L12 Statistical Rethinking Winter 2019

Lecture 12 of the Dec 2018 through March 2019 edition of Statistical Rethinking. Covers Chapter 11 and 11, generalized linear models, binomial, Poisson GLMs, survival analysis.

Transcript

1. God Spiked The Integers Statistical Rethinking Winter 2019 Lecture 12 / Week 6
2. https://ncase.me/fireflies/
5. Logit link priors UIF QSJPS QSPCBCJMJUZ PO UIF PVUDPNF TDBMF JT SBUIFS ĘBU ćJT JT QSPCBCMZ JT PQUJNBM TJODF QSPCBCJMJUJFT OFBS UIF DFOUFS BSF NPSF QMBVTJCMF #VU UIJ EFGBVMU QSJPST NPTU QFPQMF VTF NPTU PG UIF UJNF 8FMM VTF JU /PX XF OFFE UP EFUFSNJOF B QSJPS GPS UIF USFBUNFOU FČFDUT UIF β QBS EFGBVMU UP VTJOH UIF TBNF /PSNBM   QSJPS GPS UIF USFBUNFOU FČFDUT UIBU UIFZ BSF BMTP KVTU JOUFSDFQUT POF JOUFSDFQU GPS FBDI USFBUNFOU #VU XFJSEOFTT PG DPOWFOUJPOBMMZ ĘBU QSJPST MFUT TFF XIBU /PSNBM   MPP NPEFM 3 DPEF  (ǎǎǡǏ ʚǶ ,0+ǿ '$./ǿ +0'' Ǿ' !/ ʡ $)*(ǿ ǎ Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ  ʔ ȁ/- /( )/Ȃ Ǣ  ʡ )*-(ǿ Ǎ Ǣ ǎǡǒ ȀǢ ȁ/- /( )/Ȃ ʡ )*-(ǿ Ǎ Ǣ ǎǍ Ȁ Ȁ Ǣ /ʙ Ȁ • What about treatments?
6. 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 prior prob pull left Density a ~ dnorm(0,10) a ~ dnorm(0,1.5) 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 12 14 prior diff between treatments Density b ~ dnorm(0,10) b ~ dnorm(0,0.5) 'ĶĴłĿIJ ƉƉƋ 1SJPS QSFEJDUJWF TJNVMBUJPOT GPS UIF NPTU CBTJD MPHJTUJD SFHSFT TJPO #MBDL EFOTJUZ " ĘBU /PSNBM   QSJPS PO UIF JOUFSDFQU QSPEVDFT B WFSZ OPOĘBU QSJPS EJTUSJCVUJPO PO UIF PVUDPNF TDBMF #MVF EFOTJUZ " NPSF DPODFOUSBUFE /PSNBM   QSJPS QSPEVDFT TPNFUIJOH NPSF SFBTPOBCMF Figure 11.3
7. ȕ +-/$' . $) ǎǎǶ$( ).$*)' .+ (ǎǎǡǑ ʚǶ 0'(ǿ '$./ǿ +0'' Ǿ' !/ ʡ $)*(ǿ ǎ Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ ȁ/*-Ȃ ʔ ȁ/- /( )/Ȃ Ǣ ȁ/*-Ȃ ʡ )*-(ǿ Ǎ Ǣ ǎǡǒ ȀǢ ȁ/- /( )/Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǒ Ȁ Ȁ Ǣ /ʙ/Ǿ'$./ Ǣ #$).ʙǑ Ȁ +- $.ǿ (ǎǎǡǑ Ǣ  +/#ʙǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ )Ǿ !! #/ ȁǎȂ ǶǍǡǑǑ ǍǡǐǑ ǶǍǡǖǔ Ǎǡǎǎ ǔǐǓ ǎ ȁǏȂ ǐǡǖǍ Ǎǡǔǔ ǏǡǔǕ ǒǡǏǏ ǖǏǎ ǎ ȁǐȂ ǶǍǡǔǒ ǍǡǐǑ ǶǎǡǏǖ ǶǍǡǏǍ ǕǕǓ ǎ ȁǑȂ ǶǍǡǔǑ ǍǡǐǑ ǶǎǡǏǕ ǶǍǡǎǖ ǔǔǍ ǎ ȁǒȂ ǶǍǡǑǑ ǍǡǐǑ ǶǍǡǖǕ ǍǡǍǕ ǕǐǏ ǎ ȁǓȂ ǍǡǑǕ ǍǡǐǑ ǶǍǡǍǕ ǎǡǍǏ ǕǒǑ ǎ ȁǔȂ ǎǡǖǓ ǍǡǑǎ ǎǡǏǖ ǏǡǓǎ ǕǑǔ ǎ ȁǎȂ ǶǍǡǍǒ ǍǡǏǖ ǶǍǡǒǎ ǍǡǑǏ ǔǕǎ ǎ ȁǏȂ ǍǡǑǕ ǍǡǏǖ ǍǡǍǐ ǍǡǖǑ Ǔǒǔ ǎ ȁǐȂ ǶǍǡǐǖ ǍǡǏǕ ǶǍǡǕǐ ǍǡǍǔ ǓǓǖ ǎ ȁǑȂ ǍǡǐǓ ǍǡǏǖ ǶǍǡǎǎ ǍǡǕǎ ǔǐǏ ǎ Chimpanzees { Treatments RN LN RP LP
8. Individual differences ȁǑȂ ǍǡǐǓ ǍǡǏǖ ǶǍǡǎǎ ǍǡǕǎ ǔǐǏ ǎ ćJT JT UIF HVUT PG UIF UJEF QSFEJDUJPO FOHJOF 8FMM OFFE UP EP B MJUUMF XP ĕSTU  QBSBNFUFST BSF UIF JOUFSDFQUT VOJRVF UP FBDI DIJNQBO[FF &BDI P UFOEFODZ PG FBDI JOEJWJEVBM UP QVMM UIF MFę MFWFS -FUT MPPL BU UIFTF PO 3 DPEF  +*./ ʚǶ 3/-/ǡ.(+' .ǿ(ǎǎǡǑȀ +Ǿ' !/ ʚǶ $)1Ǿ'*"$/ǿ +*./ɶ Ȁ +'*/ǿ +- $.ǿ .ǡ/ǡ!-( ǿ+Ǿ' !/Ȁ Ȁ Ǣ 3'$(ʙǿǍǢǎȀ Ȁ V7 V6 V5 V4 V3 V2 V1 0.0 0.2 0.4 0.6 0.8 Value &BDI SPX JT B DIJNQBO[FF UIF OVNCFST DPSSFTQPOEJOH UP UIF WBMVFT JO JOEJWJEVBMT‰OVNCFST    BOE ‰TIPXB QSFGFSFODF GPSUIFSJHIU MFWF JT UIF HVUT PG UIF UJEF QSFEJDUJPO FOHJOF 8FMM OFFE UP EP B MJUUMF XPSL UP JOUFSQSFU JU  QBSBNFUFST BSF UIF JOUFSDFQUT VOJRVF UP FBDI DIJNQBO[FF &BDI PG UIFTF FYQSFTTFT FODZ PG FBDI JOEJWJEVBM UP QVMM UIF MFę MFWFS -FUT MPPL BU UIFTF PO UIF PVUDPNF TDBM ʚǶ 3/-/ǡ.(+' .ǿ(ǎǎǡǑȀ !/ ʚǶ $)1Ǿ'*"$/ǿ +*./ɶ Ȁ ǿ +- $.ǿ .ǡ/ǡ!-( ǿ+Ǿ' !/Ȁ Ȁ Ǣ 3'$(ʙǿǍǢǎȀ Ȁ V7 V6 V5 V4 V3 V2 V1 0.0 0.2 0.4 0.6 0.8 1.0 Value SPX JT B DIJNQBO[FF UIF OVNCFST DPSSFTQPOEJOH UP UIF WBMVFT JO /*- 'PVS PG WJEVBMT‰OVNCFST    BOE ‰TIPXB QSFGFSFODF GPSUIFSJHIU MFWFS 5XPJOEJWJEVB “Lefty”
9. Treatments UFOEFODJFT ćJT JT FYBDUMZ UIF LJOE PG FČFDU UIBU NBLFT QVSF FYQFSJNFOUT EJďD CFIBWJPSBM TDJFODFT )BWJOH SFQFBU NFBTVSFNFOUT MJLF JO UIJT FYQFSJNFOU BOE N UIFN JT WFSZ VTFGVM /PX MFUT DPOTJEFS UIF USFBUNFOU FČFDUT IPQFGVMMZ FTUJNBUFE NPSF QSFDJTFMZ C NPEFM DPVME TVCUSBDU PVU UIF IBOEFEOFTT WBSJBUJPO BNPOH BDUPST 0O UIF MPHJU TD '. ʚǶ ǿǫȅǫǢǫ ȅǫǢǫȅǫǢǫ ȅǫȀ +'*/ǿ +- $.ǿ (ǎǎǡǑ Ǣ  +/#ʙǏ Ǣ +-.ʙǫǫ Ȁ Ǣ ' '.ʙ'. Ȁ L/P R/P L/N R/N -0.5 0.0 0.5 1.0 Value *WF BEEFE USFBUNFOU MBCFMT JO QMBDF PG UIF QBSBNFUFS OBNFT -/ NFBOT iQSPTPDJ OP QBSUOFSw 31 NFBOT wQSPTPDJBM PO SJHIU  QBSUOFSw 5P VOEFSTUBOE UIFTF EJTUSJCV IFMQ UP DPOTJEFS PVS FYQFDUBUJPOT 8IBU XF BSF MPPLJOH GPS JT FWJEFODF UIBU UIF DIJ DIPPTF UIF QSPTPDJBM PQUJPO NPSF XIFO B QBSUOFS JT QSFTFOU ćJT JNQMJFT DPNQ ĕSTU SPX XJUI UIF UIJSE SPX BOE UIF TFDPOE SPX XJUI UIF GPVSUI SPX :PV DBO QSP BMSFBEZ UIBU UIFSF JTOU NVDI FWJEFODF PG QSPTPDJBM JOUFOUJPO JO UIFTF EBUB #VU MFU OEFODJFT ćJT JT FYBDUMZ UIF LJOE PG FČFDU UIBU NBLFT QVSF FYQFSJNFOUT EJďDVMU JO U IBWJPSBM TDJFODFT )BWJOH SFQFBU NFBTVSFNFOUT MJLF JO UIJT FYQFSJNFOU BOE NFBTVSJ FN JT WFSZ VTFGVM /PX MFUT DPOTJEFS UIF USFBUNFOU FČFDUT IPQFGVMMZ FTUJNBUFE NPSF QSFDJTFMZ CFDBVTF U PEFM DPVME TVCUSBDU PVU UIF IBOEFEOFTT WBSJBUJPO BNPOH BDUPST 0O UIF MPHJU TDBMF . ʚǶ ǿǫȅǫǢǫ ȅǫǢǫȅǫǢǫ ȅǫȀ */ǿ +- $.ǿ (ǎǎǡǑ Ǣ  +/#ʙǏ Ǣ +-.ʙǫǫ Ȁ Ǣ ' '.ʙ'. Ȁ L/P R/P L/N R/N -0.5 0.0 0.5 1.0 Value F BEEFE USFBUNFOU MBCFMT JO QMBDF PG UIF QBSBNFUFS OBNFT -/ NFBOT iQSPTPDJBM PO MF QBSUOFSw 31 NFBOT wQSPTPDJBM PO SJHIU  QBSUOFSw 5P VOEFSTUBOE UIFTF EJTUSJCVUJPOT J Q UP DPOTJEFS PVS FYQFDUBUJPOT 8IBU XF BSF MPPLJOH GPS JT FWJEFODF UIBU UIF DIJNQBO[F PPTF UIF QSPTPDJBM PQUJPO NPSF XIFO B QBSUOFS JT QSFTFOU ćJT JNQMJFT DPNQBSJOH U
10. proportion left lever 0 0.5 1 actor 1 actor 2 actor 3 actor 4 actor 5 actor 6 actor 7 R/N L/N R/P L/P observed proportions proportion left lever 0 0.5 1 actor 1 actor 2 actor 3 actor 4 actor 5 actor 6 actor 7 posterior predictions Figure 11.4
11. proportion left lever 0 0.5 1 actor 1 actor 2 actor 3 actor 4 actor 5 actor 6 actor 7 R/N L/N R/P L/P observed proportions proportion left lever 0 0.5 1 actor 1 actor 2 actor 3 actor 4 actor 5 actor 6 actor 7 posterior predictions Figure 11.4
12. Comparing no-interaction .$ ʙ ɶ.$ Ǣ *) ʙ ɶ*) Ȁ (ǎǎǡǒ ʚǶ 0'(ǿ '$./ǿ +0'' Ǿ' !/ ʡ $)*(ǿ ǎ Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ ȁ/*-Ȃ ʔ .ȁ.$ Ȃ ʔ ȁ*)Ȃ Ǣ ȁ/*-Ȃ ʡ )*-(ǿ Ǎ Ǣ ǎǡǒ ȀǢ .ȁ.$ Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǒ ȀǢ ȁ*)Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǒ Ȁ Ȁ Ǣ /ʙ/Ǿ'$./Ǐ Ǣ #$).ʙǑ Ǣ '*"Ǿ'$&ʙ Ȁ (P CBDL UP NPEFM (ǎǎǡǑ BOE BEE '*"Ǿ'$&ʙ ćFO XF DBO DPNQ IFSF VTJOH -00*4 3 DPEF  *(+- ǿ (ǎǎǡǒ Ǣ (ǎǎǡǑ Ǣ !0)ʙ  Ȁ  +    2 $"#/   (ǎǎǡǒ ǒǐǎǡǏ ǔǡǖ ǍǡǍ ǍǡǓǓ ǎǖǡǎǔ  (ǎǎǡǑ ǒǐǏǡǓ Ǖǡǔ ǎǡǑ ǍǡǐǑ ǎǖǡǍǎ ǎǡǏǕ 8"*$ QSPEVDFT JEFOUJDBM SFTVMUT "T XF HVFTTFE UIF NPEFM XJUIPVU UIF OP XPSTF JO FYQFDUFE QSFEJDUJWF BDDVSBDZ UIBO UIF NPEFM XJUI JU :P 3 DPEF  /Ǿ'$./Ǐ ʚǶ '$./ǿ +0'' Ǿ' !/ ʙ ɶ+0'' Ǿ' !/Ǣ /*- ʙ ɶ/*-Ǣ .$ ʙ ɶ.$ Ǣ *) ʙ ɶ*) Ȁ (ǎǎǡǒ ʚǶ 0'(ǿ '$./ǿ +0'' Ǿ' !/ ʡ $)*(ǿ ǎ Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ ȁ/*-Ȃ ʔ .ȁ.$ Ȃ ʔ ȁ*)Ȃ Ǣ ȁ/*-Ȃ ʡ )*-(ǿ Ǎ Ǣ ǎǡǒ ȀǢ .ȁ.$ Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǒ ȀǢ ȁ*)Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǒ Ȁ Ȁ Ǣ /ʙ/Ǿ'$./Ǐ Ǣ #$).ʙǑ Ǣ '*"Ǿ'$&ʙ Ȁ (P CBDL UP NPEFM (ǎǎǡǑ BOE BEE '*"Ǿ'$&ʙ ćFO XF DBO DPNQBSF UIF U IFSF VTJOH -00*4 3 DPEF  *(+- ǿ (ǎǎǡǒ Ǣ (ǎǎǡǑ Ǣ !0)ʙ  Ȁ  +    2 $"#/   (ǎǎǡǒ ǒǐǎǡǏ ǔǡǖ ǍǡǍ ǍǡǓǓ ǎǖǡǎǔ  (ǎǎǡǑ ǒǐǏǡǓ Ǖǡǔ ǎǡǑ ǍǡǐǑ ǎǖǡǍǎ ǎǡǏǕ
13. Relative and absolute effects • Parameters on relative effect scale • Predictions on absolute effect scale • Proportional odds: Relative effect measure • Good for scaring people, getting published • Not so good for public health, scientific progress • But needed for causal inference  3FMBUJWF TIBSL BOE BCTPMVUF QFOHVJO *O UIF BOBMZTJT BCPWF * NPTUMZ GP DIBOHFT JO QSFEJDUJPOT PO UIF PVUDPNF TDBMF‰IPX NVDI EJČFSFODF EPFT UIF USFBUN JO UIF QSPCBCJMJUZ PG QVMMJOH B MFWFS ćJT WJFX PG QPTUFSJPS QSFEJDUJPO GPDVTFT PO Į IJijijIJİŁŀ UIF EJČFSFODF B DPVOUFSGBDUVBM DIBOHF JO B WBSJBCMF NJHIU NBLF PO BO TDBMF PG NFBTVSFNFOU MJLF UIF QSPCBCJMJUZ PG BO FWFOU *U JT NPSF DPNNPO UP TFF MPHJTUJD SFHSFTTJPOT JOUFSQSFUFE UISPVHI ĿIJĹĮŁĶŃIJ 3FMBUJWF FČFDUT BSF QSPQPSUJPOBM DIBOHFT JO UIF PEET PG BO PVUDPNF *G XF DIBOHF BOE TBZ UIF PEET PG BO PVUDPNF EPVCMF UIFO XF BSF EJTDVTTJOH SFMBUJWF FČFDUT :PV MBUF UIFTF ĽĿļĽļĿŁĶļĻĮĹ ļııŀ SFMBUJWF FČFDU TJ[FT CZ TJNQMZ FYQPOFOUJBUJOH UIF Q PG JOUFSFTU 'PS FYBNQMF UP DBMDVMBUF UIF QSPQPSUJPOBM PEET PG TXJUDIJOH GSPN USFBU USFBUNFOU  BEEJOH B QBSUOFS  3 DPEF  +*./ ʚǶ 3/-/ǡ.(+' .ǿ(ǎǎǡǑȀ ( )ǿ 3+ǿ+*./ɶȁǢǑȂǶ+*./ɶȁǢǏȂȀ Ȁ ȁǎȂ ǍǡǖǏǍǓǑǔǖ 0O BWFSBHF UIF TXJUDI NVMUJQMFT UIF PEET PG QVMMJOH UIF MFę MFWFS CZ  BO  S JO PEET ćJT JT XIBU JT NFBOU CZ QSPQPSUJPOBM PEET ćF OFX PEET BSF DBMDVMBUFE UIF PME PEET BOE NVMUJQMZJOH UIFN CZ UIF QSPQPSUJPOBM PEET XIJDI JT  JO UIJT ćF SJTL PG GPDVTJOH PO SFMBUJWF FČFDUT TVDI BT QSPQPSUJPOBM PEET JT UIBU UI FOPVHI UP UFMM VT XIFUIFS B WBSJBCMF JT JNQPSUBOU PS OPU *G UIF PUIFS QBSBNFUFST JO U NBLF UIF PVUDPNF WFSZ VOMJLFMZ UIFO FWFO B MBSHF QSPQPSUJPOBM PEET MJLF  XPVME UIF PVUDPNF GSFRVFOU $POTJEFS GPS FYBNQMF B SBSF EJTFBTF XIJDI PDDVST JO  QFS PO QFPQMF 4VQQPTF BMTP UIBU SFBEJOH UIJT UFYUCPPL JODSFBTFE UIF PEET PG UIF EJTFBTF G
15. Relative and absolute effects • Parameters on relative effect scale • Predictions on absolute effect scale • Using relative effects may exaggerate importance of predictor • Good for scaring people, getting published • Not so good for public health, scientific progress • But needed for causal inference relative shark absolute penguin
16. Risk communication • Many people mistake relative risk for absolute risk • Example: • 1/1000 women develop blood clots • 3/1000 women on birth control develop blood clots • => 200% increase in blood clots! • Change in probability is only 0.002 • Pregnancy much more dangerous than blood clots
17. Aggregated Binomial • Numbers accepted/rejected to 6 PhD programs at UC Berkeley (largest depts in 1973) • Evidence of gender discrimination? Dean was afraid of lawsuit. • Call in the statisticians! 0'( NPEFMT EP OPU CFDBVTF UIFZ USVTU UIBU ZPV LOPX XIBU ZPV BSF EPJOH  "HHSFHBUFE CJOPNJBM (SBEVBUF TDIPPM BENJTTJPOT *O UIF BHHSFHBUFE CJO BNQMF BCPWF UIF OVNCFS PG USJBMT XBT BMXBZT  PO FWFSZ SPX ćJT JT PęFO OPU UIF XBZ UP IBOEMF UIJT JT UP JOTFSU B WBSJBCMF GSPN UIF EBUB JO QMBDF PG UIF iǎǕw -FUT XPSL BO FYBNQMF 'JSTU MPBE UIF EBUB 3 DPEF  '$--4ǿ- /#$)&$)"Ȁ /ǿ($/Ȁ  ʚǶ ($/ ćJT EBUB UBCMF POMZ IBT  SPXT TP MFUT MPPL BU UIF FOUJSF UIJOH  +/ ++'$)/ǡ" ) - ($/ - % / ++'$/$*). ǎ  (' ǒǎǏ ǐǎǐ ǕǏǒ Ǐ  ! (' Ǖǖ ǎǖ ǎǍǕ ǐ  (' ǐǒǐ ǏǍǔ ǒǓǍ Ǒ  ! (' ǎǔ Ǖ Ǐǒ ǒ  (' ǎǏǍ ǏǍǒ ǐǏǒ Ǔ  ! (' ǏǍǏ ǐǖǎ ǒǖǐ ǔ  (' ǎǐǕ Ǐǔǖ Ǒǎǔ Ǖ  ! (' ǎǐǎ ǏǑǑ ǐǔǒ ǖ  (' ǒǐ ǎǐǕ ǎǖǎ ǎǍ  ! (' ǖǑ Ǐǖǖ ǐǖǐ ǎǎ  (' ǏǏ ǐǒǎ ǐǔǐ ǎǏ  ! (' ǏǑ ǐǎǔ ǐǑǎ
18. UCB admissions dept applicant.gender admit reject applications 1 A male 512 313 825 2 A female 89 19 108 3 B male 353 207 560 4 B female 17 8 25 5 C male 120 205 325 6 C female 202 391 593 7 D male 138 279 417 8 D female 131 244 375 9 E male 53 138 191 10 E female 94 299 393 11 F male 22 351 373 12 F female 24 317 341 0'( NPEFMT EP OPU CFDBVTF UIFZ USVTU UIBU ZPV LOPX XIBU ZPV BSF EPJOH  "HHSFHBUFE CJOPNJBM (SBEVBUF TDIPPM BENJTTJPOT *O UIF BHHSFHBUFE CJO BNQMF BCPWF UIF OVNCFS PG USJBMT XBT BMXBZT  PO FWFSZ SPX ćJT JT PęFO OPU UIF XBZ UP IBOEMF UIJT JT UP JOTFSU B WBSJBCMF GSPN UIF EBUB JO QMBDF PG UIF iǎǕw -FUT XPS BO FYBNQMF 'JSTU MPBE UIF EBUB 3 DPEF  '$--4ǿ- /#$)&$)"Ȁ /ǿ($/Ȁ  ʚǶ ($/ ćJT EBUB UBCMF POMZ IBT  SPXT TP MFUT MPPL BU UIF FOUJSF UIJOH  +/ ++'$)/ǡ" ) - ($/ - % / ++'$/$*). ǎ  (' ǒǎǏ ǐǎǐ ǕǏǒ Ǐ  ! (' Ǖǖ ǎǖ ǎǍǕ ǐ  (' ǐǒǐ ǏǍǔ ǒǓǍ Ǒ  ! (' ǎǔ Ǖ Ǐǒ ǒ  (' ǎǏǍ ǏǍǒ ǐǏǒ Ǔ  ! (' ǏǍǏ ǐǖǎ ǒǖǐ ǔ  (' ǎǐǕ Ǐǔǖ Ǒǎǔ Ǖ  ! (' ǎǐǎ ǏǑǑ ǐǔǒ ǖ  (' ǒǐ ǎǐǕ ǎǖǎ ǎǍ  ! (' ǖǑ Ǐǖǖ ǐǖǐ ǎǎ  (' ǏǏ ǐǒǎ ǐǔǐ ǎǏ  ! (' ǏǑ ǐǎǔ ǐǑǎ
19. 8F XJMM NPEFM UIF BENJTTJPO EFDJTJPOT GPDVTJOH PO BQQMJDBOU HFOEFS BT B QSFEJDUPS WBSJBCM 4P XF XBOU UP ĕU B CJOPNJBM SFHSFTTJPO UIBU NPEFMT ($/ BT B GVODUJPO PG FBDI BQQMJDBO HFOEFS ćJT XJMM FTUJNBUF UIF BTTPDJBUJPO CFUXFFO HFOEFS BOE QSPCBCJMJUZ PG BENJTTJPO ć JT XIBU UIF NPEFM MPPLT MJLF JO NBUIFNBUJDBM GPSN "J ∼ #JOPNJBM(/J, QJ) MPHJU(QJ) = αĴĶı[J] αK ∼ /PSNBM(, .) ćF WBSJBCMF /J JOEJDBUFT ++'$/$*).ȁ$Ȃ UIF OVNCFS PG BQQMJDBUJPOT PO SPX J ćF JOEF WBSJBCMF ĴĶı[J] JOEFYFT HFOEFS PG BO BQQMJDBOU  JOEJDBUFT NBMF BOE  JOEJDBUFT GFNBMF 8F DPOTUSVDU JU KVTU CFGPSF ĕUUJOH UIF NPEFM MJLF UIJT ɶ"$ ʚǶ $! '. ǿ ɶ++'$)/ǡ" ) -ʙʙǫ(' ǫ Ǣ ǎ Ǣ Ǐ Ȁ (ǎǎǡǔ ʚǶ ,0+ǿ '$./ǿ ($/ ʡ $)*(ǿ ++'$/$*). Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ ȁ"$Ȃ Ǣ ȁ"$Ȃ ʡ )*-(ǿ Ǎ Ǣ ǎǡǒ Ȁ Ȁ Ǣ /ʙ Ȁ +- $.ǿ (ǎǎǡǔ Ǣ  +/#ʙǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ ȁǎȂ ǶǍǡǏǏ ǍǡǍǑ ǶǍǡǏǕ ǶǍǡǎǓ ȁǏȂ ǶǍǡǕǐ ǍǡǍǒ ǶǍǡǖǎ ǶǍǡǔǒ ćF QPTUFSJPS GPS NBMF BQQMJDBOUT ȁǎȂ JT IJHIFS UIBO UIBU PG GFNBMF BQQMJDBOUT )PX NVD Trials vary by row EBUB ćFO UIFSF XPVME CF  SPXT JO UIF EBUB 0VS KPC JT UP FWBMVBUF XIFUIFS UIFTF EBUB DPOUBJO FWJEFODF PG HFOEFS CJBT JO BENJTTJPOT 8F XJMM NPEFM UIF BENJTTJPO EFDJTJPOT GPDVTJOH PO BQQMJDBOU HFOEFS BT B QSFEJDUPS WBSJBCMF 4P XF XBOU UP ĕU B CJOPNJBM SFHSFTTJPO UIBU NPEFMT ($/ BT B GVODUJPO PG FBDI BQQMJDBOUT HFOEFS ćJT XJMM FTUJNBUF UIF BTTPDJBUJPO CFUXFFO HFOEFS BOE QSPCBCJMJUZ PG BENJTTJPO ćJT T XIBU UIF NPEFM MPPLT MJLF JO NBUIFNBUJDBM GPSN "J ∼ #JOPNJBM(/J, QJ) MPHJU(QJ) = αĴĶı[J] αK ∼ /PSNBM(, .) ćF WBSJBCMF /J JOEJDBUFT ++'$/$*).ȁ$Ȃ UIF OVNCFS PG BQQMJDBUJPOT PO SPX J ćF JOEFY WBSJBCMF ĴĶı[J] JOEFYFT HFOEFS PG BO BQQMJDBOU  JOEJDBUFT NBMF BOE  JOEJDBUFT GFNBMF 8FMM DPOTUSVDU JU KVTU CFGPSF ĕUUJOH UIF NPEFM MJLF UIJT 3 DPEF  ɶ"$ ʚǶ $! '. ǿ ɶ++'$)/ǡ" ) -ʙʙǫ(' ǫ Ǣ ǎ Ǣ Ǐ Ȁ (ǎǎǡǔ ʚǶ ,0+ǿ '$./ǿ ($/ ʡ $)*(ǿ ++'$/$*). Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ ȁ"$Ȃ Ǣ ȁ"$Ȃ ʡ )*-(ǿ Ǎ Ǣ ǎǡǒ Ȁ Ȁ Ǣ /ʙ Ȁ +- $.ǿ (ǎǎǡǔ Ǣ  +/#ʙǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ ȁǎȂ ǶǍǡǏǏ ǍǡǍǑ ǶǍǡǏǕ ǶǍǡǎǓ ȁǏȂ ǶǍǡǕǐ ǍǡǍǒ ǶǍǡǖǎ ǶǍǡǔǒ
20. Posterior contrast • Compute the contrast between genders • On both logit (shark) and prob (penguin) scales Ȁ Ǣ /ʙ Ȁ +- $.ǿ (ǎǎǡǔ Ǣ  +/#ʙǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ ȁǎȂ ǶǍǡǏǏ ǍǡǍǑ ǶǍǡǏǕ ǶǍǡǎǓ ȁǏȂ ǶǍǡǕǐ ǍǡǍǒ ǶǍǡǖǎ ǶǍǡǔǒ ćF QPTUFSJPS GPS NBMF BQQMJDBOUT ȁǎȂ JT IJHIFS UIBO UIBU PG GFNBMF BQQMJDBOUT )PX IJHIFS 8F OFFE UP DPNQVUF UIF DPOUSBTU -FUT DBMDVMBUF UIF DPOUSBTU PO UIF MPHJU TIBSL BT XFMM BT UIF DPOUSBTU PO UIF PVUDPNF TDBMF QFOHVJO  +*./ ʚǶ 3/-/ǡ.(+' .ǿ(ǎǎǡǔȀ $!!Ǿ ʚǶ +*./ɶȁǢǎȂ Ƕ +*./ɶȁǢǏȂ $!!Ǿ+ ʚǶ $)1Ǿ'*"$/ǿ+*./ɶȁǢǎȂȀ Ƕ $)1Ǿ'*"$/ǿ+*./ɶȁǢǏȂȀ +- $.ǿ '$./ǿ $!!Ǿʙ$!!Ǿ Ǣ $!!Ǿ+ʙ$!!Ǿ+ Ȁ Ȁ Ǫ/ǡ!-( Ǫǣ ǎǍǍǍǍ *.ǡ *! Ǐ 1-$' .ǣ ( ) . ǒǡǒʉ ǖǑǡǒʉ #$./*"-( $!!Ǿ ǍǡǓǎ ǍǡǍǓ Ǎǡǒǎ Ǎǡǔǎ ΤΤΤΦΪΪΨΥΤΤΤ $!!Ǿ+ ǍǡǎǑ ǍǡǍǎ ǍǡǎǏ ǍǡǎǓ ΤΤΤΥΦΪΪΨΥΤΤΤΤ ćF MPHPEET EJČFSFODF JT DFSUBJOMZ QPTJUJWF DPSSFTQPOEJOH UP B IJHIFS QSPCBCJMJUZ PG B TJPO GPS NBMF BQQMJDBOUT 0O UIF QSPCBCJMJUZ TDBMF JUTFMG UIF EJČFSFODF JT TPNFXIFSF CFU  BOE 
21. 0.0 0.2 0.4 0.6 0.8 1.0 case admit 1 2 3 4 5 6 7 8 9 10 11 12 Posterior validation check A B C D E F 'ĶĴłĿIJ ƉƉƍ 1PTUFSJPS WBMJEBUJPO GPS NPEFM (ǎǎǡǔ #MVF QPJOUT BSF PC TFSWFE QSPQPSUJPOT BENJUUFE GPS FBDI SPX JO UIF EBUB XJUI QPJOUT GSPN UIF TBNF EFQBSUNFOU DPOOFDUFE CZ B CMVF MJOF 0QFO QPJOUT UIF UJOZ WFSUJDBM m f Females admitted more in all but 2 departments! Figure 11.5
22. Backdoor admissions • Backdoor path through department • Use unique intercepts to adjust for that path PG BQQMJDBOUT 4P XIJMF JU JT USVF PWFSBMM UIBU GFNBMFT IBE B MPXFS QSPCBCJMJUZ PG JU JT DMFBSMZ OPU USVF XJUIJO NPTU EFQBSUNFOUT "OE OPUF UIBU KVTU EJTUSJCVUJPO BMPOF XPVME OFWFS IBWF SFWFBMFE UIBU GBDU UP VT 8F IBE PVUTJEF UIF ĕU NPEFM *O UIJT DBTF JU XBT B TJNQMF QPTUFSJPS WBMJEBUJ *OTUFBE PG BTLJOH i8IBU BSF UIF BWFSBHF QSPCBCJMJUJFT PG BENJTTJ BDSPTT BMM EFQBSUNFOUT w XF XBOU UP BTL i8IBU JT UIF BWFSBHF EJČFSFO TJPO CFUXFFO GFNBMFT BOE NBMFT XJUIJO EFQBSUNFOUT w *O PSEFS UP BTL FTUJNBUF VOJRVF GFNBMF BOE NBMF BENJTTJPO SBUFT JO FBDI EFQBSUN BTLT UIJT OFX RVFTUJPO "J ∼ #JOPNJBM(/J, QJ) MPHJU(QJ) = αĴĶı[J] + δıIJĽŁ[J] αK ∼ /PSNBM(, .) δL ∼ /PSNBM(, .) XIFSF ıIJĽŁ JOEFYFT EFQBSUNFOU JO L = .. 4P OPX FBDI EFQBSUNFO PG BENJTTJPO δL CVUUIF NPEFMTUJMM FTUJNBUFT VOJWFSTBM BEKVTUNFOUT‰ GPS NBMF BOE GFNBMF BQQMJDBUJPOT     Ǎǡǐǒ Ǎǡǒǐ Ǎǡǐǐ ǍǡǒǏ ǍǡǓǒ ǍǡǑǔ ǍǡǓǔ ǍǡǑǕ SUJPOT PG BMM BQQMJDBUJPOT JO FBDI EFQBSUNFOU UIBU BSF FJUIFS NBMF UPQ PN SPX  %FQBSUNFOU " SFDFJWFT  PG JUT BQQMJDBUJPOT GSPN NBMFT FT  GSPN NBMFT /PX MPPL CBDL BU UIF  '/ QPTUFSJPS NFBOT JO UIF (ǎǎǡǕ ćF EFQBSUNFOUT XJUI B MBSHFS QSPQPSUJPO PG GFNBMF BQQMJDBOUT PXFS PWFSBMM BENJTTJPOT SBUFT DPOGPVOE (FOEFS JOĘVFODFT DIPJDF PG EFQBSUNFOU BOE EFQBSUNFOU BENJTTJPO $POUSPMMJOH GPS EFQBSUNFOU SFWFBMT B NPSF QMBVTJCMF DBVTBM *O %"( GPSN A D G EFS % JT EFQBSUNFOU BOE " JT BDDFQUBODF ćFSF JT B CBDLEPPS QJQF ( → UP BDDFQUBODF 4P UP JOGFS UIF EJSFDU FČFDU ( → " XF OFFE UP DPOEJUJPO BDLEPPS .PEFM (ǎǎǡǕ EPFT UIBU *G ZPV JOTQFDU +*./# &ǿ(ǎǎǡǕȀ
23. Stratification by department Stat Q: What are the average probabilities of admission for females and males across all departments? Causal Q: What is the TOTAL influence of gender? XIFUIFS UIFTF EBUB DPOUBJO FWJEFODF PG HFOEFS CJBT JO BENJTTJPOT PO EFDJTJPOT GPDVTJOH PO BQQMJDBOU HFOEFS BT B QSFEJDUPS WBSJBCMF JBM SFHSFTTJPO UIBU NPEFMT ($/ BT B GVODUJPO PG FBDI BQQMJDBOUT UIF BTTPDJBUJPO CFUXFFO HFOEFS BOE QSPCBCJMJUZ PG BENJTTJPO ćJT LF JO NBUIFNBUJDBM GPSN "J ∼ #JOPNJBM(/J, QJ) MPHJU(QJ) = αĴĶı[J] αK ∼ /PSNBM(, .) ++'$/$*).ȁ$Ȃ UIF OVNCFS PG BQQMJDBUJPOT PO SPX J ćF JOEFY OEFS PG BO BQQMJDBOU  JOEJDBUFT NBMF BOE  JOEJDBUFT GFNBMF 8FMM JOH UIF NPEFM MJLF UIJT 3 DPEF  '$)/ǡ" ) -ʙʙǫ(' ǫ Ǣ ǎ Ǣ Ǐ Ȁ ǿ ++'$/$*). Ǣ + Ȁ Ǣ "$Ȃ Ǣ ǿ Ǎ Ǣ ǎǡǒ Ȁ
24. Stratification by department Stat Q: What are the average probabilities of admission for females and males across all departments? Causal Q: What is the TOTAL influence of gender? Stat Q: What is the average difference in probability of admission for females and males within departments? Causal Q: What is the DIRECT influence of gender? XIFUIFS UIFTF EBUB DPOUBJO FWJEFODF PG HFOEFS CJBT JO BENJTTJPOT PO EFDJTJPOT GPDVTJOH PO BQQMJDBOU HFOEFS BT B QSFEJDUPS WBSJBCMF JBM SFHSFTTJPO UIBU NPEFMT ($/ BT B GVODUJPO PG FBDI BQQMJDBOUT UIF BTTPDJBUJPO CFUXFFO HFOEFS BOE QSPCBCJMJUZ PG BENJTTJPO ćJT LF JO NBUIFNBUJDBM GPSN "J ∼ #JOPNJBM(/J, QJ) MPHJU(QJ) = αĴĶı[J] αK ∼ /PSNBM(, .) ++'$/$*).ȁ$Ȃ UIF OVNCFS PG BQQMJDBUJPOT PO SPX J ćF JOEFY OEFS PG BO BQQMJDBOU  JOEJDBUFT NBMF BOE  JOEJDBUFT GFNBMF 8FMM JOH UIF NPEFM MJLF UIJT 3 DPEF  '$)/ǡ" ) -ʙʙǫ(' ǫ Ǣ ǎ Ǣ Ǐ Ȁ ǿ ++'$/$*). Ǣ + Ȁ Ǣ "$Ȃ Ǣ ǿ Ǎ Ǣ ǎǡǒ Ȁ BDSPTT BMM EFQBSUNFOUT w XF XBOU UP BTL i8IBU JT UIF BWFSBHF EJČFSFOD TJPO CFUXFFO GFNBMFT BOE NBMFT XJUIJO EFQBSUNFOUT w *O PSEFS UP BTL FTUJNBUF VOJRVF GFNBMF BOE NBMF BENJTTJPO SBUFT JO FBDI EFQBSUN BTLT UIJT OFX RVFTUJPO "J ∼ #JOPNJBM(/J, QJ) MPHJU(QJ) = αĴĶı[J] + δıIJĽŁ[J] αK ∼ /PSNBM(, .) δL ∼ /PSNBM(, .) XIFSF ıIJĽŁ JOEFYFT EFQBSUNFOU JO L = .. 4P OPX FBDI EFQBSUNFO PG BENJTTJPO δL CVUUIF NPEFMTUJMM FTUJNBUFT VOJWFSTBM BEKVTUNFOUT‰ GPS NBMF BOE GFNBMF BQQMJDBUJPOT 'JUUJOH UIJT NPEFM JT TUSBJHIUGPSXBSE 8FMM VTF UIF JOEFYJOH OP BO JOUFSDFQU GPS FBDI EFQBSUNFOU #VU ĕSTU XF BMTP OFFE UP DPOTUSVD OVNCFST UIF EFQBSUNFOUT  UISPVHI  ćF GVODUJPO * - Ǿ$) 3 UIF  +/ GBDUPS BT JOQVU )FSFT UIF DPEF UP DPOTUSVDU UIF JOEFY BOE ɶ +/Ǿ$ ʚǶ - +ǿǎǣǓǢ #ʙǏȀ
25. Stratification by department GPS NBMF BOE GFNBMF BQQMJDBUJPOT 'JUUJOH UIJT NPEFM JT TUSBJHIUGPSXBSE 8FMM VTF UIF JOEFYJOH OPUBUJPO BHBJO BO JOUFSDFQU GPS FBDI EFQBSUNFOU #VU ĕSTU XF BMTP OFFE UP DPOTUSVDU B OVNFSJD OVNCFST UIF EFQBSUNFOUT  UISPVHI  ćF GVODUJPO * - Ǿ$) 3 DBO EP UIJT UIF  +/ GBDUPS BT JOQVU )FSFT UIF DPEF UP DPOTUSVDU UIF JOEFY BOE ĕU CPUI NP ɶ +/Ǿ$ ʚǶ - +ǿǎǣǓǢ #ʙǏȀ (ǎǎǡǕ ʚǶ ,0+ǿ '$./ǿ ($/ ʡ $)*(ǿ ++'$/$*). Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ ȁ"$Ȃ ʔ  '/ȁ +/Ǿ$Ȃ Ǣ ȁ"$Ȃ ʡ )*-(ǿ Ǎ Ǣ ǎǡǒ Ȁ Ǣ  '/ȁ +/Ǿ$Ȃ ʡ )*-(ǿ Ǎ Ǣ ǎǡǒ Ȁ Ȁ Ǣ /ʙ Ȁ +- $.ǿ (ǎǎǡǕ Ǣ  +/#ʙǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ ȁǎȂ ǶǍǡǒǐ Ǎǡǒǐ ǶǎǡǐǕ ǍǡǐǏ ȁǏȂ ǶǍǡǑǐ Ǎǡǒǐ ǶǎǡǏǕ ǍǡǑǏ  '/ȁǎȂ ǎǡǎǎ ǍǡǒǑ ǍǡǏǒ ǎǡǖǓ  '/ȁǏȂ ǎǡǍǓ ǍǡǒǑ ǍǡǏǍ ǎǡǖǏ  '/ȁǐȂ ǶǍǡǎǒ Ǎǡǒǐ ǶǎǡǍǍ ǍǡǔǍ  '/ȁǑȂ ǶǍǡǎǕ ǍǡǒǑ ǶǎǡǍǑ ǍǡǓǔ  '/ȁǒȂ ǶǍǡǓǏ ǍǡǒǑ ǶǎǡǑǕ ǍǡǏǐ  '/ȁǓȂ ǶǏǡǎǔ Ǎǡǒǒ ǶǐǡǍǒ ǶǎǡǐǍ ćF JOUFSDFQU GPS NBMF BQQMJDBOUT ȁǎȂ JT OPX B MJUUMF TNBMMFS PO BWFSBHF UIBO
26. Again with shark & penguin • Difference on logit and probability scales   (0% 41*,&% 5)& */5&(&34 3 DPEF  +*./ ʚǶ 3/-/ǡ.(+' .ǿ(ǎǎǡǕȀ $!!Ǿ ʚǶ +*./ɶȁǢǎȂ Ƕ +*./ɶȁǢǏȂ $!!Ǿ+ ʚǶ $)1Ǿ'*"$/ǿ+*./ɶȁǢǎȂȀ Ƕ $)1Ǿ'*"$/ǿ+*./ɶȁǢǏȂȀ +- $.ǿ '$./ǿ $!!Ǿʙ$!!Ǿ Ǣ $!!Ǿ+ʙ$!!Ǿ+ Ȁ Ȁ Ǫ/ǡ!-( Ǫǣ ǎǍǍǍǍ *.ǡ *! Ǐ 1-$' .ǣ ( ) . ǒǡǒʉ ǖǑǡǒʉ #$./*"-( $!!Ǿ ǶǍǡǎǍ ǍǡǍǕ ǶǍǡǏǏ ǍǡǍǐ ΤΤΤΤΥΨΪΪΨΥΤΤΤΤ $!!Ǿ+ ǶǍǡǍǏ ǍǡǍǏ ǶǍǡǍǒ ǍǡǍǎ ΤΤΤΥΪΪΥΤΤ *G NBMF BQQMJDBOUT IBWF JU XPSTF JU JT POMZ CZ B WFSZ TNBMM BNPVOU BCPVU  PO B 8IZ EJE BEEJOH EFQBSUNFOUT UP UIF NPEFM DIBOHF UIF JOGFSFODF BCPVU HFOE ćF FBSMJFS ĕHVSF HJWFT ZPV B IJOU‰UIF SBUFT PG BENJTTJPO WBSZ B MPU BDSPTT E 'VSUIFSNPSF GFNBMFT BOE NBMFT UFOE UP BQQMZ UP EJČFSFOU EFQBSUNFOUT -FUT
27. Backdoor admissions • What happened? Females apply more to most selective departments. So overall rate of admission lower. • Proportions of m/f applications by department: $!!Ǿ ʚǶ +*./ɶȁǢǎȂ Ƕ +*./ɶȁǢǏȂ $!!Ǿ+ ʚǶ $)1Ǿ'*"$/ǿ+*./ɶȁǢǎȂȀ Ƕ $)1Ǿ'*"$/ǿ+*./ɶȁǢǏȂȀ +- $.ǿ '$./ǿ $!!Ǿʙ$!!Ǿ Ǣ $!!Ǿ+ʙ$!!Ǿ+ Ȁ Ȁ Ǫ/ǡ!-( Ǫǣ ǎǍǍǍǍ *.ǡ *! Ǐ 1-$' .ǣ ( ) . ǒǡǒʉ ǖǑǡǒʉ #$./*"-( $!!Ǿ ǶǍǡǎǍ ǍǡǍǕ ǶǍǡǏǏ ǍǡǍǐ ΤΤΤΤΥΨΪΪΨΥΤΤΤΤ $!!Ǿ+ ǶǍǡǍǏ ǍǡǍǏ ǶǍǡǍǒ ǍǡǍǎ ΤΤΤΥΪΪΥΤΤ *G NBMF BQQMJDBOUT IBWF JU XPSTF JU JT POMZ CZ B WFSZ TNBMM BNPVOU BCPVU  PO BWFSBHF 8IZ EJE BEEJOH EFQBSUNFOUT UP UIF NPEFM DIBOHF UIF JOGFSFODF BCPVU HFOEFS TP N ćF FBSMJFS ĕHVSF HJWFT ZPV B IJOU‰UIF SBUFT PG BENJTTJPO WBSZ B MPU BDSPTT EFQBSUN 'VSUIFSNPSF GFNBMFT BOE NBMFT UFOE UP BQQMZ UP EJČFSFOU EFQBSUNFOUT -FUT EP B R UBCVMBUJPO UP TIPX UIBU 3 DPEF  +" ʚǶ .++'4ǿ ǎǣǓ Ǣ !0)/$*)ǿ&Ȁ ɶ++'$/$*).ȁɶ +/Ǿ$ʙʙ&Ȃȅ.0(ǿɶ++'$/$*).ȁɶ +/Ǿ$ʙʙ&ȂȀ Ȁ -*2)( .ǿ+"Ȁ ʚǶ ǿǫ(' ǫǢǫ! (' ǫȀ *')( .ǿ+"Ȁ ʚǶ 0)$,0 ǿɶ +/Ȁ -*0)ǿ +" Ǣ Ǐ Ȁ       (' ǍǡǕǕ ǍǡǖǓ Ǎǡǐǒ Ǎǡǒǐ Ǎǡǐǐ ǍǡǒǏ ! (' ǍǡǎǏ ǍǡǍǑ ǍǡǓǒ ǍǡǑǔ ǍǡǓǔ ǍǡǑǕ ćFTF BSF UIF QSPQPSUJPOT PG BMM BQQMJDBUJPOT JO FBDI EFQBSUNFOU UIBU BSF FJUIFS NBMF SPX PS GFNBMF CPUUPN SPX  %FQBSUNFOU " SFDFJWFT  PG JUT BQQMJDBUJPOT GSPN N
28. Backdoor admissions • Careful about casual interpretation • No evidence for direct path G –> A • Lots of evidence of indirect path G –> D –> A • Total causal influence of G still strong • Still results in disenfranchisement • But effective intervention very different ( .ǿ+"Ȁ ʚǶ 0)$,0 ǿɶ +/Ȁ ǿ +" Ǣ Ǐ Ȁ       ǍǡǕǕ ǍǡǖǓ Ǎǡǐǒ Ǎǡǒǐ Ǎǡǐǐ ǍǡǒǏ ǍǡǎǏ ǍǡǍǑ ǍǡǓǒ ǍǡǑǔ ǍǡǓǔ ǍǡǑǕ BSF UIF QSPQPSUJPOT PG BMM BQQMJDBUJPOT JO FBDI EFQBSUNFOU UIBU BSF FJUIFS NBMF UPQ PS GFNBMF CPUUPN SPX  %FQBSUNFOU " SFDFJWFT  PG JUT BQQMJDBUJPOT GSPN NBMFT SUNFOU & SFDFJWFT  GSPN NBMFT /PX MPPL CBDL BU UIF  '/ QPTUFSJPS NFBOT JO UIF $. PVUQVU GSPN (ǎǎǡǕ ćF EFQBSUNFOUT XJUI B MBSHFS QSPQPSUJPO PG GFNBMF BQQMJDBOUT TP UIPTF XJUI MPXFS PWFSBMM BENJTTJPOT SBUFT %FQBSUNFOU JT B DPOGPVOE (FOEFS JOĘVFODFT DIPJDF PG EFQBSUNFOU BOE EFQBSUNFOU ODFT DIBODF PG BENJTTJPO $POUSPMMJOH GPS EFQBSUNFOU SFWFBMT B NPSF QMBVTJCMF DBVTB ODF PG HFOEFS *O %"( GPSN A D G direct effect indirect effect
29. Poisson GLMs • Counts without upper limit, constant expected value • Single parameter: events per unit time/distance • Variance equal to mean Z ∼ 1PJTTPO(λ) Z J ∼ /PSNBM(µJ, σ), µJ = α + βYJ &(Z J |YJ) = α + βYJ ∂ ∂YJ &(Z J |YJ) = β Z J ∼ #JOPNJBM(Q J, O), &(Z) = λ WBS(Z) = λ Z J ∼ /PSNBM(µJ, σ), µJ = α + βYJ &(Z J |YJ) = α + βYJ ∂ ∂YJ &(Z J |YJ) = β 0 5 10 15 0 500 1000 1500 Count Frequency 0 5 10 20 30 0 200 400 600 800 Count Frequency
30. Poisson GLMs • Examples: Soccer goals, fission events, photons striking a detector, DNA mutations, soldiers killed by horses Siméon Denis Poisson (1781–1840) Abraham de Moivre (1667–1754)
31. Oceanic tool complexity   (&/&3"-*;&% -*/&"3 . 3 DPEF  GORJSRS  ORJ G3RSXODWLRQ GFRQWDFWKLJK  LIHOVH G&RQWDFW 8FÔMM DPOTJEFS B TFSJFT PG GPVS NPEFM G  B 1PJTTPO NPEFM XJUI B DPOTUBOU NFB EFQFOET VQPO MPHQPQVMBUJPO  B NPE DPOUBDU SBUF BOE  B NPEFM UIBU JOUFS culture population contact total_tools mean_TU Malekula 1100 low 13 3.2 Tikopia 1500 low 22 4.7 Santa Cruz 3600 low 24 4.0 Yap 4791 high 43 5.0 Lau Fiji 7400 high 33 5.0 Trobriand 8000 high 19 4.0 Chuuk 9200 high 40 3.8 Manus 13000 low 28 6.6 Tonga 17500 high 55 5.4 0 Hawaii 275000 low 71 6.6 Dr. Michelle Kline (Simon Fraser U) (1) Complexity of toolkit proportional to magnitude of population? (2) Contact with other islands moderates impact?
32. Anatomy of Poisson GLM -FUT CVJME OPX 'JSTU XF NBLF TPNF OFX DPMVNOT XJUI UIF TUBOEBSEJ[FE MPH $*) BOE BO JOEFY WBSJBCMF GPS *)// ʚǶ .' ǿ '*"ǿɶ+*+0'/$*)Ȁ Ȁ *)//Ǿ$ ʚǶ $! '. ǿ ɶ*)//ʙʙǫ#$"#ǫ Ǣ Ǐ Ǣ ǎ Ȁ NPEFM UIBU DPOGPSNT UP UIF SFTFBSDI IZQPUIFTJT JODMVEFT BO JOUFSBDUJPO CFUX VMBUJPO BOE DPOUBDU SBUF *O NBUI GPSN UIJT JT 5J ∼ 1PJTTPO(λJ) MPH λJ = αİĶı[J] + βİĶı[J] MPH 1J αK ∼ UP CF EFUFSNJOFE βK ∼ UP CF EFUFSNJOFE SF 1 JT +*+0'/$*) BOE İĶı JT *)//Ǿ$ 8F OFFE UP ĕHVSF PVU TPNF TFOTJCMF QSJPST "T XJUI CJOPNJBM NPEFMT UIF USBOTGP BMF CFUXFFO UIF TDBMF PG UIF MJOFBS NPEFM BOE UIF DPVOU TDBMF PG UIF PVUDPNF NF FUIJOH ĘBU PO UIF MJOFBS NPEFM TDBMF XJMM OPU CF ĘBU PO UIF PVUDPNF TDBMF -FUT YBNQMF KVTU B NPEFM XJUI BO JOUFSDFQU BOE B WBHVF /PSNBM   QSJPS PO JU log link total_tools (outcome) expected tools for case i
33. Log link • Goal: Map linear model to positive reals • Log link maps all negative numbers to [0,1] • All positive numbers to [1,infinity]   #*( &/5301: "/% 5)& (&/&3"-*;&% -*/&"3 .0%&- -1.0 -0.5 0.0 0.5 1.0 -3 -2 -1 0 1 2 3 x log measurement -1.0 -0.5 0.0 0.5 1.0 x 0 2 4 6 8 10 original measurement 'ĶĴłĿIJ ƑƏ ćF MPH MJOL USBOTGPSNT B MJOFBS NPEFM MFę JOUP B TUSJDUMZ QPT
34. Priors & the log link • Log link not intuitive — simulate βK ∼ UP CF EFUFSNJOFE İĶı JT *)//Ǿ$ NF TFOTJCMF QSJPST "T XJUI CJOPNJBM NPEFMT UIF USBOTGPSNBUJPO UIF MJOFBS NPEFM BOE UIF DPVOU TDBMF PG UIF PVUDPNF NFBOT UIBU NPEFM TDBMF XJMM OPU CF ĘBU PO UIF PVUDPNF TDBMF -FUT DPOTJEFS UI BO JOUFSDFQU BOE B WBHVF /PSNBM   QSJPS PO JU 5J ∼ 1PJTTPO(λJ) MPH λJ = α α ∼ /PSNBM(, ) LF PO UIF PVUDPNF TDBMF λ *G α IBT B OPSNBM EJTUSJCVUJPO UIFO JPO 4P MFUT QMPU B MPHOPSNBM XJUI UIFTF WBMVFT GPS UIF OPSNBM O 3 DPEF  ǎǍ Ȁ Ǣ !-*(ʙǍ Ǣ /*ʙǎǍǍ Ǣ )ʙǏǍǍ Ȁ O 'ĶĴłĿIJ ƉƉƏ BT UIF CMBDL DVSWF *WF VTFE B SBOHF GSPN  UP    (0% 41*,&% 0 20 40 60 80 100 0.00 0.02 0.04 0.06 0.08 mean number of tools Density a ~ dnorm(0,10) a ~ dnorm(3,0.5) MPH λJ = α α ∼ /PSNBM(, ) 8IBU EPFT UIJT QSJPS MPPL MJLF PO UIF PVUDPNF TDBMF λ *G α IBT B OPSNBM EJTUSJCVUJP λ IBT B MPHOPSNBM EJTUSJCVUJPO 4P MFUT QMPU B MPHOPSNBM XJUI UIFTF WBMVFT GPS UIF O NFBO BOE TUBOEBSE EFWJBUJPO 0-1 ǿ ')*-(ǿ 3 Ǣ Ǎ Ǣ ǎǍ Ȁ Ǣ !-*(ʙǍ Ǣ /*ʙǎǍǍ Ǣ )ʙǏǍǍ Ȁ ćF EJTUSJCVUJPO JT TIPXO JO 'ĶĴłĿIJ ƉƉƏ BT UIF CMBDL DVSWF *WF VTFE B SBOHF GSPN PO UIF IPSJ[POUBM BYJT SFĘFDUJOH UIF OPUJPO UIBU XF LOPX BMM IJTUPSJDBM UPPM LJUT JO UIF XFSF JO UIJT SBOHF 'PS UIF α ∼ /PSNBM(, ) QSJPS UIFSF JT B IVHF TQJLF SJHIU [FSP‰UIBU NFBOT [FSP UPPMT PO BWFSBHF‰BOE B WFSZ MPOH UBJM )PX MPOH 8FMM UIF N B MPHOPSNBM EJTUSJCVUJPO JT FYQ(µ+σ/) XIJDI FWBMVBUFT UP FYQ() XIJDI JT JN MBSHF *G ZPV EPVCU UIJT KVTU TJNVMBUF JU  ʚǶ -)*-(ǿǎ ǑǢǍǢǎǍȀ '( ʚǶ 3+ǿȀ ( )ǿ '( Ȁ ȁǎȂ ǖǡǓǏǏǖǖǑ ʔǎǏ ćBUT B MPU PG UPPMT FOPVHI UP DPWFS BO FOUJSF JTMBOE 8F DBO EP CFUUFS UIBO UIJT * FODPVSBHF ZPV UP QMBZ BSPVOE XJUI UIF 0-1 DPEF BCPWF USZJOH EJČFSFOU NF
35. Priors & the log link • Slopes equally unintuitive  10*440/ 3&(3&44*0/  -2 -1 0 1 2 0 20 40 60 80 100 log population (std) total tools b ~ dnorm( 0 , 10 ) -2 -1 0 1 2 0 20 40 60 80 100 log population (std) total tools b ~ dnorm( 0 , 0.2 )
36. Tools models UIFTF JTTVFT 0LBZ ĕOBMMZ XF DBO BQQSPYJNBUF TPNF QPTUFSJPS EJTUSJCVUJPOT *N HPJOH UP DPEF CPUI UIF JOUFSBDUJPO NPEFM QSFTFOUFE BCPWF BT XFMM BT B WFSZ TJNQMF JOUFSDFQUPOMZ NPEFM ćF JOUFSDFQU POMZ NPEFM JT IFSF CFDBVTF * XBOU UP TIPX ZPV TPNFUIJOH JOUFSFTUJOH BCPVU 1PJTTPO NPEFMT BOE IPX QBSBNFUFST SFMBUF UP NPEFM DPNQMFYJUZ )FSFT UIF DPEF GPS CPUI NPEFMT 3 DPEF  / ʚǶ '$./ǿ  ʙ ɶ/*/'Ǿ/**'. Ǣ  ʙ ɶ Ǣ $ ʙ ɶ*)//Ǿ$ Ȁ ȕ $)/ - +/ *)'4 (ǎǎǡǖ ʚǶ 0'(ǿ '$./ǿ  ʡ +*$.ǿ '( ȀǢ '*"ǿ'(Ȁ ʚǶ Ǣ  ʡ )*-(ǿǐǢǍǡǒȀ ȀǢ /ʙ/ Ǣ #$).ʙǑ Ǣ '*"Ǿ'$&ʙ Ȁ  10*440/ 3&(3&44*0/  ȕ $)/ -/$*) (* ' (ǎǎǡǎǍ ʚǶ 0'(ǿ '$./ǿ  ʡ +*$.ǿ '( ȀǢ '*"ǿ'(Ȁ ʚǶ ȁ$Ȃ ʔ ȁ$ȂȉǢ ȁ$Ȃ ʡ )*-(ǿ ǐ Ǣ Ǎǡǒ ȀǢ ȁ$Ȃ ʡ )*-(ǿ Ǎ Ǣ ǍǡǏ Ȁ ȀǢ /ʙ/ Ǣ #$).ʙǑ Ǣ '*"Ǿ'$&ʙ Ȁ -FUT MPPL BU UIF -00*4 DPNQBSJTPO RVJDLMZ KVTU UP ĘBH UXP JNQPSUBOU GBDUT 3 DPEF
37. Compare using PSIS-LOO • Warning indicates strongly influential points • Look at those pLOO values: No relationship to parameter count — not unusual & not a mistake '*"ǿ'(Ȁ ʚǶ ȁ$Ȃ ʔ ȁ$ȂȉǢ ȁ$Ȃ ʡ )*-(ǿ ǐ Ǣ Ǎǡǒ ȀǢ ȁ$Ȃ ʡ )*-(ǿ Ǎ Ǣ ǍǡǏ Ȁ ȀǢ /ʙ/ Ǣ #$).ʙǑ Ǣ '*"Ǿ'$&ʙ Ȁ -FUT MPPL BU UIF -00*4 DPNQBSJTPO RVJDLMZ KVTU UP ĘBH UXP JNQPSUBOU GBDUT 3 DPEF  *(+- ǿ (ǎǎǡǖ Ǣ (ǎǎǡǎǍ Ǣ !0)ʙ  Ȁ  +    2 $"#/   (ǎǎǡǎǍ Ǖǒǡǒ ǔǡǎ ǍǡǍ ǎ ǎǐǡǏǏ  (ǎǎǡǖ ǎǑǎǡǎ ǕǡǍ ǒǒǡǒ Ǎ ǐǐǡǐǐ ǐǏǡǔǕ -)$)" ( .." .ǣ ǎǣ *( - /* & $")*./$ 1'0 . - /** #$"#ǡ  # '+ǿǪ+- /*Ƕ&Ƕ$")*./$ǪȀ !*-  /$'.ǡ 'JSTU OPUF UIBU XF HFU UIF 1BSFUP L XBSOJOH BHBJO ćJT JOEJDBUFT TPNF IJHIMZ JOĘVFOUJBM QPJOUT ćBU TIPVMEOU CF TVSQSJTJOH‰UIJT JT B TNBMM EBUBTFU #VU JU NFBOT XFMM XBOU UP UBLF B MPPL BU UIF QPTUFSJPS QSFEJDUJPOT XJUI UIBU JO NJOE 4FDPOE XIJMF JUT OP TVSQSJTF UIBU UIF JOUFSDFQUPOMZ NPEFM (ǎǎǡǖ IBT B XPSTF TDPSF UIBO UIF JOUFSBDUJPO NPEFM (ǎǎǡǎǍ JU NJHIU CF WFSZ TVSQSJTJOH UIBU UIF iFČFDUJWF OVNCFS PG QBSBNFUFSTw +  JT BDUVBMMZ MBSHFS GPS UIF NPEFM XJUI GFXFS QBSBNFUFST .PEFM (ǎǎǡǖ IBT POMZ POF QBSBNFUFS .PEFM (ǎǎǡǎǍ IBT GPVS QBSBNFUFST ćJT JTOU TPNF XFJSE UIJOH BCPVU -00*4‰8"*$ UFMMT ZPV UIF TBNF TUPSZ 8IBU JT HPJOH PO IFSF ćF POMZ QMBDF UIBU NPEFM DPNQMFYJUZ‰B NPEFMT UFOEFODZ UP PWFSĕU‰BOE QBSBNFUFS DPVOU IBWF B DMFBS SFMBUJPOTIJQ JT JO B TJNQMF MJOFBS SFHSFTTJPO XJUI ĘBU QSJPST 0ODF B EJTUSJ CVUJPO JT CPVOEFE GPS FYBNQMF UIFO QBSBNFUFS WBMVFT OFBS UIF CPVOEBSZ QSPEVDF MFTT PWFS ĕUUJOH UIBO UIPTF GBS GSPN UIF CPVOEBSZ ćF TBNF QSJODJQMF BQQMJFT UP EBUB EJTUSJCVUJPOT "OZ DPVOU OFBS [FSP JT IBSEFS UP PWFSĕU 4P PWFSĕUUJOH SJTL EFQFOET CPUI VQPO TUSVDUVSBM EFUBJMT PG UIF NPEFM BOE UIF DPNQPTJUJPO PG UIF TBNQMF *O UIJT TBNQMF B NBKPS TPVSDF PG PWFSĕUUJOH SJTL JT UIF IJHIMZ JOĘVFOUJBM QPJOU ĘBHHFE CZ
38. Hawaii has leverage   (0% 41*,&% 5)& */5&(&34 -1 0 1 2 0 20 40 60 log population (std) total tools Yap (0.6) Trobriand (0.56) Tonga (0.69) Hawaii (1.01) 0 50000 150000 250000 0 20 40 60 population total tools 'ĶĴłĿIJ ƉƉƑ 1PTUFSJPS QSFEJDUJPOT GPS UIF 0DFBOJD UPPMT NPEFM 'JMMFE QPJOUT BSF TPDJFUJFT XJUI IJTUPSJDBMMZ IJHI DPOUBDU 0QFO QPJOUT BSF UIPTF XJUI MPX DPOUBDU 1PJOU TJ[F JT TDBMFE CZ SFMBUJWF -00*4 1BSFUP L WBMVFT -BSHFS QPJOUT BSF NPSF JOĘVFOUJBM ćF TPMJE DVSWF JT UIF QPTUFSJPS NFBO • Point size proportional to Pareto-k diagnostic value
39. Generalized Linear Madness • This model is terrible: • Intercepts don’t pass through origin • Zero population = zero tools • We can do better by thinking scientifically instead of statistically   (0% 41*,&% 5)& */5&(&34 -1 0 1 2 0 20 40 60 log population (std) total tools Yap (0.6) Trobriand (0.56) Tonga (0.69) Hawaii (1.01) 0 50000 150000 250000 0 20 40 60 population total tools 'ĶĴłĿIJ ƉƉƑ 1PTUFSJPS QSFEJDUJPOT GPS UIF 0DFBOJD UPPMT NPEFM 'JMMFE QPJOUT BSF TPDJFUJFT XJUI IJTUPSJDBMMZ IJHI DPOUBDU 0QFO QPJOUT BSF UIPTF XJUI MPX DPOUBDU 1PJOU TJ[F JT TDBMFE CZ SFMBUJWF -00*4 1BSFUP L WBMVFT
40. Scientific model • Change in tools per unit time: IPX UIF TDJFOUJĕD NPEFM DPNQBSFT UP UIF HFPDFOUSJD NPE NQMF XIFUIFS ZPV VTF -00*4 PS 8"*$ JT B GFX QPJOUT *U JT TUJMM UVHHFE BSPVOE CZ )BXBJJ BOE 5POHB 8FMM SFUV OE BQQSPBDI DPOUBDU SBUF B EJČFSFOU XBZ CZ UBLJOH BDDP UP POF BOPUIFS FMJOH UPPM JOOPWBUJPO 5BLJOH UIF WFSCBM NPEFM JO UIF NBJO U JO UIF FYQFDUFE OVNCFS PG UPPMT JO POF UJNF TUFQ JT ∆5 = α1β − γ5 UJPO TJ[F 5 JT UIF OVNCFS PG UPPMT BOE α β BOE γ BSF QBSBNFUF N OVNCFS PG UPPMT 5 KVTU TFU UIF FRVBUJPO BCPWF FRVBM UP [FS ˆ 5 = α1β
41. Scientific model • Change in tools per unit time: IPX UIF TDJFOUJĕD NPEFM DPNQBSFT UP UIF HFPDFOUSJD NPE NQMF XIFUIFS ZPV VTF -00*4 PS 8"*$ JT B GFX QPJOUT *U JT TUJMM UVHHFE BSPVOE CZ )BXBJJ BOE 5POHB 8FMM SFUV OE BQQSPBDI DPOUBDU SBUF B EJČFSFOU XBZ CZ UBLJOH BDDP UP POF BOPUIFS FMJOH UPPM JOOPWBUJPO 5BLJOH UIF WFSCBM NPEFM JO UIF NBJO U JO UIF FYQFDUFE OVNCFS PG UPPMT JO POF UJNF TUFQ JT ∆5 = α1β − γ5 UJPO TJ[F 5 JT UIF OVNCFS PG UPPMT BOE α β BOE γ BSF QBSBNFUF N OVNCFS PG UPPMT 5 KVTU TFU UIF FRVBUJPO BCPWF FRVBM UP [FS ˆ 5 = α1β Innovation rate Diminishing returns (“elasticity”) Population
42. Scientific model • Change in tools per unit time: IPX UIF TDJFOUJĕD NPEFM DPNQBSFT UP UIF HFPDFOUSJD NPE NQMF XIFUIFS ZPV VTF -00*4 PS 8"*$ JT B GFX QPJOUT *U JT TUJMM UVHHFE BSPVOE CZ )BXBJJ BOE 5POHB 8FMM SFUV OE BQQSPBDI DPOUBDU SBUF B EJČFSFOU XBZ CZ UBLJOH BDDP UP POF BOPUIFS FMJOH UPPM JOOPWBUJPO 5BLJOH UIF WFSCBM NPEFM JO UIF NBJO U JO UIF FYQFDUFE OVNCFS PG UPPMT JO POF UJNF TUFQ JT ∆5 = α1β − γ5 UJPO TJ[F 5 JT UIF OVNCFS PG UPPMT BOE α β BOE γ BSF QBSBNFUF N OVNCFS PG UPPMT 5 KVTU TFU UIF FRVBUJPO BCPWF FRVBM UP [FS ˆ 5 = α1β Tools Innovation rate Diminishing returns (“elasticity”) Population Loss rate
43. Scientific model • Solve for steady state expected number of tools • Where ∆T = 0 OPWBUJPO 5BLJOH UIF WFSCBM NPEFM JO UIF NBJO UFYU BCPWF XF DBO DUFE OVNCFS PG UPPMT JO POF UJNF TUFQ JT ∆5 = α1β − γ5 T UIF OVNCFS PG UPPMT BOE α β BOE γ BSF QBSBNFUFST UP CF FTUJNBUFE PG UPPMT 5 KVTU TFU UIF FRVBUJPO BCPWF FRVBM UP [FSP BOE TPMWF GPS 5 ˆ 5 = α1β γ No ad hoc link function!  10*440/ 3&(3&44*0/ 8FSF HPJOH UP VTF UIJT JOTJEF B 1PJTTPO NPEFM OPX ćF OPJTF BSPVOE UIF PV CFDBVTF UIBU JT TUJMM UIF NBYJNVN FOUSPQZ EJTUSJCVUJPO JO UIJT DPOUFYU‰/*/ OP DMFBS VQQFS CPVOE #VU UIF MJOFBS NPEFM JT HPOF 5J ∼ 1PJTTPO(λJ) λJ = α1β J /γ /PUJDF UIBU UIFSF JT OP MJOL GVODUJPO "MM XF IBWF UP EP UP FOTVSF UIBU λ SF TVSF UIF QBSBNFUFST BSF QPTJUJWF *O UIF DPEF CFMPX *MM VTF FYQPOFOUJBM QS /PSNBM GPS α ćFO UIFZ BMM IBWF UP CF QPTJUJWF *O CVJMEJOH UIF NPEFM XF B BMM PG UIF QBSBNFUFST UP WBSZ CZ DPOUBDU SBUF 4JODF DPOUBDU SBUF JT TVQQPTF U QPQVMBUJPO TJ[F MFUT BMMPX α BOE β *U DPVME BMTP JOĘVFODF γ CFDBVTF USBE UPPMT GSPN WBOJTIJOH PWFS UJNF #VU XFMM MFBWF UIBU BT BO FYFSDJTF GPS UIF SF
44. Scientific model TF UIJT JOTJEF B 1PJTTPO NPEFM OPX ćF OPJTF BSPVOE UIF PVUDPNF XJMM TUJMM C TUJMM UIF NBYJNVN FOUSPQZ EJTUSJCVUJPO JO UIJT DPOUFYU‰/*/'Ǿ/**'. JT B D CPVOE #VU UIF MJOFBS NPEFM JT HPOF 5J ∼ 1PJTTPO(λJ) λJ = α1β J /γ SF JT OP MJOL GVODUJPO "MM XF IBWF UP EP UP FOTVSF UIBU λ SFNBJOT QPTJUJWF FUFST BSF QPTJUJWF *O UIF DPEF CFMPX *MM VTF FYQPOFOUJBM QSJPST GPS β BOE γ ćFO UIFZ BMM IBWF UP CF QPTJUJWF *O CVJMEJOH UIF NPEFM XF BMTP XBOU UP BMMP FUFST UP WBSZ CZ DPOUBDU SBUF 4JODF DPOUBDU SBUF JT TVQQPTF UP NFEJBUF UIF JO MFUT BMMPX α BOE β *U DPVME BMTP JOĘVFODF γ CFDBVTF USBEF OFUXPSLT NJH TIJOH PWFS UJNF #VU XFMM MFBWF UIBU BT BO FYFSDJTF GPS UIF SFBEFS )FSFT UIF ǿ ʙɶ/*/'Ǿ/**'.Ǣ ʙɶ+*+0'/$*)Ǣ $ʙɶ*)//Ǿ$ Ȁ (ǿ +*$.ǿ '( ȀǢ 8FSF HPJOH UP VTF UIJT JOTJEF B 1PJTTPO NPEFM OPX ćF OPJTF BSPVOE UIF PVUDPNF XJMM TUJMM CF 1PJTTPO CFDBVTF UIBU JT TUJMM UIF NBYJNVN FOUSPQZ EJTUSJCVUJPO JO UIJT DPOUFYU‰/*/'Ǿ/**'. JT B DPVOU XJUI OP DMFBS VQQFS CPVOE #VU UIF MJOFBS NPEFM JT HPOF 5J ∼ 1PJTTPO(λJ) λJ = α1β J /γ /PUJDF UIBU UIFSF JT OP MJOL GVODUJPO "MM XF IBWF UP EP UP FOTVSF UIBU λ SFNBJOT QPTJUJWF JT UP NBLF TVSF UIF QBSBNFUFST BSF QPTJUJWF *O UIF DPEF CFMPX *MM VTF FYQPOFOUJBM QSJPST GPS β BOE γ BOE B MPH /PSNBM GPS α ćFO UIFZ BMM IBWF UP CF QPTJUJWF *O CVJMEJOH UIF NPEFM XF BMTP XBOU UP BMMPX TPNF PS BMM PG UIF QBSBNFUFST UP WBSZ CZ DPOUBDU SBUF 4JODF DPOUBDU SBUF JT TVQQPTF UP NFEJBUF UIF JOĘVFODF PG QPQVMBUJPO TJ[F MFUT BMMPX α BOE β *U DPVME BMTP JOĘVFODF γ CFDBVTF USBEF OFUXPSLT NJHIU QSFWFOU UPPMT GSPN WBOJTIJOH PWFS UJNF #VU XFMM MFBWF UIBU BT BO FYFSDJTF GPS UIF SFBEFS )FSFT UIF DPEF 3 DPEF  /Ǐ ʚǶ '$./ǿ ʙɶ/*/'Ǿ/**'.Ǣ ʙɶ+*+0'/$*)Ǣ $ʙɶ*)//Ǿ$ Ȁ (ǎǎǡǎǎ ʚǶ 0'(ǿ '$./ǿ  ʡ +*$.ǿ '( ȀǢ '( ʚǶ 3+ǿȁ$ȂȀȉʟȁ$Ȃȅ"Ǣ ȁ$Ȃ ʡ )*-(ǿǎǢǎȀǢ ȁ$Ȃ ʡ  3+ǿǎȀǢ " ʡ  3+ǿǎȀ ȀǢ /ʙ/Ǐ Ǣ #$).ʙǑ Ǣ '*"Ǿ'$&ʙ Ȁ *WF JOWFOUFE UIF FYBDU QSJPST CFIJOE UIF TDFOFT -FUT OPU HFU EJTUSBDUFE XJUI UIPTF * FODPVSBHF ZPV UP QMBZ BSPVOE ćF MFTTPO IFSF JT JO IPX XF CVJME JO UIF QSFEJDUPS WBSJBCMFT 6TJOH QSJPS TJNVMBUJPOT UP EFTJHO UIF QSJPST JT UIF TBNF BMUIPVHI FBTJFS OPX UIBU UIF QBSBNFUFST NFBO TPNFUIJOH 'JOBMMZ UIF DPEF UP QSPEVDF QPTUFSJPS QSFEJDUJPOT JT OP EJČFSFOU UIBO UIF DPEF JO UIF NBJO UFYU VTFE UP QMPU QSFEJDUJPOT GPS (ǎǎǡǎǍ
45. Science pays   (0% 41*,&% 5)& */5&(&34 0 50000 150000 250000 20 30 40 50 60 70 population total tools low contact high contact 'ĶĴłĿIJ ƉƉƉƈ 1PTUFSJPS QSFEJDUJPOT GPS TDJFOUJĕD NPEFM PG UIF 0DFBOJD UPPM DPVO $PNQBSF UP UIF SJHIU IBOE QMPU JO ' łĿIJ ƉƉƑ 4JODF UIJT NPEFM GPSDFT UIF USFOE QBTT UISPVHI UIF PSJHJO BT JU NVTU JUT CFIBW JT NPSF TFOTJCMF JO BEEJUJPO UP IBWJOH QBSB FUFST XJUI NFBOJOH PVUTJEF B MJOFBS NPEFM 8IBU XF XBOU JT B EZOBNJD NPEFM PG UIF DVMUVSBM FWPMVUJPO PG UPPMT 5PPMT BSFOU DSFB   (0% 41*,&% 5)& */5&(&34 -1 0 1 2 0 20 40 60 log population (std) total tools Yap (0.6) Trobriand (0.56) Tonga (0.69) Hawaii (1.01) 0 50000 150000 250000 0 20 40 60 population total tools 'ĶĴłĿIJ ƉƉƑ 1PTUFSJPS QSFEJDUJPOT GPS UIF 0DFBOJD UPPMT NPEFM 'JMMFE QPJOUT BSF TPDJFUJFT XJUI IJTUPSJDBMMZ IJHI DPOUBDU 0QFO QPJOUT BSF UIPTF XJUI MPX DPOUBDU 1PJOU TJ[F JT TDBMFE CZ SFMBUJWF -00*4 1BSFUP L WBMVFT Scientific model Statistical model Model violations now mean something. Parameters now mean something.
46. Poisson exposure (offsets) • Poisson outcome: events per unit time/distance • Q: What if time/distance varies across cases? • A: Use an exposure, aka offset IFOPNFOB DBO SFNBJO FYQPOFOUJBM GPS MPOH 4P POF UIJOH UP BMXBZT IFUIFS JU NBLFT TFOTF BU BMM SBOHFT PG UIF QSFEJDUPS WBSJBCMFT F FYQFDUFE WBMVF CVU JUT BMTP DPNNPOMZ UIPVHIU PG BT B SBUF #PUI U BOE SFBMJ[JOH UIJT BMMPXT VT UP NBLF 1PJTTPO NPEFMT GPS XIJDI UIF FT J 4VQQPTF GPS FYBNQMF UIBU B OFJHICPSJOH NPOBTUFSZ QFSGPSNT E NBOVTDSJQUT XIJMF ZPVS NPOBTUFSZ EPFT EBJMZ UPUBMT *G ZPV DPNF UT PG SFDPSET IPX DPVME ZPV BOBMZ[F CPUI JO UIF TBNF NPEFM HJWFO HBUFE PWFS EJČFSFOU BNPVOUT PG UJNF EJČFSFOU FYQPTVSFT Z λ JT FRVBM UP BO FYQFDUFE OVNCFS PG FWFOUT µ QFS VOJU UJNF PS IBU λ = µ/τ XIJDI MFUT VT SFEFĕOF UIF MJOL ZJ ∼ 1PJTTPO(λJ) MPH λJ = MPH µJ τJ = α + βYJ BUJP JT UIF TBNF BT B EJČFSFODF PG MPHBSJUINT XF DBO BMTP XSJUF MPH λJ = MPH µJ − MPH τJ = α + βYJ YQPTVSFTw 4P JG EJČFSFOU PCTFSWBUJPOT J IBWF EJČFSFOU FYQPTVSFT FYQFDUFE WBMVF PO SPX J JT HJWFO CZ exposure expected count  10*440/ 3&(3&44*0/ NPEFM MJLF ZJ ∼ 1PJTTPO(µJ) MPH µJ = MPH τJ + α + βYJ XIFSF τ JT B DPMVNO JO UIF EBUB 4P UIJT JT KVTU MJLF BEEJOH B QSF FYQPTVSF XJUIPVU BEEJOH B QBSBNFUFS GPS JU ćFSF XJMM CF BO FY  &YBNQMF 0DFBOJD UPPM DPNQMFYJUZ )FSFT BO FYBNQMF JTMBOE TPDJFUJFT PG 0DFBOJB QSPWJEF B OBUVSBM FYQFSJNFOU JO UFDI
47. Additional count distributions • Multinomial/categorical: generalized binomial, more than 2 un-ordered outcomes • Geometric: number of trials until specific event • Mixtures, coping with heterogeneity: • Beta-binomial: varying probabilities • gamma-Poisson: aka negative-Binomial, varying rates • others (e.g. Dirichlet-multinomial)
48. Survival Analysis • Count models are fundamentally about rates • Rate of heads per coin toss • Rate of tools per person • Can also estimate rates by modeling time-to-event • Tricky, because cannot ignore censored cases • Left-censored: Don’t know when time started • Right-censored: Something cut observation off before event occurred • Ignoring censored cases leads to inferential error • Imagine estimating time-to-PhD but ignoring people who drop out • Time in program before dropping out is info about rate
49. Survival Analysis • Example: Cat adoptions • data(AustinCats) • 20-thousand cats • time-to-event • Event either: (1) adopted or (2) something else • Something else could be: death, escape, censored
50. Un-censored observations • For observed adoptions, just need: OU UIFO XBJUJOH UJNFT DBO FOE VQ MPPLJOH WFSZ (BVTTJBO Y JPOT QSPCBCJMJUZ PG PCTFSWFE XBJUJOH UJNF JT TJNQMZ %J ∼ &YQPOFOUJBM(λJ) Q(%J|λJ) = λJ FYQ(−λJ %J) UT UIBU BSF USJDLZ *G TPNFUIJOH FMTF IBQQFOFE CFGPSF B DBU DPVME CF IBTOU CFFO BEPQUFE ZFU UIFO XF OFFE UIF QSPCBCJMJUZ PG OPU CFJOH O UIF PCTFSWBUJPO UJNF TP GBS 0OF XBZ UP NPUJWBUF UIJT JT UP JNBHF KPJOJOH UIF TIFMUFS PO UIF TBNF EBZ *G IBMG IBWF CFFO BEPQUFE BęFS BCJMJUZ PG XBJUJOH  EBZT BOE TUJMM OPU CFJOH BEPQUFE JT  *G BęFS O UIFO UIF QSPCBCJMJUZ PG XBJUJOH  EBZT BOE OPU ZFU CFJOH BEPQUFE PG BEPQUJPO JNQMJFT B QSPQPSUJPO PG UIF DPIPSU PG  DBUT UIBU XJMM 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 x dexp(x) probability event happens at time x λ = 0.5 λ = 1.0  4JNVMBUFE DBUT Y  "DUVBM DBUT Y 'PS PCTFSWFE BEPQUJPOT QSPCBCJMJUZ PG PCTFSWFE XBJUJOH UJNF JT %J ∼ &YQPOFOUJBM(λJ) *O SBXFS GPSN Q(%J|λJ) = λJ FYQ(−λJ %J) *UT UIF DFOTPSFE DBUT UIBU BSF USJDLZ *G TPNFUIJOH FMTF IBQQFOF BEPQUFE PS JU TJNQMZ IBTOU CFFO BEPQUFE ZFU UIFO XF OFFE UIF Q BEPQUFE DPOEJUJPOBM PO UIF PCTFSWBUJPO UJNF TP GBS 0OF XBZ UP N B DPIPSU PG  DBUT BMM KPJOJOH UIF TIFMUFS PO UIF TBNF EBZ *G IBMG I  EBZT UIFO UIF QSPCBCJMJUZ PG XBJUJOH  EBZT BOE TUJMM OPU CFJOH  EBZT POMZ  SFNBJO UIFO UIF QSPCBCJMJUZ PG XBJUJOH  EBZT BOE JT  "OZ HJWFO SBUF PG BEPQUJPO JNQMJFT B QSPQPSUJPO PG UIF DPI SFNBJO BęFS BOZ HJWFO OVNCFS PG EBZT
51. Censored cats • Cumulative distribution (CDF): Probability event before-or-at time x • Complementary cumulative distribution (CCDF): Probability not-event-yet 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 x pexp(x) probability event before-or-at time x 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 x exp(-x) probability not-event before-or-at time x λ = 0.5 λ = 1.0 CDF CCDF  (0% 41*,&% 5)& */5&(&34 XF DPNFT GSPN UIF İłĺłĹĮŁĶŃIJ ĽĿļįĮįĶĹĶŁņ ıĶŀŁĿĶįłŁĶļĻ " DV HJWFT UIF QSPQPSUJPO PG DBUT BEPQUFE CFGPSF PS BU B DFSUBJO OVNCFS PG UIF DVNVMBUJWF EJTUSJCVUJPO HJWFT UIF QSPCBCJMJUZ B DBU JT OPU BEPQUFE PG EBZT ćBU JT UIF QSPCBCJMJUZ UIBU XF OFFE ćJT EJTUSJCVUJPO‰POF F QSPCBCJMJUZ EJTUSJCVUJPO‰JT DBMMFE UIF İļĺĽĹIJĺIJĻŁĮĿņ İłĺłĹĮ ĶŀŁĿĶįłŁĶļĻ *O UIF DBTF PG UIF FYQPOFOUJBM EJTUSJCVUJPO UIF DVNVMB 1S(%J|λJ) =  − FYQ(−λJ %J) KVTU 1S(%J|λJ) = FYQ(−λJ %J) E JO PVS NPEFM TJODF JU JT QSPCBCJMJUZ PG XBJUJOH %J EBZT XJUIPVU CFJOH Ȁ ćF QSPCBCJMJUZ XF DPNFT GSPN UIF İłĺłĹĮŁĶŃIJ ĽĿļįĮįĶĹ NVMBUJWF EJTUSJCVUJPO HJWFT UIF QSPQPSUJPO PG DBUT BEPQUFE CFGPSF EBZT 4P POFNJOVT UIF DVNVMBUJWF EJTUSJCVUJPO HJWFT UIF QSPCB CZ UIF TBNF OVNCFS PG EBZT ćBU JT UIF QSPCBCJMJUZ UIBU XF OFFE NJOVT UIF DVNVMBUJWF QSPCBCJMJUZ EJTUSJCVUJPO‰JT DBMMFE UIF İļ ŁĶŃIJ ĽĿļįĮįĶĹĶŁņ ıĶŀŁĿĶįłŁĶļĻ *O UIF DBTF PG UIF FYQPOFOUJB UJWF JT 1S(%J|λJ) =  − FYQ(−λJ %J) 4P UIF DPNQMFNFOU JT KVTU 1S(%J|λJ) = FYQ(−λJ %J) 4P UIBUT XIBU XF OFFE JO PVS NPEFM TJODF JU JT QSPCBCJMJUZ PG XBJU BEPQUFE ZFU 3 DPEF  '$--4ǿ- /#$)&$)"Ȁ /ǿ0./$)/.Ȁ  ʚǶ 0./$)/. ɶ*+/ ʚǶ $! '. ǿ ɶ*0/Ǿ 1 )/ʙʙǫ*+/$*)ǫ Ǣ ǎ Ǣ Ǎ Ȁ ɶ'& ʚǶ $! '. ǿ ɶ*'*-ʙʙǫ'&ǫ Ǣ ǎ Ǣ Ǎ Ȁ
52. Cat code 4P UIBUT XIBU XF OFFE JO PVS NPEFM TJODF JU JT UIF QSPCBCJMJUZ PG XBJUJOH %J EBZT XJUIPVU CFJOH BEPQUFE ZFU ćF NPEFM %J|"J =  ∼ &YQPOFOUJBM(λJ) %J|"J =  ∼ &YQPOFOUJBM$$%'(λJ) λJ = /µJ MPH µJ = αİĶı[J] '$--4ǿ- /#$)&$)"Ȁ /ǿ0./$)/.Ȁ  ʚǶ 0./$)/. ɶ*+/ ʚǶ $! '. ǿ ɶ*0/Ǿ 1 )/ʙʙǫ*+/$*)ǫ Ǣ ǎ Ǣ Ǎ Ȁ / ʚǶ '$./ǿ 4.Ǿ/*Ǿ 1 )/ ʙ .ǡ)0( -$ǿ ɶ4.Ǿ/*Ǿ 1 )/ ȀǢ *'*-Ǿ$ ʙ $! '. ǿ ɶ*'*-ʙʙǫ'&ǫ Ǣ ǎ Ǣ Ǐ Ȁ Ǣ *+/  ʙ ɶ*+/ Ȁ (ǎǎǡǎǑ ʚǶ 0'(ǿ '$./ǿ EF  '$--4ǿ- /#$)&$)"Ȁ /ǿ0./$)/.Ȁ  ʚǶ 0./$)/. ɶ*+/ ʚǶ $! '. ǿ ɶ*0/Ǿ 1 )/ʙʙǫ*+/$*)ǫ Ǣ ǎ Ǣ Ǎ Ȁ / ʚǶ '$./ǿ 4.Ǿ/*Ǿ 1 )/ ʙ .ǡ)0( -$ǿ ɶ4.Ǿ/*Ǿ 1 )/ ȀǢ *'*-Ǿ$ ʙ $! '. ǿ ɶ*'*-ʙʙǫ'&ǫ Ǣ ǎ Ǣ Ǐ Ȁ Ǣ *+/  ʙ ɶ*+/ Ȁ (ǎǎǡǎǑ ʚǶ 0'(ǿ '$./ǿ 4.Ǿ/*Ǿ 1 )/Ȇ*+/ ʙʙǎ ʡ 3+*) )/$'ǿ '( ȀǢ 4.Ǿ/*Ǿ 1 )/Ȇ*+/ ʙʙǍ ʡ 0./*(ǿ 3+*) )/$'Ǿ'!ǿ Ǧ Ȇ '( ȀȀǢ '( ʚǶ ǎǡǍȅ(0Ǣ '*"ǿ(0Ȁ ʚǶ ȁ*'*-Ǿ$ȂǢ ȁ*'*-Ǿ$Ȃ ʡ )*-('ǿǍǢǎȀ ȀǢ /ʙ/ Ǣ #$).ʙǑ Ǣ *- .ʙǑ Ȁ +- $.ǿ (ǎǎǡǎǑ Ǣ Ǐ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ )Ǿ !! #/ ȁǎȂ ǑǡǍǒ ǍǡǍǐ ǑǡǍǎ ǑǡǍǖ ǎǑǍǒ ǎ ȁǏȂ ǐǡǕǕ ǍǡǍǎ ǐǡǕǔ ǐǡǖǍ ǎǑǍǐ ǎ
53. Other cats Posterior survival curves 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Days Proportion remaining Black cats Other cats  $&/403*/( "/% +*./ ʚǶ 3/-/ǡ.(+' .ǿ (ǎǎǡǎǑ Ȁ +*./ɶ ʚǶ 3+ǿ+*./ɶȀ +- $.ǿ +*./ Ǣ Ǐ Ȁ Ǫ/ǡ!-( Ǫǣ ǏǍǍǍ *.ǡ *! Ǒ 1-$' .ǣ ( ) . ǒǡǒʉ ǖǑǡǒʉ #$./*"-( ȁǎȂ ǑǡǍǒ ǍǡǍǐ ǑǡǍǎ ǑǡǍǖ ΤΤΤΥΨΪΨΥΤΤ ȁǏȂ ǐǡǕǕ ǍǡǍǎ ǐǡǕǔ ǐǡǖǍ ΤΤΥΪΪΦΤΤ ȁǎȂ ǒǔǡǑǑ ǎǡǑǔ ǒǒǡǎǎ ǒǖǡǔǔ ΤΤΤΦΪΪΨΥΤΤΤ ȁǏȂ ǑǕǡǑǑ ǍǡǑǖ Ǒǔǡǔǎ ǑǖǡǏǏ ΤΤΦΪΪΥΤΤ 0WFSUIJOLJOH $VTUPN EJTUSJCVUJPOT JO 4UBO ćF TVS BTTJHONFOU UP EFĕOF UIF DPNQMFNFOUBSZ DVNVMBUJWF E FBTZ UP BEE BOZ DVTUPN QSPCBCJMJUZ EJTUSJCVUJPO UIJT XB EJTUSJCVUJPO JO 0'( BDUVBM EPFT JO UIF 4UBO DPEF TP ZPV PO UIF NPEFM CMPDL PG ./)* ǿ(ǍǍȀ (* 'ȃ 1 /*-ȁǏǏǐǒǓȂ '(Ǥ  ʡ )*-('ǿ Ǎ Ǣ ǎ ȀǤ  ʡ )*-('ǿ Ǎ Ǣ ǎ ȀǤ
54. Homework • 3 problems, 2 data sets, multiple good DAGs • One of the data sets (NWOGrants) is new in rethinking 1.83, so update • Next week: More adventures with integers