Upgrade to Pro — share decks privately, control downloads, hide ads and more …

L09 Statistical Rethinking Winter 2019

L09 Statistical Rethinking Winter 2019

Lecture 09 of the Dec 2018 through March 2019 edition of Statistical Rethinking. Covers Chapter 8, interactions.

A0f2f64b2e58f3bfa48296fb9ed73853?s=128

Richard McElreath

January 21, 2019
Tweet

Transcript

  1. Conditional Manatees Statistical Rethinking Winter 2019 Lecture 09 / Week

    5
  2. None
  3. Stop testing, start thinking • Off-the-shelf tools • p-values •

    information criteria • linear models • Good decisions benefit from bespoke tools • bespoke risk analysis • bespoke models Bespoke Broke
  4. None
  5. None
  6. Blizzard calibration • Was it bad to predict NYC blizzard

    from ECMWF? • Other models were more accurate • But welfare enhanced by being prepared => use extreme forecasts • Accuracy always matters, but it isn’t all that matters
  7. None
  8. None
  9. 'ĶĴłĿIJ ƏƉ ŁļĽ %PSTBM TDBST GPS  BEVMU 'MPSJEB NBOBUFFT

    3PXT PG Figure 8.1
  10.  */5&3"$5*0/4 'ĶĴłĿIJ ƏƉ ŁļĽ %PSTBM TDBST GPS  BEVMU

    'MPSJEB NBOBUFFT 3PXT PG TIPSU TDBST GPS FYBNQMF PO UIF JOEJWJEVBMT "GSJDB BOE 'MBTI BSF JOEJDBUJWF PG QSPQFMMFS MBDFSBUJPO įļŁŁļĺ ćSFF FYFNQMBST PG EBNBHF PO "8 CPNCFST SFUVSOJOH GSPN NJTTJPOT OE B MBSHF QBSU PG UIF QPXFS PG TUBUJTUJDBM NPEFMJOH DPNFT GSPN DSFBUJOH EFWJDFT MPX QSPCBCJMJUZ UP CF DPOEJUJPOBM PG BTQFDUT PG FBDI DBTF ćF MJOFBS NPEFMT ZPVWF Figure 8.1 rotor/wing survival keel/engine Observe only: undamaged rotor/wing damaged
  11. Manatees and bombers • Conditioning: Dependence on state • Everything

    is conditional • On data • On model • On information state • Influence of variable conditional on other variable(s)
  12. Interaction effects • Interactions: Influence of predictor conditional on other

    predictor(s) • Influence of sugar in coffee depends on stirring • Influence of gene on phenotype depends on environment • Influence of skin color on cancer depends on latitude • Generalized linear models (GLMs): All predictors interact to some degree • Multilevel models: Massive interaction engines
  13. Interaction effects in DAGs • In DAG, interaction doesn’t look

    special: • This just means: • We have to figure out the function f. coffee sweet sugar stirred coffee sweet = f (sugar, stirred)
  14. coffee sweet sugar stirred 0.0 0.2 0.4 0.6 0.8 1.0

    0.0 0.2 0.4 0.6 0.8 1.0 sugar sweetness not an interaction stirred not stirred 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 sugar sweetness interaction stirred not stirred
  15. None
  16. Figure 8.2   $0/%*5*0/"- ."/"5&&4 0.0 0.2 0.4 0.6

    0.8 1.0 0.8 0.9 1.0 1.1 ruggedness (standardized) log GDP (as proportion of mean) African nations Lesotho Seychelles 0.0 0.2 0.4 0.6 0.8 1.0 0.8 0.9 1.0 1.1 1.2 1.3 ruggedness (standardized) log GDP (as proportion of mean) Non-African nations Switzerland Tajikistan 'ĶĴłĿIJ ƐƊ 4FQBSBUF MJOFBS SFHSFTTJPOT JOTJEF BOE PVUTJEF PG "GSJDB GPS MPH (%1 BHBJOTU UFSSBJO SVHHFEOFTT ćF TMPQF JT QPTJUJWF JOTJEF "GSJDB CVU  .BLJOH UXP NPEFMT 5P HFU TUBSUFE MPBE UIF EBUB VTFE JO UIJT ĕHVSF BOE TQMJU JU J "GSJDB BOE OPO"GSJDB XJUI UIJT DPEF '$--4ǿ- /#$)&$)"Ȁ /ǿ-0"" Ȁ  ʚǶ -0""  ȕ (& '*" 1 -.$*) *! *0/*( ɶ'*"Ǿ"+ ʚǶ '*"ǿ ɶ-"++ǾǏǍǍǍ Ȁ ȕ 3/-/ *0)/-$ . 2$/#  /  ʚǶ ȁ *(+' / ǡ. .ǿɶ-"++ǾǏǍǍǍȀ Ǣ Ȃ ȕ - .' 1-$' . ɶ'*"Ǿ"+Ǿ./ ʚǶ ɶ'*"Ǿ"+ ȅ ( )ǿɶ'*"Ǿ"+Ȁ ɶ-0"" Ǿ./ ʚǶ ɶ-0""  ȅ (3ǿɶ-0"" Ȁ ȕ .+'$/ *0)/-$ . $)/* !-$ ) )*/Ƕ!-$ ǡǎ ʚǶ ȁ ɶ*)/Ǿ!-$ʙʙǎ Ǣ Ȃ ȕ !-$ ǡǍ ʚǶ ȁ ɶ*)/Ǿ!-$ʙʙǍ Ǣ Ȃ ȕ )*/ !-$ &BDI SPX JO UIFTF EBUB JT B DPVOUSZ BOE UIF WBSJPVT DPMVNOT BSF FDPOPNJD HFPHSBQIJD B IJTUPSJDBM GFBUVSFT $IBODFT BSF UIBU SBX NBHOJUVEFT PG (%1 BOE UFSSBJO SVHHFEOFTT BSFOU NFBOJOHGV ZPV 4P *WF TDBMFE UIF WBSJBCMFT UP NBLF UIF VOJUT NPSF VTFGVM ćF VTVBM TUBOEBSEJ[BU JT UP TVCUSBDU UIF NFBO BOE EJWJEF CZ UIF TUBOEBSE EFWJBUJPO ćJT NBLFT B WBSJBCMF J [TDPSFT 8F EPOU XBOU UP EP UIBU IFSF CFDBVTF [FSP JT NFBOJOHGVM 4P JOTUFBE UFSS
  17. The sermon on priors  #6*-%*/( "/ */5&3"$5*0/  0.0

    0.2 0.4 0.6 0.8 1.0 0.6 0.8 1.0 1.2 1.4 ruggedness log GDP (prop of mean) a ~ dnorm(1, 1) b ~ dnorm(0, 1) 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.8 1.0 1.2 1.4 ruggedness log GDP (prop of mean) a ~ dnorm(1, 0.1) b ~ dnorm(0, 0.3) 'ĶĴłĿIJ ƐƋ 4JNVMBUJOH JO TFBSDI PG SFBTPOBCMF QSJPST GPS UIF UFSSBJO SVHHFE OFTT FYBNQMF ćF EBTIFE IPSJ[POUBM MJOFT JOEJDBUF UIF NJOJNVN BOE NBY JNVN PCTFSWFE (%1 WBMVFT -Fę ćF ĕSTU HVFTT XJUI WFSZ WBHVF QSJPST Figure 8.3
  18. • Splitting the data is a bad idea: • No

    inference for how you split the data • Does not pool information • How about adding a categorical variable for Africa? The value of being rugged   $0/%*5*0/" 0.0 0.2 0.4 0.6 0.8 1.0 0.8 0.9 1.0 1.1 ruggedness (standardized) log GDP (as proportion of mean) African nations Lesotho Seychelles 'ĶĴłĿIJ ƐƊ 4FQBSBUF MJOFBS SFHSFTTJPO (%1 BHBJOTU UFSSBJO SVHHFEOFTT ćF OFHBUJWF PVUTJEF )PX DBO XF SFDPWF DPNCJOFE EBUB MPTFT OP JOGPSNBUJPO *U KVTU DIBOHFT XIBU XF TPDJBUJPO CFUXFFO WBSJBCMFT *O UIJT DBTF SBX ( CFDBVTF PG JUT FYQPOFOUJBM QBUUFSO #VU MPH (% 8IBU JT HPJOH PO JO UIJT ĕHVSF *U NBLFT TF DPVOUSJFT JO NPTU PG UIF XPSME 3VHHFE UFSSBJ NBSLFU BDDFTT JT IBNQFSFE 8IJDI NFBOT SFEV SFMBUJPOTIJQ XJUIJO "GSJDB JT QV[[MJOH 8IZ TIP   $0/%*5*0/"- ."/"5&&4 0.0 0.2 0.4 0.6 0.8 1.0 0.8 0.9 1.0 1.1 ruggedness (standardized) log GDP (as proportion of mean) African nations Lesotho Seychelles 0.0 0.2 0.4 0.6 0.8 1.0 0.8 0.9 1.0 1.1 1.2 1.3 ruggedness (standardized) log GDP (as proportion of mean) Non-African nations Switzerland Tajikistan
  19. Category doesn’t work • Index variable for continent: Figure 8.4

    B JT NPSF VODFSUBJO CFGPSF TFFJOH UIF EBUB UIBO µJ PVUTJEF "GSJDB TF ćJT JT UIF TBNF JTTVF XF DPOGSPOUFE CBDL JO $IBQUFS  XIFO * WBSJBCMFT *O UIF QSPCMFNT BU UIF FOE PG UIJT DIBQUFS *MM BTL ZPV UP PSF EFUBJM BJHIU UP B TPMVUJPO /BUJPOT JO "GSJDB XJMM HFU POF JOUFSDFQU BOE UIPTF ćJT JT XIBU µJ MPPLT MJLF OPX µJ = αİĶı[J] + β(SJ − ¯ S) SJBCMF DPOUJOFOU *% *U UBLFT UIF WBMVF  GPS "GSJDBO OBUJPOT BOE  GPS NFBOT UIFSF BSF UXP QBSBNFUFST α BOE α POF GPS FBDI VOJRVF JOEFY ı[J] KVTU NFBOT UIF WBMVF PG İĶı PO SPX J * VTF UIF CSBDLFU OPUBUJPO DBVTF JU JT FBTJFS UP SFBE UIBO BEEJOH B TFDPOE MFWFM PG TVCTDSJQU αİĶıJ  PVSTFMWFT 3  ) 3 !-$ ǿǎȀ *- )*/ ǿǏȀ ɶ*)/Ǿ!-$ʙʙǎ Ǣ ǎ Ǣ Ǐ Ȁ   $0/%*5*0/"- ."/"5&&4 NJOE UIBU UIJT JT TUSVDUVSBMMZ UIF TBNF NPEFM ZPVE HFU JO UIF DPOWFOUJPOBM BQQSPBDI *U JT KVTU NVDI FBTJFS UIJT XBZ UP BTTJHO TFOTJCMF QSJPST 5P EFĕOF UIF NPEFM JO ,0+ XF OFFE UP BEE CSBDLFUT JO UIF MJOFBS NPEFM BOE UIF QSJPS F  (ǕǡǑ ʚǶ ,0+ǿ '$./ǿ '*"Ǿ"+Ǿ./ ʡ )*-(ǿ (0 Ǣ .$"( Ȁ Ǣ (0 ʚǶ ȁ$Ȃ ʔ ȉǿ -0"" Ǿ./ Ƕ ǍǡǏǎǒ Ȁ Ǣ ȁ$Ȃ ʡ )*-(ǿ ǎ Ǣ Ǎǡǎ Ȁ Ǣ  ʡ )*-(ǿ Ǎ Ǣ Ǎǡǐ Ȁ Ǣ .$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ Ȁ /PX UP DPNQBSF UIFTF NPEFMT VTJOH 8"*$ F  *(+- ǿ (Ǖǡǐ Ǣ (ǕǡǑ Ȁ  #6*-%*/( 0.0 0.2 0.4 0.6 0.8 1.0 0.7 0.8 0.9 1.0 1.1 1.2 1.3 ruggedness (standardized) log GDP (as proportion of mean) m8.4 Africa Not Africa ȕ *(+0/ (0 *1 - .(+' .Ǣ !$3$)" $
  20. Interaction • Need to allow effect of ruggedness to depend

    upon continent N JT µJ = αİĶı[J] + β(SJ − ¯ S) MM EPVCMFEPXO PO PVS JOEFYJOH UP NBLF UIF TMPQF DPOEJUJPOBM BT µJ = αİĶı[J] + βİĶı[J] (SJ − ¯ S) IFSF JT B DPOWFOUJPOBM BQQSPBDI UP TQFDJGZJOH BO JOUFSBDUJPO UIBU V OE B OFX JOUFSBDUJPO QBSBNFUFS *U XPVME MPPL MJLF UIJT µJ = αİĶı[J] + (β + γ"J)(SJ − ¯ S) B  JOEJDBUPS GPS "GSJDBO OBUJPOT ćJT JT FRVJWBMFOU UP PVS JOE I IBSEFS UP TUBUF TFOTJCMF QSJPST "OZ QSJPS XF QVU PO γ NBLFT UI VODFSUBJO UIBO UIF TMPQF PVUTJEF "GSJDB "OE BHBJO UIBU NBLFT O
  21. 3 → ( ← $ XIFSF 3 JT SVHHFEOFTT (

    JT (%1 BOE $ JT DPOUJOFOU %"(T BSF IFVSJTUJD TPNF WBSJBCMF JT B GVODUJPO PG TPNF PUIFST ( = G(3, $) ćFZ EPOU TBZ JT 4P XIFO XF TUBSU CVJMEJOH JOUFSBDUJPO NPEFMT XF BSF HPJOH CFZPOE UIF %"( /FJUIFS TUBUJTUJDBM NPEFMT OPS %"(T BSF DPNQMFUF SFQSFTFOUBUJ 5P BQQSPYJNBUF UIF QPTUFSJPS PG UIJT OFX NPEFM ZPV DBO KVTU VTF ,0 UIF DPEF UIBU JODMVEFT BO JOUFSBDUJPO CFUXFFO SVHHFEOFTT BOE CFJOH JO " 3 DPEF  (Ǖǡǒ ʚǶ ,0+ǿ '$./ǿ '*"Ǿ"+Ǿ./ ʡ )*-(ǿ (0 Ǣ .$"( Ȁ Ǣ (0 ʚǶ ȁ$Ȃ ʔ ȁ$Ȃȉǿ -0"" Ǿ./ Ƕ ǍǡǏǎǒ Ȁ Ǣ ȁ$Ȃ ʡ )*-(ǿ ǎ Ǣ Ǎǡǎ Ȁ Ǣ ȁ$Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǐ Ȁ Ǣ .$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ Ȁ -FUT JOTQFDU UIF NBSHJOBM QPTUFSJPS EJTUSJCVUJPOT 3 DPEF  +- $.ǿ (Ǖǡǒ Ǣ  +/#ʙǏ Ȁ Interaction µJ = αİĶı[J] + β(SJ − ¯ S) MM EPVCMFEPXO PO PVS JOEFYJOH UP NBLF UIF TMPQF DPOEJUJPOBM BT µJ = αİĶı[J] + βİĶı[J] (SJ − ¯ S) IFSF JT B DPOWFOUJPOBM BQQSPBDI UP TQFDJGZJOH BO JOUFSBDUJPO UIBU V OE B OFX JOUFSBDUJPO QBSBNFUFS *U XPVME MPPL MJLF UIJT µJ = αİĶı[J] + (β + γ"J)(SJ − ¯ S) B  JOEJDBUPS GPS "GSJDBO OBUJPOT ćJT JT FRVJWBMFOU UP PVS JOE I IBSEFS UP TUBUF TFOTJCMF QSJPST "OZ QSJPS XF QVU PO γ NBLFT UI VODFSUBJO UIBO UIF TMPQF PVUTJEF "GSJDB "OE BHBJO UIBU NBLFT O OH BQQSPBDI XF DBO FBTJMZ BTTJHO UIF TBNF QSJPS UP UIF TMPQF OP TBM HSBQI MJLF B %"( BO JOUFSBDUJPO JT KVTU UXP BSSPXT FOUFSJOH
  22. UIF DPEF UIBU JODMVEFT BO JOUFSBDUJPO CFUXFFO SVHHFEOFTT BOE CFJOH

    JO " 3 DPEF  (Ǖǡǒ ʚǶ ,0+ǿ '$./ǿ '*"Ǿ"+Ǿ./ ʡ )*-(ǿ (0 Ǣ .$"( Ȁ Ǣ (0 ʚǶ ȁ$Ȃ ʔ ȁ$Ȃȉǿ -0"" Ǿ./ Ƕ ǍǡǏǎǒ Ȁ Ǣ ȁ$Ȃ ʡ )*-(ǿ ǎ Ǣ Ǎǡǎ Ȁ Ǣ ȁ$Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǐ Ȁ Ǣ .$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ Ȁ -FUT JOTQFDU UIF NBSHJOBM QPTUFSJPS EJTUSJCVUJPOT 3 DPEF  +- $.ǿ (Ǖǡǒ Ǣ  +/#ʙǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ ȁǎȂ ǍǡǕǖ ǍǡǍǏ ǍǡǕǓ Ǎǡǖǎ ȁǏȂ ǎǡǍǒ ǍǡǍǎ ǎǡǍǐ ǎǡǍǔ ȁǎȂ Ǎǡǎǐ ǍǡǍǔ ǍǡǍǎ ǍǡǏǒ ȁǏȂ ǶǍǡǎǑ ǍǡǍǒ ǶǍǡǏǐ ǶǍǡǍǓ .$"( Ǎǡǎǎ ǍǡǍǎ ǍǡǎǍ ǍǡǎǏ UIF DPEF UIBU JODMVEFT BO JOUFSBDUJPO CFUXFFO SVHHFEOFTT BOE CFJOH JO "GSJDB 3 DPEF  (Ǖǡǒ ʚǶ ,0+ǿ '$./ǿ '*"Ǿ"+Ǿ./ ʡ )*-(ǿ (0 Ǣ .$"( Ȁ Ǣ (0 ʚǶ ȁ$Ȃ ʔ ȁ$Ȃȉǿ -0"" Ǿ./ Ƕ ǍǡǏǎǒ Ȁ Ǣ ȁ$Ȃ ʡ )*-(ǿ ǎ Ǣ Ǎǡǎ Ȁ Ǣ ȁ$Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǐ Ȁ Ǣ .$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ Ȁ -FUT JOTQFDU UIF NBSHJOBM QPTUFSJPS EJTUSJCVUJPOT 3 DPEF  +- $.ǿ (Ǖǡǒ Ǣ  +/#ʙǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ ȁǎȂ ǍǡǕǖ ǍǡǍǏ ǍǡǕǓ Ǎǡǖǎ ȁǏȂ ǎǡǍǒ ǍǡǍǎ ǎǡǍǐ ǎǡǍǔ ȁǎȂ Ǎǡǎǐ ǍǡǍǔ ǍǡǍǎ ǍǡǏǒ ȁǏȂ ǶǍǡǎǑ ǍǡǍǒ ǶǍǡǏǐ ǶǍǡǍǓ .$"( Ǎǡǎǎ ǍǡǍǎ ǍǡǎǍ ǍǡǎǏ ćF TMPQF JT FTTFOUJBMMZ SFWFSTFE JOTJEF "GSJDB  JOTUFBE PG −.
  23.  4:..&53: 0' */5&3"$5*0/4  0.0 0.2 0.4 0.6 0.8

    1.0 0.8 0.9 1.0 1.1 ruggedness (standardized) log GDP (as proportion of mean) African nations Burundi Equatorial Guinea Lesotho Rwanda Swaziland Seychelles South Africa 0.0 0.2 0.4 0.6 0.8 1.0 0.8 0.9 1.0 1.1 1.2 1.3 ruggedness (standardized) log GDP (as proportion of mean) Non-African nations Switzerland Greece Lebanon Luxembourg Nepal Tajikistan Yemen 'ĶĴłĿIJ Ɛƍ 1PTUFSJPS QSFEJDUJPOT GPS UIF UFSSBJO SVHHFEOFTT NPEFM JODMVE JOH UIF JOUFSBDUJPO CFUXFFO "GSJDB BOE SVHHFEOFTT 4IBEFE SFHJPOT BSF  QPTUFSJPS JOUFSWBMT PG UIF NFBO Figure 8.5
  24. Interpreting interactions • Is hard • Add interaction => other

    parameters change meaning • Influence of predictor depends upon multiple parameters and their covariation JT 4P XIFO XF TUBSU CVJMEJOH JOUFSBDUJPO NPEFMT XF BSF HPJOH CFZPOE UIF JOGPSNB UIF %"( /FJUIFS TUBUJTUJDBM NPEFMT OPS %"(T BSF DPNQMFUF SFQSFTFOUBUJPOT 8F OFF 5P BQQSPYJNBUF UIF QPTUFSJPS PG UIJT OFX NPEFM ZPV DBO KVTU VTF ,0+ BT CFGPSF UIF DPEF UIBU JODMVEFT BO JOUFSBDUJPO CFUXFFO SVHHFEOFTT BOE CFJOH JO "GSJDB 3 DPEF  (Ǖǡǒ ʚǶ ,0+ǿ '$./ǿ '*"Ǿ"+Ǿ./ ʡ )*-(ǿ (0 Ǣ .$"( Ȁ Ǣ (0 ʚǶ ȁ$Ȃ ʔ ȁ$Ȃȉǿ -0"" Ǿ./ Ƕ ǍǡǏǎǒ Ȁ Ǣ ȁ$Ȃ ʡ )*-(ǿ ǎ Ǣ Ǎǡǎ Ȁ Ǣ ȁ$Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǐ Ȁ Ǣ .$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ Ȁ -FUT JOTQFDU UIF NBSHJOBM QPTUFSJPS EJTUSJCVUJPOT 3 DPEF  +- $.ǿ (Ǖǡǒ Ǣ  +/#ʙǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ ȁǎȂ ǍǡǕǖ ǍǡǍǏ ǍǡǕǓ Ǎǡǖǎ ȁǏȂ ǎǡǍǒ ǍǡǍǎ ǎǡǍǐ ǎǡǍǔ ȁǎȂ Ǎǡǎǐ ǍǡǍǔ ǍǡǍǎ ǍǡǏǒ ȁǏȂ ǶǍǡǎǑ ǍǡǍǒ ǶǍǡǏǐ ǶǍǡǍǓ .$"( Ǎǡǎǎ ǍǡǍǎ ǍǡǎǍ ǍǡǎǏ ćF TMPQF JT FTTFOUJBMMZ SFWFSTFE JOTJEF "GSJDB  JOTUFBE PG −.
  25. Interactions are symmetric • Effect of ruggedness depends upon continent:

    • Effect of continent depends upon ruggedness: )PX NVDI EPFT UIF BTTPDJBUJPO PG "GSJDB XJUI MPH (%1 EFQFOE VQPO SV FTF UXP QPTTJCJMJUJFT TPVOE EJČFSFOU UP NPTU IVNBOT ZPVS HPMFN UIJOL T TFDUJPO XFMM FYBNJOF UIJT GBDU ĕSTU NBUIFNBUJDBMMZ ćFO XFMM QMPU UI (%1 FYBNQMF BHBJO CVU XJUI UIF SFWFSTF QISBTJOH‰UIF BTTPDJBUJPO CFUXF EFQFOET VQPO SVHHFEOFTT JEFS ZFU BHBJO UIF NPEFM GPS µJ  µJ = αİĶı[J] + βİĶı[J] (SJ − ¯ S) QSFUBUJPO QSFWJPVTMZ IBT CFFO UIBU UIF TMPQF JT DPOEJUJPOBM PO DPOUJOFOU # Z UIBU UIF JOUFSDFQU JT DPOEJUJPOBM PO SVHHFEOFTT *UT FBTJFS UP TFF UIJT J FYQSFTTJPO BOPUIFS XBZ µJ = ( − İĶıJ)(α + β(SJ − ¯ S)) İĶı[J]= + (İĶıJ − )(α + β(SJ − ¯ S)) İĶı[J]= T XFJSE CVU JUT UIF TBNF NPEFM 8IFO İĶıJ =  POMZ UIF ĕSTU UFSN ST SFNBJOT ćF TFDPOE UFSN WBOJTIFT UP [FSP 8IFO JOTUFBE İĶı =   )PX NVDI EPFT UIF BTTPDJBUJPO CFUXFFO SVHHFEOFTT BOE MPH (%1 EFQFOE VQ XIFUIFS UIF OBUJPO JT JO "GSJDB  )PX NVDI EPFT UIF BTTPDJBUJPO PG "GSJDB XJUI MPH (%1 EFQFOE VQPO SVHHFEOF JMF UIFTF UXP QPTTJCJMJUJFT TPVOE EJČFSFOU UP NPTU IVNBOT ZPVS HPMFN UIJOLT UIFZ OUJDBM *O UIJT TFDUJPO XFMM FYBNJOF UIJT GBDU ĕSTU NBUIFNBUJDBMMZ ćFO XFMM QMPU UIF SVHH T BOE (%1 FYBNQMF BHBJO CVU XJUI UIF SFWFSTF QISBTJOH‰UIF BTTPDJBUJPO CFUXFFO "GS (%1 EFQFOET VQPO SVHHFEOFTT $POTJEFS ZFU BHBJO UIF NPEFM GPS µJ  µJ = αİĶı[J] + βİĶı[J] (SJ − ¯ S) JOUFSQSFUBUJPO QSFWJPVTMZ IBT CFFO UIBU UIF TMPQF JT DPOEJUJPOBM PO DPOUJOFOU #VU JUT B UP TBZ UIBU UIF JOUFSDFQU JT DPOEJUJPOBM PO SVHHFEOFTT *UT FBTJFS UP TFF UIJT JG XF XS BCPWF FYQSFTTJPO BOPUIFS XBZ µJ = ( − İĶıJ)(α + β(SJ − ¯ S)) İĶı[J]= + (İĶıJ − )(α + β(SJ − ¯ S)) İĶı[J]= T MPPLT XFJSE CVU JUT UIF TBNF NPEFM 8IFO İĶıJ =  POMZ UIF ĕSTU UFSN UIF "GS BNFUFST SFNBJOT ćF TFDPOE UFSN WBOJTIFT UP [FSP 8IFO JOTUFBE İĶıJ =  UIF ĕ N WBOJTIFT UP [FSP BOE POMZ UIF TFDPOE UFSN SFNBJOT /PX JG XF JNBHJOF TXJUDIJO
  26. Interactions are symmetric   $0/%*5*0/"- ."/"5&&4 0.0 0.2 0.4

    0.6 0.8 1.0 -0.3 -0.2 -0.1 0.0 0.1 0.2 ruggedness expected difference log GDP Africa higher GDP Africa lower GDP 'ĶĴłĿIJ ƐƎ ćF PUI CFUXFFO SVHHFEOFTT UJDBM BYJT JT UIF EJČF UJPOBM MPH (%1 GPS B PVUTJEF "GSJDB "U MP iNPWJOHw B OBUJPO U PNZ #VU BU IJHI SV USVF ćF BTTPDJBUJP FDPOPNZ EFQFOET V NVDI BT UIF BTTPDJB BOE FDPOPNZ EFQFO Figure 8.6
  27. Continuous interactions • data(tulips): 27 replicate blooms across three levels

    of both water and shade blooms 1.0 2.0 3.0 0 100 300 1.0 2.0 3.0 water 0 100 300 1.0 2.0 3.0 1.0 2.0 3.0 shade
  28. Tulip blooms No interaction: water and shade have independent effects

    Interaction: water and shade have interdependent effects FMT *N HPJOH UP GPDVT PO KVTU UXP NPEFMT  UIF NPEFM XJUI CPUI 2/ - OP JOUFSBDUJPO BOE  UIF NPEFM UIBU BMTP DPOUBJOT UIF JOUFSBDUJPO PG 2/ - V DPVME BMTP JOTQFDU NPEFMT UIBU DPOUBJO POMZ POF PG UIFTF WBSJBCMFT 2/ - FODPVSBHF UIF SFBEFS UP USZ UIBU BU UIF FOE BOE NBLF TVSF ZPV VOEFSTUBOE F PG NPEFMT EFM DPOUBJOJOH POMZ iNBJO FČFDUT w CFHJOT UIJT XBZ CJ ∼ /PSNBM(µJ, σ) µJ = α + βX(XJ − ¯ X) + βT(TJ −¯ T) BMVF PG '**(. PO SPX J XJ JT UIF WBMVF PG 2/ - BOE TJ JT UIF WBMVF PG .#  BOE¯ T BSF UIF NFBOT PG XBUFS BOE TIBEF SFTQFDUJWFMZ "MM UPHFUIFS UIJT JT KVTU PO XJUI UXP QSFEJDUPST FBDI DFOUFSFE CZ TVCUSBDUJOH JUT NFBO OUFS UIFTF WBSJBCMFT BT XFMM BT TDBMF UIF PVUDPNF CZ JUT NBYJNVN ʚǶ ɶ'**(. ȅ (3ǿɶ'**(.Ȁ ʚǶ ɶ2/ - Ƕ ( )ǿɶ2/ -Ȁ ʚǶ ɶ.# Ƕ ( )ǿɶ.# Ȁ / SBOHFT GSPN  UP  BOE CPUI 2/ -Ǿ )/ BOE .# Ǿ )/ SBOHF GSPN γX,J JT UIF TMPQF EFĕOJOH IPX RVJDLMZ CMPPNT DIBOHF XJUI XBUFS MFWFM ćF QBSBNFUFS UIF SBUF PG DIBOHF XIFO TIBEF JT BU JUT NFBO WBMVF "OE βXT JT UIF SBUF DIBOHF JO γX,J BT F DIBOHFT‰UIF TMPQF GPS TIBEF PO UIF TMPQF PG XBUFS 3FNFNCFS JUT UVSUMFT BMM UIF XBZ O /PUF UIF J JO γX,J ‰JU EFQFOET VQPO UIF SPX J CFDBVTF JU IBT 4J JO JU 8F BMTP XBOU UP BMMPX UIF BTTPDJBUJPO XJUI TIBEF βT UP EFQFOE VQPO XBUFS -VDLJMZ VTF PG UIF TZNNFUSZ PG TJNQMF JOUFSBDUJPOT XF HFU UIJT GPS GSFF ćFSF JT KVTU OP XBZ QFDJGZ B TJNQMF MJOFBS JOUFSBDUJPO JO XIJDI ZPV DBO TBZ UIF FČFDU PG TPNF WBSJBCMF Y OET VQPO [ CVU UIF FČFDU PG [ EPFT OPU EFQFOE VQPO Y * FYQMBJO UIJT JO NPSF EFUBJM JO 0WFSUIJOLJOH CPY BU UIF FOE PG UIJT TFDUJPO ćF JNQBDU PG UIJT JT UIBU JU JT DPOWFOUJPOBM CTUJUVUF γX,J JOUP UIF FRVBUJPO GPS µJ BOE KVTU TUBUF µJ = α + (βX + βXT 4J) γX,J 8J + βT 4J = α + βX 8J + βT 4J + βXT 4J 8J UIBUT UIF DPOWFOUJPOBM GPSN PG B DPOUJOVPVT JOUFSBDUJPO XJUI UIF FYUSB UFSN PO UIF GBS FOE IPMEJOH UIF QSPEVDU PG UIF UXP WBSJBCMFT -FUT QVU UIJT UP XPSL PO UIF UVMJQT ćF JOUFSBDUJPO NPEFM JT CJ ∼ /PSNBM(µJ, σ) µJ = α + βX(XJ − ¯ X) + βT(TJ −¯ T) + βXT(XJ − ¯ X)(TJ −¯ T) BTU UIJOH XF OFFE JT B QSJPS GPS UIJT OFX JOUFSBDUJPO QBSBNFUFS βXT  ćJT JT IBSE #VU MFUT 4VQQPTF UIF TUSPOHFTU QMBVTJCMF JOUFSBDUJPO JT POF JO XIJDI IJHI FOPVHI TIBEF NBLFT S IBWF [FSP FČFDU ćBU JNQMJFT
  29. How is interaction formed? µJ = α + (βX +

    βXT 4J) γX,J 8J + βT 4J = α + βX 8J + βT 4J + βXT 4J 8J OE UIBUT UIF DPOWFOUJPOBM GPSN PG B DPOUJOVPVT JOUFSBDUJPO XJUI UIF FYUSB UFSN PO UIF GBS HIU FOE IPMEJOH UIF QSPEVDU PG UIF UXP WBSJBCMFT -FUT QVU UIJT UP XPSL PO UIF UVMJQT ćF JOUFSBDUJPO NPEFM JT CJ ∼ /PSNBM(µJ, σ) µJ = α + βX(XJ − ¯ X) + βT(TJ −¯ T) + βXT(XJ − ¯ X)(TJ −¯ T) ćF MBTU UIJOH XF OFFE JT B QSJPS GPS UIJT OFX JOUFSBDUJPO QBSBNFUFS βXT  ćJT JT IBSE #VU MFUT Z 4VQQPTF UIF TUSPOHFTU QMBVTJCMF JOUFSBDUJPO JT POF JO XIJDI IJHI FOPVHI TIBEF NBLFT BUFS IBWF [FSP FČFDU ćBU JNQMJFT γX,J = βX + βXT 4J =  XF TFU 4J =  UIF NBYJNVN JO UIF TBNQMF UIFO UIJT NFBOT UIF JOUFSBDUJPO OFFET UP CF F TBNF NBHOJUVEF BT UIF NBJO FČFDU CVU SFWFSTFE βXT = −βX  ćBU JT MBSHFTU DPODFJWBCMF UFSBDUJPO 4P JG XF TFU UIF QSJPS GPS βXT UP IBWF UIF TBNF TUBOEBSE EFWJBUJPO BT βX NBZCF BU JTOU SJEJDVMPVT "MM UPHFUIFS OPX JO DPEF GPSN ǡǔ ʚǶ ,0+ǿ '$./ǿ '**(.Ǿ./ ʡ )*-(ǿ (0 Ǣ .$"( Ȁ Ǣ (0 ʚǶ  ʔ 2ȉ2/ -Ǿ )/ ʔ .ȉ.# Ǿ )/ ʔ 2.ȉ2/ -Ǿ )/ȉ.# Ǿ )/ Ǣ  ʡ )*-(ǿ Ǎǡǒ Ǣ ǍǡǏǒ Ȁ Ǣ
  30. How is interaction formed? µJ = α + (βX +

    βXT 4J) γX,J 8J + βT 4J = α + βX 8J + βT 4J + βXT 4J 8J OE UIBUT UIF DPOWFOUJPOBM GPSN PG B DPOUJOVPVT JOUFSBDUJPO XJUI UIF FYUSB UFSN PO UIF GBS HIU FOE IPMEJOH UIF QSPEVDU PG UIF UXP WBSJBCMFT -FUT QVU UIJT UP XPSL PO UIF UVMJQT ćF JOUFSBDUJPO NPEFM JT CJ ∼ /PSNBM(µJ, σ) µJ = α + βX(XJ − ¯ X) + βT(TJ −¯ T) + βXT(XJ − ¯ X)(TJ −¯ T) ćF MBTU UIJOH XF OFFE JT B QSJPS GPS UIJT OFX JOUFSBDUJPO QBSBNFUFS βXT  ćJT JT IBSE #VU MFUT Z 4VQQPTF UIF TUSPOHFTU QMBVTJCMF JOUFSBDUJPO JT POF JO XIJDI IJHI FOPVHI TIBEF NBLFT BUFS IBWF [FSP FČFDU ćBU JNQMJFT γX,J = βX + βXT 4J =  XF TFU 4J =  UIF NBYJNVN JO UIF TBNQMF UIFO UIJT NFBOT UIF JOUFSBDUJPO OFFET UP CF F TBNF NBHOJUVEF BT UIF NBJO FČFDU CVU SFWFSTFE βXT = −βX  ćBU JT MBSHFTU DPODFJWBCMF UFSBDUJPO 4P JG XF TFU UIF QSJPS GPS βXT UP IBWF UIF TBNF TUBOEBSE EFWJBUJPO BT βX NBZCF BU JTOU SJEJDVMPVT "MM UPHFUIFS OPX JO DPEF GPSN ǡǔ ʚǶ ,0+ǿ '$./ǿ '**(.Ǿ./ ʡ )*-(ǿ (0 Ǣ .$"( Ȁ Ǣ (0 ʚǶ  ʔ 2ȉ2/ -Ǿ )/ ʔ .ȉ.# Ǿ )/ ʔ 2.ȉ2/ -Ǿ )/ȉ.# Ǿ )/ Ǣ  ʡ )*-(ǿ Ǎǡǒ Ǣ ǍǡǏǒ Ȁ Ǣ FSFE QBSU &WFO JG XF POMZ DBSFE BCPVU UIF UISFF PCTFSWFE WBMVFT XFE TUJMM OFFE U F UIF PSEFSJOH XIJDI JT CJHHFS UIBO XIJDI 4P XIBU UP EP F DPOWFOUJPOBM BOTXFS JT UP SFBQQMZ UIF PSJHJOBM HFPDFOUSJTN UIBU KVTUJĕFT B MJOFBS SF O 8IFO XF IBWF UXP WBSJBCMF BO PVUDPNF BOE B QSFEJDUPS BOE XF XJTI UP NPEF BO PG UIF PVUDPNF TVDI UIBU JU JT DPOEJUJPOBM PO UIF WBMVF PG B DPOUJOVPVT QSFEJDUPS Y VTF B MJOFBS NPEFM µJ = α + βYJ  /PX JO PSEFS UP NBLF UIF TMPQF β DPOEJUJPOBM PO UIFS WBSJBCMF XF DBO KVTU SFDVSTJWFMZ BQQMZ UIF TBNF USJDL S CSFWJUZ MFU 8J = XJ − ¯ X BOE 4J = TJ −¯ T ćFO JG XF EFĕOF UIF TMPQF βX XJUI JUT PXO NPEFM γX  µJ = α + γX,J 8J + βT 4J γX,J = βX + βXT 4J X,J JT UIF TMPQF EFĕOJOH IPX RVJDLMZ CMPPNT DIBOHF XJUI XBUFS MFWFM ćF QBSBNFUF F SBUF PG DIBOHF XIFO TIBEF JT BU JUT NFBO WBMVF "OE βXT JT UIF SBUF DIBOHF JO γX,J B IBOHFT‰UIF TMPQF GPS TIBEF PO UIF TMPQF PG XBUFS 3FNFNCFS JUT UVSUMFT BMM UIF XB /PUF UIF J JO γX,J ‰JU EFQFOET VQPO UIF SPX J CFDBVTF JU IBT 4J JO JU BMTP XBOU UP BMMPX UIF BTTPDJBUJPO XJUI TIBEF βT UP EFQFOE VQPO XBUFS -VDLJMZ F PG UIF TZNNFUSZ PG TJNQMF JOUFSBDUJPOT XF HFU UIJT GPS GSFF ćFSF JT KVTU OP XB JGZ B TJNQMF MJOFBS JOUFSBDUJPO JO XIJDI ZPV DBO TBZ UIF FČFDU PG TPNF WBSJBCMF T VQPO [ CVU UIF FČFDU PG [ EPFT OPU EFQFOE VQPO Y * FYQMBJO UIJT JO NPSF EFUBJM JO FSUIJOLJOH CPY BU UIF FOE PG UIJT TFDUJPO ćF JNQBDU PG UIJT JT UIBU JU JT DPOWFOUJPOB
  31. How is interaction formed? µJ = α + (βX +

    βXT 4J) γX,J 8J + βT 4J = α + βX 8J + βT 4J + βXT 4J 8J OE UIBUT UIF DPOWFOUJPOBM GPSN PG B DPOUJOVPVT JOUFSBDUJPO XJUI UIF FYUSB UFSN PO UIF GBS HIU FOE IPMEJOH UIF QSPEVDU PG UIF UXP WBSJBCMFT -FUT QVU UIJT UP XPSL PO UIF UVMJQT ćF JOUFSBDUJPO NPEFM JT CJ ∼ /PSNBM(µJ, σ) µJ = α + βX(XJ − ¯ X) + βT(TJ −¯ T) + βXT(XJ − ¯ X)(TJ −¯ T) ćF MBTU UIJOH XF OFFE JT B QSJPS GPS UIJT OFX JOUFSBDUJPO QBSBNFUFS βXT  ćJT JT IBSE #VU MFUT Z 4VQQPTF UIF TUSPOHFTU QMBVTJCMF JOUFSBDUJPO JT POF JO XIJDI IJHI FOPVHI TIBEF NBLFT BUFS IBWF [FSP FČFDU ćBU JNQMJFT γX,J = βX + βXT 4J =  XF TFU 4J =  UIF NBYJNVN JO UIF TBNQMF UIFO UIJT NFBOT UIF JOUFSBDUJPO OFFET UP CF F TBNF NBHOJUVEF BT UIF NBJO FČFDU CVU SFWFSTFE βXT = −βX  ćBU JT MBSHFTU DPODFJWBCMF UFSBDUJPO 4P JG XF TFU UIF QSJPS GPS βXT UP IBWF UIF TBNF TUBOEBSE EFWJBUJPO BT βX NBZCF BU JTOU SJEJDVMPVT "MM UPHFUIFS OPX JO DPEF GPSN ǡǔ ʚǶ ,0+ǿ '$./ǿ '**(.Ǿ./ ʡ )*-(ǿ (0 Ǣ .$"( Ȁ Ǣ (0 ʚǶ  ʔ 2ȉ2/ -Ǿ )/ ʔ .ȉ.# Ǿ )/ ʔ 2.ȉ2/ -Ǿ )/ȉ.# Ǿ )/ Ǣ  ʡ )*-(ǿ Ǎǡǒ Ǣ ǍǡǏǒ Ȁ Ǣ FSFE QBSU &WFO JG XF POMZ DBSFE BCPVU UIF UISFF PCTFSWFE WBMVFT XFE TUJMM OFFE U F UIF PSEFSJOH XIJDI JT CJHHFS UIBO XIJDI 4P XIBU UP EP F DPOWFOUJPOBM BOTXFS JT UP SFBQQMZ UIF PSJHJOBM HFPDFOUSJTN UIBU KVTUJĕFT B MJOFBS SF O 8IFO XF IBWF UXP WBSJBCMF BO PVUDPNF BOE B QSFEJDUPS BOE XF XJTI UP NPEF BO PG UIF PVUDPNF TVDI UIBU JU JT DPOEJUJPOBM PO UIF WBMVF PG B DPOUJOVPVT QSFEJDUPS Y VTF B MJOFBS NPEFM µJ = α + βYJ  /PX JO PSEFS UP NBLF UIF TMPQF β DPOEJUJPOBM PO UIFS WBSJBCMF XF DBO KVTU SFDVSTJWFMZ BQQMZ UIF TBNF USJDL S CSFWJUZ MFU 8J = XJ − ¯ X BOE 4J = TJ −¯ T ćFO JG XF EFĕOF UIF TMPQF βX XJUI JUT PXO NPEFM γX  µJ = α + γX,J 8J + βT 4J γX,J = βX + βXT 4J X,J JT UIF TMPQF EFĕOJOH IPX RVJDLMZ CMPPNT DIBOHF XJUI XBUFS MFWFM ćF QBSBNFUF F SBUF PG DIBOHF XIFO TIBEF JT BU JUT NFBO WBMVF "OE βXT JT UIF SBUF DIBOHF JO γX,J B IBOHFT‰UIF TMPQF GPS TIBEF PO UIF TMPQF PG XBUFS 3FNFNCFS JUT UVSUMFT BMM UIF XB /PUF UIF J JO γX,J ‰JU EFQFOET VQPO UIF SPX J CFDBVTF JU IBT 4J JO JU BMTP XBOU UP BMMPX UIF BTTPDJBUJPO XJUI TIBEF βT UP EFQFOE VQPO XBUFS -VDLJMZ F PG UIF TZNNFUSZ PG TJNQMF JOUFSBDUJPOT XF HFU UIJT GPS GSFF ćFSF JT KVTU OP XB JGZ B TJNQMF MJOFBS JOUFSBDUJPO JO XIJDI ZPV DBO TBZ UIF FČFDU PG TPNF WBSJBCMF T VQPO [ CVU UIF FČFDU PG [ EPFT OPU EFQFOE VQPO Y * FYQMBJO UIJT JO NPSF EFUBJM JO FSUIJOLJOH CPY BU UIF FOE PG UIJT TFDUJPO ćF JNQBDU PG UIJT JT UIBU JU JT DPOWFOUJPOB µJ = α + γX,J 8J + βT 4J γX,J = βX + βXT 4J X,J JT UIF TMPQF EFĕOJOH IPX RVJDLMZ CMPPNT DIBOHF XJUI XBUFS MFWFM ćF QBSB F SBUF PG DIBOHF XIFO TIBEF JT BU JUT NFBO WBMVF "OE βXT JT UIF SBUF DIBOHF JO IBOHFT‰UIF TMPQF GPS TIBEF PO UIF TMPQF PG XBUFS 3FNFNCFS JUT UVSUMFT BMM UI /PUF UIF J JO γX,J ‰JU EFQFOET VQPO UIF SPX J CFDBVTF JU IBT 4J JO JU BMTP XBOU UP BMMPX UIF BTTPDJBUJPO XJUI TIBEF βT UP EFQFOE VQPO XBUFS - F PG UIF TZNNFUSZ PG TJNQMF JOUFSBDUJPOT XF HFU UIJT GPS GSFF ćFSF JT KVTU O JGZ B TJNQMF MJOFBS JOUFSBDUJPO JO XIJDI ZPV DBO TBZ UIF FČFDU PG TPNF WBSJ T VQPO [ CVU UIF FČFDU PG [ EPFT OPU EFQFOE VQPO Y * FYQMBJO UIJT JO NPSF EF FSUIJOLJOH CPY BU UIF FOE PG UIJT TFDUJPO ćF JNQBDU PG UIJT JT UIBU JU JT DPOWFO UJUVUF γX,J JOUP UIF FRVBUJPO GPS µJ BOE KVTU TUBUF µJ = α + (βX + βXT 4J) γX,J 8J + βT 4J = α + βX 8J + βT 4J + βXT 4J 8J BUT UIF DPOWFOUJPOBM GPSN PG B DPOUJOVPVT JOUFSBDUJPO XJUI UIF FYUSB UFSN PO OE IPMEJOH UIF QSPEVDU PG UIF UXP WBSJBCMFT T QVU UIJT UP XPSL PO UIF UVMJQT ćF JOUFSBDUJPO NPEFM JT
  32. Tulip model – no interaction GSPN FJUIFS WBSJBCMF‰. × 

    =  4P JG XF BTTJHO B TUBOEBSE EFWJBUJPO PG  UP FBDI UIFO  PG UIF QSJPS TMPQFT BSF GSPN −. UP . TP FJUIFS WBSJBCMF DPVME JO QSJODJQMF BDDPVOU GPS UIF FOUJSF SBOHF CVU JU XPVME CF VOMJLFMZ 3FNFNCFS UIF HPBMT IFSF BSF UP BTTJHO XFBLMZ JOGPSNBUJWF QSJPST UP EJTDPVSBHF PWFSĕUUJOH‰JNQPTTJCMZ MBSHF FČFDUT TIPVME CF BTTJHOFE MPX QSJPS QSPCBCJMJUZ‰BOE BMTP UP GPSDF PVSTFMWFT UP UIJOL BCPVU XIBU UIF NPEFM NFBOT "MM UPHFUIFS OPX JO DPEF GPSN 3 DPEF  (ǕǡǓ ʚǶ ,0+ǿ '$./ǿ '**(.Ǿ./ ʡ )*-(ǿ (0 Ǣ .$"( Ȁ Ǣ (0 ʚǶ  ʔ 2ȉ2/ -Ǿ )/ ʔ .ȉ.# Ǿ )/ Ǣ  ʡ )*-(ǿ Ǎǡǒ Ǣ ǍǡǏǒ Ȁ Ǣ 2 ʡ )*-(ǿ Ǎ Ǣ ǍǡǏǒ Ȁ Ǣ . ʡ )*-(ǿ Ǎ Ǣ ǍǡǏǒ Ȁ Ǣ .$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ Ȁ *UT B HPPE JEFB BU UIJT QPJOU UP TJNVMBUF MJOFT GSPN UIF QSJPS #VU CFGPSF EPJOH UIBU MFUT EFĕOF UIF JOUFSBDUJPO NPEFM BT XFMM ćFO XF DBO UBML BCPVU IPX UP QMPU QSFEJDUJPOT GSPN JOUFSBDUJPOT BOE TFF CPUI QSJPS BOE QPTUFSJPS QSFEJDUJPOT UPHFUIFS 5P CVJME JO BO JOUFSBDUJPO CFUXFFO XBUFS BOE TIBEF XF OFFE UP DPOTUSVDU µ TP UIBU UIF JNQBDU PG DIBOHJOH FJUIFS XBUFS PS TIBEF EFQFOET VQPO UIF WBMVF PG UIF PUIFS WBSJBCMF 'PS FYBNQMF JG XBUFS JT MPX UIFO EFDSFBTJOH UIF TIBEF JODSFBTF MJHIU DBOU IFMQ BT NVDI BT XIFO XBUFS JT IJHI 4P XF XBOU UIF TMPQF PG XBUFS βX UP CF DPOEJUJPOBM PO TIBEF -JLFXJTF GPS TIBEF CFJOH DPOEJUJPOBM PO XBUFS SFNFNCFS #VSJEBOT JOUFSBDUJPO   )PX DBO XF EP UIJT
  33. Plotting interaction • Slope changes with values of other predictor,

    so use more than one plot • Here, need three plots, triptych Lewis Powell (1844–1865), before his hanging for conspiracy to assassinate Abraham Lincoln.
  34. Prior predictions No interaction Interaction Figure 8.8   $0/%*5*0/"-

    ."/"5&&4 water blooms -1 0 1 0 0.5 1 m8.6 prior: shade = -1 water blooms -1 0 1 0 0.5 1 m8.6 prior: shade = 0 water blooms -1 0 1 0 0.5 1 m8.6 prior: shade = 1 water blooms -1 0 1 0 0.5 1 m8.7 prior: shade = -1 water blooms -1 0 1 0 0.5 1 m8.7 prior: shade = 0 water blooms -1 0 1 0 0.5 1 m8.7 prior: shade = 1
  35. Posterior predictions   $0/%*5*0/"- ."/"5&&4 water blooms -1 0

    1 0 0.5 1 m8.6 post: shade = -1 water blooms -1 0 1 0 0.5 1 m8.6 post: shade = 0 water blooms -1 0 1 0 0.5 1 m8.6 post: shade = 1 water blooms -1 0 1 0 0.5 1 m8.7 post: shade = -1 water blooms -1 0 1 0 0.5 1 m8.7 post: shade = 0 water blooms -1 0 1 0 0.5 1 m8.7 post: shade = 1 No interaction Interaction Figure 8.7
  36. Tulip model – interaction XBUFS IBWF [FSP FČFDU ćBU JNQMJFT

    γX,J = βX + βXT 4J =  *G XF TFU 4J =  UIF NBYJNVN JO UIF TBNQMF UIFO UIJT NFBOT UIF JOUFSBDUJPO OFFET UP CF UIF TBNF NBHOJUVEF BT UIF NBJO FČFDU CVU SFWFSTFE βXT = −βX  ćBU JT MBSHFTU DPODFJWBCMF JOUFSBDUJPO 4P JG XF TFU UIF QSJPS GPS βXT UP IBWF UIF TBNF TUBOEBSE EFWJBUJPO BT βX NBZCF UIBU JTOU SJEJDVMPVT "MM UPHFUIFS OPX JO DPEF GPSN 3 DPEF  (Ǖǡǔ ʚǶ ,0+ǿ '$./ǿ '**(.Ǿ./ ʡ )*-(ǿ (0 Ǣ .$"( Ȁ Ǣ (0 ʚǶ  ʔ 2ȉ2/ -Ǿ )/ ʔ .ȉ.# Ǿ )/ ʔ 2.ȉ2/ -Ǿ )/ȉ.# Ǿ )/ Ǣ  ʡ )*-(ǿ Ǎǡǒ Ǣ ǍǡǏǒ Ȁ Ǣ  $0/5*/6064 */5&3"$5*0/4  2 ʡ )*-(ǿ Ǎ Ǣ ǍǡǏǒ Ȁ Ǣ . ʡ )*-(ǿ Ǎ Ǣ ǍǡǏǒ Ȁ Ǣ 2. ʡ )*-(ǿ Ǎ Ǣ ǍǡǏǒ Ȁ Ǣ .$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ Ȁ 4P NVDI GPS UIF TUSVDUVSF PG B TJNQMF DPOUJOVPVT JOUFSBDUJPO /FYU MFUT ĕHVSF PVU IPX UP QMPU UIFTF DSFBUVSFT 0WFSUIJOLJOH )PX JT JOUFSBDUJPO GPSNFE "T JO UIF NBJO UFYU JG ZPV TVCTUJUVUF γX,J JOUP µJ BCPWF BOE FYQBOE µJ = α + (βX + βXT 4J)8J + βT 4J = α + βX 8J + βT 4J + βXT 4J 8J /PX JUT QPTTJCMF UP SFGBDUPS UIJT UP DPOTUSVDU B γT,J UIBU NBLFT UIF BTTPDJBUJPO PG TIBEF XJUI CMPPNT Interpreting parameters very hard! Plot.
  37. Prior predictions No interaction Interaction Figure 8.8   $0/%*5*0/"-

    ."/"5&&4 water blooms -1 0 1 0 0.5 1 m8.6 prior: shade = -1 water blooms -1 0 1 0 0.5 1 m8.6 prior: shade = 0 water blooms -1 0 1 0 0.5 1 m8.6 prior: shade = 1 water blooms -1 0 1 0 0.5 1 m8.7 prior: shade = -1 water blooms -1 0 1 0 0.5 1 m8.7 prior: shade = 0 water blooms -1 0 1 0 0.5 1 m8.7 prior: shade = 1
  38. Posterior predictions   $0/%*5*0/"- ."/"5&&4 water blooms -1 0

    1 0 0.5 1 m8.6 post: shade = -1 water blooms -1 0 1 0 0.5 1 m8.6 post: shade = 0 water blooms -1 0 1 0 0.5 1 m8.6 post: shade = 1 water blooms -1 0 1 0 0.5 1 m8.7 post: shade = -1 water blooms -1 0 1 0 0.5 1 m8.7 post: shade = 0 water blooms -1 0 1 0 0.5 1 m8.7 post: shade = 1 No interaction Interaction Figure 8.7
  39. Causal thinking • Tulip experiment: • Tulip reality: S W

    B S W B
  40. Interactions not always linear • Suppose all tulip data collected

    under “cool” temperatures • Under “hot” temperature, tulips do not bloom • Interaction, but not a linear one • blooms goes to zero at threshold
  41. Higher order interactions • Just keep multiplying: Z J ∼

    /PSNBM(µ, σ) β →  γSJ|"J= ≈ −. + .() = . Z J ∼ /PSNBM(µJ, σ), µJ = α + β YJ + β YJ + β YJ + β YJ YJ + β YJ YJ + β YJ YJ + β YJ YJ YJ. main effects 2-way interactions 3-way interaction
  42. Higher order interactions • Dangers of high-order interactions • Hard

    to interpret: “The extent to which the effect of x1 depends upon the value of x2 depends upon the value of x3, dude.” • Hard to estimate: need lots of data, must regularize • But you might really need them, because conditionality runs deep The Dude abides high-order interactions
  43. Higher order interactions • data(Wines2012) • Judgment of Princeton, 2012

    • New Jersey wines vs fine French wines • Outcome variable: score • Predictors: • region (NJ/FR) • nationality of judge (USA/FR-BE) • flight (red/white)
  44. Higher order interactions • Predictors: region, nationality of judge, flight

    • Consider interactions: • Interaction of region and judge is bias. Bias depends upon flight. • Interaction of judge and flight is preference. Preference depends upon region. • Interaction of region and flight is comparative advantage. Advantage depends upon judge.
  45. Interaction everywhere • Interaction, regularization, responsibility • Next time: Markov

    Chain Monte Carlo