EXAMINATION QUESTION SVSOS3003 Fall 2004 Some suggestions for answering the questions

Transcription

1 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr EXAMINATION QUESTION SVSOS3003 Fall 2004 Som suggstions for answring th qustions Erling Brg Dpartmnt of sociology and political scinc, Norwgian Univrsity of Scinc and Tchnology Rad m In trying to answr th qustions at th xaminations it should b kpt in mind that th qustions oftn ar problmatic in rlation to th modl rquirmnt of bing basd on th bst availabl thory. Th lack of thortical foundation can b dfndd on two accounts. Most important is simply a lack of tim and suitabl data for construction of th ralistic xamination qustions w want. But if it is takn for grantd that th qustions sldom ar wll groundd in thory, this will of cours furnish th studnts will good argumnts in th ffort to critically valuat th spcification rquirmnt of th modls. As you rad th suggstions for answrs prsntd hr it is important to undrstand that it is not th only way of answring th qustions. Most qustions can b answrd in many ways. Evn if th tchnical qustions hav prcis answrs, th many valuations ncssary (.g. Is th distribution of th rsiduals clos nough to th normal distribution for th tsts to b blivabl? ) ar prcisly valuations. And for th valuation, th argumnts for or against ar th ssntial parts of th answr. During th xaminations tim is scarc. Fw ar abl to answr xhaustivly on all qustions. In this not on th answrs to th qustions thr has bn don much mor than w xpct to find at th xamination. Som sctions contain mor dtails of computation or stuff that may b rlvant or rlatd to th qustion askd, but not ncssary to answr th qustion. Howvr, th lvl of dtail and additions varis. I hav to warn against prsntation of rrors and rash conclusions. This author has as much capacity to rr as othr popl. Critical rading by studnts and collagus is th bst quality control thr is. Anyon finding an rror or thinking som othr valuation mor appropriat is ncouragd to writ m, for xampl by -mail: Erling.Brg@svt.ntnu.no, I will apprciat that. Erling Brg 2004

2 2 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 QUESTION 1 (OLS-rgrssion, wight 0,5) In a Norwgian study of trust in fllow citizns diffrncs btwn rgions wr invstigatd by OLS rgrssion. Six modls wr stimatd. a) Explain what Modl 1 tlls about rgional diffrncs in trust. b) Dtrmin which of th six modls bst prdicts th lvl of trust a prson xprsss. Find th F-statistic of a tst of th bst modl against modl 2. c) Evaluat th hypothsis Th rlationship btwn ag and trust in fllow citizns is linar. Find a 90% confidnc intrval for th impact of ducation in th bst modl. d) Formulat th modl idntifid as th bst. ) Basd on th bst modl writ up th formula for producing conditional ffct plots according to ag, sx and location in Oslo/Akrshus or Trøndlag. f) Discuss th dgr to which th assumptions of OLS rgrssion ar mt in th bst modl. a) Explain what Modl 1 tlls about rgional diffrncs in trust. Th dpndnt variabl is scor on th trust indx dfind in th appndix tabls. In modl 1 w find th dummis of a rgional cod. W s that Oslo/ Akrshus is th rfrnc catgory. Th rgrssion cofficints ar stimats of th xpctd diffrnc btwn th rfrnc catgory and th prsons locatd in th indicatd rgion. Whn all rgion variabls hav th valu zro, th constant givs us th prdictd valu of th trust indx for Oslo/ Akrshus. On th scal from 0 to 10, th popl of Oslo/ Akrshus xprss an avrag trust of Modl Unstandardizd Cofficints t Sig. Collinarity Statistics B Std. Error Tolra nc VIF 1 (Constant) HdOppl SouthEast AgdrRog WstNorw Trondl NorthNor Popl living in Hdmark/ Oppland will on avrag scor 0.34 points abov thos living in Oslo/ Akrshus. Th diffrncs btwn thos living in Oslo/ Akrshus and thos living in th South East of th country or in Agdr/ Rogaland ar ngligibl. Th sam is th diffrnc btwn Oslo/ Akrshus and North Norway. Thos living in Trøndlag hav a scor 0.33 abov thos from Oslo/ Akrshus or about th sam as thos living in Hdmark/ Oppland. For thos living in Wst

3 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr Norway th modl stimats a diffrnc of 0.19 to thos from Oslo/ Akrshus. This is about half th diffrnc of Trøndlag and Hdmark/ Oppland. Howvr, it is not quit significant with a p-valu of Th pictur mrging from this is that popl living in Trøndlag/ Hdmark/ Oppland show a highr lvl of trust than th rst of th country xcpt thos living in Wst Norway whr th lvl of trust probably lis somwhr in btwn. b) Dtrmin which of th six modls bst prdicts th lvl of trust a prson xprsss. Find th F-statistic of a tst of th bst modl against modl 2. Modl R R Squar Adjustd R Squar Std. Error of th Estimat R Squar Chang Chang Statistics F Chang df1 df2 Sig. F Chang 1.086(a) (b) (c) (d) () (f) From th chang statistics in th tabl Modl Summary w s that th modls 1, 2, 3, and 4 all ar improvmnts on th prvious modl. Modl 5 is not an improvmnt and probably nithr is modl 6 an improvmnts. Basd on a critrion of parsimony modl 4 will b sn as th bst modl. But to b sur w may tst modl 6 against modl 4. F H nk RSS RSS K H K H RSS K n K = ((4187, ,206)/3)/(4182,206/( ))=0,788 In th tabl of th F-distribution w s that with a tru null hypothsis finding a F > 2.60 will hav a probability of lss than 5%. Th F-valu of found hr will hav a much largr probability than 0.05 and hnc w rjct th null hypothsis.

4 4 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 b) continud Find th F-statistic of a tst of th bst modl against modl 2. In a comparison of two modls stimatd on th sam sampl of n cass, on modl with K paramtrs and on modl with K-H paramtrs, th statistic F H nk RSS RSS KH K H RSSK n K follows a F-distribution with H and n-k dgrs of frdom if it is tru that th H xtra variabls includd in th big modl hav no ffct (if H 0 No impact of th nw variabls is tru) and th assumptions of OLS rgrssion ar mt. In this formula th RSS [K] is th sum of squars of th rsiduals of th big modl with K paramtrs (or K-1 variabls) and RSS [K-H] is th sum of squard rsiduals in th small modl whr th H nw variabls ar not includd. W rjct th nullhypothsis that th H nw variabls do not hav an impact with lvl of significanc if F H n-k is largr than th critical valu for lvl of significanc in th tabl of th F-distribution with H and n-k dgrs of frdom. ANOVA(g) Modl Sum of Squars df Man Squar 1 Rgrssion (a) Rsidual Total Rgrssion (b) Rsidual Total Rgrssion (c) Rsidual Total Rgrssion (d) Rsidual Total Rgrssion () Rsidual Total Rgrssion (f) Rsidual Total F Sig.

5 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr Basd on th ANOVA tabl w find that in comparing modl 4 to modl 2 F H nk RSS RSS K H K H RSS K n K = (( )/3) / ( /( )) = (36,559/3) / ( /1988) = 12,186 / 2,106 = 5,786 In th tabl of th F-distribution w s that with a tru null hypothsis finding a F > 2.60 will hav a probability of lss than 5%. Th F-valu of found hr will lad to rjction of th null hypothsis. c) Evaluat th hypothsis Th rlationship btwn ag and trust in fllow citizns is linar. Find a 90% confidnc intrval for th impact of ducation in th bst modl. Th diffrnc btwn modl 2 and modl 3 is th scond ordr trm in th ag polynomial, Ag2. Th addition contributs significantly to th modl with a t- valu of and a p-valu of This lads to a rjction of th nullhypothsis that th rlationship btwn trust in fllow citizns and ag is linar. Th curvilinar rlationship appars vn strongr in modl 4 whr an intraction trm btwn ag and gndr is introducd. In modl 3 ag is rlatd to th dpndnt variabl through th rlationship (ag) (ag) 2 Plotting this polynomial w s that it rachs a low valu around th ag of Applying calculus to th rlationship [d/d(ag)( ag ag 2 )] = 0 W find that for ag = 31,43 trust rachs its lowst valu.

6 6 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 c) continud Find a 90% confidnc intrval for th impact of ducation in th bst modl. In modl 4 w s that b duc = with a standard rror of If w for th prsnt computation assum that th rsiduals ar normally distributd w can find a 90% confidnc intrval as b duc SE duc *t 0.1 < duc < b duc + SE duc *t 0.1 whr b duc is th stimatd rgrssion cofficint of ducation, SE duc is th standard rror, and t 0.1 is th critical valu in th t-distribution in a two-sidd tst with =0.1 and df = n-k = = 1988, dgrs of frdom: In tabl A4.1 in Hamilton (1992:350) w find that for df > 120 t 0.1 = This mans that a 90% confidnc intrval is givn by *1.645 < duc < * < duc < < duc < In modl 6 whr ducation is includd as a curvilinar variabl, th computation of confidnc intrval is mor complicatd. d) Formulat th modl idntifid as th bst. To dfin a modl thr ar thr typs of lmnts that nd to b considrd: 1. Dfinitions of th lmnts of th modl (variabls, rror trm, population and sampl) 2. Dfinitions of th rlationships among th lmnts of th modl (th quation linking variabls and rror trm, th sampling procdur linking sampl and population, thoris and tim squncs of vnts and obsrvations linking causs and ffcts) 3. Dfinitions of th assumptions that hav to b mt in ordr to us a particular mthod (such as th OLS mthod for linar rgrssion) for stimating th modl (modl spcification, distribution and proprtis of th rror trm) At a minimum th formulation will includ variabl dfinitions, th formula linking variabls and rror trm, and th assumptions ndd to mak valid infrncs from th stimats of a particular procdur. For th prsnt problm w ar told that data com from a random sampl from th Norwgian population usd by th Europan Social Survy in thir 2002 invstigation of th rlations btwn institutional conditions and th attituds,

7 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr valus and opinions of citizns of Europan countris. Basd on ths data th following variabls hav bn dfind: Y Trustindx Rgion in Norway X 1 HdmarkOppland X 2 SouthEast X 3 AgdrRogaland X 4 WstNorway X 5 Trondlag X 6 NorthNorway Ag X 7 Ag in yars Ag X 11 Ag in yars squard Ag2 X 8 Fmal Education X 9 Education in yars Educ X 14 Education in yars squard Educ2 X 10 Numbr of houshold mmbrs NoHHmmbrs Intraction trms X 12 Fmal by ag FmalAg X 13 Fmal by ag squard FmalAg2 X 15 Fmal by ducation FmalEduc X 16 Fmal by ducation squard FmalEduc2 Sinc modls 5 and 6 ar not improvmnts on modl 4, th variabls X 14, X 15, and X 16 ar irrlvant and will not b considrd furthr. Th main objctiv of th modl is to invstigat th rgional variations in trust as masurd by th Trustindx. If thr in th Norwgian population is a linar or curvilinar rlationship btwn th Trustindx and th dfind indpndnt variabls w can writ th quation linking th variabls by Y i = X 1i + 2 X 2i + 3 X 3i + 4 X 4i + 5 X 5i + 6 X 6i + 7 X 7i + 8 X 8i + 9 X 9i + 10 X 10i + 11 X 11i + 12 X 12i + 13 X 13i + i. Hr i runs ovr th whol of th Norwgian population. If w lt k=0, 1, 2, 3,,13 k will b th unknown paramtrs showing how many masurmnt units

8 8 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 of y will b addd to y pr unit incras in X k. i is th rror trm, a variabl that compriss all rlvant factors not obsrvd as wll as random nois in th masurmnt of y. Th quation for th modl can also b writtn y i = E[y i ] + i whr E[y i ] = x 1i + 2 x 2i + 3 x 3i x 13i (For E[y i ] rad xpctd valu of y i ) In Norway, 2036 prsons wr intrviwd. A total of 34 prsons hav missing on on or mor of th variabls of th modl and wr rmovd. Sinc nothing mor is indicatd it must b assumd thy wr rmovd by listwis dltion. Only 2 wr missing on th dpndnt variabl, 32 wr missing on ducation. This lft 2002 for th analysis. Thr ar no indications that th missing cass ar nonrandom in rlation to th dpndnt variabl. Assuming that thy ar missing at random (MAR), listwis dltion is a propr procdur as long as it lavs nough cass to prform th analysis. Th Trustindx is a gnralizd masur of th confidnc a prsons puts in his or hr fllow mn. Attituds as xprssd in an intrviw situation ar prsumd to b basd on mor basic valus shapd by socialization and accumulation of xprincs within th social positions ach individual hav occupid. It is usually assumd that rgion will b a proxy for variations in such socialization and accumulatd xprincs. By th sam rasoning it is also xpctd that ag, gndr and ducation will affct th gnral lvl of trust. It is also possibl that th siz of th houshold might affct this variabl. An OLS stimat of th modl paramtrs dfind abov can b stimatd as th b- valus of ŷ i = b 0 + b 1 x 1i + b 2 x 2i + b 3 x 3i b 13 x 13i that minimizs th sum of squard rsiduals, RSS = i (y i - ŷ i ) 2 2 = i i (For ŷ i rad stimatd or prdictd valu of y i or just y-hat.) OLS stimats will b unbiasd and fficint with a known sampling distribution if th following assumptions ar tru: I: Th modl is corrct, that is All rlvant variabls ar includd No irrlvant variabls ar includd Th modl is linar in th paramtrs II: Th Gauss-Markov rquirmnts for Bst Linar Unbiasd Estimats (BLUE) Fixd x-valus (no random componnt in thir masurmnt)

9 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr Th rror trms hav an xpctd valu of 0 for all cass i o E( i ) = 0 for all i Th rror trms hav constant varianc for all cass i (homoscdasticity) for all i o var( i ) = 2 for all i Th rror trms do not corrlat with ach othr across cass (no autocorrlation) for all i j o cov( i, j ) = 0 for all i j III: Th rror trms ar normally distributd Th rror trms ar normally distributd (and with th sam varianc) for all cass for all i o i ~ N(0, 2 ) for all i ) Basd on th bst modl writ up th formula for producing conditional ffct plots according to ag, sx and location in Oslo/Akrshus or Trøndlag. Modl 4 has bn idntifid as th bst modl. B Variabl Conditions Minimum Maximum Man valus (Constant),00 10,00 6, HdmarkOppland 0,00 1,00, SouthEast 0,00 1,00, AgdrRogaland 0,00 1,00, WstNorway 0,00 1,00, Trondlag 1 or 0,00 1,00, NorthNorway 0,00 1,00, Ag X 17,00 93,00 45, Fmal 1 or 0,00 1,00, Educ 13 5,00 20,00 12, NoHHmmbrs 3 1,00 9,00 2, Ag2 X 2 289, , , FmalAg (1 or 0)X 25,00 400,00 170, FmalAg2 (1 or 0)X 2,00 93,00 21,1109 Thr is not said anything about Educ and NoHHmmbrs. To produc a conditional ffct plot ths variabls ar givn rasonabl valus clos to thir avrag in th population. Educ is st to 13 and NoHHmmbrs to 3. Choosing othr numbrs will shift th rgrssion lin up or down a bit but not altr th pattrn according to ag. On may xprss th conditional ffct of both sx and location in on quation sinc Trøndlag is a dummy with Oslo/ Akrshus as rfrnc catgory.

10 10 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 Th conditional ffct plot is producd by making a graph of Y = Trondlag 0.046X 0.814Fmal * * X Fmal*X Fmal*X 2 = Trondlag 0.046X X Fmal Fmal*X Fmal*X 2 Thr ar howvr othr ways of rprsnting th rlationships that may b mor informativ. Th abov xprssion may for xampl b rstatd as Y=0.434Trondlag+(6, X X 2 )+Fmal( X X 2 ) If Trondlag = 1 w gt th graphs for Trøndlag and if Trondlag = 0 w gt th graphs of Oslo/ Akrshus (sinc Oslo/ Akrshus is th xcludd rfrnc catgory). Th trust indx will for prsons in Trøndlag b points highr than for prsons in Oslo/ Akrshus. y= ( x x 2 )+1( x x 2 ) y= ( x x 2 )+1( x x 2 ) y= ( x x 2 )+0( x x 2 ) y= ( x x 2 )+0( x x 2 ) From th diagram it appars that th diffrnc btwn mn and womn in th trust thy put in thir fllow citizns ar at its largst about th ag of 50. Th ag ffct among womn is vry clos to linar whil it is curvilinar for mn. Looking back at th formula it is sn that th FmalAg2 cofficint narly cancls th Ag2 cofficint for mn giving ag a linar ffct for womn.

11 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr f) Discuss th dgr to which th assumptions of OLS rgrssion ar mt in th bst modl. Th assumptions that hav to b mt hav bn statd abov. Th first of th rquirmnts, th spcification of th modl, cannot b invstigatd byond stating that thr ar no irrlvant variabls in th modl and that it is linar in its paramtrs. Th Gauss-Markov rquirmnts of fixd x-valus and an xpctd valu of 0 for th rror trms cannot b tstd. For th rquirmnts of homoscdasticity and no autocorrlation thr ar tsts. Th sampl is assumd to b a simpl random sampl of th Norwgian population. In such sampls autocorrlation will not occur. Homoscdasticity can b valuatd in a scattr plot of rsidual against prdictd valu: 4, , Unstandardizd Rsidual 0, , , , ,00000 Loss Lin (20% of th points) and idntification numbr addd 5, , , , , ,00000 Unstandardizd Prdictd Valu

12 12 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 Th rsiduals ar fairly vnly distributd around th man valu but with mor ngativ than positiv valus. Th distribution of th rsidual is slightly skwd. Th loss lin addd to th plot dips down a bit at th uppr valus of prdictd y possibly indicating a low lvl of htroscdasticity. Both of ths problms may b rlatd to outlirs such as th cass 2604, 2563, and Th assumption of normally distributd rrors rquirs a symmtrical distribution of th rsiduals. From th scattr plot abov w s that is not th cas. Th distribution can furthr b invstigatd in a comparison of th quantils of th distributions of th rsiduals to th quantils of th normal distribution: Normal Q-Q Plot of Unstandardizd Rsidual 5,0 2,5 Obsrvd Valu 0,0-2,5-5,0-7, Expctd Normal Valu Th plot confirms our conclusions abov. Th distribution of th rsiduals is ngativly skwd with a fw ngativ outlirs. This casts doubt on th tsts prformd. Thy ar not trustworthy. Mayb modl 4 is not th bst modl aftr all. On of th problms may b th outlirs. Th thr largst rsiduals ar found for rspondnts no Rspondnt's idntification numbr TrustIndx Prdictd Valu Rsidual 2563,00 6,7964-6, ,00 6,6035-6, ,67 6,8194-6,15273

13 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr Looking at th DFBETAs it is sms that rspondnts with id no 2703, 2604, 1988 ar candidats for possibly unwantd high impact. Cook s D indicats rspondnt no 2563 as th on with highst influnc. 0, , , , , , , Cook's Distanc Looking closr at th most influntial cas, 2563, w s it is a 84 yar old lady living in Wst Norway without any trust in hr fllow citizns at all. idno TrustIndx 0,000 0,000 0,667 HdmarkOppland SouthEast AgdrRogaland WstNorway Trondlag NorthNorway Ag Fmal Educ NoHHmmbrs RES_1-6,816-6,672-6,174 ZRE_1-4,697-4,597-4,254 COO_1 0,028 0,013 0,009 LEV_1 0,017 0,008 0,006 DFB4_1 (WstNorway) -0,018 0,001 0,001 DFB5_1 (Trondlag) 0,000-0,034-0,032 DFB8_1 (Fmal) -0,106 0,014 0,025 Many of us hav ncountrd such prsons. Thr do not sm to b any invalid data, and hnc no rason to xclud cass. Th most obvious thing th thr prsons listd hr hav in common is a vry low trust in thir fllow citizns.

14 14 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 From th distribution of th trust indx it is sn that thr ar vry fw prsons with a valu blow Frquncy Man = 6,5331 Std. Dv. = 1,48977 N = ,00 2,00 4,00 6,00 8,00 10,00 TrustIndx With non-normal rsiduals and a distribution lik this a transformation of th dpndnt variabl may b a solution to th problm of valid tsts. For xampl th squar root of th trustindx valu or vn th natural logarithm of it may b candidats.

15 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr QUESTION 2 In th sam study of trust, rgional diffrncs in viwing politicians as vot maximizrs wr also invstigatd. Basd on th information availabl in th tabls attachd, plas answr th following: a) Discuss what th study says about th hypothss H1: Th rlationship btwn Politicians intrstd in vots rathr than popls opinions and Ag is curvilinar H2: Th impact of Ag on Politicians intrstd in vots rathr than popls opinions dpnds on th sx of th prson b) Basd on modl 4 writ up th quation dtrmining th rlationship btwn dpndnt and indpndnt variabls in th population studid and prsnt th assumptions that hav to b mt to draw valid infrncs from th stimatd rlationships c) Find in modl 2 th odds ratio btwn womn and mn for thinking that politicians ar vot maximizrs. Discuss rgional variation in Politicians intrstd in vots rathr than popls opinions d) Discuss th dgr to which th assumptions of logistic rgrssion hav bn mt in modl 5 stimatd for this qustion ) Basd on modl 4 writ up th quation for a conditional ffct plot according to ag of rspondnt that will maximis th probability of obsrving Y=1 on th variabl Politicians intrstd in vots rathr than popls opinions f) Find from modl 5 an xprssion for th odds ratio for obsrving Y=1 on Politicians intrstd in vots rathr than popls opinions btwn groups of mn with on yar diffrnc in ag a) Discuss what th study says about th hypothss H1: Th rlationship btwn Politicians intrstd in vots rathr than popls opinions and Ag is curvilinar H2: Th impact of Ag on Politicians intrstd in vots rathr than popls opinions dpnds on th sx of th prson It is assumd that th hypothss apply to th logit rlationships. H1 is thn tstd by comparing th Modl of block 5 to th modl stimatd in block 4. Th tst statistic is 2 H = -2{log L K-H - log L K } In this tst H=2, K=16, -2log L K = 2302,313, and -2log L K-H = 2306,050 This mans that 2 H = 2306, ,313 = 3,737. Th critical valu of th Chisquar distribution with 2 dgrs of frdom and a lvl of significanc 5% is Th two trms with ag squard do not add significantly to th modl.

16 16 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 Hnc w hav to rjct th H1 hypothsis and conclud that ag is linarly rlatd to y. Th sam rsult follows from th omnibus tst of Block 5. Th tst statistic of block 5 in th omnibus tst of modl cofficints is Th p-valu is givn as and with a significanc lvl of th tst of 0.05 w hav to rjct H1. H2 is tstd by including an intraction trm btwn ag and sx in th modl. But sinc sx also is includd in an intraction trm with ducation, th lvl of multicollinarity will incras and t-tsts of th significanc of th intraction trm alon will not b prcis nough. Th two variabls Ag and FmalAg includd in block 4 ar significant togthr sinc 2 H = -2{log L K-H - log L K } = 2328, ,050 = 22,414 is largr than th critical lvl for 2 dgrs of frdom (H=2). Hnc w hav to rjct a null hypothsis of no impact of Ag and FmalAg. Again, th sam rsult follows from th omnibus tst of Block 4. Until furthr invstigations show othrwis w accpt H2. b) Basd on modl 4 writ up th quation dtrmining th rlationship btwn dpndnt and indpndnt variabls in th population studid and prsnt th assumptions that hav to b mt to draw valid infrncs from th stimatd rlationships In th population invstigatd thr is assumd to b a logistic rlationship btwn th probability of a valu of 1 on th dpndnt variabl Y and th indpndnt X-variabls. Th modl can b writtn as Pr[Y i =1] = E[Y i ], whr Y i =[1/(1+xp{-L * i })] + i, i=1,,n, i is th rror trm and L i is th logit dfind as L K 1 X i 0 k k i k 1 whr k is an indx indicating th K-1 indpndnt variabls. Th rlationship btwn dpndnt and indpndnt variabls is supposd to b valid for all individuals i in th Norwgian population. For modl 4 th following variabls can b dfind:

17 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10 X 11 X 12 X 13 HdmarkOppland SouthEast AgdrRogaland WstNorway Trondlag NorthNorway Votd LftRightScal Fmal Educ FmalEduc Ag FmalAg With K=14 and I running ovr th sampl of n=1954 prsons, th paramtrs of this modl can b stimatd by th maximum liklihood mthod, and valid infrncs can b mad if th following assumptions ar mt: Th modl is corrctly spcifid, i..: o All conditional probabilitis for Y=1 ar logistic functions of th x-variabls (this mans th logit is linar in its paramtrs) o Thr ar no irrlvant variabls includd in th modl o Thr ar no rlvant variabls xcludd from th modl All indpndnt variabls hav bn masurd without rrors All cass ar indpndnt In addition it should b obsrvd that th mthod also rquir No prfct multicollinarity No prfct discrimination And that th prcision of th stimats ar affctd by High dgr of multicollinarity High dgr of discrimination Small sampl If th assumptions ar mt, th stimats of th paramtrs will b unbiasd, fficint (minimum varianc) and normally distributd. Th liklihood ratio tst can b usd and in larg sampls b k / SE bk will asymptotically follow a normal distribution.

18 18 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 c) Find in modl 2 th odds ratio btwn womn and mn for thinking that politicians ar vot maximizrs. Discuss rgional variation in Politicians intrstd in vots rathr than popls opinions Th odds ratio btwn womn and mn is dfind as th ratio of xp(logit(womn))/xp(logit(mn)). In modl two th odds ratio btwn womn and mn is found to b OR(womn/mn) = xp(-0,279) = 0,757. This mans that th odds for finding Y=1 among womn ar 24,3% lss than for comparabl mn. If w try to dtrmin th odds ratio btwn womn and mn in modl 3 it bcoms a bit mor complicatd. In that cas it will b bst to us th dfinition of th odds ratio. In modl 3 th logit is dfind as L i = 1,092-0,646 HdmarkOppland i1 +0,168 SouthEast i2-0,084 AgdrRogaland i3 -,320 WstNorway i4-0,408 Trondlag i5-0,003 NorthNorway i6-0,296 Votd i7 +0,026 LftRightScal i8 + 1,148 Fmal i9-0,118 Educ i10-0,118 FmalEduc i11 Th w find th odds as Odds(womn) = xp{ 1,092-0,646 HdmarkOppland i1 +0,168 SouthEast i2-0,084 AgdrRogaland i3 -,320 WstNorway i4-0,408 Trondlag i5-0,003 NorthNorway i6-0,296 Votd i7 +0,026 LftRightScal i8 + 1,148 Fmal i9-0,118 Educ i10-0,118 FmalEduc i11 } Hnc w find th odds ratio of womn to mn as OR (womn/mn) = 1,092-0,646HdOpp +0,168SEast -0,084AgdRog -,320WNor -0,408Trond -0,003NNor -0,296Vot +0,026LRScal + 1,148(Fmal=1) -0,118Educ -0,118(Fmal=1)Educ 1,092-0,646HdOpp +0,168SEast -0,084AgdRog-,3 20WNor -0,408Trond-0,003NNor -0,296Vot +0,026LRScal + 1,148(Fmal=1) -0,118Educ-0,118(Fmal=1)Educ With lmnts of th logit that ar th sam for womn and mn collctd as Const th odds ratio bcoms OR Const + 1,148(Fmal=1) -0,118(Fmal=1)Educ Const +1,148(Fmal=0) -0,118(Fmal=0)Educ 1,148(Fmal=1) -0,118(Fmal=1)Educ 0 1,148(Fmal=1) -0,118(Fmal=1)Educ 1,148-0,118Educ

19 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr Th odds ratio btwn womn and mn in modl 3 dpnds on th amount of ducation thy hav. From th plot blow w s that womn with mor than 9 yar of ducation incrasingly hav smallr odds for thinking politicians ar vot maximizrs compard to mn with th sam amount of ducation. 1 y= x In modl 4 w will gt an additional trm making th odds ratio dpndnt on both ducation and ag. c) continud: 2 c) continud Discuss rgional variation in Politicians intrstd in vots rathr than popls opinions Sinc th first part of 2c is rstrictd to modl 2 it may b a rasonabl assumption that this part should b qually rstrictd. That is not a ncssary implication but must b accptd. Th commnts about rgional variations can b mad both basd on odds ratios and basd on logit cofficints. Th conclusions in broad trms will b th sam. Th commnts hr ar basd on logit cofficints for all modls. Th stimats of th logit cofficints of th rgion dummis tll us how diffrnt thy ar on avrag, th prsons living in this rgion compard to prsons in th rfrnc rgion, Oslo/ Akrshus. Only Hdmark/ Oppland is at significanc lvl 0.05 diffrnt from th rfrnc rgion consistntly across all modls. Variabls in th Equation B block 1* B block 2 B block 3 B block 4 B block 5 HdmarkOppland -,435 -,444 -,646 -,655 -,638 SouthEast,335,329,168,176,184 AgdrRogaland,098,079 -,084 -,052 -,044 WstNorway -,147 -,166 -,320 -,282 -,294 Trondlag -,258 -,286 -,408 -,391 -,385 NorthNorway,152,141 -,003,014,021 * Not that boldfac mans significant at lvl 0.1 and boldfac mans significant at lvl 0.05 Th sign of th cofficint tlls if th probability of thinking politicians ar vot

20 20 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 maximizrs incrass (+) of dcrass (-). In modl 2 only SouthEast givs a highr probability than Oslo/ Akrshus of obsrving Y=1 and th diffrnc is significant at lvl Likwis Hdmark/ Oppland givs a lowr probability significant at lvl As variabls ar addd w s that th impact of rgion changs. Th addition of votd, LftRightScal and Fmal in block 2 incrass th diffrnc btwn Hdmark/ Oppland and th rfrnc catgory Oslo/ Akrshus by about 50%. And whn ducation and th intraction btwn gndr and ducation ar addd in block 3, th South East rgion cass to b significantly diffrnt from Oslo/ Akrshus whil Wst Norway and Trøndlag bcoms significantly diffrnt. From block 3 to block 5 th cofficint stimats do not chang much. It is intrsting to not that th basic pattrn of trust/ confidnc in politicians is th sam hr as in th mor gnral trust qustion of 1a. W know that ducation and political sympathis ar not vnly distributd across rgions. Hnc it must b xpctd that to assss th tru impact of rgion, political sympathis and ducation has to b controlld for. W also know that gndr and ag is vnly distributd across rgions and should not affct stimats of th impact of rgion. Th pattrn of changs of th cofficint stimats is consistnt with this. Basd on th stimats from modl 5 th pattrn of th cofficints suggsts two rgions with somwhat diffrnt viws on politicians. Prsons living in Oslo/ Akrshus, Sout East, Agdr/ Rogaland, and North Norway shar on viw whil thos living in Hdmakr/ Oppland, Trondlag and Wst Norway shar anothr viw with a lowr probability of thinking politicians ar vot maximizrs. In th diffrnt rgions th probabilitis of thinking politicians ar vot maximizrs for womn that votd, with 13 yar of ducation and placing thmslvs at 6 on th lft-right scal ar shown blow. Th lft out rgions Agdr/ Rogaland and North Norway ar vry clos to Oslo/ Akrshus.

21 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr SouthEast OsloAkrshus WstNorway Trondlag HdmarkOppland y=1/(1+ -( x x x x 2) ) y=1/(1+ -( x x x x 2) ) y=1/(1+ -( x x x x 2) ) y=1/(1+ -( x x x x 2) ) y=1/(1+ -( x x x x 2) ) SouthEast OsloAkrshus WstNorway Trondlag HdmarkOppland d) Discuss th dgr to which th assumptions of logistic rgrssion hav bn mt in modl 5 Th assumptions hav bn statd abov. Th discussion will assum that th tabls rfr to modl 5. It has alrady bn dtrmind that in modl 5 Ag2 and FmalAg2 ar irrlvant variabls. From th stimat of modl 5 it also is apparnt that th LftRightScal do not contribut significantly to th modl and must b considrd irrlvant basd on th information prsntd hr. Howvr, thr may concivably b good thortical argumnts for kping th LftRightScal in th modl. Th possibl absnc of rlvant variabls cannot b commntd upon. Th tst of ag as a curvilinar variabl in th logit rjctd th possibility of curvilinarity for ag. Th possibility that Educ and LftRightScal might b curvilinarly rlatd to th logit ought to b invstigatd. Plots of th logits computd in th tabl of probability of ln(p/(1-p)) according to LftRightScal, Ag and Education do not giv clar indications of curvilinarity. Th most probabl curvilinar rlationship is for ag, but this has alrady bn rjctd.

22 22 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 Chart of Logit for groups on th LftRightScal 1,50 1,25 1,00 Logit 0,75 0,50 0,25 0, Rows Chart of Logit for Ag groups 1,5 1,0 0,5 Logit 0,0-0,5-1,0-1, Rows

23 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr Chart of Logit for Education groups 1,5 1,0 Logit 0,5 0, Rows Th conclusion on th spcification rquirmnt must b that thr ar 3 irrlvant variabls. In othr rspcts th modl must b sn as corrct. Th two othr rquirmnt that cass ar indpndnt and that th x-variabls ar masurd without rror cannot b tstd. But th sourc of th data, Th Europan Social Survy assurs that th data hav bn collctd using th bst availabl procdurs. Th absnc of prfct multicollinarity and prfct discrimination is dmonstratd by th xistnc of th modl stimats. Strictly spaking th abov discussion covrs th assumptions of th modl. But th possibility for svr statistical problms du to multicollinarity, discrimination, small sampl and influntial cass might also b invstigatd. Th ffctiv sampl is 1954 cass, and th dpndnt variabl has about 31,5% in th Y=1 catgory. Th sampl siz should b mor than larg nough. Thr ar no cross tabulations of th dpndnt variabl and th dichotomous indpndnt variabls. Hnc th possibility for som dgr of discrimination cannot b commntd upon.

24 24 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 Th lvl of multicollinarity is high in modl 5 du to intraction trms and ag squard as a part of th tst of ag as a curvilinar variabl. As long as this is considrd in th tst procdur mployd, multicollinarity du to modl spcification cannot b considrd a problm. Th possibl xistnc of influntial cass can b invstigatd in th plots of th DltaParsonChi2 and th DltaDviancChi2. As a rul of thumb valus of ths poornss of fit statistics abov 4 ar considrd significant sinc thir distribution is asymptotically with 1 dgr of frdom. Th ultimat tst of influnc is to compar two rgrssions, on with th possibly influntial cas includd and on with it xcludd. Basd on th figurs blow 3 cass may b considrd for such an invstigation: cass no 2082, 2625, and 651. From ths cass and down to th nxt in lin thr is a clar gap. From inspction of th plot of th analogu of Cook s D statistic a fourth cas no 884 may b addd. 12,00 10, PoliticiansVotMaximisrs,00 1,00 DltaParsonChi2 8,00 6,00 4, , ,00 0, , , , ,80000 Prdictd probability

25 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr ,00 PoliticiansVotMaximisrs,00 1,00 5, DltaDviancChi2 4,00 3,00 2, , ,00 0, , , , ,80000 Prdictd probability Th four cass idntifid abov sm to hav svral charactristics in common. Thy ar all fmal with high ducation. Thy hav all votd and thy all considr politicians to b vot maximizrs. idno Ag Fmal Educ OsloAkrshus 1 HdmarkOppland 1 SouthEast AgdrRogaland WstNorway 1 Trondlag 1 NorthNorway Politicians VotMaximisrs Votd LftRightScal Th 884 cas has th highst analogu to Cook s D statistic; th othr thr ar idntifid by high valus on th DltaParsonChi2 and th DltaDviancChi2 statistics. Rgrssion whr ths two groups of cass wr dltd should hav bn compard to th rportd rgrssion to dtrmin thir actual impact.

26 26 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 ) Basd on modl 4 writ up th quation for a conditional ffct plot according to ag of rspondnt that will maximis th probability of obsrving Y=1 on th variabl Politicians intrstd in vots rathr than popls opinions B Minimum Maximum Valu maximizing logit,514 Constant -,655 HdmarkOppland,00 1,00 0,176 SouthEast,00 1,00 1 -,052 AgdrRogaland,00 1,00 0 -,282 WstNorway,00 1,00 0 -,391 Trondlag,00 1,00 0,014 NorthNorway,00 1,00 0 -,494 Votd,00 1,00 0,032 LftRightScal,00 10,00 10,254 Fmal,00 1,00 1 -,102 Educ 5,00 20,00 5 -,080 FmalEduc,00 18,00 5,011 Ag 17,00 93,00 x,009 FmalAg,00 93,00 x To maximiz th probability on has to choos variabl valus maximizing th logit. That mans choosing maximum variabl valus whr th cofficint is positiv and minimum valus whr th cofficint is ngativ. It is sn that th prson maximizing will b a non-votr placing him or hrslf at th xtrm right on th LftRightScal and locatd in th Southast rgion. Fmal, ag and ducation ar linkd togthr and it is not at th outst obvious how to maximiz th logit for ths. Sinc th cofficint for both Educ and FmalEduc is ngativ, Educ should hav its minimum valu of 5. And sinc th cofficints for Fmal, Ag and FmalAg ar all positiv on should choos th4 highst valu of Fmal (fmal=1). Hnc a prson maximizing th probability of thinking politicians ar vot maximizrs will b a fmal prson with a minimum of ducation living in th SouthEast who do not vot but plac hrslf at th xtrm right of th LftRightScal. Th quation for th logit maximizing th probability according to ag can thn b writtn L i = 0,514 +0,176(SouthEast i) +0,032(LftRightScal i) + 0,254(Fmal i) -0,102(Educ i) -0,08(Fmal i*educ i) +0,011(Ag i) +0,009(Fmal i*ag i) = 0,514 +0,176(SouthEast i =1) +0,032(LftRightScal i =10) + 0,254(Fmal i =1) -0,102(Educ i =5) -0,08(Fmal i =1)(Educ i =5) +0,011(Ag i =x) +0,009(Fmal i =1)(Ag i =x) = 0, ,02x

27 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr Sinc th cofficint of x is positiv, w s that th probability incrass with ag. W can plot th prdictd P = 1/(1+xp(-[0,354+0,02x])). For mn th formula bcoms P = 1/(1+xp(-[0,5+0,011x])) y=1/( x ) y=1/( x ) 80 Womn Mn If w would do th sam for modl 5 th problm bcom mor complx sinc ag is curvilinarly rlatd to th logit. W hav to dtrmin what is mor important: th ag polynomial for mn or th ag polynomial for womn? B Variabl Minimum Maximum Valu maximizing logit 1,063 Constant -,638 HdmarkOppland,00 1,00 0,184 SouthEast,00 1,00 1 -,044 AgdrRogaland,00 1,00 0 -,294 WstNorway,00 1,00 0 -,385 Trondlag,00 1,00 0,021 NorthNorway,00 1,00 0 -,477 Votd,00 1,00 0,032 LftRightScal,00 10, ,703 Fmal,00 1,00 1 -,090 Educ 5,00 20,00 5 -,100 FmalEduc,00 18,00 1*5 -,023 Ag 17,00 93,00 X,065 FmalAg,00 93,00 1*X,00035 Ag2 289, ,00 X 2 -,00058 FmalAg2, ,00 1* X 2 Womn will maximiz th logit if th diffrnc btwn th two polynomials, -0,703(Fmal i ) + 0,065(FmalAg i ) -0,00058(FmalAg2 i ), is largr than zro. From a plot of this polynomial w s that for rasonabl valus of ag, 15< ag < 95, womn will hav highr logit than mn.

28 28 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 y= x x Hnc th logit for th conditional ffct plot that will maximiz th probability according to ag is L i = 1,063 +0,184(SouthEast i =1) +0,032(LftRightScal i =10) -0,703 (Fmal i =1) -0,09(Educ i =5) -0,1(Fmal i =1)(Educ i =5) -0,023(Ag i =x) + 0,065(Fmal i =1)(Ag i =x) + 0,00035(Ag2 i =x 2 ) - 0,00058(Fmal i =1)(Ag2 i =x 2 ) = - 0, ,042x 0,00023x 2 W can plot th prdictd P = 1/(1+xp(-[- 0, ,042x 0,00023x 2 ])) and finds that th probability dclins with ag. 0.2 y=1/( x x2 ) f) Find from modl 5 an xprssion for th odds ratio for obsrving Y=1 on Politicians intrstd in vots rathr than popls opinions btwn groups of mn with on yar diffrnc in ag Th odds ratio btwn ag groups with on yar diffrnc is dfind as th ratio of xp(logit(ag=x+1))/xp(logit(ag=x)). B Ag -,023 FmalAg,065 Ag2,00035 FmalAg2 -,00058 As argud abov thos parts of th logit quation that dos not involv ag can b sn as a constant calld Const in th xprssion blow. In modl 5 four lmnts involv ag:

29 Erling Brg SOS3003 Applid statistical data analysis for th social scincs 10 dsmbr 2004 OR(ag+1/ag) = = 2 2 Const -0,023(Ag+1) + 0,065(Fmal=1)(Ag+1) +0,00035(Ag+1) -0,00058(Fmal=1)(Ag+1) 2 2 Const -0,023Ag + 0,065(Fmal=1)Ag +0,00035Ag -0,00058(Fmal=1)Ag Const -0,023Ag -0, ,065(Fmal=1)Ag+0,065(Fmal=1) +0,00035(Ag+1) -0,00058(Fmal=1)(Ag+1) Const -0,023Ag + 0,065(Fmal=1)Ag +0,00035Ag -0,00058(Fmal=1)Ag -0,023 +0,065(Fmal=1) +0,00035(Ag+1) -0,00058(Fmal=1)(Ag+1) ,00035Ag -0,00058(Fmal=1)Ag Th odds ratio for mn and th odds ratio for womn ar thn givn as OR mn (ag+1/ag) = = -0,023 +0,00035(2Ag+1) = = 0 OR = 2 2-0,023 +0,00035(Ag+1) -0,023 +0,00035(Ag +2Ag+1) womn ,00035Ag +0,00035Ag (ag+1/ag) -0, ,0007Ag ,023 +0,065*1 +0,00035(Ag+1) -0,00058*1*(Ag+1) ,00035Ag -0,00058*1*Ag 2 2 0,042-0,00023(Ag+1) 0,042-0,00023(Ag +2Ag+1) 2 2-0,00023Ag -0,00023Ag 0,042-0,00023(2Ag+1) 0 0, ,00046Ag