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Multinomial Logistic Regression Inanimate objects can be classified scientifically into three major categories; those that don't work, those that break down and those that get lost” (Russell Baker)

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Multinomial Logistic Regression “ Inanimate objects can be classified scientifically into three major categories; those that don't work, those that break down and those that get lost” (Russell Baker). Multinomial Logistic Regression. - PowerPoint PPT Presentation

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  • Multinomial Logistic Regression

    Inanimate objects can be classified scientifically into three major categories; those that don't work, those that break down and those that get lost (Russell Baker)

  • Multinomial Logistic RegressionAlso known as polytomous or nominal logistic or logit regression or the discrete choice model

    Generalization of binary logistic regression to a polytomous DVWhen applied to a dichotomous DV identical to binary logistic regression

  • Polytomous VariablesThree or more unordered categoriesCategories mutually exclusive and exhaustiveSometimes called multicategorical or sometimes multinomial variables

  • Polytomous DVsReason for leaving welfare:marriage, stable employment, move to another state, incarceration, or deathStatus of foster home application:licensed to foster, discontinued application process prior to licensure, or rejected for licensureChanges in living arrangements of the elderly:newly co-residing with their children, no longer co-residing, or residing in institutions

  • Single (Dichotomous) IV ExampleDV = interview tracking efforteasy-to-interview and track mothers (Easy); difficult-to-track mothers who required more telephone calls (MoreCalls); difficult-to-track mothers who required more unscheduled home visits (MoreVisits) IV = race, 0 = European-American, 1 = African-AmericanN = 246 mothersWhat is the relationship between race and interview tracking effort?

  • CrosstabulationTable 3.1

    Relationship between race and tracking effort is statistically significant [2(2, N = 246) = 8.69, p = .013]

  • Reference CategoryIn binary logistic regression category of the DV coded 0 implicitly serves as the reference category Known as baseline, base, or comparison categoryNecessary to explicitly select reference categoryEasy selected

  • ProbabilitiesTable 3.1More Calls (vs. Easy)European-American: .24 = 30 / (30 + 96) African-American: .31 = 24 / (24 + 53) More Visits (vs. Easy)European-American: .15 = 17 / (17 + 96) African-American: .33 = 26 / (26 +53)

  • Odds & Odds RatioMore Calls (vs. Easy)European-American: .3125 (.2098 / .6713)African-American: .4528 (.2330 / .5146)Odds Ratio = 1.45 (.4528 / .3125)45% increase in the oddsMore Visits (vs. Easy)European-American: .1771 (.1189 / .6713)African-American: .4905 (.2524 / .5146). Odds Ratio = 2.77 (.4905 / .1771)177% increase in the odds

  • Question & AnswerWhat is the relationship between race and interview tracking effort?

    The odds of requiring more calls, compared to being easy-to-track, are higher for African-Americans by a factor of 1.45 (45%). The odds of requiring more visits, compared to being easy-to-track, are higher for African-Americans by a factor of 2.77 (177%).

  • Multinomial Logistic RegressionSet of binary logistic regression models estimated simultaneouslyNumber of non-redundant binary logistic regression equations equals the number of categories of the DV minus one

  • Statistical Significance Table 3.2(Race, More Calls vs. Easy) = (Race, More Visits vs. Easy) = 0RejectTable 3.3(Race, More Calls vs. Easy) = (Race, More Visits vs. Easy) = 0RejectTable 3.4(Race, More Calls vs. Easy) = 0Dont Reject(Race, More Visits vs. Easy) = 0Reject

  • Odds RatiosOR(More Calls vs. Easy) = 1.45The odds of requiring more calls, compared to being easy-to-track, are not significantly different for European- and African-Americans.OR(More Visits vs. Easy) = 2.77The odds of requiring more visits, compared to being easy-to-track, are higher for African-Americans by a factor of 2.77 (177%).

  • Estimated Logits (L) Table 3.4

    L(More Calls vs. Easy) = a + BRaceXRaceL(More Calls vs. Easy) = -1.163 + (.371)(XRace)

    L(More Visits vs. Easy) = a + BRaceXRaceL(More Visits vs. Easy) = -1.731 + (1.019)(XRace)

  • Logits to OddsAfrican-Americans (X = 1)

    L(More Calls vs. Easy) = -.792 = -1.163 + (.371)(1)Odds = e-.792 = .45

    L(More Visits vs. Easy) = -.712 = -1.731 + (1.019)(1)Odds = e-.712 = .49

  • Logits to Probabilities

    African-Americans, L(More Calls vs. Easy) = -.792

    African-Americans, L(More Visits vs. Easy) = -.712

  • Question & AnswerWhat is the relationship between race and interview tracking effort?

    The odds of requiring more calls, compared to being easy-to-track, are not significantly different for European- and African-Americans.The odds of requiring more visits, compared to being easy-to-track, are higher for African-Americans by a factor of 2.77 (177%).

  • Single (Quantitative) IV ExampleDV = interview tracking efforteasy-to-interview and track mothers (Easy); difficult-to-track mothers who required more telephone calls (MoreCalls); difficult-to-track mothers who required more unscheduled home visits (MoreVisits) IV = years of educationN = 246 mothersWhat is the relationship between education and interview tracking effort?

  • Statistical Significance Table 3.6(Education, More Calls vs. Easy) = (Education, More Visits vs. Easy) = 0RejectTable 3.7(Education, More Calls vs. Easy) = 0Dont Reject(Education, More Visits vs. Easy) = 0Reject

  • Odds RatiosOR(More Calls vs. Easy) = .88The odds of requiring more calls, compared to being easy-to-track, are not significantly associated with education.OR(More Visits vs. Easy) = .76For every additional year of education the odds of needing more visits, compared to being easy-to-track, decrease by a factor of .76 (i.e., -24.1%).

  • FiguresEducation.xls

  • Estimated Logits (L) Table 3.7

    X = 12 (high school education)

    L(More Calls vs. Easy) = -.977 = .583 + (-.130)(12)L(More Visits vs. Easy) = -1.235 = 2.077 + (-.276)(12)

  • Effect of Education on Tracking Effort (Logits)

    Chart1

    -0.4550586931-0.1326143619

    -0.584770778-0.4088679673

    -0.7144828629-0.6851215727

    -0.8441949479-0.9613751781

    -0.9739070328-1.2376287835

    -1.1036191177-1.5138823889

    -1.2333312026-1.7901359943

    -1.3630432875-2.0663895997

    -1.4927553724-2.3426432051

    -1.6224674574-2.6188968105

    More Calls

    More Visits

    Years of Education

    Logits

    Data

    More Calls

    aLogitOddsp

    0.583-0.1308-0.460.630.39

    0.583-0.1309-0.580.560.360.03

    0.583-0.13010-0.710.490.330.03

    0.583-0.13011-0.840.430.300.03

    0.583-0.13012-0.970.380.270.03

    0.583-0.13013-1.100.330.250.03

    0.583-0.13014-1.230.290.230.02

    0.583-0.13015-1.360.260.200.02

    0.583-0.13016-1.490.220.180.02

    0.583-0.13017-1.620.200.160.02

    More Visits

    2.077-0.2768-0.130.880.47

    2.077-0.2769-0.410.660.400.07

    2.077-0.27610-0.690.500.340.06

    2.077-0.27611-0.960.380.280.06

    2.077-0.27612-1.240.290.220.05

    2.077-0.27613-1.510.220.180.04

    2.077-0.27614-1.790.170.140.04

    2.077-0.27615-2.070.130.110.03

    2.077-0.27616-2.340.100.090.02

    2.077-0.27617-2.620.070.070.02

    Logits

    Logits

    -0.4550586931-0.1326143619

    -0.584770778-0.4088679673

    -0.7144828629-0.6851215727

    -0.8441949479-0.9613751781

    -0.9739070328-1.2376287835

    -1.1036191177-1.5138823889

    -1.2333312026-1.7901359943

    -1.3630432875-2.0663895997

    -1.4927553724-2.3426432051

    -1.6224674574-2.6188968105

    More Calls

    More Visits

    Years of Education

    Logits

    Odds

    Odds

    0.63441073130.8758027699

    0.55723357740.6644019493

    0.48944515660.5040289497

    0.42990331350.3823667019

    0.37760483780.2900712248

    0.33166856140.2200539822

    0.29132051190.1669374656

    0.25588087180.1266421863

    0.22475252470.0960733607

    0.19741099440.0728832225

    More Calls

    More Visits

    Years of Education

    Odds

    Probabilities

    Probabilities

    0.38815869180.4668949124

    0.35783557810.3991835924

    0.32860904910.3351191809

    0.30065201570.2766029458

    0.2741024330.224849

    0.24906239510.1803641359

    0.22559891920.143056051

    0.20374613350.1124067497

    0.18350852120.0876523088

    0.16486485870.0679321113

    More Calls

    More Visits

    Years of Education

    Probabilities

  • Logits to OddsX = 12 (high school education)

    Odds(More Calls vs. Easy) = e-.977 = .38Odds(More Visits vs. Easy) = e-1.235 = .29

  • Effect of Education on Tracking Effort (Odds)

    Chart2

    0.63441073130.8758027699

    0.55723357740.6644019493

    0.48944515660.5040289497

    0.42990331350.3823667019

    0.37760483780.2900712248

    0.33166856140.2200539822

    0.29132051190.1669374656

    0.25588087180.1266421863

    0.22475252470.0960733607

    0.19741099440.0728832225

    More Calls

    More Visits

    Years of Education

    Odds

    Data

    More Calls

    aLogitOddsp

    0.583-0.1308-0.460.630.39

    0.583-0.1309-0.580.560.360.03

    0.583-0.13010-0.710.490.330.03

    0.583-0.13011-0.840.430.300.03

    0.583-0.13012-0.970.380.270.03

    0.583-0.13013-1.100.330.250.03

    0.583-0.13014-1.230.290.230.02

    0.583-0.13015-1.360.260.200.02

    0.583-0.13016-1.490.220.180.02

    0.583-0.13017-1.620.200.160.02

    More Visits

    2.077-0.2768-0.130.880.47

    2.077-0.2769-0.410.660.400.07

    2.077-0.27610-0.690.500.340.06

    2.077-0.27611-0.960.380.280.06

    2.077-0.27612-1.240.290.220.05

    2.077-0.27613-1.510.220.180.04

    2.077-0.27614-1.790.170.140.04

    2.077-0.27615-2.070.130.110.03

    2.077-0.27616-2.340.100.090.02

    2.077-0.27617-2.620.070.070.02

    Logits

    Logits

    -0.4550586931-0.1326143619

    -0.584770778-0.4088679673

    -0.7144828629-0.6851215727

    -0.8441949479-0.9613751781

    -0.9739070328-1.2376287835

    -1.1036191177-1.5138823889

    -1.2333312026-1.7901359943

    -1.3630432875-2.0663895997

    -1.4927553724-2.3426432051

    -1.6224674574-2.6188968105

    More Calls

    More Visits

    Years of Education

    Logits

    Odds

    Odds

    0.63441073130.8758027699

    0.55723357740.6644019493

    0.48944515660.5040289497

    0.42990331350.3823667019

    0.37760483780.2900712248

    0.33166856140.2200539822

    0.29132051190.1669374656

    0.25588087180.1266421863

    0.22475252470.0960733607

    0.19741099440.0728832225

    More Calls

    More Visits

    Years of Education

    Odds

    Probabilities

    Probabilities

    0.38815869180.4668949124

    0.35783557810.3991835924

    0.32860904910.3351191809

    0.30065201570.2766029458

    0.2741024330.224849

    0.24906239510.1803641359

    0.22559891920.143056051

    0.20374613350.1124067497

    0.18350852120.0876523088

    0.16486485870.0679321113

    More Calls

    More Visits

    Years of Education

    Probabilities

  • Logits to ProbabilitiesX = 12 (high school education)

  • Effect of Education on Tracking Effort (Probabilities)

    Chart3

    0.38815869180.4668949124

    0.35783557810.3991835924

    0.32860904910.3351191809

    0.30065201570.2766029458

    0.2741024330.224849

    0.24906239510.1803641359

    0.22559891920.143056051

    0.20374613350.1124067497

    0.18350852120.0876523088

    0.16486485870.0679321113

    More Calls

    More Visits

    Years of Education

    Probabilities

    Data

    More Calls

    aLogitOddsp

    0.583-0.1308-0.460.630.39

    0.583-0.1309-0.580.560.360.03

    0.583-0.13010-0.710.490.330.03

    0.583-0.13011-0.840.430.300.03

    0.583-0.13012-0.970.380.270.03

    0.583-0.13013-1.100.330.250.03

    0.583-0.13014-1.230.290.230.02

    0.583-0.13015-1.360.260.200.02

    0.583-0.13016-1.490.220.180.02

    0.583-0.13017-1.620.200.160.02

    More Visits

    2.077-0.2768-0.130.880.47

    2.077-0.2769-0.410.660.400.07

    2.077-0.27610-0.690.500.340.06

    2.077-0.27611-0.960.380.280.06

    2.077-0.27612-1.240.290.220.05

    2.077-0.27613-1.510.220.180.04

    2.077-0.27614-1.790.170.140.04

    2.077-0.27615-2.070.130.110.03

    2.077-0.27616-2.340.100.090.02

    2.077-0.27617-2.620.070.070.02

    Logits

    Logits

    -0.4550586931-0.1326143619

    -0.584770778-0.4088679673

    -0.7144828629-0.6851215727

    -0.8441949479-0.9613751781

    -0.9739070328-1.2376287835

    -1.1036191177-1.5138823889

    -1.2333312026-1.7901359943

    -1.3630432875-2.0663895997

    -1.4927553724-2.3426432051

    -1.6224674574-2.6188968105

    More Calls

    More Visits

    Years of Education

    Logits

    Odds

    Odds

    0.63441073130.8758027699

    0.55723357740.6644019493

    0.48944515660.5040289497

    0.42990331350.3823667019

    0.37760483780.2900712248

    0.33166856140.2200539822

    0.29132051190.1669374656

    0.25588087180.1266421863

    0.22475252470.0960733607

    0.19741099440.0728832225

    More Calls

    More Visits

    Years of Education

    Odds

    Probabilities

    Probabilities

    0.38815869180.4668949124

    0.35783557810.3991835924

    0.32860904910.3351191809

    0.30065201570.2766029458

    0.2741024330.224849

    0.24906239510.1803641359

    0.22559891920.143056051

    0.20374613350.1124067497

    0.18350852120.0876523088

    0.16486485870.0679321113

    More Calls

    More Visits

    Years of Education

    Probabilities

  • Question & AnswerWhat is the relationship between education and interview tracking effort?

    The odds of requiring more calls, compared to being easy-to-track, are not significantly associated with education. For every additional year of education the odds of needing more visits, compared to being easy-to-track, decrease by a factor of .76 (i.e., -24.1%).

  • Multiple IV ExampleDV = interview tracking efforteasy-to-interview and track mothers (Easy); difficult-to-track mothers who required more telephone calls (MoreCalls); difficult-to-track mothers who required more unscheduled home visits (MoreVisits) IV = race, 0 = European-American, 1 = African-AmericanIV = years of educationN = 246 mothers

  • Multiple IV Example (contd)What is the relationship between race and interview tracking effort, when controlling for education?

  • Statistical Significance Table 3.8(Race, More Calls vs. Easy) = (Race, More Visits vs. Easy) = (Ed, More Calls vs. Easy) = (Ed, More Visits vs. Easy) = 0Reject Table 3.9(Race, More Calls vs. Easy) = (Race, More Visits vs. Easy) = 0Reject(Ed, More Calls vs. Easy) = (Ed, More Visits vs. Easy) = 0Reject

  • Statistical Significance (contd) Table 3.10(Race, More Calls vs. Easy) = 0Dont reject(Race, More Visits vs. Easy) = 0Reject(Ed, More Calls vs. Easy) = 0Dont reject(Ed, More Visits vs. Easy) = 0Reject

  • Odds Ratios: RaceOR(More Calls vs. Easy) = 1.36The odds of requiring more calls, compared to being easy-to-track, are not significantly different for European- and African-Americans.OR(More Visits vs. Easy) = 2.48The odds of requiring more visits, compared to being easy-to-track, are higher for African-Americans by a factor of 2.48 (148%).

  • Odds Ratios: EducationOR(More Calls vs. Easy) = .89The odds of requiring more calls, compared to being easy-to-track, are not significantly associated with education.OR(More Visits vs. Easy) = .77For every additional year of education the odds of needing more visits, compared to being easy-to-track, decrease by a factor of .77 (i.e., -23%), when controlling for race.

  • FiguresRace & Education.xls

  • Effect of Education on Tracking Effort for African-Americans (Odds)

    Chart1

    0.73225909611.3007208427

    0.64922211321.0050460465

    0.57560138820.7765828935

    0.5103291330.6000530947

    0.45245864460.4636513622

    0.40115057490.3582559403

    0.35566075630.2768185952

    0.31532940870.2138932702

    0.27957156990.1652718851

    0.24786861150.1277029238

    More Calls

    More Visits

    Years of Education

    Odds

    Data

    More Calls

    aBEdXEdBRaceXRaceLogitOddsp

    0.345-0.12080.3071-0.310.730.42

    0.345-0.12090.3071-0.430.650.39

    0.345-0.120100.3071-0.550.580.37

    0.345-0.120110.3071-0.670.510.34

    0.345-0.120120.3071-0.790.450.31

    0.345-0.120130.3071-0.910.400.29

    0.345-0.120140.3071-1.030.360.26

    0.345-0.120150.3071-1.150.320.24

    0.345-0.120160.3071-1.270.280.22

    0.345-0.120170.3071-1.390.250.20

    More Visits

    1.416-0.25880.91010.261.300.57

    1.416-0.25890.91010.011.010.50

    1.416-0.258100.9101-0.250.780.44

    1.416-0.258110.9101-0.510.600.38

    1.416-0.258120.9101-0.770.460.32

    1.416-0.258130.9101-1.030.360.26

    1.416-0.258140.9101-1.280.280.22

    1.416-0.258150.9101-1.540.210.18

    1.416-0.258160.9101-1.800.170.14

    1.416-0.258170.9101-2.060.130.11

    Logits

    Logits

    -0.6182500365-0.6475540385

    -0.738609547-0.9054392858

    -0.8589690575-1.163324533

    -0.979328568-1.4212097802

    -1.0996880785-1.6790950275

    -1.220047589-1.9369802747

    -1.3404070995-2.194865522

    -1.46076661-2.4527507692

    -1.5811261205-2.7106360165

    -1.701485631-2.9685212637

    More Calls

    More Visits

    Years of Education

    Estimated Logits

    Figure x. Effect of Education on Tracking Effort

    Odds

    Odds

    0.73225909611.3007208427

    0.64922211321.0050460465

    0.57560138820.7765828935

    0.5103291330.6000530947

    0.45245864460.4636513622

    0.40115057490.3582559403

    0.35566075630.2768185952

    0.31532940870.2138932702

    0.27957156990.1652718851

    0.24786861150.1277029238

    More Calls

    More Visits

    Years of Education

    Estimated Odds

    Figure x. Effect of Education on Tracking Effort for African-Americans

    Probabilities

    Probabilities

    0.35017955790.3435409406

    0.32330827230.2879340063

    0.29755478410.2380637251

    0.27302503080.1944719983

    0.24979834370.1572153394

    0.22792807590.1259799802

    0.20744311920.1002125154

    0.18835010220.0792376233

    0.1706360550.0623486587

    0.15427133270.0488684093

    More Calls

    More Visits

    Years of Education

    Estimated Probabilities

    Figure x. Effect of Education on Tracking Effort

  • Effect of Education on Tracking Effort for African-Americans (Probabilities)

    Chart4

    0.4227191520.5653536138

    0.3936535340.5012583368

    0.36532170670.4371216769

    0.33789266310.3750207394

    0.31151223910.3167771876

    0.28630083170.2637617327

    0.26235232870.2168033864

    0.23973417350.1762043464

    0.21848841950.1418311788

    0.19863358140.1132416358

    More Calls

    More Visits

    Years of Education

    Probabilities

    Data

    More Calls

    aBEdXEdBRaceXRaceLogitOddsp

    0.345-0.12080.3071-0.310.730.42

    0.345-0.12090.3071-0.430.650.39

    0.345-0.120100.3071-0.550.580.37

    0.345-0.120110.3071-0.670.510.34

    0.345-0.120120.3071-0.790.450.31

    0.345-0.120130.3071-0.910.400.29

    0.345-0.120140.3071-1.030.360.26

    0.345-0.120150.3071-1.150.320.24

    0.345-0.120160.3071-1.270.280.22

    0.345-0.120170.3071-1.390.250.20

    More Visits

    1.416-0.25880.91010.261.300.57

    1.416-0.25890.91010.011.010.50

    1.416-0.258100.9101-0.250.780.44

    1.416-0.258110.9101-0.510.600.38

    1.416-0.258120.9101-0.770.460.32

    1.416-0.258130.9101-1.030.360.26

    1.416-0.258140.9101-1.280.280.22

    1.416-0.258150.9101-1.540.210.18

    1.416-0.258160.9101-1.800.170.14

    1.416-0.258170.9101-2.060.130.11

    Logits

    Logits

    -0.6182500365-0.6475540385

    -0.738609547-0.9054392858

    -0.8589690575-1.163324533

    -0.979328568-1.4212097802

    -1.0996880785-1.6790950275

    -1.220047589-1.9369802747

    -1.3404070995-2.194865522

    -1.46076661-2.4527507692

    -1.5811261205-2.7106360165

    -1.701485631-2.9685212637

    More Calls

    More Visits

    Years of Education

    Estimated Logits

    Figure x. Effect of Education on Tracking Effort

    Odds

    Odds

    0.53888664490.5233242435

    0.47777778140.4043642146

    0.4235985630.3124457162

    0.37556317930.2414217729

    0.33297492950.1865427157

    0.29521611750.1441385521

    0.2617390930.1113735378

    0.23205830830.0860565389

    0.2057432760.0664945017

    0.18241232520.0513792305

    More Calls

    More Visits

    Years of Education

    Estimated Odds

    Figure x. Effect of Education on Tracking Effort

    Probabilities

    Probabilities

    0.4227191520.5653536138

    0.3936535340.5012583368

    0.36532170670.4371216769

    0.33789266310.3750207394

    0.31151223910.3167771876

    0.28630083170.2637617327

    0.26235232870.2168033864

    0.23973417350.1762043464

    0.21848841950.1418311788

    0.19863358140.1132416358

    More Calls

    More Visits

    Years of Education

    Estimated Probabilities

    Figure x. Effect of Education on Tracking Effort for African-Americans

  • Question & AnswerWhat is the relationship between race and interview tracking effort, when controlling for education?

    The odds of requiring more calls, compared to being easy-to-track, are not significantly different for European- and African-Americans, when controlling for education. The odds of requiring more visits, compared to being easy-to-track, are higher for African-Americans by a factor of 2.48 (148%), when controlling for education.

  • Assumptions Necessary for Testing HypothesesAssumptions discussed in GZLM lectureIndependence of irrelevant alternatives (IIA)Odds of one outcome (e.g., More Calls) relative to another (e.g., Easy) are not influenced by other alternatives (e.g., More Visits)

  • Model EvaluationCreate a set of binary DVs from the polytomous DV

    recode TrackCat (1=0) (2=1) (3=sysmis) into MoreCalls.recode TrackCat (1=0) (2=sysmis) (3=1) into MoreVisits.

    Run separate binary logistic regressionsUse binary logistic regression methods to detect outliers and influential observations

  • Model Evaluation (contd)Index plotsLeverage valuesStandardized or unstandardized deviance residualsCooks DGraph and compare observed and estimated counts

  • Analogs of R2None in standard use and each may give different resultsTypically much smaller than R2 values in linear regressionDifficult to interpret

  • MulticollinearitySPSS multinomial logistic regression doesnt compute multicollinearity statisticsUse SPSS linear regressionProblematic levelsTolerance < .10 or VIF > 10

  • Additional TopicsPolytomous IVsCurvilinear relationshipsInteractions

  • Additional Regression Models for Polytomous DVsMultinomial probit regressionSubstantive results essentially indistinguishable from binary logistic regressionChoice between this and binary logistic regression largely one of convenience and discipline-specific convention Many researchers prefer binary logistic regression because it provides odds ratios whereas probit regression does not, and binary logistic regression comes with a wider variety of fit statistics

  • Additional Regression Models for Polytomous DVs (contd)Discriminant analysisLimited to continuous IVs