Statistical Rethinking - Lecture 16 (part 2)

Lecture 16 (part 2) - Multilevel model introduction, varying intercepts - Statistical Rethinking: A Bayesian Course with R Examples

Transcript

1. Week 8.5: Amnesia & the Multilevel Model Richard McElreath Statistical Rethinking
2. Anterograde amnesia • Musicologist and conductor Clive Wearing • Lost parts of prefrontal and hippocampus • Can still play piano • Can’t remember what happened 1 min ago
3. Learning, forward and back
4. Depends upon variation
5. Anterograde amnesia • Statistical models sometimes have amnesia  $06/5*/( "/% $-"44*'*$"5*0/ 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
6. Anterograde amnesia • “Fixed effects” models have anterograde amnesia • Every new cluster (individual, pond, road, classroom) is a new world • No information passed among clusters • Multilevel models remember and pool information • Properties of clusters come from a population • Inferred population defines pooling
7. cit-consent onor with- omics view, e to the de- nd too little ew has led a regulated ased (8, 9), nors’ fami- estions that perty upon to change ad. In clas- have a lim- t consistent choose an s from re- onstructed, e minds of (14–16). If fault (18). Finally, defaults often represent the existing state or status quo, and change usually involves a trade-off. Psychologists have shown that losses loom larger than the equivalent gains, a phenomenon known as loss aversion (19). Thus, changes in the de- fault may result in a change of choice. cantly lower. In the last two decades, a number of European countries have had opt-in or opt- out default options for individuals’ deci- sions to become organ donors. Actual deci- sions about organ donation may be affected by governmental educational programs, the ion Sciences, 27, USA. be addressed: 4.25 27.5 17.17 12 99.98 98 99.91 99.97 99.5 99.64 85.9 0 10 20 30 40 50 60 70 80 90 100 Denmark Netherlands Effective consent percentage United Kingdom Germany Austria Belgium France Hungary Poland Portugal Sweden Effective consent rates, by country. Explicit consent (opt-in, gold) and presumed consent (opt- out, blue). 21 NOVEMBER 2003 VOL 302 SCIENCE www.sciencemag.org opt-in opt-out organ donation consent percentage
8. Multilevel should be default • Defaults are powerful things • Single-level regression is default • People justify multilevel models • This is backwards • Multilevel estimates usually better • Should have to justify not using multilevel model
9. Goals • Introduce multilevel models • How shrinkage and pooling work • Why they produce better estimates • How to fit with map2stan • Methods of plotting and comparing • Advanced: Continuous categories and Gaussian process regression
10. Multilevel models • Usual use is to model clustering • Classrooms within schools • Students within classrooms • Grades within students • Questions within exams • Repeat measures of units • Imbalance in sampling • “pseudoreplication”
11. • Examples from earlier: • !Kung individuals in families • Species in clades • Nations in continents • Pulls in chimpanzees in blocks • Applicants in departments Multilevel models
12. Example: Tadpole predation • Numbers of surviving tadpoles • Different densities/sizes • With and without predators • We’ll focus on variation across tanks (for now) β BOE τ BT QBSBNFUFST 5P HFU UIF QPTUFSJPS GPS θ ZPVE KVTU FYQPOFOUJB GPS τ *MM VTF UIJT LJOE PG MJOL JO POF PG UIF NPEFM ĕUT UP DPNF TP ZPVMM H MPPLT MJLF JO DPEF GPSN  &YBNQMF #PMLFST 3FFEGSPHT 8FMM VTF BO FYBNQMF BMTP GSPN # UBMJUZ EBUB PO SFFE GSPH UBEQPMFT WBSJBCMZ FYQPTFE UP BRVBUJD QSFEBUPST BU EFUFSNJOFE EFOTJUJFT :PV DBO MPBE UIF QBDLBHF BOE EBUB XJUI )&//6ǯ/"1%&+(&+$ǰ !1ǯ/""!#/,$0ǰ ! ʆǦ /""!#/,$0 ćF EBUB GSBNF IBT  SPXT BOE  DPMVNOT 8FSF HPJOH UP CF JOUFSFTUFE JO Q UIF OVNCFS PG UBEQPMFT UIBU TVSWJWFE UIF EVSBUJPO PG UIF FYQFSJNFOU PVS P BU UIF TUBSU ćF QSFEJDUPS WBSJBCMFT PG JOUFSFTU XJMM CF !"+0&16 -/"! BCTFODF PG QSFEBUPST BOE 0&7" UIF TJ[F PG UBEQPMFT *OUFSDFQUPOMZ NPEFMT *UMM CF VTFGVM UP CFHJO XJUI TJNQMF NPEFMT UI QSFEJDUPST TP ZPV DBO HFU B TFOTF GPS IPX B CFUBCJOPNJBM NPEFM DPNQ UJPO UP B QMBJO CJOPNJBM NPEFM 4P MFUT ĕU CPUI UP UIFTF EBUB BO PMEGBTI SFHSFTTJPO BOE B CFUBCJOPNJBM SFHSFTTJPO 'JSTU UIF SFHVMBS CJOPNJBM NPEFM ćJT DPEF JT KVTU MJLF ZPVE FYQFDU B *ƾƿǑǁ ʆǦ *-ǯ )&01ǯ
13. Tadpole models • Structure: • Tadpoles in tanks, different treatments • Outcome: number surviving • Can fit two basic models: 1. Dummy variable for each tank 2. Multilevel model with varying intercepts by tank
14. Regularized intercepts number surviving, tank i   .6-5*-&7&- .0%&-4 )FSF JT B NPEFM GPS QSFEJDUJOH UBEQPMF NPSUBMJUZ JO FBDI UBOL VTJOH QSJPST PG FBSMJFS DIBQUFST TJ ∼ #JOPNJBM(OJ, QJ) MPHJU(QJ) = αŁĮĻĸ[J] >XQLTXHORJR αŁĮĻĸ ∼ /PSNBM(, ) >ZHDN "OE ZPV DBO ĕU UIJT UP UIF EBUB JO UIF TUBOEBSE XBZ VTJOH (+ PS (+Ǐ (+Ǐ./) GSPN IFSF POXBSET CFDBVTF UIF OFYU NPEFM XJMM OPU XPSL JO ( EBMMJFE JO JOTUBMMJOH 4UBO NDTUBOPSH UBLF B CSFBL OPX BOE HP EP UIBU (P PMMPX UIF JOTUSVDUJPOT GPS ZPVS QMBUGPSN ćFO FYFDVUF UIJT DPEF '$--4ǿ- /#$)&$)"Ȁ /ǿ- !-*".Ȁ regularizing prior
15. Regularized intercepts )FSF JT B NPEFM GPS QSFEJDUJOH UBEQPMF NPSUBMJUZ JO FBDI UBOL VTJOH QSJPST PG FBSMJFS DIBQUFST TJ ∼ #JOPNJBM(OJ, QJ) MPHJU(QJ) = αŁĮĻĸ[J] >XQLTXHORJR αŁĮĻĸ ∼ /PSNBM(, ) >ZHDN "OE ZPV DBO ĕU UIJT UP UIF EBUB JO UIF TUBOEBSE XBZ VTJOH (+ PS (+Ǐ (+Ǐ./) GSPN IFSF POXBSET CFDBVTF UIF OFYU NPEFM XJMM OPU XPSL JO ( EBMMJFE JO JOTUBMMJOH 4UBO NDTUBOPSH UBLF B CSFBL OPX BOE HP EP UIBU (P PMMPX UIF JOTUSVDUJPOT GPS ZPVS QMBUGPSN ćFO FYFDVUF UIJT DPEF '$--4ǿ- /#$)&$)"Ȁ /ǿ- !-*".Ȁ  ʚǶ - !-*". ȕ (& /# /)& '0./ - 1-$' ɶ/)& ʚǶ ǎǣ)-*2ǿȀ )FSF JT B NPEFM GPS QSFEJDUJOH UBEQPMF NPSUBMJUZ JO FBDI UBOL VTJOH UIF SFHVMBSJ[JOH QSJPST PG FBSMJFS DIBQUFST TJ ∼ #JOPNJBM(OJ, QJ) >OLNHOLKRRG@ MPHJU(QJ) = αŁĮĻĸ[J] >XQLTXHORJRGGVIRUHDFKWDQN J@ αŁĮĻĸ ∼ /PSNBM(, ) >ZHDNO\UHJXODUL]LQJSULRU@ "OE ZPV DBO ĕU UIJT UP UIF EBUB JO UIF TUBOEBSE XBZ VTJOH (+ PS (+Ǐ./) 8FMM VTF (+Ǐ./) GSPN IFSF POXBSET CFDBVTF UIF OFYU NPEFM XJMM OPU XPSL JO (+ 4P JG ZPVWF EBMMJFE JO JOTUBMMJOH 4UBO NDTUBOPSH UBLF B CSFBL OPX BOE HP EP UIBU (P UP UIF 63- BOE GPMMPX UIF JOTUSVDUJPOT GPS ZPVS QMBUGPSN ćFO FYFDVUF UIJT DPEF 3 DPEF  '$--4ǿ- /#$)&$)"Ȁ /ǿ- !-*".Ȁ  ʚǶ - !-*". ȕ (& /# /)& '0./ - 1-$' ɶ/)& ʚǶ ǎǣ)-*2ǿȀ ȕ !$/ (ǎǏǡǎ ʚǶ (+Ǐ./)ǿ '$./ǿ .0-1 ʡ $)*(ǿ  ).$/4 Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ Ǿ/)&ȁ/)&Ȃ Ǣ Ǿ/)&ȁ/)&Ȃ ʡ )*-(ǿ Ǎ Ǣ ǒ Ȁ ȀǢ /ʙ Ȁ
16. Adaptive regularization *G ZPV JOTQFDU UIF FTUJNBUFT +- $.ǿ(ǎǏǡǎǢ +/#ʙǏȀ ZPVMM TFF  EJČFS TFUT POF GPS FBDI UBOL 5P HFU FBDI UBOLT FYQFDUFE TVSWJWBM QSPCBCJMJUZ KVT Ǿ/)& WBMVFT BOE UIFO VTF UIF MPHJTUJD USBOTGPSN 4P GBS OPUIJOH OFX /PX MFUT ĕU UIF NVMUJMFWFM NPEFM XIJDI BEBQUJWFMZ QPPMT JOGPSNBUJPO UIBU JT SFRVJSFE UP FOBCMF BEBQUJWF QPPMJOH JT UP NBLF UIF QSJPS GPS UIF Ǿ/ GVODUJPO PG JUT PXO QBSBNFUFST )FSF JT UIF NVMUJMFWFM NPEFM JO NBUIFNBU TJ ∼ #JOPNJBM(OJ, QJ) MPHJU(QJ) = αŁĮĻĸ[J] >ORJRG αŁĮĻĸ ∼ /PSNBM(α, σ) >YDU\ α ∼ /PSNBM(, ) >SU σ ∼ )BMG$BVDIZ(, ) >SULRUIRUVWDQGDU /PUJDF UIBU UIF QSJPS GPS UIF αŁĮĻĸ JOUFSDFQUT JT OPX B GVODUJPO PG UXP QBSB ćJT JT XIFSF UIF iNVMUJw JO NVMUJMFWFM BSJTFT ćF (BVTTJBO EJTUSJCVUJPO X TUBOEBSE EFWJBUJPO σ JT UIF QSJPS GPS FBDI UBOLT JOUFSDFQU #VU UIBU QSJPS JUT α BOE σ 4P UIFSF BSF UXP MFWFMT JO UIF NPEFM FBDI SFTFNCMJOH B TJNQMFS N MFWFM UIF PVUDPNF JT T UIF QBSBNFUFST BSF αŁĮĻĸ BOE UIF QSJPS JT αŁĮĻĸ ∼ *O UIF TFDPOE MFWFM UIF iPVUDPNFw WBSJBCMF JT UIF WFDUPS PG JOUFSDFQU QBSBN varying intercepts
17. Terminology • Varying intercepts also called random intercepts • Neither of these terms makes much sense • “random”? Sometimes associated with research design, but design irrelevant • Ordinary dummy variables also “vary” across clusters • Distinctive because individual intercepts learn from one another • mnestic: opposite of amnestic
18. Adaptive regularization *G ZPV JOTQFDU UIF FTUJNBUFT +- $.ǿ(ǎǏǡǎǢ +/#ʙǏȀ ZPVMM TFF  EJČFS TFUT POF GPS FBDI UBOL 5P HFU FBDI UBOLT FYQFDUFE TVSWJWBM QSPCBCJMJUZ KVT Ǿ/)& WBMVFT BOE UIFO VTF UIF MPHJTUJD USBOTGPSN 4P GBS OPUIJOH OFX /PX MFUT ĕU UIF NVMUJMFWFM NPEFM XIJDI BEBQUJWFMZ QPPMT JOGPSNBUJPO UIBU JT SFRVJSFE UP FOBCMF BEBQUJWF QPPMJOH JT UP NBLF UIF QSJPS GPS UIF Ǿ/ GVODUJPO PG JUT PXO QBSBNFUFST )FSF JT UIF NVMUJMFWFM NPEFM JO NBUIFNBU TJ ∼ #JOPNJBM(OJ, QJ) MPHJU(QJ) = αŁĮĻĸ[J] >ORJRG αŁĮĻĸ ∼ /PSNBM(α, σ) >YDU\ α ∼ /PSNBM(, ) >SU σ ∼ )BMG$BVDIZ(, ) >SULRUIRUVWDQGDU /PUJDF UIBU UIF QSJPS GPS UIF αŁĮĻĸ JOUFSDFQUT JT OPX B GVODUJPO PG UXP QBSB ćJT JT XIFSF UIF iNVMUJw JO NVMUJMFWFM BSJTFT ćF (BVTTJBO EJTUSJCVUJPO X TUBOEBSE EFWJBUJPO σ JT UIF QSJPS GPS FBDI UBOLT JOUFSDFQU #VU UIBU QSJPS JUT α BOE σ 4P UIFSF BSF UXP MFWFMT JO UIF NPEFM FBDI SFTFNCMJOH B TJNQMFS N MFWFM UIF PVUDPNF JT T UIF QBSBNFUFST BSF αŁĮĻĸ BOE UIF QSJPS JT αŁĮĻĸ ∼ *O UIF TFDPOE MFWFM UIF iPVUDPNFw WBSJBCMF JT UIF WFDUPS PG JOUFSDFQU QBSBN mean standard deviation varying intercepts
19. Adaptive regularization *G ZPV JOTQFDU UIF FTUJNBUFT +- $.ǿ(ǎǏǡǎǢ +/#ʙǏȀ ZPVMM TFF  EJČFS TFUT POF GPS FBDI UBOL 5P HFU FBDI UBOLT FYQFDUFE TVSWJWBM QSPCBCJMJUZ KVT Ǿ/)& WBMVFT BOE UIFO VTF UIF MPHJTUJD USBOTGPSN 4P GBS OPUIJOH OFX /PX MFUT ĕU UIF NVMUJMFWFM NPEFM XIJDI BEBQUJWFMZ QPPMT JOGPSNBUJPO UIBU JT SFRVJSFE UP FOBCMF BEBQUJWF QPPMJOH JT UP NBLF UIF QSJPS GPS UIF Ǿ/ GVODUJPO PG JUT PXO QBSBNFUFST )FSF JT UIF NVMUJMFWFM NPEFM JO NBUIFNBU TJ ∼ #JOPNJBM(OJ, QJ) MPHJU(QJ) = αŁĮĻĸ[J] >ORJRG αŁĮĻĸ ∼ /PSNBM(α, σ) >YDU\ α ∼ /PSNBM(, ) >SU σ ∼ )BMG$BVDIZ(, ) >SULRUIRUVWDQGDU /PUJDF UIBU UIF QSJPS GPS UIF αŁĮĻĸ JOUFSDFQUT JT OPX B GVODUJPO PG UXP QBSB ćJT JT XIFSF UIF iNVMUJw JO NVMUJMFWFM BSJTFT ćF (BVTTJBO EJTUSJCVUJPO X TUBOEBSE EFWJBUJPO σ JT UIF QSJPS GPS FBDI UBOLT JOUFSDFQU #VU UIBU QSJPS JUT α BOE σ 4P UIFSF BSF UXP MFWFMT JO UIF NPEFM FBDI SFTFNCMJOH B TJNQMFS N MFWFM UIF PVUDPNF JT T UIF QBSBNFUFST BSF αŁĮĻĸ BOE UIF QSJPS JT αŁĮĻĸ ∼ *O UIF TFDPOE MFWFM UIF iPVUDPNFw WBSJBCMF JT UIF WFDUPS PG JOUFSDFQU QBSBN varying intercepts Survival across tanks has a distribution. This distribution is the prior for each tank. Distribution needs its own prior.
20. GVODUJPO PG JUT PXO QBSBNFUFST )FSF JT UIF NVMUJMFWFM NPEFM JO NBUIFNBU TJ ∼ #JOPNJBM(OJ, QJ) MPHJU(QJ) = αŁĮĻĸ[J] >ORJRG αŁĮĻĸ ∼ /PSNBM(α, σ) >YDU\ α ∼ /PSNBM(, ) >SU σ ∼ )BMG$BVDIZ(, ) >SULRUIRUVWDQGDU /PUJDF UIBU UIF QSJPS GPS UIF αŁĮĻĸ JOUFSDFQUT JT OPX B GVODUJPO PG UXP QBSB ćJT JT XIFSF UIF iNVMUJw JO NVMUJMFWFM BSJTFT ćF (BVTTJBO EJTUSJCVUJPO X TUBOEBSE EFWJBUJPO σ JT UIF QSJPS GPS FBDI UBOLT JOUFSDFQU #VU UIBU QSJPS JUT α BOE σ 4P UIFSF BSF UXP MFWFMT JO UIF NPEFM FBDI SFTFNCMJOH B TJNQMFS N MFWFM UIF PVUDPNF JT T UIF QBSBNFUFST BSF αŁĮĻĸ BOE UIF QSJPS JT αŁĮĻĸ ∼ *O UIF TFDPOE MFWFM UIF iPVUDPNFw WBSJBCMF JT UIF WFDUPS PG JOUFSDFQU QBSBN QBSBNFUFST BSF α BOE σ BOE UIFJS QSJPST BSF α ∼ /PSNBM(, ) BOE σ ∼ ) 4P JG ZPV IBWF B HPPE SFBTPO UP VTF BOPUIFS EJTUSJCVUJPO UIFO EP TP ćF QSBDUJDF QSPCMFNT BU UIF FOE PG UIF DIBQUFS QSPWJEF BO FYBNQMF 'JUUJOH UIF NPEFM UP EBUB FTUJNBUFT CPUI MFWFMT TJNVMUBOFPVTMZ JO UIF TBNF XBZ UIBU PVS SPCPU BU UIF TUBSU PG UIF DIBQUFS MFBSOFE CPUI BCPVU FBDI DBGÏ BOE UIF WBSJBUJPO BNPOH DBGÏT #VU ZPV DBOOPU ĕU UIJT NPEFM XJUI (+ 8IZ #FDBVTF UIF MJLFMJIPPE NVTU OPX BWFSBHF PWFS UIF MFWFM  QBSBNFUFST α BOE σ BOE (+ KVTU IJMM DMJNCT VTJOH TUBUJD WBMVFT GPS BMM PG UIF QBSBNFUFST *U DBOU TFF UIF MFWFMT 'PS NPSF FYQMBOBUJPO TFF UIF 0WFSUIJOLJOH CPY GVSUIFS EPXO :PV DBO IPXFWFS ĕU UIJT NPEFM XJUI (+Ǐ./) 3 DPEF  (ǎǏǡǏ ʚǶ (+Ǐ./)ǿ '$./ǿ .0-1 ʡ $)*(ǿ  ).$/4 Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ Ǿ/)&ȁ/)&Ȃ Ǣ Ǿ/)&ȁ/)&Ȃ ʡ )*-(ǿ  Ǣ .$"( Ȁ Ǣ  ʡ )*-(ǿǍǢǎȀ Ǣ .$"( ʡ 0#4ǿǍǢǎȀ ȀǢ /ʙ Ǣ $/ -ʙǑǍǍǍ Ǣ #$).ʙǑ Ȁ ćJT NPEFM ĕU QSPWJEFT FTUJNBUFT GPS  QBSBNFUFST POF PWFSBMM TBNQMF JOUFSDFQU α UIF WBSJ BODF BNPOH UBOLT σ BOE UIFO  QFSUBOL JOUFSDFQUT -FUT DIFDL 8"*$ UIPVHI UP TFF UIF FČFDUJWF OVNCFS PG QBSBNFUFST 3 DPEF   ǿ(ǎǏǡǏȀ
21. :PV DBO IPXFWFS ĕU UIJT NPEFM XJUI (+Ǐ./) 3 DPEF  (ǎǏǡǏ ʚǶ (+Ǐ./)ǿ '$./ǿ .0-1 ʡ $)*(ǿ  ).$/4 Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ Ǿ/)&ȁ/)&Ȃ Ǣ Ǿ/)&ȁ/)&Ȃ ʡ )*-(ǿ  Ǣ .$"( Ȁ Ǣ  ʡ )*-(ǿǍǢǎȀ Ǣ .$"( ʡ 0#4ǿǍǢǎȀ ȀǢ /ʙ Ǣ $/ -ʙǑǍǍǍ Ǣ #$).ʙǑ Ȁ ćJT NPEFM ĕU QSPWJEFT FTUJNBUFT GPS  QBSBNFUFST POF PWFSBMM TBNQMF JOUFSDFQU α UIF WBSJ BODF BNPOH UBOLT σ BOE UIFO  QFSUBOL JOUFSDFQUT -FUT DIFDL 8"*$ UIPVHI UP TFF UIF FČFDUJWF OVNCFS PG QBSBNFUFST 3 DPEF   ǿ(ǎǏǡǏȀ ȁǎȂ ǎǍǎǍǡǏǐǐ //-ǿǢǫ'++ǫȀ ȁǎȂ ǶǑǓǔǡǍǓǐǔ //-ǿǢǫ+ ǫȀ ȁǎȂ ǐǕǡǍǒǏǔ //-ǿǢǫ. ǫȀ ȁǎȂ ǐǕǡǍǐǍǎǎ *U IBT MFTT UIBO  FČFDUJWF QBSBNFUFST ćFSF BSF  GFXFS FČFDUJWF QBSBNFUFST UIBO BDUVBM  '$./ǿ .0-1 ʡ $)*(ǿ  ).$/4 Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ Ǿ/)&ȁ/)&Ȃ Ǣ Ǿ/)&ȁ/)&Ȃ ʡ )*-(ǿ  Ǣ .$"( Ȁ Ǣ  ʡ )*-(ǿǍǢǎȀ Ǣ .$"( ʡ 0#4ǿǍǢǎȀ ȀǢ /ʙ Ǣ $/ -ʙǑǍǍǍ Ǣ #$).ʙǑ Ȁ ćJT NPEFM ĕU QSPWJEFT FTUJNBUFT GPS  QBSBNFUFST POF PWFSBMM TBNQMF JOUFSDFQU α UIF WBSJ BODF BNPOH UBOLT σ BOE UIFO  QFSUBOL JOUFSDFQUT -FUT DIFDL 8"*$ UIPVHI UP TFF UIF FČFDUJWF OVNCFS PG QBSBNFUFST 3 DPEF   ǿ(ǎǏǡǏȀ ȁǎȂ ǎǍǎǍǡǏǐǐ //-ǿǢǫ'++ǫȀ ȁǎȂ ǶǑǓǔǡǍǓǐǔ //-ǿǢǫ+ ǫȀ ȁǎȂ ǐǕǡǍǒǏǔ //-ǿǢǫ. ǫȀ ȁǎȂ ǐǕǡǍǐǍǎǎ *U IBT MFTT UIBO  FČFDUJWF QBSBNFUFST ćFSF BSF  GFXFS FČFDUJWF QBSBNFUFST UIBO BDUVBM QBSBNFUFST CFDBVTF UIF QSJPS BTTJHOFE UP FBDI JOUFSDFQU TISJOLT UIFN BMM UPXBSET UIF NFBO α *O UIJT DBTF UIF QSJPS JT SFBTPOBCMZ TUSPOH $IFDL UIF NFBO PG .$"( XJUI +- $. PS * ! BOE ZPVMM TFF JUT BSPVOE  ćJT JT B ĿIJĴłĹĮĿĶŇĶĻĴ ĽĿĶļĿ MJLF ZPVWF VTFE JO QSFWJPVT 48 tanks + a + sigma => 50 parameters
22. Regularizing distribution   .6-5*-&7&- .0%&-4 -3 -2 -1 0 1 2 3 4 0.00 0.10 0.20 0.30 log-odds survive Density 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 probability survive Density 'ĶĴłĿIJ ƉƊƊ ćF JOGFSSFE QPQVMBUJPO PG TVSWJWBM BDSPTT UBOLT -Fę  (BVTTJBO EJTUSJCVUJPOT PG UIF MPHPEET PG TVSWJWBM TBNQMFE GSPN UIF QPTUF SJPS PG (ǎǏǡǏ 3JHIU 4VSWJWBM QSPCBCJMJUJFT GPS UIPVTBOE OFX TJNVMBUFE UBOLT BWFSBHJOH PWFS UIF QPTUFSJPS EJTUSJCVUJPO PO UIF MFę
23. 0.0 0.2 0.4 0.6 0.8 1.0 tank proportion survival 1 16 32 48 small tanks medium tanks large tanks 'ĶĴłĿIJ ƉƊƉ &NQJSJDBM QSPQPSUJPOT PG TVSWJWPST JO FBDI UBEQPMF UBOL TIPXO CZ UIF ĕMMFE CMVF QPJOUT QMPUUFE XJUI UIF  QFSUBOL FTUJNBUFT GSPN UIF NVMUJMFWFM NPEFM TIPXO CZ UIF CMBDL DJSDMFT ćF EBTIFE MJOF MPDBUFT UIF PWFSBMM BWFSBHF QSPQPSUJPO PG TVSWJWPST BDSPTT BMM UBOLT ćF WFSUJDBM Don’t expect predictions to match observations exactly. Instead expect shrinkage. Fixed estimate Multilevel estimate