binary logistic regression

Click here to load reader

Post on 18-Jan-2016

57 views

Category:

Documents

3 download

Embed Size (px)

DESCRIPTION

Binary Logistic Regression “To be or not to be, that is the question..”(William Shakespeare, “Hamlet”). Binary Logistic Regression. Also known as “logistic” or sometimes “logit” regression Foundation from which more complex models derived - PowerPoint PPT Presentation

TRANSCRIPT

  • Binary Logistic Regression

    To be or not to be, that is the question..(William Shakespeare, Hamlet)

  • Binary Logistic RegressionAlso known as logistic or sometimes logit regression

    Foundation from which more complex models derivede.g., multinomial regression and ordinal logistic regression

  • Dichotomous VariablesTwo categories indicating whether an event has occurred or some characteristic is present

    Sometimes called binary or binomial variables

  • Dichotomous DVsPlaced in foster care or notDiagnosed with a disease or notAbused or notPregnant or notService provided or not

  • Single (Dichotomous) IV ExampleDV = continue fostering, 0 = no, 1 = yesCustomary to code category of interest 1 and the other category 0IV = married, 0 = not married, 1 = marriedN = 131 foster families

    Are two-parent families more likely to continue fostering than one-parent families?

  • CrosstabulationTable 2.1

    Relationship between marital status and continuation is statistically significant [2(1, N = 131) = 5.65, p = .017]

    A higher percentage of two-parent families (62.20%) than single-parent families (40.82%) planned to continue fostering

  • Strength & Direction of RelationshipsDifferent ways to quantify the relationship between IV(s) and DVProbabilitiesOddsOdds Ratio (OR)Also abbreviated as eB, Exp(B) (on SPSS output), or exp(B)% change

  • Roadmap to Computations

  • ProbabilitiesPercentages in Table 2.1 as probabilities (e.g., 62.20% as .6220)

    pProbability that event will occur (continue)e.g., probability that one-parent families plan to continue is .4082

    1 pProbability that event will not occur (not continue)e.g., probability that one-parent families do not plan to continue is .5918 (1 - .4082)

  • OddsRatio of probability that event will occur to probability that it will not

    e.g., odds of continuation for one-parent families are .69 (.4082 / .5918)

    Can range from 0 to positive infinity

  • Probabilities and OddsTable 2.2Odds = 1Both outcomes equally likelyOdds > 1Probability that event will occur greater than probability that it will notOdds < 1Probability that event will occur less than probability that it will not

  • Odds Ratio (OR)Odds of the event for one value of the IV (two-parent families) divided by the odds for a different value of the IV, usually a value one unit lower (one-parent families)

    e.g., odds of continuing for two-parent families more than double the odds for one-parent familiesOR = 1.6455 / .6898 = 2.39

  • OR (contd)Plays a central role in quantifying the strength and direction of relationships between IVs and DVs in binary, multinomial, and ordinal logistic regression

    OR < 1 indicates a negative relationshipOR > 1 indicates a positive relationshipOR = 1 indicates no linear relationship

  • ORs > 1e.g., OR of 2.39

    A one-unit increase in the independent variable increases the odds of continuing by a factor of 2.39

    The odds of continuing are 2.39 times higher for two-parent compared to one-parent families

  • ORs < 1e.g., OR = .50

    A one-unit increase in the independent variable decreases the odds of continuing by a factor of .50

    The odds that two-parent families will continue are .50 (or one-half) of the odds that one-parent families will continue

  • ORs < 1 (contd)Compute reciprocal (i.e., 1 / .50 = 2.00)Express relationship as opposite event of interest (e.g., discontinuing)

    A one-unit increase in the independent variable increases the odds of discontinuing by a factor of 2.00

    The odds that two-parent families will discontinue are 2.00 times (or twice) the odds of one-parent families

  • OR to Percentage Change% change = 100(OR 1)Alternative way to express OR

    e.g., A one-unit increase in the independent variable increases the odds of continuing by 139.00% 100(2.39 1) = 139.00

    e.g., A one-unit increase in the independent variable decreases the odds of continuing by 50.00%100(.50 1) = -50.00

  • Comparing OR > 1 and OR < 1Compute reciprocal of one of the ORs

    e.g., OR of 2.00 and an OR of .50

    Reciprocal of .50 is 2.00 (1 / .50 = 2.00)ORs are equal in size (but not in direction of the relationship)

  • Qualitative Descriptors for OR Table 2.3Use cautiously with IVs that arent dichotomous

  • Question & AnswerAre two-parent families more likely to continue fostering than one-parent families?Yes. The odds of continuing are 2.39 times (139%) higher for two-parent compared to one-parent families. The probability of continuing is .41 for one-parent families and .62 for two-parent families.

  • Binary Logistic Regression ExampleDV = continue fostering, 0 = no, 1 = yesCustomary to code category of interest 1 and the other category 0IV = married, 0 = not married, 1 = marriedN = 131 foster families

    Are two-parent families more likely to continue fostering than one-parent families?

  • Statistical SignificanceTable 2.4Relationship between marital status and continuation is statistically significant (Wald 2 = 5.544, p = .019)

  • Direction of RelationshipB = slopePositive slope, positive relationshipOR > 1Negative slope, negative relationshipOR < 10 slope, no linear relationshipOR = 1

  • Direction/Strength of RelationshipPositive relationship between marital status and continuationTwo-parent families more likely to continueB = .869Exp(B) = OR = 2.385% change = 100(2.385 - 1) = 139%The odds of continuing are 2.39 times (139%) higher for two-parent compared to one-parent families

  • Roadmap to Computations

  • Binary Logistic Regression Modelln(/ (1 - )) = + 1X1 + 1X2 + kXk, orln( / (1 - )) =

    is the probability of the event (eta) is the abbreviation for the linear predictor (right hand side of this equation)k = number of independent variables

  • Logit Linkln( / (1 - ))Log of the odds that the DV equals 1 (event occurs)Connects (i.e., links) DV to linear combination of IVs

  • Estimated Logits (L) ln(p / 1 - p) = a + B1X1 + B1X2 + BkXk

    ln(p / 1 p)Log of the odds that the DV equals 1 (event occurs)Estimated logit, LDoes not have intuitive or substantive meaning Useful for examining curvilinear relationships and interaction effectsPrimarily useful for estimating probabilities, odds, and ORs

  • Estimated Logits (L) L(Continue) = a + BMarriedXMarried

    L(Continue) = -.372 + (.869)(XMarried)

    a = interceptB = slope

  • Logit to OddsIf L = 0:Odds = eL = e0 = 1.00

    If L = .50:Odds = eL = e.50 = 1.65

    If L = 1.00:Odds = eL = e1.00 = 2.72

  • Logits to Odds (contd)Table 2.4One-parent familiesL(Continue) = -.372 = -.372 + (.869)(0)Odds of continuing = e-.372 = .69

    Two-parent familiesL(Continue) = .497 = -.372 + (.869)(1)Odds of continuing = e.497 = 1.65

  • Odds to OROR = 1.65 / .69 = 2.39, or

    e.869 = 2.39, labeled Exp(B)Table 2.4

  • OR to Percentage Change% change = 100(OR 1)

    e.g., A one-unit increase in the independent variable increases the odds of continuing by 139.00% 100(2.39 1) = 139.00

    e.g., A one-unit increase in the independent variable decreases the odds of continuing by 50.00%100(.50 1) = -50.00

  • Logits to Probabilities

    One-parent families, L(Continue) = -.372

    Two-parent families, L(Continue) = .497

  • Question & AnswerAre two-parent families more likely to continue fostering than one-parent families?Yes. The odds of continuing are 2.39 times (139%) higher for two-parent compared to one-parent families. The probability of continuing is .41 for one-parent families and .62 for two-parent families.

  • Single (Quantitative) IV ExampleDV = continue fostering, 0 = no, 1 = yesCustomary to code category of interest 1 and other category 0IV = number of resourcesN = 131 foster families

    Are foster families with more resources more likely to continue fostering?

  • Statistical SignificanceTable 2.5Relationship between resources and continuation is statistically significant (Wald 2 = 4.924, p = .026)

    H0: = 0, 0, 0, same asH0: OR = 1, OR 1, OR 1Likelihood ratio 2 better than Wald

  • Direction/Strength of RelationshipPositive relationship between resources and continuationFamilies with more resources are more likely to continueB = .212Exp(B) = OR = 1.237% change = 100(1.237 1) = 24%The odds of continuing are 1.24 times (24%) higher for each additional resource

  • Estimated LogitsL(Continue) = -1.227 + (.212)(X)

  • FiguresResources.xls

  • Effect of Resources on Continuation (Logits)

    Chart1

    -1.014362424

    -0.8019574168

    -0.5895524096

    -0.3771474024

    -0.1647423952

    0.047662612

    0.2600676192

    0.4724726264

    0.6848776336

    0.8972826408

    1.109687648

    Logits

    Resources

    Logits

    Data

    aBResourcesXResourcesLogitOddsp

    -1.2270.2121-1.010.360.27

    -1.2270.2122-0.800.450.31

    -1.2270.2123-0.590.550.36

    -1.2270.2124-0.380.690.41

    -1.2270.2125-0.160.850.46

    -1.2270.21260.051.050.51

    -1.2270.21270.261.300.56

    -1.2270.21280.471.600.62

    -1.2270.21290.681.980.66

    -1.2270.212100.902.450.71

    -1.2270.212111.113.030.75

    Logits

    Logits

    -1.014362424

    -0.8019574168

    -0.5895524096

    -0.3771474024

    -0.1647423952

    0.047662612

    0.2600676192

    0.4724726264

    0.6848776336

    0.8972826408

    1.109687648

    Logits

    Resources

    Predicted Logits

    Odds

    Odds

    0.3626335626

    0.4484503003

    0.5545754518

    0.6858149757

    0.8481121537

    1.0488167375

    1.297017787

    1.6039552763

    1.9835291034

    2.4529285588

    3.0334107546

    Odds

    Resources

    Predicted Odds

    Probabilities

    Probabilities

    0.2661269857

    0.3096069642

    0.3567375589

    0.4068150927

    0.4589072974

    0.5119133978

    0.564652914

    0.6159688267

    0.6648264638

    0.7103907645

    0.7520708748

    Probabilities

    Resources

    Predicted Probabilities

  • Effect of Resources on Continuation (Odds)

    Chart2

    0.3626335626

    0.4484503003

    0.5545754518

    0.6858149757

    0.8481121537

    1.0488167375

    1.297017787

    1.6039552763

    1.9835291034

    2.4529285588

    3.0334107546

    Odds

    Resources

    Odds

    Data

    aBResourcesXResourcesLogitOddsp

    -1.2270.2121-1.010.360.27

    -1.2270.2122-0.800.450.31

    -1.2270.2123-0.590.550.36

    -1.2270.2124-0.380.690.41

    -1.2270.2125-0.160.850.46

    -1.2270.21260.051.050.51

    -1.2270.21270.261.300.56

    -1.2270.21280.471.600.62

    -1.2270.21290.681.980.66

    -1.2270.212100.902.450.71

    -1.2270.212111.113.030.75

    Logits

    Logits

    -1.014362424

    -0.8019574168

    -0.5895524096

    -0.3771474024

    -0.1647423952

    0.047662612

    0.2600676192

    0.4724726264

    0.6848776336

    0.8972826408

    1.109687648

    Logits

    Resources

    Predicted Logits

    Odds

    Odds

    0.3626335626

    0.4484503003

    0.5545754518

    0.6858149757

    0.8481121537

    1.0488167375

    1.297017787

    1.6039552763

    1.9835291034

    2.4529285588

    3.0334107546

    Odds

    Resources

    Predicted Odds

    Probabilities

    Probabilities

    0.2661269857

    0.3096069642

    0.3567375589

    0.4068150927

    0.4589072974

    0.5119133978

    0.564652914

    0.6159688267

    0.6648264638

    0.7103907645

    0.7520708748

    Probabilities

    Resources

    Predicted Probabilities

  • Effect of Resources on Continuation (Probabilities)

    Chart3

    0.2661269857

    0.3096069642

    0.3567375589

    0.4068150927

    0.4589072974

    0.5119133978

    0.564652914

    0.6159688267

    0.6648264638

    0.7103907645

    0.7520708748

    Probabilities

    Resources

    Probabilities

    Data

    aBResourcesXResourcesLogitOddsp

    -1.2270.2121-1.010.360.27

    -1.2270.2122-0.800.450.31

    -1.2270.2123-0.590.550.36

    -1.2270.2124-0.380.690.41

    -1.2270.2125-0.160.850.46

    -1.2270.21260.051.050.51

    -1.2270.21270.261.300.56

    -1.2270.21280.471.600.62

    -1.2270.21290.681.980.66

    -1.2270.212100.902.450.71

    -1.2270.212111.113.030.75

    Logits

    Logits

    -1.014362424

    -0.8019574168

    -0.5895524096

    -0.3771474024

    -0.1647423952

    0.047662612

    0.2600676192

    0.4724726264

    0.6848776336

    0.8972826408

    1.109687648

    Logits

    Resources

    Predicted Logits

    Odds

    Odds

    0.3626335626

    0.4484503003

    0.5545754518

    0.6858149757

    0.8481121537

    1.0488167375

    1.297017787

    1.6039552763

    1.9835291034

    2.4529285588

    3.0334107546

    Odds

    Resources

    Predicted Odds

    Probabilities

    Probabilities

    0.2661269857

    0.3096069642

    0.3567375589

    0.4068150927

    0.4589072974

    0.5119133978

    0.564652914

    0.6159688267

    0.6648264638

    0.7103907645

    0.7520708748

    Probabilities

    Resources

    Predicted Probabilities

  • Question & AnswerAre foster families with more resources more likely to continue fostering?Yes. The odds of continuing are 1.24 times (24%) higher for each additional resource. The probability of continuing is .31 for families with two resources, .51 for families with 6 resources, and .71 for families with 10 resources.

  • Relationship of Linear Predictor to Logits, Odds & pRelationship between linear predictor and logits is linear

    Relationship between linear predictor and odds is non-linear

    Relationship between linear predictor and p is non-linearChallenge is to summarize changes in odds and probabilities associated with changes in IVs in the most meaningful and parsimonious way

  • Logit as Function of Linear Predictor

    Chart4

    -3

    -2.5

    -2

    -1.5

    -1

    -0.5

    0

    0.5

    1

    1.5

    2

    2.5

    3

    Linear Predictor

    Logit

    Logit

    xlogitoddsp

    -3.00-3.00.05.05

    -2.50-2.50.08.08

    -2.00-2.00.14.12

    -1.50-1.50.22.18

    -1.00-1.00.37.27

    -.50-.50.61.38

    .00.001.00.50

    .50.501.65.62

    1.001.002.72.73

    1.501.504.48.82

    2.002.007.39.88

    2.502.5012.18.92

    3.003.0020.09.95

    Logit

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    X

    Logit

    Odds

    xlogitoddsp

    -3.00-3.00.05.05

    -2.50-2.50.08.08

    -2.00-2.00.14.12

    -1.50-1.50.22.18

    -1.00-1.00.37.27

    -.50-.50.61.38

    .00.001.00.50

    .50.501.65.62

    1.001.002.72.73

    1.501.504.48.82

    2.002.007.39.88

    2.502.5012.18.92

    3.003.0020.09.95

    Odds

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    X

    Odds

    p

    xlogitoddsp

    -3.00-3.00.05.05

    -2.50-2.50.08.08

    -2.00-2.00.14.12

    -1.50-1.50.22.18

    -1.00-1.00.37.27

    -.50-.50.61.38

    .00.001.00.50

    .50.501.65.62

    1.001.002.72.73

    1.501.504.48.82

    2.002.007.39.88

    2.502.5012.18.92

    3.003.0020.09.95

    p

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    X

    Probability

  • Odds as Function of Linear Predictor

    Chart5

    0.0497870684

    0.0820849986

    0.1353352832

    0.2231301601

    0.3678794412

    0.6065306597

    1

    1.6487212707

    2.7182818285

    4.4816890703

    7.3890560989

    12.1824939607

    20.0855369232

    Linear Predictor

    Odds

    Logit

    xlogitoddsp

    -3.00-3.00.05.05

    -2.50-2.50.08.08

    -2.00-2.00.14.12

    -1.50-1.50.22.18

    -1.00-1.00.37.27

    -.50-.50.61.38

    .00.001.00.50

    .50.501.65.62

    1.001.002.72.73

    1.501.504.48.82

    2.002.007.39.88

    2.502.5012.18.92

    3.003.0020.09.95

    Logit

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    X

    Logit

    Odds

    xlogitoddsp

    -3.00-3.00.05.05

    -2.50-2.50.08.08

    -2.00-2.00.14.12

    -1.50-1.50.22.18

    -1.00-1.00.37.27

    -.50-.50.61.38

    .00.001.00.50

    .50.501.65.62

    1.001.002.72.73

    1.501.504.48.82

    2.002.007.39.88

    2.502.5012.18.92

    3.003.0020.09.95

    Odds

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    X

    Odds

    p

    xlogitoddsp

    -3.00-3.00.05.05

    -2.50-2.50.08.08

    -2.00-2.00.14.12

    -1.50-1.50.22.18

    -1.00-1.00.37.27

    -.50-.50.61.38

    .00.001.00.50

    .50.501.65.62

    1.001.002.72.73

    1.501.504.48.82

    2.002.007.39.88

    2.502.5012.18.92

    3.003.0020.09.95

    p

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    X

    Probability

  • Probabilities as Function of Linear Predictor

    Chart1

    0.0474258732

    0.07585818

    0.119202922

    0.1824255238

    0.2689414214

    0.3775406688

    0.5

    0.6224593312

    0.7310585786

    0.8175744762

    0.880797078

    0.92414182

    0.9525741268

    Linear Predictor

    Probability

    Logit

    xlogitoddsp

    -3.00-3.00.05.05

    -2.50-2.50.08.08

    -2.00-2.00.14.122.72

    -1.50-1.50.22.182.72

    -1.00-1.00.37.272.72

    -.50-.50.61.382.72

    .00.001.00.502.72

    .50.501.65.622.72

    1.001.002.72.732.72

    1.501.504.48.822.72

    2.002.007.39.882.72

    2.502.5012.18.922.72

    3.003.0020.09.952.72

    Logit

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    IV

    Logit

    Odds

    xlogitoddsp

    -3.00-3.00.05.05

    -2.50-2.50.08.08

    -2.00-2.00.14.12

    -1.50-1.50.22.18

    -1.00-1.00.37.27

    -.50-.50.61.38

    .00.001.00.50

    .50.501.65.62

    1.001.002.72.73

    1.501.504.48.82

    2.002.007.39.88

    2.502.5012.18.92

    3.003.0020.09.95

    Odds

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    IV

    Odds

    p

    xlogitoddsp

    -3.00-3.00.05.05

    -2.50-2.50.08.08

    -2.00-2.00.14.12

    -1.50-1.50.22.18

    -1.00-1.00.37.27

    -.50-.50.61.38

    .00.001.00.50

    .50.501.65.62

    1.001.002.72.73

    1.501.504.48.82

    2.002.007.39.88

    2.502.5012.18.92

    3.003.0020.09.95

    p

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    IV

    Probability

  • IVs to z-scoresz-scores (standard scores)Only the IV (not DV)--semi-standardized slopesOne-unit increase in the IV refers to a one-standard-deviation increaseOR interpreted as expected change in the odds associated with a one standard deviation increase in the IVConversion to z-scores changes intercept, slope, and OR, but not associated test statisticsTable 2.6 (compare to Table 2.5)

  • FigureszResources.xls

  • Effect of zResources on Continuation (Probabilities)

    Chart1

    0.2580746016

    0.3438914934

    0.441272322

    0.5433905808

    0.641987286

    0.7298772729

    0.802817861

    Probabilities

    Standardized Resources

    Probabilities

    Data

    aBzResourcesXzResourcesLogitOddsp

    0.1740.410-3-1.060.350.26

    0.1740.410-2-0.650.520.34

    0.1740.410-1-0.240.790.44

    0.1740.41000.171.190.54

    0.1740.41010.581.790.64

    0.1740.41020.992.700.73

    0.1740.41031.404.070.80

    Logits

    Logits

    -1.056

    -0.646

    -0.236

    0.174

    0.584

    0.994

    1.404

    Logits

    Standardized Resources

    Predicted Logits

    Odds

    Odds

    0.3478444089

    0.5241381418

    0.7897806739

    1.1900555658

    1.7931968919

    2.7020209688

    4.0714532516

    Odds

    Standardized Resources

    Predicted Odds

    Probabilities

    Probabilities

    0.2580746016

    0.3438914934

    0.441272322

    0.5433905808

    0.641987286

    0.7298772729

    0.802817861

    Probabilities

    Standardized Resources

    Predicted Probabilities

  • Question & AnswerAre foster families with more resources more likely to continue fostering?Yes. The odds of continuing are 1.51 times (51%) higher for each one standard deviation (1.93) increase in resources. The probability of continuing is .34 for families with resources two standard deviations below the mean, .54 for families with the mean number of resources (6.60), and .73 for families with resources two standard deviations above the mean.

  • IVs CenteredCenteringTypically center on meanUseful when testing interactions, curvilinear relationships, or when no meaningful 0 point (e.g., no family with 0 resources)Centering doesnt change slope, OR, or associated test statistics, but does change the interceptTable 2.7 (compare to Table 2.5)

  • FigurescResources.xls

  • Effect of cResources on Continuation (Probabilities)

  • Question & AnswerAre foster families with more resources more likely to continue fostering?Yes. The odds of continuing are 1.24 times (24%) higher for each additional resource. The probability of continuing is .34 for families with 4 resources below the mean, .54 for families with the mean number of resources (6.60), and .74 for families with 4 resources above the mean.

  • Multiple IV ExampleDV = continue fostering, 0 = no, 1 = yesCustomary to code the category of interest as 1 and the other category as 0IV = married, 0 = not married, 1 = marriedIV = number of resources (z-scores)N = 131 foster families

    Are foster families with more resources more likely to continue fostering, controlling for marital status?

  • Statistical SignificanceTable 2.12Relationship between set of IVs and continuation is statistically significant (2 = 6.58, p = .037)

    H0: 1 = 2 = k = 0, same asH0: 1 = 2 = k = 1 (psi) is symbol for population value of OR

  • Statistical Significance (contd)Table 2.13Relationship between resources and continuation is not statistically significant, controlling for marital status (2 = .92, p = .338)Relationship between marital status and continuation is not statistically significant, controlling for resources (2 = 1.42, p = .234)

    H0: = 0, 0, 0, same asH0: = 1, 1, 1 (psi) is symbol for population value of ORLikelihood ratio 2 better than Wald

  • Statistical Significance (contd)Table 2.9Relationship between resources and continuation is not statistically significant, controlling for marital status (2 = .91, p = .340)Relationship between marital status and continuation is not statistically significant, controlling for resources (2 = 1.41, p = .235)

    H0: = 0, 0, 0, same asH0: = 1, 1, 1 (psi) is symbol for population value of ORWald 2, but likelihood ratio 2 better

  • Estimated LogitsL(Continue) = -.183 + (.228)(XzResources) + (.570)(XMarried)

  • ORs & Percentage ChangeORzResources = 1.256 (ns)The odds of continuing are 1.26 times (26%) higher for each one standard deviation (1.93) increase in resources, controlling for marital statusORMarried = 1.769 (ns)The odds of continuing are 1.77 times (77%) higher for two-parent compared to one-parent families, controlling for marital status

  • FiguresMarried & zResources.xls

  • Effect of Resources and Marital Status on Plans to Continue Fostering (Odds)

    Chart1

    0.42021029110.7430440124

    0.52781998020.9333266801

    0.66298693161.1723379467

    0.83276815571.4725564913

    1.04602785991.8496565996

    1.31390024482.3233265118

    1.65037081662.9182963377

    One-Parent

    Two-Parent

    Standardized Resources

    Odds

    Data

    aBMarriedXMarriedBzResourcesXzResourcesLogitOddsp

    -0.1830.57000.228-3-0.870.420.30

    -0.1830.57000.228-2-0.640.530.35

    -0.1830.57000.228-1-0.410.660.40

    -0.1830.57000.2280-0.180.830.45

    -0.1830.57000.22810.051.050.51

    -0.1830.57000.22820.271.310.57

    -0.1830.57000.22830.501.650.62

    -0.1830.57010.228-3-0.300.740.43

    -0.1830.57010.228-2-0.070.930.48

    -0.1830.57010.228-10.161.170.54

    -0.1830.57010.22800.391.470.60

    -0.1830.57010.22810.621.850.65

    -0.1830.57010.22820.842.320.70

    -0.1830.57010.22831.072.920.74

    Logits

    Logits

    -0.867-0.297

    -0.639-0.069

    -0.4110.159

    -0.1830.387

    0.0450.615

    0.2730.843

    0.5011.071

    One-Parent

    Two-Parent

    zResources

    Predicted Logits

    Odds

    Odds

    0.42021029110.7430440124

    0.52781998020.9333266801

    0.66298693161.1723379467

    0.83276815571.4725564913

    1.04602785991.8496565996

    1.31390024482.3233265118

    1.65037081662.9182963377

    One-Parent

    Two-Parent

    zResources

    Predicted Odds

    Probabilities

    Probabilities

    0.29587892280.4262910214

    0.34547262570.4827568407

    0.39867236420.5396664679

    0.45437725070.5955603023

    0.51124810190.6490805242

    0.56782925180.6990966742

    0.62269430610.7447870417

    One-Parent

    Two-Parent

    zResources

    Predicted Probabilities

  • Effect of Resources and Marital Status on Plans to Continue Fostering (Probabilities)

    Chart2

    0.29587892280.4262910214

    0.34547262570.4827568407

    0.39867236420.5396664679

    0.45437725070.5955603023

    0.51124810190.6490805242

    0.56782925180.6990966742

    0.62269430610.7447870417

    One-Parent

    Two-Parent

    Standardized Resources

    Probabilities

    Data

    aBMarriedXMarriedBzResourcesXzResourcesLogitOddsp

    -0.1830.57000.228-3-0.870.420.30

    -0.1830.57000.228-2-0.640.530.35

    -0.1830.57000.228-1-0.410.660.40

    -0.1830.57000.2280-0.180.830.45

    -0.1830.57000.22810.051.050.51

    -0.1830.57000.22820.271.310.57

    -0.1830.57000.22830.501.650.62

    -0.1830.57010.228-3-0.300.740.43

    -0.1830.57010.228-2-0.070.930.48

    -0.1830.57010.228-10.161.170.54

    -0.1830.57010.22800.391.470.60

    -0.1830.57010.22810.621.850.65

    -0.1830.57010.22820.842.320.70

    -0.1830.57010.22831.072.920.74

    Logits

    Logits

    -0.867-0.297

    -0.639-0.069

    -0.4110.159

    -0.1830.387

    0.0450.615

    0.2730.843

    0.5011.071

    One-Parent

    Two-Parent

    zResources

    Predicted Logits

    Odds

    Odds

    0.42021029110.7430440124

    0.52781998020.9333266801

    0.66298693161.1723379467

    0.83276815571.4725564913

    1.04602785991.8496565996

    1.31390024482.3233265118

    1.65037081662.9182963377

    One-Parent

    Two-Parent

    zResources

    Predicted Odds

    Probabilities

    Probabilities

    0.29587892280.4262910214

    0.34547262570.4827568407

    0.39867236420.5396664679

    0.45437725070.5955603023

    0.51124810190.6490805242

    0.56782925180.6990966742

    0.62269430610.7447870417

    One-Parent

    Two-Parent

    zResources

    Predicted Probabilities

  • Presenting Odds and Probabilities in TablesTables 2.10 and 2.11

  • Question & AnswerAre foster families with more resources more likely to continue fostering, controlling for marital status?No (ns). The odds of continuing are 1.26 times (26%) higher for each one standard deviation (1.93) increase in resources, controlling for marital status.

    Contd

  • Question & Answer (contd)For one-parent families the probability of continuing is .35 for families with resources two standard deviations below the mean, .45 for families with the mean number of resources, and .57 for families with resources two standard deviations above the mean. For two-parent families the probability of continuing is .48 for families with resources two standard deviations below the mean, .60 for families with the mean number of resources, and .70 for families with resources two standard deviations above the mean.

  • Comparing the Relative Strength of IVsSize of slope and OR depend on how the IV is measuredWhen IVs measured the same way (e.g., two dichotomous IVs or two continuous IVs transformed to z-scores) relative strength can be compared

    Nothing comparable to standardized slope (Beta)

  • Nested Models

  • Nested Models (contd)One regression model is nested within another if it contains a subset of variables included in the model within which its nested, and same cases are analyzed in both modelsThe more complex model called the full modelThe nested model called the reduced model. Comparison of full and reduced models allows you to examine whether one or more variable(s) in the full model contribute to explanation of the DV

  • Sequential Entry of IVsUsed to compare full and reduced modelse.g., family resources entered first, and then marital status

    Fchange used in linear regression

  • Sequential Entry of IVs (contd)SPSS GZLM doesnt allow sequential of IVsEstimate models separately and compare omnibus likelihood ratio 2 values

    Reduced model 2(1) = 5.168Full model 2(2) = 6.585

    2 difference = 6.585 5.168 = 1.417df difference = 2 1p = .234Chi-square Difference.xls

  • Assumptions Necessary for Testing HypothesesNo assumptions unique to binary logistic regression other than ones discussed in GZLM lecture

  • Model EvaluationEvaluate your model before you test hypotheses or interpret substantive resultsOutliersAnalogs of R2

  • OutliersAtypical casesCan lead to flawed conclusionsCan provide theoretical insightsCommon causesData entry errorsModel misspecificationRare events

  • Outliers (contd)Leverage

    ResidualsStandardized or unstandardized deviance residuals

    InfluenceCooks D

  • LeverageThink of a seesawLeverage value for each caseCases with greater leverage can exert a disproportionately large influenceLeverage value for each caseNo clear benchmarksIdentify cases with substantially different leverage values than those of other cases

  • ResidualsDifference between actual and estimated values of the DV for a case Residual for each caseLarge residual indicates a case for which model fits poorly

  • Residuals (contd)Standardized or unstandardized deviance residualsNot normally distributedValues less than -2 or greater than +2 warrant some concernValues less than -3 or greater than +3 merit close inspection

  • InfluenceCases whose deletion result in substantial changes to regression coefficientsCooks D for each caseApproximate aggregate change in regression parameters resulting from deletion of a caseValues of 1.0 or more indicate a problematic degree of influence for an individual case

  • Index PlotScatterplot

    Horizontal axis (X)Case id

    Vertical axis (Y)Leverage values, orResiduals, orCooks D

  • Index Plot: Leverage Values

  • Index Plot: Standardized Deviance Residuals

  • Index Plot: Cooks D

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

  • MulticollinearitySPSS GZLM doesnt compute multicollinearity statistics

    Use SPSS linear regression

    Problematic levelsTolerance < .10 or VIF > 10

  • Additional TopicsPolychotomous IVsCurvilinear relationshipsInteractions

  • Overview of the ProcessSelect IVs and decide whether to test curvilinear relationships or interactionsCarefully screen and clean dataTransform and code variables as neededEstimate regression modelExamine assumptions necessary to estimate binary regression model, examine model fit, and revise model as needed

  • Overview of the Process (contd)Test hypotheses about the overall model and specific model parameters, such as ORsCreate tables and graphs to present results in the most meaningful and parsimonious wayInterpret results of the estimated model in terms of logits, probabilities, odds, and odds ratios, as appropriate

  • Additional Regression Models for Dichotomous DVsBinary 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 Dichotomous DVs (contd)Complementary log-log (clog-log) and log-log models Probability of the event is very small or largeLoglinear regressionLimited to categorical IVsDiscriminant analysisLimited to continuous IVs