# binary logistic regression

Click here to load reader

Post on 18-Jan-2016

57 views

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 PresentationTRANSCRIPT

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