1.2 1.4 AK DC E RI SD VT WY -2 -1 0 1 2 3 -2 -1 0 1 2 median age marriage (std) divorce rate (std) AR ID ME MN ND RI WY SJOLBHF SFTVMUJOH GSPN NPEFMJOH UIF NFBTVSFNFOU FS IF PSJHJOBM NFBTVSFNFOU UIF MFTT TISJOLBHF JO UIF QPT
Many procedures invented • errors-in-variables • reduced major axis • total least squares • Our approach will be logical • State information • Deduce implications • Garbage in? You know what comes out. 0 1 2 3 15 20 25 30 log population Marriage rate
but uncertainty discarded at analysis • Examples: • Predicting with averages • Parentage analysis • Phylogenetics: distribution of trees • Archaeology/paleontology/forensics: identification, sexing, aging, dating • Propagate uncertainty
analysis • drop all cases with any missing values • Discards a lot of information • Alternatives • replace missing with mean of column: NEVER DO THIS • Multiple imputation • Bayesian imputation • others
of R_B • Do not need to condition on anything for R_B not to be a confound • On right, no path through R_B, conditioning on B_obs • Do not NEED to impute • But imputation adds precision K XIFSF . JT CPEZ NBTT # JT OFPDPSUFY QFSDFOU , JT NJML FOFSHZ BOE WBSJBCMF UIBU SFOEFST . BOE # QPTJUJWFMZ DPSSFMBUFE 8F XBOU UP B HSBQI 8IBU UIBU NFBOT JT SFBMJ[JOH UIBU XF IBWFOU PCTFSWFE # OF JOTUFBE PCTFSWFE #PCT B QBSUJBMMZ PCTFSWFE TFU PG WBMVFT HFOFSBUFE -FUT OBNF UIF QSPDFTT UIBU HFOFSBUFT NJTTJOH WBMVFT 3# BOE OPX BE B B_obs K M R_B U ćF XBZ UP SFBE UIJT JT UP UIJOL PG UIF PCTFSWFEXJUINJTTJOHOFTT # PG UIF DPNQMFUFCVUVOPCTFSWFE # BOE UIF NJTTJOHOFTT QSPDFTT 3# BOPUIFS QPTTJCMF DPOGPVOE ćFO XF DBO VTF PVS GSJFOE UIF CBDLEPP XIFO XF OFFE JOGPSNBUJPO BCPVU OFFE UP DPOEJUJPO PO 3# *O UI JOGFS UIF JOĘVFODF PG # PO , 5P ĕHVSF PVU XIFO UIF FTUJNBUF JT DPO QBUIT GSPN #ļįŀ UP , *G BOZ PG UIFN BSF CBDLEPPST XF OFFE UP DMPT
WBSJBCMF JOĘVFODFT UIF NJTTJOHOFTT QSPDFTT B B_obs K M R_B U /PX . JOĘVFODFT 3# XIJDI NFBOT GPS FYBNQMF UIBU TQFDJFT XJUI TNBMMFS CPEJFT BSF NPS PS MFTT MJLFMZ UP IBWF NJTTJOH WBMVFT JO #ļįŀ ćJT DPVME IBQQFO JG SFTFBSDIFST BSF MFTT JOUF TUFE JO TNBMM TQFDJFT BOE TP EP OPU PęFO HP UISPVHI UIF USPVCMF PG NBLJOH EFUBJMFE CSBJ NFBTVSFNFOUT GPS UIFN 8IBU IBQQFOT JO UIJT DBTF ćFSF JT OPX B CBDLEPPS QBUI GSPN ļįŀ UIPVHI 3# UP , 4P UIF NJTTJOHOFTT QSPDFTT DBO DPOGPVOE PVS JOGFSFODF VOMFTT XF DB MPTF UIF CBDLEPPS *O UIJT DBTF XF DBO TIVU UIF CBDLEPPS CZ DPOEJUJPOJOH PO . 8F NJHI Missing At Random Missingness more likely for specific values of M. How can this happen?
of R_B • Must to condition on M for R_B not to be a confound • Still must impute to de-bias estimates • Why? If you delete cases of M/K where B is missing, missingness obscures causation. .*44*/( %"5" "OPUIFS QPTTJCJMJUZ JT UIBU TPNF PUIFS WBSJBCMF JOĘVFODFT UI B B_obs K M R_B U /PX . JOĘVFODFT 3# XIJDI NFBOT GPS FYBNQMF UIBU TQFDJFT XJ PS MFTT MJLFMZ UP IBWF NJTTJOH WBMVFT JO #ļįŀ ćJT DPVME IBQQFO FTUFE JO TNBMM TQFDJFT BOE TP EP OPU PęFO HP UISPVHI UIF USPVC NFBTVSFNFOUT GPS UIFN 8IBU IBQQFOT JO UIJT DBTF ćFSF JT O #ļįŀ UIPVHI 3# UP , 4P UIF NJTTJOHOFTT QSPDFTT DBO DPOGPVOE P DMPTF UIF CBDLEPPS *O UIJT DBTF XF DBO TIVU UIF CBDLEPPS CZ DPO
RANDOM H* A D H H* A D H H* A D H DOG EATS ANY HOMEWORK DOG EATS STUDENTS’ HOMEWORK DOG EATS BAD HOMEWORK H: Homework H*: Homework with missing values A: Attribute of student D: Dog (missingness mechanism)
What is your best guess of each missing value? • A: Posterior distribution derived from remaining data neocortex.perc 1 55.16 2 NA 3 NA 4 NA 5 NA 6 64.54 7 64.54 8 67.64 9 NA 10 68.85 11 58.85 12 61.69 13 60.32 14 NA 15 NA 16 69.97 17 NA 18 70.41 19 NA 20 73.40 21 NA 22 67.53 23 NA 24 71.26 25 72.60 26 NA 27 70.24 28 76.30 29 75.49
another technique (see text) • Extends to many model types: • Mark-recapture, occupancy (presence/absence) • Latent-state models (hidden Markov models)
risen to life to protect us, can easily change into a destructive force. Therefore let us treat carefully that which is strong, just as we bow kindly and patiently to that which is weak.” Rabbi Judah Loew ben Bezalel (1512–1609) From Breath of Bones: A Tale of the Golem