UIBU  BOE ( BSF DPSSFMBUFE XJUI POF BOPUIFS 8IFO XPSLJOH XJUI TBNQMFT UIF QSPDFEVSF JT UIF TBNF OP NBUUFS XIBU UIF DPSSFMBUJPO
25. More than two categories • For k categories, need k–1 dummy variables season spring summer fall 1 winter 0 0 0 2 spring 1 0 0 3 summer 0 1 0 4 fall 0 0 1
26. Milk again... clade species clade.NWM clade.OWM clade.S 1 Strepsirrhine Eulemur fulvus 0 0 1 2 Strepsirrhine E macaco 0 0 1 3 Strepsirrhine E mongoz 0 0 1 4 Strepsirrhine E rubriventer 0 0 1 5 Strepsirrhine Lemur catta 0 0 1 6 New World Monkey Alouatta seniculus 1 0 0 7 New World Monkey A palliata 1 0 0 8 New World Monkey Cebus apella 1 0 0 9 New World Monkey Saimiri boliviensis 1 0 0 10 New World Monkey S sciureus 1 0 0 11 New World Monkey Cebuella pygmaea 1 0 0 12 New World Monkey Callimico goeldii 1 0 0 13 New World Monkey Callithrix jacchus 1 0 0 14 New World Monkey Leontopithecus rosalia 1 0 0 15 Old World Monkey Chlorocebus pygerythrus 0 1 0 16 Old World Monkey Miopithecus talpoin 0 1 0 17 Old World Monkey M fuscata 0 1 0 18 Old World Monkey M mulatta 0 1 0 19 Old World Monkey M sinica 0 1 0 20 Old World Monkey Papio spp 0 1 0 21 Ape Nomascus concolor 0 0 0 22 Ape Hylobates lar 0 0 0 23 Ape Symphalangus syndactylus 0 0 0 24 Ape Pongo pygmaeus 0 0 0 25 Ape Gorilla gorilla gorilla 0 0 0 26 Ape G gorilla beringei 0 0 0 27 Ape Pan paniscus 0 0 0 28 Ape P troglodytes 0 0 0 29 Ape Homo sapiens 0 0 0
27. Milk again... clade species clade.NWM clade.OWM clade.S 1 Strepsirrhine Eulemur fulvus 0 0 1 2 Strepsirrhine E macaco 0 0 1 3 Strepsirrhine E mongoz 0 0 1 4 Strepsirrhine E rubriventer 0 0 1 5 Strepsirrhine Lemur catta 0 0 1 6 New World Monkey Alouatta seniculus 1 0 0 7 New World Monkey A palliata 1 0 0 8 New World Monkey Cebus apella 1 0 0 9 New World Monkey Saimiri boliviensis 1 0 0 10 New World Monkey S sciureus 1 0 0 11 New World Monkey Cebuella pygmaea 1 0 0 12 New World Monkey Callimico goeldii 1 0 0 13 New World Monkey Callithrix jacchus 1 0 0 14 New World Monkey Leontopithecus rosalia 1 0 0 15 Old World Monkey Chlorocebus pygerythrus 0 1 0 16 Old World Monkey Miopithecus talpoin 0 1 0 17 Old World Monkey M fuscata 0 1 0 18 Old World Monkey M mulatta 0 1 0 19 Old World Monkey M sinica 0 1 0 20 Old World Monkey Papio spp 0 1 0 21 Ape Nomascus concolor 0 0 0 22 Ape Hylobates lar 0 0 0 23 Ape Symphalangus syndactylus 0 0 0 24 Ape Pongo pygmaeus 0 0 0 25 Ape Gorilla gorilla gorilla 0 0 0 26 Ape G gorilla beringei 0 0 0 27 Ape Pan paniscus 0 0 0 28 Ape P troglodytes 0 0 0 29 Ape Homo sapiens 0 0 0 d$clade.NWM <- ifelse( d$clade=="New World Monkey" , 1 , 0 ) d$clade.OWM <- ifelse( d$clade=="Old World Monkey" , 1 , 0 ) d$clade.S <- ifelse( d$clade=="Strepsirrhine" , 1 , 0 ) d$clade.A <- ifelse( d$clade=="Ape" , 1 , 0 )
28.   .6-5*7"3*"5& -*/&"3 .0%&-4 ćF NPEFM XF BJN UP ĕU UP UIF EBUB JT ( )Ǒ-"/Ǒ$ SFHSFTTFE PO UIF EVNN BSJBCMFT GPS )!" LJ ∼ /PSNBM(µJ, σ) µJ = α + β/8. /8.J + β08. 08.J + β4 4J MJOFBS NPEFM MJLF UIJT SFBMMZ EFĕOFT  EJČFSFOU MJOFBS NPEFMT FBDI DPSSFTQPOE OH UP B EJČFSFOU DBUFHPSZ " UBCMF NJHIU IFMQ $BUFHPSZ /8.J 08.J 4J µJ "QF    µJ = α /FX 8PSME NPOLFZ    µJ = α + β/8. 0ME 8PSME NPOLFZ    µJ = α + β08. 4USFQTJSSIJOF    µJ = α + β4 BDI DBUFHPSZ JNQMJFT B EJČFSFOU TFU PG T BOE T JO UIF EVNNZ WBSJBCMFT XIJD O UVSO JNQMZ B EJČFSFOU FRVBUJPO GPS µJ PODF TJNQMJĕFE 'JUUJOH UIF NPEFM JT TUSBJHIUGPSXBSE 0/1 dummy variables
29.   .6-5*7"3*"5& -*/&"3 .0%&-4 ćF NPEFM XF BJN UP ĕU UP UIF EBUB JT ( )Ǒ-"/Ǒ$ SFHSFTTFE PO UIF EVNN BSJBCMFT GPS )!" LJ ∼ /PSNBM(µJ, σ) µJ = α + β/8. /8.J + β08. 08.J + β4 4J MJOFBS NPEFM MJLF UIJT SFBMMZ EFĕOFT  EJČFSFOU MJOFBS NPEFMT FBDI DPSSFTQPOE OH UP B EJČFSFOU DBUFHPSZ " UBCMF NJHIU IFMQ $BUFHPSZ /8.J 08.J 4J µJ "QF    µJ = α /FX 8PSME NPOLFZ    µJ = α + β/8. 0ME 8PSME NPOLFZ    µJ = α + β08. 4USFQTJSSIJOF    µJ = α + β4 BDI DBUFHPSZ JNQMJFT B EJČFSFOU TFU PG T BOE T JO UIF EVNNZ WBSJBCMFT XIJD O UVSO JNQMZ B EJČFSFOU FRVBUJPO GPS µJ PODF TJNQMJĕFE 'JUUJOH UIF NPEFM JT TUSBJHIUGPSXBSE 0/1 dummy variables difference of each category from APE category
30. ćF NPEFM XF BJN UP ĕU UP UIF EBUB JT ( )Ǒ-"/Ǒ$ SFHSFTTFE PO UIF EVNN BSJBCMFT GPS )!" LJ ∼ /PSNBM(µJ, σ) µJ = α + β/8. /8.J + β08. 08.J + β4 4J MJOFBS NPEFM MJLF UIJT SFBMMZ EFĕOFT  EJČFSFOU MJOFBS NPEFMT FBDI DPSSFTQPOE OH UP B EJČFSFOU DBUFHPSZ " UBCMF NJHIU IFMQ $BUFHPSZ /8.J 08.J 4J µJ "QF    µJ = α /FX 8PSME NPOLFZ    µJ = α + β/8. 0ME 8PSME NPOLFZ    µJ = α + β08. 4USFQTJSSIJOF    µJ = α + β4 BDI DBUFHPSZ JNQMJFT B EJČFSFOU TFU PG T BOE T JO UIF EVNNZ WBSJBCMFT XIJD O UVSO JNQMZ B EJČFSFOU FRVBUJPO GPS µJ PODF TJNQMJĕFE 'JUUJOH UIF NPEFM JT TUSBJHIUGPSXBSE ǂǑƾƽ ʆǦ *-ǯ )&01ǯ   .6-5*7"3*"5& -*/&"3 .0%&-4 ćF NPEFM XF BJN UP ĕU UP UIF EBUB JT ( )Ǒ-"/Ǒ$ SFHSFTTFE PO UIF EVNNZ WBSJBCMFT GPS )!" LJ ∼ /PSNBM(µJ, σ) µJ = α + β/8. /8.J + β08. 08.J + β4 4J " MJOFBS NPEFM MJLF UIJT SFBMMZ EFĕOFT  EJČFSFOU MJOFBS NPEFMT FBDI DPSSFTQPOE JOH UP B EJČFSFOU DBUFHPSZ " UBCMF NJHIU IFMQ $BUFHPSZ /8.J 08.J 4J µJ "QF    µJ = α /FX 8PSME NPOLFZ    µJ = α + β/8. 0ME 8PSME NPOLFZ    µJ = α + β08. 4USFQTJSSIJOF    µJ = α + β4 &BDI DBUFHPSZ JNQMJFT B EJČFSFOU TFU PG T BOE T JO UIF EVNNZ WBSJBCMFT XIJDI JO UVSO JNQMZ B EJČFSFOU FRVBUJPO GPS µJ PODF TJNQMJĕFE 'JUUJOH UIF NPEFM JT TUSBJHIUGPSXBSE 3 DPEF  *ǂǑƾƽ ʆǦ *-ǯ )&01ǯ ( )Ǒ-"/Ǒ$ ʍ !+,/*ǯ *2 ǒ 0&$* ǰ ǒ *2 ʍ  ʀ Ǒǹ )!"Ǒ ʀ Ǒǹ )!"Ǒ ʀ Ǒǹ )!"Ǒ ǰ ǒ !1ʅ! ǒ 01/1ʅ)&01ǯʅƽǒǑʅƽǒǑʅƽǒǑʅƽǒ
31. ćF NPEFM XF BJN UP ĕU UP UIF EBUB JT ( )Ǒ-"/Ǒ$ SFHSFTTFE PO UIF EVNN BSJBCMFT GPS )!" LJ ∼ /PSNBM(µJ, σ) µJ = α + β/8. /8.J + β08. 08.J + β4 4J MJOFBS NPEFM MJLF UIJT SFBMMZ EFĕOFT  EJČFSFOU MJOFBS NPEFMT FBDI DPSSFTQPOE OH UP B EJČFSFOU DBUFHPSZ " UBCMF NJHIU IFMQ $BUFHPSZ /8.J 08.J 4J µJ "QF    µJ = α /FX 8PSME NPOLFZ    µJ = α + β/8. 0ME 8PSME NPOLFZ    µJ = α + β08. 4USFQTJSSIJOF    µJ = α + β4 BDI DBUFHPSZ JNQMJFT B EJČFSFOU TFU PG T BOE T JO UIF EVNNZ WBSJBCMFT XIJD O UVSO JNQMZ B EJČFSFOU FRVBUJPO GPS µJ PODF TJNQMJĕFE 'JUUJOH UIF NPEFM JT TUSBJHIUGPSXBSE ǂǑƾƽ ʆǦ *-ǯ )&01ǯ ( )Ǒ-"/Ǒ$ ʍ !+,/*ǯ *2 ǒ 0&$* ǰ ǒ differences from apes }   .6-5*7"3*"5& -*/&"3 .0%&-4 3 DPEF  (ǀǏƼƿ ʄǤ (+ǭ '$./ǭ &'Ǐ+ -Ǐ" ʋ )*-(ǭ (0 ǐ .$"( Ǯ ǐ (0 ʄǤ  ɾ ǏǷ' Ǐ ɾ ǏǷ' Ǐ ɾ ǏǷ' Ǐ ǐ  ʋ )*-(ǭ ƻǏǁ ǐ Ƽƻ Ǯ ǐ Ǐ ʋ )*-(ǭ ƻ ǐ Ƽ Ǯ ǐ Ǐ ʋ )*-(ǭ ƻ ǐ Ƽ Ǯ ǐ Ǐ ʋ )*-(ǭ ƻ ǐ Ƽ Ǯ ǐ .$"( ʋ 0)$!ǭ ƻ ǐ Ƽƻ Ǯ Ǯ ǐ /ʃ Ǯ +- $.ǭ(ǀǏƼƿǮ  ) / 1 ƽǏǀɳ DŽǂǏǀɳ  ƻǏǀǀ ƻǏƻƿ ƻǏƿǂ ƻǏǁƽ Ǐ ƻǏƼǂ ƻǏƻǀ ƻǏƻǁ ƻǏƽǂ Ǐ ƻǏƽƿ ƻǏƻǁ ƻǏƼƽ ƻǏƾǁ Ǐ ǤƻǏƻƿ ƻǏƻǁ ǤƻǏƼǁ ƻǏƻDŽ .$"( ƻǏƼƼ ƻǏƻƽ ƻǏƻDŽ ƻǏƼƿ ćF FTUJNBUF  JT UIF BWFSBHF NJML FOFSHZ GPS BQFT BOE UIF FTUJNBUFT GPS UIF PUIFS DBUFHPSJFT
32. Compute mu for each taxon Ǐ ǤƻǏƻƿ ƻǏƻǁ ǤƻǏƼǁ ƻǏƻDŽ .$"( ƻǏƼƼ ƻǏƻƽ ƻǏƻDŽ ƻǏƼƿ ćF FTUJNBUF  JT UIF BWFSBHF NJML FOFSHZ GPS BQFT BOE UIF FTUJNBUFT GPS UIF PUIFS DBUFHPSJFT BSF EJČFSFODFT GSPN BQFT 4P UP HFU QPTUFSJPS EJTUSJCVUJPOT PG UIF BWFSBHF NJML FOFSHZ JO FBDI DBUFHPSZ ZPV DBO BHBJO VTF TBNQMFT 3 DPEF  ȃ .(+' +*./ -$*- +*./ ʄǤ 3/-/Ǐ.(+' .ǭ(ǀǏƼƿǮ ȃ *(+0/ 1 -" . !*- # / "*-4 (0Ǐ+ ʄǤ +*./ɠ (0Ǐ ʄǤ +*./ɠ ɾ +*./ɠǏ (0Ǐ ʄǤ +*./ɠ ɾ +*./ɠǏ (0Ǐ ʄǤ +*./ɠ ɾ +*./ɠǏ ȃ .0((-$5 0.$)" +- $. +- $.ǭ /Ǐ!-( ǭ(0Ǐ+ ǐ(0Ǐǐ(0Ǐǐ(0ǏǮ Ǯ  ) / 1 '*2 - ƻǏDŽǀ 0++ - ƻǏDŽǀ (0Ǐ+ ƻǏǀǀ ƻǏƻƿ ƻǏƿǂ ƻǏǁƽ (0Ǐ ƻǏǂƼ ƻǏƻƿ ƻǏǁƿ ƻǏǂDŽ (0Ǐ ƻǏǂDŽ ƻǏƻǀ ƻǏǂƻ ƻǏǃǃ (0Ǐ ƻǏǀƼ ƻǏƻǀ ƻǏƿƼ ƻǏǁƼ ćFTF UFMM VT UIBU UIF NPTU QMBVTJCMF DPOEJUJPOBM PO EBUB BOE NPEFM BWFSBHF NJML FOFSHJFT JO FBDI DBUFHPSZ BSF    BOE  ćF )1%* GPS FBDI JT TIPXO JO UIF UXP SJHIUNPTU DPMVNOT 0ODF ZPV HFU BDDVTUPNFE UP NBOJQVMBUJOH FTUJNBUFT JO UIJT XBZ ZPV DBO FČFDUJWFMZ SF
33. Compute mu for each taxon 0.4 0.6 0.8 1.0 clade mean kcal.per.g Ape NWM OWM Strep OLQHV F   PXSL2:0 PEF  ȃ .(+' +*./ -$*- +*./ ʄǤ 3/-/Ǐ.(+' .ǭ(ǀǏƼƿǮ ȃ *(+0/ 1 -" . !*- # / "*-4 (0Ǐ+ ʄǤ +*./ɠ (0Ǐ ʄǤ +*./ɠ ɾ +*./ɠǏ (0Ǐ ʄǤ +*./ɠ ɾ +*./ɠǏ (0Ǐ ʄǤ +*./ɠ ɾ +*./ɠǏ ȃ .0((-$5 0.$)" +- $. +- $.ǭ /Ǐ!-( ǭ(0Ǐ+ ǐ(0Ǐǐ(0Ǐǐ(0ǏǮ Ǯ  ) / 1 '*2 - ƻǏDŽǀ 0++ - ƻǏDŽǀ (0Ǐ+ ƻǏǀǀ ƻǏƻƿ ƻǏƿǂ ƻǏǁƽ (0Ǐ ƻǏǂƼ ƻǏƻƿ ƻǏǁƿ ƻǏǂDŽ (0Ǐ ƻǏǂDŽ ƻǏƻǀ ƻǏǂƻ ƻǏǃǃ (0Ǐ ƻǏǀƼ ƻǏƻǀ ƻǏƿƼ ƻǏǁƼ ćFTF UFMM VT UIBU UIF NPTU QMBVTJCMF DPOEJUJPOBM PO EBUB BOE NPEFM BWFSBHF NJML FOFSHJFT JO FBDI DBUFHPSZ BSF    BOE  ćF )1%* GPS FBDI JT TIPXO JO UIF UXP SJHIUNPTU DPMVNOT 0ODF ZPV HFU BDDVTUPNFE UP NBOJQVMBUJOH FTUJNBUFT JO UIJT XBZ ZPV DBO FČFDUJWFMZ SF QBSBNFUFSJ[F ZPVS NPEFM BęFS ZPVWF BMSFBEZ ĕU JU UP UIF EBUB 'PS FYBNQMF BCPWF XF ĕU NPEFM (ǀǏƼƿ TVDI UIBU FBDI  QBSBNFUFS JT B EJČFSFODF GSPN BQFT ćJT NBLFT JU IBSE UP
34. Difference and uncertainty -6 -4 -2 0 2 4 0.00 0.25 estimate likelihood 1 2 diff
35. What about differences? • Differences between clades (contrasts) have their own distributions • Difference between NWM and OWM:  $"5&(03*$"- 7"3*"#-&4  SFBTPO BCPVU UIF EJČFSFODFT BNPOH PUIFS DBUFHPSJFT 4P TVQQPTF ZPV XBOU UP LOPX UIF FTUJ NBUFE EJČFSFODF CFUXFFO UIF UXP NPOLFZ HSPVQT ćFO KVTU TVCUSBDU UIF FTUJNBUFE NFBOT UP HFU B EJČFSFODF 3 DPEF  $!!ǏǏ ʄǤ (0Ǐ Ǥ (0Ǐ ,0)/$' ǭ $!!ǏǏ ǐ +-*.ʃǭƻǏƻƽǀǐƻǏǀǐƻǏDŽǂǀǮ Ǯ ƽǏǀɳ ǀƻɳ DŽǂǏǀɳ ǤƻǏƼDŽƾƽƼDŽƼǂ ǤƻǏƻǂƽƼƿƼƼƻ ƻǏƻƿǀDŽǂƼǂǂ ćPTF WBMVFT BSF UIF QPTUFSJPS MPXFS  CPVOEBSZ UIF NFEJBO BOE UIF VQQFS  CPVOE BSZ GPS UIF EJČFSFODF CFUXFFO /FX 8PSME BOE 0ME 8PSME NPOLFZT 4JODF ZPVSF TUJMM XPSL JOH XJUI TBNQMFT GSPN UIF QPTUFSJPS ZPV HFU B QPTUFSJPS EJTUSJCVUJPO GPS UIF EJČFSFODF CF UXFFO /8. BOE 08. /POF PG UIF VODFSUBJOUZ JO UIF PSJHJOBM QPTUFSJPS EJTUSJCVUJPO JT EJTDBSEFE
36. OLS and using lm • Ordinary Least Squares (OLS): A quick way to find MAP for additive models with flat priors • Independently described by both Legendre and Gauss around 1800
37. OLS and using lm • OLS chooses parameters that minimize the sum of squared distances from the mean (sum-of-squares) 35 40 45 50 55 130 140 150 160 170 weight height
38. 35 40 45 50 55 130 140 150 160 170 weight height SS = 787.36 • OLS chooses parameters that minimize the sum of squared distances from the mean (sum-of-squares) OLS and using lm
39. 35 40 45 50 55 130 140 150 160 170 weight height SS = 787.36 0.0 0.2 0.4 0.6 0.8 1.0 1.2 400 600 800 beta (slope) sum of squares 35 40 45 50 55 130 140 150 160 170 weight height SS = 442.84 35 40 45 50 55 130 140 150 160 170 weight height SS = 450.78 35 40 45 50 55 130 140 150 160 170 weight height SS = 375.26
40. OLS and using lm • For Gaussian, additive models, can solve for MAP • Efficient, but provides no distribution for sigma 35 40 45 50 55 130 140 150 160 170 weight height SS = 375.26
41. Design formulas QSFEJDUPS OBNFT ćFTF BSF TVďDJFOU UP EFTDSJCF UIF NPEFMT EFTJHO QSJPST BSF ĘBU 'PS FYBNQMF JG XF IBWF UIF MJOFBS SFHSFTTJPO ZJ ∼ /PSNBM(µJ, σ) µJ = α + βYJ O UIF DPSSFTQPOEJOH EFTJHO GPSNVMB JT 6 ʍ ƾ ʀ 5 F MFBEJOH ƾ PO UIF SJHIUIBOE TJEF JOEJDBUFT UIF JOUFSDFQU α "T ZPV EJDUPS WBSJBCMFT UIF EFTJHO GPSNVMB FYQBOET 'PS FYBNQMF UIJT E MB 6 ʍ ƾ ʀ 5 ʀ 7 ʀ 4 SFTQPOET UP UIF NPEFM DUPS OBNFT ćFTF BSF TVďDJFOU UP EFTDSJCF UIF NPEFMT EFTJHO Q T BSF ĘBU FYBNQMF JG XF IBWF UIF MJOFBS SFHSFTTJPO ZJ ∼ /PSNBM(µJ, σ) µJ = α + βYJ DPSSFTQPOEJOH EFTJHO GPSNVMB JT 6 ʍ ƾ ʀ 5 JOH ƾ PO UIF SJHIUIBOE TJEF JOEJDBUFT UIF JOUFSDFQU α "T ZPV B S WBSJBCMFT UIF EFTJHO GPSNVMB FYQBOET 'PS FYBNQMF UIJT EF 6 ʍ ƾ ʀ 5 ʀ 7 ʀ 4 POET UP UIF NPEFM Model: Design formula:
42. Design formulas NFT ćFTF BSF TVďDJFOU UP EFTDSJCF UIF NPEFMT EFTJHO QSPWJEFE BU F JG XF IBWF UIF MJOFBS SFHSFTTJPO ZJ ∼ /PSNBM(µJ, σ) µJ = α + βYJ QPOEJOH EFTJHO GPSNVMB JT 6 ʍ ƾ ʀ 5 PO UIF SJHIUIBOE TJEF JOEJDBUFT UIF JOUFSDFQU α "T ZPV BEE NPSF CMFT UIF EFTJHO GPSNVMB FYQBOET 'PS FYBNQMF UIJT EFTJHO GPS 6 ʍ ƾ ʀ 5 ʀ 7 ʀ 4 UIF NPEFM ćFTF BSF TVďDJFOU UP EFTDSJCF UIF NPEFMT EFTJHO QSPWJEFE XF IBWF UIF MJOFBS SFHSFTTJPO ZJ ∼ /PSNBM(µJ, σ) µJ = α + βYJ EJOH EFTJHO GPSNVMB JT 6 ʍ ƾ ʀ 5 F SJHIUIBOE TJEF JOEJDBUFT UIF JOUFSDFQU α "T ZPV BEE NPSF UIF EFTJHO GPSNVMB FYQBOET 'PS FYBNQMF UIJT EFTJHO GPS 6 ʍ ƾ ʀ 5 ʀ 7 ʀ 4 NPEFM Model: Design formula: QSFEJDUPS WBSJBCMFT UIF EFTJHO GPSNVMB FYQBOET 'PS F NVMB 6 ʍ ƾ ʀ 5 ʀ 7 ʀ 4 DPSSFTQPOET UP UIF NPEFM ZJ ∼ /PSNBM(µJ, σ) µJ = α + βY YJ + β[ [J + βX XJ %FTJHO GPSNVMBT BSF JOUJNBUFMZ SFMBUFE UP B DPNNPO UJPO PG MJOFBS NPEFMT 4FF UIF PWFSUIJOLJOH CPY PO QBHF UJPO UP UIJT BQQSPBDI  6TJOH )* 5P ĕU B MJOFBS SFHSFTTJPO VTJOH 0-4 KVT NVMB BOE UIF OBNF PG UIF EBUB GSBNF 4P GPS CPUI FYBN UP )* XPVME CF BTTVNJOH UIF EBUB BSF JO B GSBNF OBNFE *ǂǑƾƾ ʆǦ )*ǯ 6 ʍ ƾ ʀ 5 ǒ !1ʅ! ǰ *ǂǑƾƿ ʆǦ )*ǯ 6 ʍ ƾ ʀ 5 ʀ 7 ʀ 4 ǒ !1ʅ! ǰ ZJ ∼ /PSNBM(µJ, σ) µJ = α + βYJ UIFO UIF DPSSFTQPOEJOH EFTJHO GPSNVMB JT 6 ʍ ƾ ʀ 5 ćF MFBEJOH ƾ PO UIF SJHIUIBOE TJEF JOEJDBUFT UIF JOUFS QSFEJDUPS WBSJBCMFT UIF EFTJHO GPSNVMB FYQBOET 'PS NVMB 6 ʍ ƾ ʀ 5 ʀ 7 ʀ 4 DPSSFTQPOET UP UIF NPEFM ZJ ∼ /PSNBM(µJ, σ) µJ = α + βY YJ + β[ [J + βX X %FTJHO GPSNVMBT BSF JOUJNBUFMZ SFMBUFE UP B DPNNP
43. Using lm UJPO PG MJOFBS NPEFMT 4FF UIF PWFSUIJOLJOH CPY PO QBHF  GPS B TIPSU JOUSPEVD UJPO UP UIJT BQQSPBDI  6TJOH )* 5P ĕU B MJOFBS SFHSFTTJPO VTJOH 0-4 KVTU QSPWJEF UIF EFTJHO GPS NVMB BOE UIF OBNF PG UIF EBUB GSBNF 4P GPS CPUI FYBNQMFT KVTU BCPWF UIF DBMMT UP )* XPVME CF BTTVNJOH UIF EBUB BSF JO B GSBNF OBNFE !  3 DPEF  *ǂǑƾƾ ʆǦ )*ǯ 6 ʍ ƾ ʀ 5 ǒ !1ʅ! ǰ *ǂǑƾƿ ʆǦ )*ǯ 6 ʍ ƾ ʀ 5 ʀ 7 ʀ 4 ǒ !1ʅ! ǰ ćF QBSBNFUFS FTUJNBUFT XJMM CF OBNFE BęFS UIF QSFEJDUPST TJODF ZPV EJEOU QSP WJEF BOZ FYQMJDJU QBSBNFUFS OBNFT #VU PUIFSXJTF UIF PVUQVU XJMM MPPL GBNJMJBS BOE ZPV DBO TBNQMF GSPN UIF QPTUFSJPS BOE QSPDFTT FTUJNBUFT KVTU BT CFGPSF ćFSF BSF TPNF RVJSLT UP IPX )* XPSLT UIPVHI )FSF BSF TPNF PG UIF JNQPS UBOU RVJSLT  *OUFSDFQUT BSF PQUJPOBM ćFTF UXP NPEFM ĕUT SFUVSO FYBDUMZ UIF TBNF FTUJNBUFT 3 DPEF  *ǂǑƾƾ ʆǦ )*ǯ 6 ʍ ƾ ʀ 5 ǒ !1ʅ! ǰ *ǂǑƾǀ ʆǦ )*ǯ 6 ʍ 5 ǒ !1ʅ! ǰ 8IFO ZPV PNJU UIF FYQMJDJU JOUFSDFQU )* BTTVNFT ZPV XBOUFE POF *G ZPV SFBMMZ EP OPU XBOU BO JOUFSDFQU‰FRVJWBMFOU UP ĕYJOH α = ‰UIFO ZPV DBO VTF POF PG UIFTF GPSNT 3 DPEF  3 DPEF  *ǂǑƾƾ ʆǦ )*ǯ 6 ʍ ƾ ʀ 5 ǒ !1ʅ! ǰ *ǂǑƾƿ ʆǦ )*ǯ 6 ʍ ƾ ʀ 5 ʀ 7 ʀ 4 ǒ !1ʅ! ǰ ćF QBSBNFUFS FTUJNBUFT XJMM CF OBNFE BęFS UIF QSFEJDUPST TJODF ZPV EJEOU QSP WJEF BOZ FYQMJDJU QBSBNFUFS OBNFT #VU PUIFSXJTF UIF PVUQVU XJMM MPPL GBNJMJBS BOE ZPV DBO TBNQMF GSPN UIF QPTUFSJPS BOE QSPDFTT FTUJNBUFT KVTU BT CFGPSF ćFSF BSF TPNF RVJSLT UP IPX )* XPSLT UIPVHI )FSF BSF TPNF PG UIF JNQPS UBOU RVJSLT  *OUFSDFQUT BSF PQUJPOBM ćFTF UXP NPEFM ĕUT SFUVSO FYBDUMZ UIF TBNF FTUJNBUFT 3 DPEF  *ǂǑƾƾ ʆǦ )*ǯ 6 ʍ ƾ ʀ 5 ǒ !1ʅ! ǰ *ǂǑƾǀ ʆǦ )*ǯ 6 ʍ 5 ǒ !1ʅ! ǰ 8IFO ZPV PNJU UIF FYQMJDJU JOUFSDFQU )* BTTVNFT ZPV XBOUFE POF *G ZPV SFBMMZ EP OPU XBOU BO JOUFSDFQU‰FRVJWBMFOU UP ĕYJOH α = ‰UIFO ZPV DBO VTF POF PG UIFTF GPSNT 3 DPEF  Intercept optional but assumed:   .6-5*7"3*"5& -*/&"3 .0%&-4 *ǂǑƾǁ ʆǦ )*ǯ 6 ʍ ƽ ʀ 5 ǒ !1ʅ! ǰ *ǂǑƾǂ ʆǦ )*ǯ 6 ʍ 5 Ǧ ƾ ǒ !1ʅ! ǰ  $BUFHPSJDBM WBSJBCMFT #MBDL CPY GVODUJPOT MJLF )* XJMM BVUPNBUJDBMMZ FYQBOE DBUFHPSJDBM GBDUPST JOUP UIF OFDFTTBSZ OVNCFS PG EVNNZ WBSJBCMFT #VU 3 JT OPU B NJOE SFBEFS BOE TPNFUJNFT UIF TBNF WBSJBCMF DBO CF USFBUFE BT DBUFHPSJFT PS SBUIFS BT DPOUJOVPVT 'PS FYBNQMF ZPV NJHIU DPEF B WBSJBCMF MJLF 0"0,+ XJUI To remove intercept (fix alpha = 0): 3 DPEF  *ǂǑƾƾ ʆǦ )*ǯ 6 ʍ ƾ ʀ 5 ǒ !1ʅ! ǰ *ǂǑƾƿ ʆǦ )*ǯ 6 ʍ ƾ ʀ 5 ʀ 7 ʀ 4 ǒ !1ʅ! ǰ ćF QBSBNFUFS FTUJNBUFT XJMM CF OBNFE BęFS UIF QSFEJDUPST TJODF ZPV EJEOU QSP WJEF BOZ FYQMJDJU QBSBNFUFS OBNFT #VU PUIFSXJTF UIF PVUQVU XJMM MPPL GBNJMJBS BOE ZPV DBO TBNQMF GSPN UIF QPTUFSJPS BOE QSPDFTT FTUJNBUFT KVTU BT CFGPSF ćFSF BSF TPNF RVJSLT UP IPX )* XPSLT UIPVHI )FSF BSF TPNF PG UIF JNQPS UBOU RVJSLT  *OUFSDFQUT BSF PQUJPOBM ćFTF UXP NPEFM ĕUT SFUVSO FYBDUMZ UIF TBNF FTUJNBUFT 3 DPEF  *ǂǑƾƾ ʆǦ )*ǯ 6 ʍ ƾ ʀ 5 ǒ !1ʅ! ǰ *ǂǑƾǀ ʆǦ )*ǯ 6 ʍ 5 ǒ !1ʅ! ǰ 8IFO ZPV PNJU UIF FYQMJDJU JOUFSDFQU )* BTTVNFT ZPV XBOUFE POF *G ZPV SFBMMZ EP OPU XBOU BO JOUFSDFQU‰FRVJWBMFOU UP ĕYJOH α = ‰UIFO ZPV DBO VTF POF PG UIFTF GPSNT 3 DPEF 
44. glimmer of hope 1SPWJEFE ZPV EPOU DBSF BCPVU GVMM QSFEJDUJPO JOUFSWBMT UIFO UIF MBDL PG BO FT XPOU CPUIFS ZPV CFDBVTF ZPV XPOU OFFE BO FTUJNBUF GPS σ #VU JG ZPV ĕOE ZPVS JU ZPVMM IBWF UP GBMM CBDL PO B UPPM MJLF (+ TP ZPV DBO HFU TPNF TFOTF PG UIF VO UIF FTUJNBUF "T B CPOVT ZPV DBO VTF JOGPSNBUJWF QSJPST UIFO BT XFMM  #VJMEJOH(+ GPSNVMBTGSPN'( GPSNVMBT "MM BCPVU "'$(( - <TUVC UP C 3 DPEF  /ǭ-.Ǯ "'$(( -ǭ $./ ʋ .+  ǐ /ʃ-. Ǯ '$./ǭ $./ ʋ )*-(ǭ (0 ǐ .$"( Ǯǐ (0 ʄǤ )/ - +/ ɾ Ǭ.+ Ƿ.+ ǐ )/ - +/ ʋ )*-(ǭƻǐƼƻǮǐ Ǭ.+  ʋ )*-(ǭƻǐƼƻǮǐ .$"( ʋ 0#4ǭƻǐƽǮ  13"$5*$& Ǯ  4VNNBSZ ćJT DIBQUFS JOUSPEVDFE NVMUJQMF SFHSFTTJPO B XBZ PG DPOTUSVDUJOH EFTDSJ GPS IPX UIF NFBO PG B NFBTVSFNFOU JT BTTPDJBUFE XJUI NPSF UIBO POF QSFEJ ćF EFĕOJOH RVFTUJPO PG NVMUJQMF SFHSFTTJPO JT 8IBU JT UIF WBMVF PG LOPXJOH F PODF XF BMSFBEZ LOPX UIF PUIFS QSFEJDUPST *NQMJDJU JO UIJT RVFTUJPO BSF  UIF QSFEJDUPST GPS EFTDSJQUJPO PG UIF TBNQMF JOTUFBE PG GPSFDBTUJOH B GVUVSF TBN
45. Next week • Sailing between (1) the whirlpool of underfitting (2) the many-headed monster of overfitting