Statistical Rethinking - Lecture 15

Lecture 15 - Ordered logistic models - Statistical Rethinking: A Bayesian Course with R Examples

Transcript

1. Week 8: Monsters & Mixtures Richard McElreath Statistical Rethinking
3. Do not attempt at home (or anywhere near me)
4. Monsters and mixtures • More complicated GLMs: • Monsters: Specialized, complex distributions • ordered categories, ranks • Mixtures: Blends of stochastic processes • Varying means, probabilities, rates • Varying process: zero-inflation, hurdles
5. Ordered categories • How much do you like this class? (1–7) • How important is income of a potential spouse? (1–7) • How often do you see bats in Davis? (never, sometimes, frequently) • Depth harbor seals dive? (shallow, middle, deep)
6. Ordered categories • Discrete outcomes • Defined minimum and maximum • Defined order • “Distances” between categories unknown
7. Ordered categories • Hard to model • Not continuous • Not counts • Common solution: ordered (aka ordinal) logistic regression • Good example of making a monster
8. How morally permissible is it to pull the lever? never 1 2 3 4 5 6 7 always
9. How morally permissible is it to push the man? never 1 2 3 4 5 6 7 always
10. How morally permissible is it to not pull the lever? never 1 2 3 4 5 6 7 always
11. Three principles • Action: Harm caused by action is morally worse than same harm caused by inaction. • Intention: Harm intended as means to goal worse than same harm foreseen as a side effect of goal. • Contact: Harm caused by physical contact worse than same harm without physical contact.
12. action intention contact • ◦ ◦ • • • ◦ ◦ ◦
13. Moral intuitions • Cushman et al. experiments • 331 individuals, 30 scenarios, 9930 responses • How do responses vary with action, intention, contact? • Age, gender, individual? 1 2 3 4 5 6 7 0 500 1500 How permissible Frequency
14. 1 2 3 4 5 6 7 0 500 1500 How permissible Frequency 1 2 3 4 5 6 7 0 100 200 300 400 How permissible Frequency contact 1 2 3 4 5 6 7 0 200 600 1000 How permissible Frequency action 1 2 3 4 5 6 7 0 200 600 1000 How permissible Frequency intention
15. Ordered logit • A log-cumulative-odds link probability model   .0/45&34 "/% .*9563&4 1 2 3 4 5 6 7 0 500 1000 1500 2000 response Frequency 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 response cumulative proportion 1 2 3 4 5 6 7 -2 -1 0 1 response log-cumulative-odds 'ĶĴłĿIJ ƉƉƉ 3FEFTDSJCJOH B EJTDSFUF EJTUSJCVUJPO VTJOH MPHDVNVMBUJWF PEET -Fę )JTUPHSBN PG EJTDSFUF SFTQPOTF JO UIF TBNQMF .JEEMF $VNV
16. Ordered logit • A log-cumulative-odds link probability model UZ OBUVSBMMZ DPOTUSBJOT JUTFMG UP OFWFS FYDFFEJOH B UPUBM QSPCB POF "OE CFDBVTF UIJT JT BO PSEFSFE EFOTJUZ XF LOPX UIBU UIF WF MPHPEET PG UIF MBSHFTU PCTFSWBCMF WBMVF NVTU CF +∞ XIJDI NF BT DVNVMBUJWF QSPCBCJMJUZ PG POF 5IJT BODIPST UIF EJTUSJ OE TUBOEBSEJ[FT JU BU UIF TBNF UJNF *G ZPV TUBSU JOTUFBE XJUI JOEJWJEVBM QSPCBCJMJUJFT PG FBDI PVUDPNF UIFO ZPVÔE IBWF UP EBSEJ[F UIFTF QSPCBCJMJUJFT UP FOTVSF UIFZ TVN UP FYBDUMZ POF *U U UP CF FBTJFS UP KVTU TUBSU XJUI UIF DVNVMBUJWF QSPCBCJMJUZ BOE SL CBDLXBSET UP UIF JOEJWJEVBM QSPCBCJMJUJFT * LOPX UIJT TFFNT VU *ÔMM XBML ZPV UISPVHI JU U XF XBOU JT GPS UIF DVNVMBUJWF MPHPEET PG BO PCTFSWFE WBMVF Z J VBMUPPSMFTTUIBO TPNF QPTTJCMF WBMVF L UP CF MPH 1S(Z J ≤ L)  − 1S(Z J ≤ L) = φL,  L JT B DPOUJOVPVT WBMVF EJGGFSFOU GPS FBDI PCTFSWBCMF WBMVF L BLF UIJT WBMVF JOUP B MJOFBS NPEFM JO B CJU 'PS OPX JUÔT KVTU B EFS 5IF BCPWF GVODUJPO JT KVTU B EJSFDU FNCPEJNFOU PG UIF MPH E DVNVMBUJWF EFOTJUZ PCKFDUJWFT XFÔWF TUBUFE TP GBS *U BDUVBMMZ IJOH FMTF BU BMM /PX XF TPMWF GPS UIF DVNVMBUJWF QSPCBCJMJUZ 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 response cumulative proportion cumulative log-odds response category linear model
17. Ordered logit • A log-cumulative-odds link probability model WF MPHPEET PG UIF MBSHFTU PCTFSWBCMF WBMVF NVTU CF +∞ XIJDI NF BT DVNVMBUJWF QSPCBCJMJUZ PG POF 5IJT BODIPST UIF EJTUSJ OE TUBOEBSEJ[FT JU BU UIF TBNF UJNF *G ZPV TUBSU JOTUFBE XJUI JOEJWJEVBM QSPCBCJMJUJFT PG FBDI PVUDPNF UIFO ZPVÔE IBWF UP EBSEJ[F UIFTF QSPCBCJMJUJFT UP FOTVSF UIFZ TVN UP FYBDUMZ POF *U U UP CF FBTJFS UP KVTU TUBSU XJUI UIF DVNVMBUJWF QSPCBCJMJUZ BOE SL CBDLXBSET UP UIF JOEJWJEVBM QSPCBCJMJUJFT * LOPX UIJT TFFNT VU *ÔMM XBML ZPV UISPVHI JU U XF XBOU JT GPS UIF DVNVMBUJWF MPHPEET PG BO PCTFSWFE WBMVF Z J VBMUPPSMFTTUIBO TPNF QPTTJCMF WBMVF L UP CF MPH 1S(Z J ≤ L)  − 1S(Z J ≤ L) = φL,  L JT B DPOUJOVPVT WBMVF EJGGFSFOU GPS FBDI PCTFSWBCMF WBMVF L BLF UIJT WBMVF JOUP B MJOFBS NPEFM JO B CJU 'PS OPX JUÔT KVTU B EFS 5IF BCPWF GVODUJPO JT KVTU B EJSFDU FNCPEJNFOU PG UIF MPH E DVNVMBUJWF EFOTJUZ PCKFDUJWFT XFÔWF TUBUFE TP GBS *U BDUVBMMZ IJOH FMTF BU BMM /PX XF TPMWF GPS UIF DVNVMBUJWF QSPCBCJMJUZ 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 response cumulative proportion  03%&3&% $"5&(03*$"- 065$0.&4  UTFMG %P UIJT CZ UBLJOH  BOE TPMWJOH GPS 1S(Z J ≤ L) "GUFS B FCSB ZPV HFU 1S(Z J ≤ L) = FYQ(φL)  + FYQ(φL) . IU SFDPHOJ[F UIJT QSPCBCJMJUZ BT UIF MPHJTUJD TBNF BT JO UIF MBTU *U BSPTF JO UIF TBNF XBZ FTUBCMJTIJOH UIF MPHJTUJD GVODUJPO BT STF MJOL GPS UIF CJOPNJBM NPEFM #VU OPX XF IBWF B DVNVMBUJWF ODF UIF QSPCBCJMJUZ 1S(Z J ≤ L) JT DVNVMBUJWF XF TUJMM OFFE MJLFMJIPPET XIJDI BSF OPU DVNVMBUJWF 4P IPX EP
18. 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 response cumulative proportion Ordered logit • A log-cumulative-odds link probability model WF MPHPEET PG UIF MBSHFTU PCTFSWBCMF WBMVF NVTU CF +∞ XIJDI NF BT DVNVMBUJWF QSPCBCJMJUZ PG POF 5IJT BODIPST UIF EJTUSJ OE TUBOEBSEJ[FT JU BU UIF TBNF UJNF *G ZPV TUBSU JOTUFBE XJUI JOEJWJEVBM QSPCBCJMJUJFT PG FBDI PVUDPNF UIFO ZPVÔE IBWF UP EBSEJ[F UIFTF QSPCBCJMJUJFT UP FOTVSF UIFZ TVN UP FYBDUMZ POF *U U UP CF FBTJFS UP KVTU TUBSU XJUI UIF DVNVMBUJWF QSPCBCJMJUZ BOE SL CBDLXBSET UP UIF JOEJWJEVBM QSPCBCJMJUJFT * LOPX UIJT TFFNT VU *ÔMM XBML ZPV UISPVHI JU U XF XBOU JT GPS UIF DVNVMBUJWF MPHPEET PG BO PCTFSWFE WBMVF Z J VBMUPPSMFTTUIBO TPNF QPTTJCMF WBMVF L UP CF MPH 1S(Z J ≤ L)  − 1S(Z J ≤ L) = φL,  L JT B DPOUJOVPVT WBMVF EJGGFSFOU GPS FBDI PCTFSWBCMF WBMVF L BLF UIJT WBMVF JOUP B MJOFBS NPEFM JO B CJU 'PS OPX JUÔT KVTU B EFS 5IF BCPWF GVODUJPO JT KVTU B EJSFDU FNCPEJNFOU PG UIF MPH E DVNVMBUJWF EFOTJUZ PCKFDUJWFT XFÔWF TUBUFE TP GBS *U BDUVBMMZ IJOH FMTF BU BMM /PX XF TPMWF GPS UIF DVNVMBUJWF QSPCBCJMJUZ  03%&3&% $"5&(03*$"- 065$0.&4  UTFMG %P UIJT CZ UBLJOH  BOE TPMWJOH GPS 1S(Z J ≤ L) "GUFS B FCSB ZPV HFU 1S(Z J ≤ L) = FYQ(φL)  + FYQ(φL) . IU SFDPHOJ[F UIJT QSPCBCJMJUZ BT UIF MPHJTUJD TBNF BT JO UIF MBTU *U BSPTF JO UIF TBNF XBZ FTUBCMJTIJOH UIF MPHJTUJD GVODUJPO BT STF MJOL GPS UIF CJOPNJBM NPEFM #VU OPX XF IBWF B DVNVMBUJWF ODF UIF QSPCBCJMJUZ 1S(Z J ≤ L) JT DVNVMBUJWF XF TUJMM OFFE MJLFMJIPPET XIJDI BSF OPU DVNVMBUJWF 4P IPX EP
19. 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 response cumulative proportion Ordered logit • A log-cumulative-odds link probability model OE TUBOEBSEJ[FT JU BU UIF TBNF UJNF *G ZPV TUBSU JOTUFBE XJUI JOEJWJEVBM QSPCBCJMJUJFT PG FBDI PVUDPNF UIFO ZPVÔE IBWF UP EBSEJ[F UIFTF QSPCBCJMJUJFT UP FOTVSF UIFZ TVN UP FYBDUMZ POF *U U UP CF FBTJFS UP KVTU TUBSU XJUI UIF DVNVMBUJWF QSPCBCJMJUZ BOE SL CBDLXBSET UP UIF JOEJWJEVBM QSPCBCJMJUJFT * LOPX UIJT TFFNT VU *ÔMM XBML ZPV UISPVHI JU U XF XBOU JT GPS UIF DVNVMBUJWF MPHPEET PG BO PCTFSWFE WBMVF Z J VBMUPPSMFTTUIBO TPNF QPTTJCMF WBMVF L UP CF MPH 1S(Z J ≤ L)  − 1S(Z J ≤ L) = φL,  L JT B DPOUJOVPVT WBMVF EJGGFSFOU GPS FBDI PCTFSWBCMF WBMVF L BLF UIJT WBMVF JOUP B MJOFBS NPEFM JO B CJU 'PS OPX JUÔT KVTU B EFS 5IF BCPWF GVODUJPO JT KVTU B EJSFDU FNCPEJNFOU PG UIF MPH E DVNVMBUJWF EFOTJUZ PCKFDUJWFT XFÔWF TUBUFE TP GBS *U BDUVBMMZ IJOH FMTF BU BMM /PX XF TPMWF GPS UIF DVNVMBUJWF QSPCBCJMJUZ  03%&3&% $"5&(03*$"- 065$0.&4  UTFMG %P UIJT CZ UBLJOH  BOE TPMWJOH GPS 1S(Z J ≤ L) "GUFS B FCSB ZPV HFU 1S(Z J ≤ L) = FYQ(φL)  + FYQ(φL) . IU SFDPHOJ[F UIJT QSPCBCJMJUZ BT UIF MPHJTUJD TBNF BT JO UIF MBTU *U BSPTF JO UIF TBNF XBZ FTUBCMJTIJOH UIF MPHJTUJD GVODUJPO BT STF MJOL GPS UIF CJOPNJBM NPEFM #VU OPX XF IBWF B DVNVMBUJWF ODF UIF QSPCBCJMJUZ 1S(Z J ≤ L) JT DVNVMBUJWF XF TUJMM OFFE MJLFMJIPPET XIJDI BSF OPU DVNVMBUJWF 4P IPX EP IJT UIJOH 8FMM JUÔT B QSPCBCJMJUZ EFOTJUZ TP ZPV DBO VTF JU UP F MJLFMJIPPE PG BOZ PCTFSWBUJPO Z J  #Z EFGJOJUJPO JO B EJTDSFUF  03%&3&% $"5&(03*$"- 065$0.&4  MG %P UIJT CZ UBLJOH  BOE TPMWJOH GPS 1S(Z J ≤ L) "GUFS B B ZPV HFU 1S(Z J ≤ L) = FYQ(φL)  + FYQ(φL) . SFDPHOJ[F UIJT QSPCBCJMJUZ BT UIF MPHJTUJD TBNF BT JO UIF MBTU BSPTF JO UIF TBNF XBZ FTUBCMJTIJOH UIF MPHJTUJD GVODUJPO BT MJOL GPS UIF CJOPNJBM NPEFM #VU OPX XF IBWF B DVNVMBUJWF F UIF QSPCBCJMJUZ 1S(Z J ≤ L) JT DVNVMBUJWF TUJMM OFFE MJLFMJIPPET XIJDI BSF OPU DVNVMBUJWF 4P IPX EP T UIJOH 8FMM JUÔT B QSPCBCJMJUZ EFOTJUZ TP ZPV DBO VTF JU UP JLFMJIPPE PG BOZ PCTFSWBUJPO Z J  #Z EFGJOJUJPO JO B EJTDSFUF EFOTJUZ UIF MJLFMJIPPE PG BOZ PCTFSWBUJPO Z J = L NVTU CF 1S(Z J = L) = 1S(Z J ≤ L) − 1S(Z J ≤ L − ).  ZT UIBU TJODF UIF MPHJTUJD JT DVNVMBUJWF XF DBO DPNQVUF UIF CBCJMJUZ PG FYBDUMZ Z J = L CZ TVCUSBDUJOH UIF DVNVMBUJWF QSPC OF PCTFSWBCMF WBMVF MPXFS UIBO L UJOH UIF (-. JO UIF φ 8FÔSF BMNPTU SFBEZ UP XBML UISPVHI
20. • Simplest model just uses an intercept for each category: intercept unique to category cumulative probabilities of each response PVUDPNF XFMM CF JOUFSFTUFE JO JT /"0-,+0" XIJDI JT BO JOUFHFS GSPN  NPSBMMZ QFSNJTTJCMF UIF QBSUJDJQBOU GPVOE UIF BDUJPO UBLFO PS OPU U 4JODF UIJT UZQF PG SBUJOH JT DBUFHPSJDBM BOE PSEFSFE JUT FYBDUMZ UIF UZQF PVS PSEFSFE MPHJU NPEFM UP ćF QSFEJDUPS WBSJBCMFT PG JOUFSFTU BSF HPJOH UP CF  1&,+ &+1"+1 FBDI B EVNNZ WBSJBCMF DPSSFTQPOEJOH UP FBDI QSJODJQMF PVUMJOFE BCPW  ćF CBTJD NPEFM 8FMM CFHJO XJUI B CBTJD NPEFM UIBU JODMV GPS FBDI MFWFM PG UIF PVUDPNF $POWFOUJPOT GPS XSJUJOH NBUIFNBUJDBM GP MPHJU WBSZ B MPU 8FMM VTF UIJT 3J ∼ 0SEFSFE(Q) MPH QL  − QL = αL >FXPXODWLY αL ∼ /PSNBM(, ) >FRPPRQ *O DPEF GPSN GPS *- BOE *-ƿ01+ UIF MJOL GVODUJPO XJMM CF FNCFEEF GVODUJPO BMSFBEZ 4P UP ĕU UIF CBTJD NPEFM JODPSQPSBUJOH OP QSFEJDUPS DPEF  *ƾƾǑƾ ʆǦ *-ǯ )&01ǯ Ordered logit
21. 1 2 3 4 5 6 7 0.0 0.2 0.4 response cumula 1 2 3 CJMJUZ ćFTF LFFQ HSPXJOH XJUI FBDI TVDDFTTJWF PVUDPNF WBMVF ćF CMVF MJOF TFHNFOUT TIPX UIF EJTDSFUF QSPCBCJMJUZ PG FBDI JOEJWJEVBM PVUDPNF ćFTF BSF UIF MJLFMJIPPET UIBU HP JOUP #BZFT UIF PSFN TNBMM TBNQMF DPOUFYUT ZPVMM IBWF UP UIJOL NVDI IBSEFS BCPVU QSJPST $POTJEFS GPS FYBNQMF UIBU XF LOPX α < α CFGPSF XF FWFO TFF UIF EBUB *O DPEF GPSN GPS (+ BOE (+Ǐ./) UIF MJOL GVODUJPO XJMM CF FNCFEEFE JO UIF MJLFMJIPPE GVODUJPO BMSFBEZ ćJT NBLFT UIF DBMDVMBUJPOT NPSF FďDJFOU BOE QSFWFOUT IFBEBDIFT 4P UP ĕU UIF CBTJD NPEFM JODPSQPSBUJOH OP QSFEJDUPS WBSJBCMFT 3 DPEF  (ǎǎǡǎ ʚǶ (+ǿ '$./ǿ - .+*). ʡ *-'*"$/ǿ +#$ Ǣ ǿǎǢǏǢǐǢǑǢǒǢǓȀ ȀǢ +#$ ʚǶ ǍǢ ǿǎǢǏǢǐǢǑǢǒǢǓȀ ʡ )*-(ǿǍǢǎǍȀ Ȁ Ǣ /ʙ Ǣ ./-/ʙ'$./ǿǎʙǶǏǢǏʙǶǎǢǐʙǍǢǑʙǎǢǒʙǏǢǓʙǏǡǒȀ Ȁ ćF +#$ JO UIF NPEFM DPEF JT B QMBDFIPMEFS GPS UIF MJOFBS NPEFM UIBU JODPSQPSBUFT QSFEJDUPS WBSJBCMFT :PVMM VTF JU TIPSUMZ *U JT B DPOTUBOU [FSP GPS OPX CFDBVTF POMZ UIF JOUFSDFQU QBSBN FUFST BSF PG JOUFSFTU ćF ./-/ WBMVFT GPS UIF JOUFSDFQUT BSF DIPTFO KVTU UP TUBSU UIFN JO UIF SJHIU PSEFS ćF FYBDU WBMVFT BSFOU JNQPSUBOU CVU UIFJS PSEFSJOH PO UIF MPHDVNVMBUJWF PEET TDBMF JT JNQPSUBOU ćFJS QPTUFSJPS EJTUSJCVUJPO JT BMTP PO UIF MPHDVNVMBUJWFPEET TDBMF guesses for starting log-cumulative-odds  ćF CBTJD NPEFM 8FMM CFHJO XJUI B CBTJD NPEFM UIBU JODMVE GPS FBDI MFWFM PG UIF PVUDPNF $POWFOUJPOT GPS XSJUJOH NBUIFNBUJDBM GPS MPHJU WBSZ B MPU 8FMM VTF UIJT 3J ∼ 0SEFSFE(Q) MPH QL  − QL = αL >FXPXODWLYH αL ∼ /PSNBM(, ) >FRPPRQS *O DPEF GPSN GPS *- BOE *-ƿ01+ UIF MJOL GVODUJPO XJMM CF FNCFEEFE GVODUJPO BMSFBEZ 4P UP ĕU UIF CBTJD NPEFM JODPSQPSBUJOH OP QSFEJDUPS W 3 DPEF  *ƾƾǑƾ ʆǦ *-ǯ )&01ǯ /"0-,+0" ʍ !,/!),$&1ǯ ƽ ǒ ǯƾǒƿǒǀǒǁǒǂǒǃǰ ǰǒ ǯƾǒƿǒǀǒǁǒǂǒǃǰ ʍ !+,/*ǯƽǒƾƽǰ ǰ ǒ !1ʅ! ǒ 01/1ʅ)&01ǯƾʅǦƿǒƿʅǦƾǒǀʅƽǒǁʅƾǒǂʅƿǒǃʅƿǑǂǰ ǰ -/" &0ǯ*ƾƾǑƾǰ Ordered logit
22. Ordered logit  (ǎǎǡǎ ʚǶ (+ǿ '$./ǿ - .+*). ʡ *-'*"$/ǿ +#$ Ǣ ǿǎǢǏǢǐǢǑǢǒǢǓȀ ȀǢ +#$ ʚǶ ǍǢ ǿǎǢǏǢǐǢǑǢǒǢǓȀ ʡ )*-(ǿǍǢǎǍȀ Ȁ Ǣ /ʙ Ǣ ./-/ʙ'$./ǿǎʙǶǏǢǏʙǶǎǢǐʙǍǢǑʙǎǢǒʙǏǢǓʙǏǡǒȀ Ȁ ćF +#$ JO UIF NPEFM DPEF JT B QMBDFIPMEFS GPS UIF MJOFBS NPEFM UIBU JODPSQPSBUFT QSFEJDUPS WBSJBCMFT :PVMM VTF JU TIPSUMZ *U JT B DPOTUBOU [FSP GPS OPX CFDBVTF POMZ UIF JOUFSDFQU QBSBN FUFST BSF PG JOUFSFTU ćF ./-/ WBMVFT GPS UIF JOUFSDFQUT BSF DIPTFO KVTU UP TUBSU UIFN JO UIF SJHIU PSEFS ćF FYBDU WBMVFT BSFOU JNQPSUBOU CVU UIFJS PSEFSJOH PO UIF MPHDVNVMBUJWF PEET TDBMF JT JNQPSUBOU ćFJS QPTUFSJPS EJTUSJCVUJPO JT BMTP PO UIF MPHDVNVMBUJWFPEET TDBMF 3 DPEF  +- $.ǿ(ǎǎǡǎȀ  ) / 1 Ǐǡǒʉ ǖǔǡǒʉ ǎ ǶǎǡǖǏ ǍǡǍǐ Ƕǎǡǖǔ ǶǎǡǕǓ Ǐ ǶǎǡǏǔ ǍǡǍǏ Ƕǎǡǐǎ ǶǎǡǏǏ ǐ ǶǍǡǔǏ ǍǡǍǏ ǶǍǡǔǓ ǶǍǡǓǕ Ǒ ǍǡǏǒ ǍǡǍǏ ǍǡǏǎ ǍǡǏǖ ǒ ǍǡǕǖ ǍǡǍǏ ǍǡǕǒ Ǎǡǖǐ Ǔ ǎǡǔǔ ǍǡǍǐ ǎǡǔǎ ǎǡǕǐ 4JODF UIFSF JT B MPU PG EBUB IFSF UIF QPTUFSJPS GPS FBDI JOUFSDFQU JT RVJUF QSFDJTFMZ FTUJNBUFE BT ZPV DBO TFF GSPN UIF UJOZ TUBOEBSE EFWJBUJPOT 5P HFU DVNVMBUJWF QSPCBCJMJUJFT CBDL TNBMM TBNQMF DPOUFYUT ZPVMM IBWF UP UIJOL NVDI IBSEFS BCPVU QSJPST $POTJEFS GPS FYBNQMF UIBU XF LOPX α < α CFGPSF XF FWFO TFF UIF EBUB *O DPEF GPSN GPS (+ BOE (+Ǐ./) UIF MJOL GVODUJPO XJMM CF FNCFEEFE JO UIF MJLFMJIPPE GVODUJPO BMSFBEZ ćJT NBLFT UIF DBMDVMBUJPOT NPSF FďDJFOU BOE QSFWFOUT IFBEBDIFT 4P UP ĕU UIF CBTJD NPEFM JODPSQPSBUJOH OP QSFEJDUPS WBSJBCMFT 3 DPEF  (ǎǎǡǎ ʚǶ (+ǿ '$./ǿ - .+*). ʡ *-'*"$/ǿ +#$ Ǣ ǿǎǢǏǢǐǢǑǢǒǢǓȀ ȀǢ +#$ ʚǶ ǍǢ ǿǎǢǏǢǐǢǑǢǒǢǓȀ ʡ )*-(ǿǍǢǎǍȀ Ȁ Ǣ /ʙ Ǣ ./-/ʙ'$./ǿǎʙǶǏǢǏʙǶǎǢǐʙǍǢǑʙǎǢǒʙǏǢǓʙǏǡǒȀ Ȁ ćF +#$ JO UIF NPEFM DPEF JT B QMBDFIPMEFS GPS UIF MJOFBS NPEFM UIBU JODPSQPSBUFT QSFEJDUPS WBSJBCMFT :PVMM VTF JU TIPSUMZ *U JT B DPOTUBOU [FSP GPS OPX CFDBVTF POMZ UIF JOUFSDFQU QBSBN FUFST BSF PG JOUFSFTU ćF ./-/ WBMVFT GPS UIF JOUFSDFQUT BSF DIPTFO KVTU UP TUBSU UIFN JO UIF SJHIU PSEFS ćF FYBDU WBMVFT BSFOU JNQPSUBOU CVU UIFJS PSEFSJOH PO UIF MPHDVNVMBUJWF PEET TDBMF JT JNQPSUBOU ćFJS QPTUFSJPS EJTUSJCVUJPO JT BMTP PO UIF MPHDVNVMBUJWFPEET TDBMF 3 DPEF  +- $.ǿ(ǎǎǡǎȀ  ) / 1 Ǐǡǒʉ ǖǔǡǒʉ ǎ ǶǎǡǖǏ ǍǡǍǐ Ƕǎǡǖǔ ǶǎǡǕǓ Ǐ ǶǎǡǏǔ ǍǡǍǏ Ƕǎǡǐǎ ǶǎǡǏǏ ǐ ǶǍǡǔǏ ǍǡǍǏ ǶǍǡǔǓ ǶǍǡǓǕ
23. Ordered logit in Stan ǍǡǎǏǕǐǍǍǒ ǍǡǏǎǖǕǐǖǕ ǍǡǐǏǔǓǖǑǕ ǍǡǒǓǎǓǐǎǎ ǍǡǔǍǕǕǓǍǖ ǍǡǕǒǑǐǔǕǓ "OE PG DPVSTF UIPTF BSF UIF TBNF BT UIF WBMVFT JO 0(Ǿ+-Ǿ& UIBU XF DPNQVUFE FBSMJFS #VU OPX XF BMTP IBWF B QPTUFSJPS EJTUSJCVUJPO BSPVOE UIFTF WBMVFT BOE XFSF SFBEZ UP BEE QSF EJDUPS WBSJBCMFT JO UIF OFYU TFDUJPO 5P ĕU UIF TBNF NPEFM VTJOH 4UBOT ).$ FOHJOF JU JT CFUUFS UP VTF BO FYQMJDJU WFDUPS PG JOUFSDFQU QBSBNFUFST 3 DPEF  ȕ )*/ /#/ / 2$/# )( Ǫ. Ǫ )*/ ''*2  $) /) ȕ .* 2$'' +.. +-0)  / '$./ (ǎǎǡǎ./) ʚǶ (+Ǐ./)ǿ '$./ǿ - .+*). ʡ *-'*"$/ǿ +#$ Ǣ 0/+*$)/. ȀǢ +#$ ʚǶ ǍǢ 0/+*$)/. ʡ )*-(ǿǍǢǎǍȀ Ȁ Ǣ /ʙ'$./ǿ- .+*). ʙɶ- .+*). ȀǢ ./-/ʙ'$./ǿ0/+*$)/.ʙǿǶǏǢǶǎǢǍǢǎǢǏǢǏǡǒȀȀ Ȁ ȕ )   +/#ʙǏ /* .#*2 1 /*- *! +-( / -. +- $.ǿ(ǎǎǡǎ./)Ǣ +/#ʙǏȀ  ) / 1 '*2 - Ǎǡǖǒ 0++ - Ǎǡǖǒ )Ǿ !! #/ 0/+*$)/.ȁǎȂ ǶǎǡǖǏ ǍǡǍǐ Ƕǎǡǖǔ ǶǎǡǕǓ ǑǕǕ ǎ 0/+*$)/.ȁǏȂ ǶǎǡǏǔ ǍǡǍǏ Ƕǎǡǐǎ ǶǎǡǏǏ ǓǏǓ ǎ 0/+*$)/.ȁǐȂ ǶǍǡǔǏ ǍǡǍǏ ǶǍǡǔǔ ǶǍǡǓǕ ǎǍǍǍ ǎ 0/+*$)/.ȁǑȂ ǍǡǏǒ ǍǡǍǏ ǍǡǏǍ ǍǡǏǖ ǎǍǍǍ ǎ 0/+*$)/.ȁǒȂ ǍǡǕǖ ǍǡǍǏ ǍǡǕǒ ǍǡǖǑ ǎǍǍǍ ǎ 0/+*$)/.ȁǓȂ ǎǡǔǔ ǍǡǍǐ ǎǡǔǏ ǎǡǕǏ Ǖǎǒ ǎ ćF JOEJWJEVBM 0/+*$)/. QBSBNFUFST DPSSFTQPOE UP FBDI αL GSPN FBSMJFS
24.  03%&3&% $"5&(03*$"- 065$0.&4  3 DPEF  '*"$./$ǿ* !ǿ(ǎǎǡǎȀȀ ǎ Ǐ ǐ Ǒ ǒ Ǔ ǍǡǎǏǕǐǍǍǒ ǍǡǏǎǖǕǐǖǕ ǍǡǐǏǔǓǖǑǕ ǍǡǒǓǎǓǐǎǎ ǍǡǔǍǕǕǓǍǖ ǍǡǕǒǑǐǔǕǓ "OE PG DPVSTF UIPTF BSF UIF TBNF BT UIF WBMVFT JO 0(Ǿ+-Ǿ& UIBU XF DPNQVUFE FBSMJFS #VU OPX XF BMTP IBWF B QPTUFSJPS EJTUSJCVUJPO BSPVOE UIFTF WBMVFT BOE XFSF SFBEZ UP BEE QSF EJDUPS WBSJBCMFT JO UIF OFYU TFDUJPO 5P ĕU UIF TBNF NPEFM VTJOH 4UBOT ).$ FOHJOF JU JT CFUUFS UP VTF BO FYQMJDJU WFDUPS PG JOUFSDFQU QBSBNFUFST 3 DPEF  ȕ )*/ /#/ / 2$/# )( Ǫ. Ǫ )*/ ''*2  $) /) ȕ .* 2$'' +.. +-0)  / '$./ (ǎǎǡǎ./) ʚǶ (+Ǐ./)ǿ SJHIU PSEFS ćF FYBDU WBMVFT BSFOU JNQPSUBOU CVU UIFJS PSEFSJOH PO UIF MPHDVNVMBUJWF PEET TDBMF JT JNQPSUBOU ćFJS QPTUFSJPS EJTUSJCVUJPO JT BMTP PO UIF MPHDVNVMBUJWFPEET TDBMF 3 DPEF  +- $.ǿ(ǎǎǡǎȀ  ) / 1 Ǐǡǒʉ ǖǔǡǒʉ ǎ ǶǎǡǖǏ ǍǡǍǐ Ƕǎǡǖǔ ǶǎǡǕǓ Ǐ ǶǎǡǏǔ ǍǡǍǏ Ƕǎǡǐǎ ǶǎǡǏǏ ǐ ǶǍǡǔǏ ǍǡǍǏ ǶǍǡǔǓ ǶǍǡǓǕ Ǒ ǍǡǏǒ ǍǡǍǏ ǍǡǏǎ ǍǡǏǖ ǒ ǍǡǕǖ ǍǡǍǏ ǍǡǕǒ Ǎǡǖǐ Ǔ ǎǡǔǔ ǍǡǍǐ ǎǡǔǎ ǎǡǕǐ 4JODF UIFSF JT B MPU PG EBUB IFSF UIF QPTUFSJPS GPS FBDI JOUFSDFQU JT RVJUF QSFDJTFMZ FTUJNBUFE BT ZPV DBO TFF GSPN UIF UJOZ TUBOEBSE EFWJBUJPOT 5P HFU DVNVMBUJWF QSPCBCJMJUJFT CBDL a7 missing, because known to be infinity (on logit scale)
25. 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 response cumulative proportion a1 a2 a3 a4 a5 a6 logistic( coef(m11.1) ) a1 a2 a3 a4 a5 a6 0.1282980 0.2198372 0.3276928 0.5616311 0.7088620 0.8543802
26. Adding predictor variables BSF UIF NPEFM GPS UIF BEEJUJPO PG QSFEJDUPS WBSJBCMFT UIBU PCFZ UIF PSEFSFE DPOTUS PVUDPNFT JODMVEF QSFEJDUPS WBSJBCMFT XF EFĕOF UIF MPHDVNVMBUJWFPEET PG FBDI SFTQPOTF PG JUT JOUFSDFQU αL BOE B UZQJDBM MJOFBS NPEFM 4VQQPTF GPS FYBNQMF XF XBOU UP DUPS Y UP UIF NPEFM 8FMM EP UIJT CZ EFĕOJOH B MJOFBS NPEFM φJ = βYJ  ćFO F UJWF MPHJU CFDPNFT MPH 1S(ZJ ≤ L)  − 1S(ZJ ≤ L) = αL − φJ φJ = βYJ N BVUPNBUJDBMMZ FOTVSFT UIF DPSSFDU PSEFSJOH PG UIF PVUDPNF WBMVFT XIJMF TUJMM N IF MJLFMJIPPE PG FBDI JOEJWJEVBM WBMVF BT UIF QSFEJDUPS YJ DIBOHFT WBMVF 8IZ JT NPEFM φ TVCUSBDUFE GSPN FBDI JOUFSDFQU #FDBVTF JG XF EFDSFBTF UIF MPHDVNVMB $PNQBSF UIFTF UP UIF MJLFMJIPPET KVTU BCPWF BOE OPUJDF UIBU UIF WBMVFT PO UIF MFę I NJOJTIFE XIJMF UIF WBMVFT PO UIF SJHIU IBWF JODSFBTFE ćF FYQFDUFE WBMVF JT OPX .0(ǿ +&ȉǿǎǣǔȀ Ȁ ȁǎȂ ǑǡǔǏǖǔǑ "OE UIBUT XIZ XF TVCUSBDU φ UIF MJOFBS NPEFM βYJ GSPN FBDI JOUFSDFQU SBUIFS UIBO ćJT XBZ B QPTJUJWF β WBMVF JOEJDBUFT UIBU BO JODSFBTF JO UIF QSFEJDUPS WBSJBCMF Y SFTVM JODSFBTF JO UIF BWFSBHF SFTQPOTF /PX XF DBO UVSO CBDL UP PVS iUSPMMFZw EBUB BOE JODMVEF QSFEJDUPS WBSJBCMFT UP I QMBJO WBSJBUJPO JO SFTQPOTFT ćF QSFEJDUPS WBSJBCMFT PG JOUFSFTU BSF HPJOH UP CF /$* / )/$*) BOE *)// FBDI B EVNNZ WBSJBCMF DPSSFTQPOEJOH UP FBDI QSJODJQMF P FBSMJFS ćF MPHDVNVMBUJWF PEET PG FBDI SFTQPOTF L XJMM OPX CF MPH 1S(ZJ ≤ L)  − 1S(ZJ ≤ L) = αL − φJ φJ = β" "J + β* *J + β$ $J XIFSF "J JOEJDBUFT UIF WBMVF PG /$*) PO SPX J *J JOEJDBUFT UIF WBMVF PG $)/ )/$*) P BOE $J JOEJDBUFT UIF WBMVF PG *)// PO SPX J 8IBU XFWF EPOF IFSF JT EFĕOF UIF MP PG FBDI QPTTJCMF SFTQPOTF UP CF BO BEEJUJWF NPEFM PG UIF GFBUVSFT PG UIF TUPSZ DPSSFTQ UP FBDI SFTQPOTF In general: Trolley data:
27. Adding predictor variables /PX XF DBO UVSO CBDL UP PVS iUSPMMFZw EBUB BOE JODMVEF QSFEJDUPS WBSJBCMFT UP IFMQ F O WBSJBUJPO JO SFTQPOTFT ćF QSFEJDUPS WBSJBCMFT PG JOUFSFTU BSF HPJOH UP CF /$*) $ )/$*) BOE *)// FBDI B EVNNZ WBSJBCMF DPSSFTQPOEJOH UP FBDI QSJODJQMF PVUMJO JFS ćF MPHDVNVMBUJWF PEET PG FBDI SFTQPOTF L XJMM OPX CF MPH 1S(ZJ ≤ L)  − 1S(ZJ ≤ L) = αL − φJ φJ = β" "J + β* *J + β$ $J FSF "J JOEJDBUFT UIF WBMVF PG /$*) PO SPX J *J JOEJDBUFT UIF WBMVF PG $)/ )/$*) PO SPX E $J JOEJDBUFT UIF WBMVF PG *)// PO SPX J 8IBU XFWF EPOF IFSF JT EFĕOF UIF MPHPE BDI QPTTJCMF SFTQPOTF UP CF BO BEEJUJWF NPEFM PG UIF GFBUVSFT PG UIF TUPSZ DPSSFTQPOEJ BDI SFTQPOTF :PV ĕU UIJT NPEFM KVTU BT ZPVE FYQFDU CZ BEEJOH UIF TMPQFT BOE QSFEJDUPS WBSJBCMFT +#$ QBSBNFUFS JOTJEF *-'*"$/ )FSFT B XPSLJOH NPEFM ǡǏ ʚǶ (+ǿ '$./ǿ - .+*). ʡ *-'*"$/ǿ +#$ Ǣ ǿǎǢǏǢǐǢǑǢǒǢǓȀ Ȁ Ǣ +#$ ʚǶ ȉ/$*) ʔ  ȉ$)/ )/$*) ʔ ȉ*)//Ǣ ǿǢ ǢȀ ʡ )*-(ǿǍǢǎǍȀǢ / )/$*) BOE *)// FBDI B EVNNZ WBSJBCMF DPSSFTQPOEJOH UP FBDI QSJODJQMF PVUMJOFE FBSMJFS ćF MPHDVNVMBUJWF PEET PG FBDI SFTQPOTF L XJMM OPX CF MPH 1S(ZJ ≤ L)  − 1S(ZJ ≤ L) = αL − φJ φJ = β" "J + β* *J + β$ $J XIFSF "J JOEJDBUFT UIF WBMVF PG /$*) PO SPX J *J JOEJDBUFT UIF WBMVF PG $)/ )/$*) PO SPX J BOE $J JOEJDBUFT UIF WBMVF PG *)// PO SPX J 8IBU XFWF EPOF IFSF JT EFĕOF UIF MPHPEET PG FBDI QPTTJCMF SFTQPOTF UP CF BO BEEJUJWF NPEFM PG UIF GFBUVSFT PG UIF TUPSZ DPSSFTQPOEJOH UP FBDI SFTQPOTF :PV ĕU UIJT NPEFM KVTU BT ZPVE FYQFDU CZ BEEJOH UIF TMPQFT BOE QSFEJDUPS WBSJBCMFT UP UIF +#$ QBSBNFUFS JOTJEF *-'*"$/ )FSFT B XPSLJOH NPEFM 3 DPEF  (ǎǎǡǏ ʚǶ (+ǿ '$./ǿ - .+*). ʡ *-'*"$/ǿ +#$ Ǣ ǿǎǢǏǢǐǢǑǢǒǢǓȀ Ȁ Ǣ +#$ ʚǶ ȉ/$*) ʔ  ȉ$)/ )/$*) ʔ ȉ*)//Ǣ ǿǢ ǢȀ ʡ )*-(ǿǍǢǎǍȀǢ  03%&3&% $"5&(03*$"- 065$0.&4  ǿǎǢǏǢǐǢǑǢǒǢǓȀ ʡ )*-(ǿǍǢǎǍȀ Ȁ Ǣ /ʙ Ǣ ./-/ʙ'$./ǿǎʙǶǎǡǖǢǏʙǶǎǡǏǢǐʙǶǍǡǔǢǑʙǍǡǏǢǒʙǍǡǖǢǓʙǎǡǕȀ Ȁ ćF QBSBNFUFS +#$ OPX DPOUBJOT UIF BEEJUJWF GVODUJPO XJUI TMPQF QBSBNFUFST BOE QSFEJDUPS WBSJBCMFT /PUJDF UIBU UIFSF JT OP MJOL GVODUJPO BSPVOE +#$ CFDBVTF UIF MJOL JT SFBMMZ JOTJEF *-'*"$/ BMSFBEZ IFODF iMPHJUw JO JUT OBNF  /PUJDF BMTP UIBU *WF BEPQUFE UIF BQQSPYJNBUF ."1 FTUJNBUFT GSPN UIF QSFWJPVT NPEFM (ǎǎǡǎ BT TUBSUJOH WBMVFT GPS UIF JOUFSDFQUT ćJT IFMQT (+ ĕOE UIF OFX ."1 FTUJNBUFT NPSF RVJDLMZ
28. • Three models: • m11.1: intercepts only • m11.2: main effects of action, intention, contact • m11.3: interact action*intention, contact*intention • cannot interact action*contact, because “contact” is a special kind of action here Adding predictor variables
29. SBNFUFST  BOE   /PX MFUT DPNQBSF UIFTF UISFF NPEFMT :PV DBO VTF ,"#1 UP HFU B RVJDL DPNQBSJTPO PG FTUJNBUFT 3 DPEF  ,"#1ǯ*ƾƾǑƾǒ*ƾƾǑƿǒ*ƾƾǑǀǰ *ƾƾǑƾ *ƾƾǑƿ *ƾƾǑǀ ƾ ǦƾǑdžƿ ǦƿǑDžǁ ǦƿǑǃǀ ƿ ǦƾǑƿDŽ ǦƿǑƾǃ ǦƾǑdžǁ ǀ ǦƽǑDŽƿ ǦƾǑǂDŽ ǦƾǑǀǁ ǁ ƽǑƿǂ ǦƽǑǂǂ ǦƽǑǀƾ ǂ ƽǑDždž ƽǑƾƿ ƽǑǀǃ ǃ ƾǑDŽDŽ ƾǑƽƿ ƾǑƿDŽ   ǦƽǑDŽƾ ǦƽǑǁDŽ   ǦƽǑDŽƿ ǦƽǑƿDž   ǦƽǑdžǃ ǦƽǑǀǀ    ǦƽǑǁǂ    ǦƾǑƿDŽ +,0 dždžǀƽ dždžǀƽ dždžǀƽ 8IBUFWFS EP UIFTF FTUJNBUFT NFBO ćF ĕSTU TJY SPXT GSPN ƾ UP ǃ BSF KVTU UIF α JOUFS DFQUT POF GPS FBDI WBMVF CFMPX UIF NBYJNVN PG iw ćFTF SFBMMZ DBOU CF JOUFSQSFUFE PO UIFJS PXO VOMFTT ZPV BSF WFSZ VTFE UP SFBEJOH MPHPEET WBMVFT ćF OFYU  SPXT GSPN  UP  BSF UIF WBSJPVT TMPQF QBSBNFUFST UISFF NBJO FČFDUT BOE UXP JOUFSBDUJPOT ćFTF BSF JOUFSQSFUBCMF PO UIFJS PXO UP B MJNJUFE FYUFOU *U NBLFT TFOTF UP BTL ĕSTU JG UIFZ BSF WFSZ GBS GSPN [FSP :PV DBO DIFDL UIF TUBOEBSE FSSPST BOE  DPOĕEFODF JOUFSWBMT XJUI -/" &0 BOE WFSJGZ UIBU BMM PG UIF TMPQF FTUJNBUFT BSF RVJUF SFMJBCMZ OFHBUJWF 4FDPOE BMM PG UIF TMPQFT BSF OFHBUJWF XIJDI JNQMJFT UIBU FBDI GBDUPSJOUFSBDUJPO SFEVDFT UIF BWFSBHF SFTQPOTF *ODMVEJOH BDUJPO JOUFOUJPO PS DPOUBDU JO B TUPSZ MFBET QFPQMF UP KVEHF JU BT MFTT NPSBMMZ QFSNJTTJCMF #VU CZ IPX NVDI 3FNFNCFS UIFTF QBSBNFUFST intercepts } As usual, can’t easily interpret from coefficients alone    ǶǎǡǏǔ )*. ǖǖǐǍ ǖǖǐǍ ǖǖǐǍ 8IBUFWFS EP UIFTF FTUJNBUFT NFBO ćF ĕSTU TJY SPXT GSPN ǎ UP Ǔ BSF KVTU UIF α JOUFS DFQUT POF GPS FBDI WBMVF CFMPX UIF NBYJNVN PG iw ćFTF SFBMMZ DBOU CF JOUFSQSFUFE PO UIFJS PXO VOMFTT ZPV BSF WFSZ VTFE UP SFBEJOH MPHPEET WBMVFT #VU UIFZ EP EFĕOF UIF SFM BUJWF GSFRVFODJFT PG UIF PVUDPNFT XIFO BMM QSFEJDUPS WBSJBCMFT BSF TFU UP [FSP 4P UIFZ BSF iJOUFSDFQUTw BT JO TJNQMFS NPEFMT ćF OFYU  SPXT GSPN  UP  BSF UIF WBSJPVT TMPQF QBSBNFUFST UISFF NBJO FČFDUT BOE UXP JOUFSBDUJPOT ćFTF BSF JOUFSQSFUBCMF PO UIFJS PXO UP B MJNJUFE FYUFOU *U NBLFT TFOTF UP BTL ĕSTU JG UIFZ BSF WFSZ GBS GSPN [FSP :PV DBO DIFDL UIF TUBOEBSE FSSPST BOE  DPOĕEFODF JOUFSWBMT XJUI +- $. BOE WFSJGZ UIBU BMM PG UIF TMPQF FTUJNBUFT BSF RVJUF SFMJBCMZ OFHBUJWF 4FDPOE BMM PG UIF TMPQFT BSF OFHBUJWF XIJDI JNQMJFT UIBU FBDI GBDUPSJOUFSBDUJPO SFEVDFT UIF BWFSBHF SFTQPOTF *ODMVEJOH BDUJPO JOUFOUJPO PS DPOUBDU JO B TUPSZ MFBET QFPQMF UP KVEHF JU BT MFTT NPSBMMZ QFSNJTTJCMF #VU CZ IPX NVDI 3FNFNCFS UIFTF QBSBNFUFST BSF QBSU PG B GVODUJPO EFĕOJOH DVNVMBUJWF MPHPEET TP UIFZ DBO CF JOUFSQSFUFE BT DIBOHFT JO DVNVMBUJWF MPHPEET #VU VOMFTT ZPV BSF WFSZ DPNGPSUBCMF UIJOLJOH BCPVU MPHPEET BOE DVNVMBUJWF QSPCBCJMJUZ EJTUSJCVUJPOT UIBU EPFTOU IFMQ ZPV NVDI *U BMTP EPFTOU IFMQ UIBU UIJT DIBOHF BQQMJFT UP UIF DVNVMBUJWF MPHPEET PG FWFSZ WBMVF PG UIF SFTQPOTF WBSJBCMF BTJEF GSPN UIF NBYJNVN POF XIJDI JT ĕYFE BU DVNVMBUJWF MPHPEET ∞  4P XIBU UP EP 'JSTU MFUT DPNQBSF UIFTF NPEFMT VTJOH 8"*$ 3 DPEF  *(+- ǿ (ǎǎǡǎ Ǣ (ǎǎǡǏ Ǣ (ǎǎǡǐ Ǣ - !- .#ʙǍǡǎ Ȁ   +    2 $"#/   (ǎǎǡǐ ǐǓǖǏǖǡǑ ǎǎǡǏ ǍǡǍ ǎ ǕǎǡǏǖ  (ǎǎǡǏ ǐǔǍǖǍǡǐ ǖǡǏ ǎǓǍǡǕ Ǎ ǔǓǡǏǏ ǏǒǡǕǍ (ǎǎǡǎ ǐǔǕǒǑǡǔ Ǔǡǎ ǖǏǒǡǏ Ǎ ǒǔǡǔǎ ǓǏǡǓǖ /PX TJODF NPEFM (ǎǎǡǐ BCTPMVUFMZ EPNJOBUFT CBTFE VQPO 8"*$‰JU HFUT WFSZ OFBSMZ 
30. Plotting ordered logits • Oh, bother: Posterior prediction a vector of probabilities, one for each level of outcome • How to plot this?
31. 0 1 0.0 0.5 1.0 intention probability action=0, contact=0 0 0.0 0.5 1.0 inte probability action=1 1.0 action=0, contact=1 'JHVSF  1 PSEFSFE DBUFHP 1 2 3 4 5 6 7
32.   .0/45&34 "/% .*9563&4 0 1 0.0 0.5 1.0 intention probability action=0, contact=0 1 2 3 4 5 6 7 0 1 0.0 0.5 1.0 intention probability action=1, contact=0 1 2 3 4 5 6 7 0 1 0.0 0.5 1.0 intention probability action=0, contact=1 1 2 3 4 5 6 7 'ĶĴłĿIJ ƉƉƋ 1PTUFSJPS QSFEJDUJPOT PG UIF PSEFSFE DBUFHPSJDBM NPEFM XJUI JOUFSBDUJPOT (ǎǎǡǐ &BDI QMPU TIPXT IPX UIF EJTUSJCVUJPO PG QSFEJDUFE SF TQPOTFT WBSJFT CZ $)/ )/$*) -Fę &ČFDU PG $)/ )/$*) XIFO /$*) BOE *)// BSF CPUI [FSP ćF PUIFS UXP QMPUT FBDI DIBOHF FJUIFS /$*) PS *)// UP POF
33. 1 2 3 4 5 6 7 0 50 150 250 350 response Frequency intention=1, contact=1 ordered logit binomial 1 2 3 4 5 6 7 0 50 150 250 350 response Frequency intention=1, contact=1 data post predict
34. Ordered logit “GLM” • MAP estimation can be hard; choose good starting values. See notes for details. • Automated MAP fit with polr. See notes for example. Beware flat priors. • Stan handles these models fine. Will be slower than other outcome types. • Also ordered probit; uses cumulative normal link