MNLM for Nominal Outcomes

Transcription

1 MNLM for Nominal Outcomes Objectives Introduce the MNLM as an extension of the BLM Derive the model as a nonlinear probability model Illustrate the difficulties in interpretation due to the large number of parameters and comparisons Introduce graphical methods that make interpretation simpler Nominal LHS \ 1

2 (Rethinking) the BLM The BLM describes the relative probability of one outcome compared to a base or reference outcome For example, being in the labor force compared to being out of the labor force Nominal LHS \ 2

3 Think of the BLM as having two sets of β 's : One set is associated with y=1 compared to y=0 The other set is associated with y=0 compared to y=1 Only J-1 sets are estimated Nominal LHS \ 3

4 Binary Logit Model (new notation) ( y = x) ( y = x) ( y= A x) ( y= B x) ln Pr 1 = xβ Pr 0 Pr ln = xβ AB Pr For a model with three independent variables ( y= A x) ( y= B x) Pr ln = β AB + β ABx + β ABx + β ABx Pr 0, 1, 1 2, 2 3, 3 Nominal LHS \ 4

5 The probability that y=1 (or A) exp( xβ) exp( xβab ) Pr( y= 1 x) = Pr( y= A x) = 1+ exp( xβ) 1+ exp( xβ ) AB The probability that y=0 (or B) 1 1 Pr( y= 0 x) = Pr( y= B x) = 1+ exp( xβ) 1+ exp( xβab ) Question (for you) Which is the base (or reference) category? Nominal LHS \ 5

6 Three outcome categories Consider y with categories L, S, and P L-Labor S-Skilled P-Professional Nominal LHS \ 6

7 Think of this as three BLMs The effect of Ed on the odds of L versus S: ( L Ed ) ( S Ed ) ln Pr = β + β Pr 0, L S 1, L S Ed For S versus P: ( ) ( ) Pr S Ed ln Pr P Ed! = β + β Ed 0,S P 1,S P Nominal LHS \ 7

8 Question (for you) What about the remaining comparison? Nominal LHS \ 8

9 Redundancy Using the property ln( a/ b) = ln( a) ln( b ): ( L Ed ) ( P Ed ) Pr ln = lnpr lnpr + 0 Pr ( L Ed ) ( P Ed ) [ ] ( L Ed ) ( P Ed ) ( S Ed ) ( S Ed ) ( L Ed ) ( S Ed ) ( S Ed ) ( P Ed ) = lnpr lnpr + lnpr lnpr = lnpr lnpr + lnpr lnpr ( L Ed ) ( S Ed ) ( S Ed ) ( P Ed ) Pr Pr = ln + ln Pr Pr Nominal LHS \ 9

10 Thus, if we add equations 1 and 2, we get 3: ( ) ( ) ( ) ( ) ( ) ( ) Pr L Ed Pr S Ed Pr L Ed ln + ln = ln Pr S Ed Pr P Ed Pr P Ed Nominal LHS \ 10

11 Logical Relationship You can find a coefficient for any comparison from a pair of other coefficients: β = β + β L P L S S P β = β β L S L P S P β = β β S P L P L S Nominal LHS \ 11

12 But Why won't the results from separate BLMs match those from MNLM exactly? Nominal LHS \ 12

13 A Minimal Set of Coefficients For J outcomes, J-1 comparisons Different software might compute different minimal sets Question (for you) Is this a problem? Nominal LHS \ 13

14 The MNLM as a Probability Model Let y have J nominal outcomes numbered 1 through J ( = x) Pr y m is a function of x β mj Take the exponential to ensure that the probabilities are non-negative J Divide by exp( ) x iβ j to make the probabilities sum to 1 J j= 1 Which results in: exp x Pr = = ( y m x ) ( ) iβmj ( x β ) i i J j= 1 exp i j J Nominal LHS \ 14

15 Identification One of the β's is constrained to equal zero For example, ( y m x ) ( x ) iβmj ( x β ) exp Pr = = where β = 0 i i J j= 1 exp i j J 1 Can be written as: ( y x ) Pr i = 1 i = J = ( x ) 1 exp ( x β ) j 2 i j ( x β ) exp Pr y = m = for m> 1 i m i i J j= 2exp i j ( x β ) Nominal LHS \ 15

16 The Data 1982 General Social Survey A sample of 337 currently employed men Nominal LHS \ 16

17 Outcome Respondents were asked to indicate their occupation These occupations were recoded to correspond to Schmidt and Strauss (1975) in an early application of the MNLM Five occupation categories: Menial jobs Blue-collar jobs Craft jobs White-collar jobs Professional jobs Nominal LHS \ 17

18 Descriptive Information. usecda cda_nomocc2. codebook, compact Variable Obs Unique Mean Min Max Label occ Occupation white Race: 1=white 0=nonwhite ed Years of education exper Years of work experience Nominal LHS \ 18

19 . sum Variable Obs Mean Std. Dev. Min Max occ white ed exper tab occ Occupation Freq. Percent Cum Menial BlueCol Craft WhiteCol Prof Total Nominal LHS \ 19

20 Descriptive Table Name Mean StdDev Min Max Description OCC Occupation: M1=menial; B2=blue collar; C3=craft; W4=white collar; P5=professional WHITE Race: 1= white; 0=another race ED Education: Number of years of formal education EXP Possible years of work experience: Age minus years of education minus 5 Note: N=337. OCC has categories: M1=menial; B2=blue collar; C3=craft; W4=white collar; P5=professional with marginal percentages 9, 21, 25, 12, and 33, respectively. Nominal LHS \ 20

21 Estimating the MNLM. mlogit occ i.white ed exper, base(1) nolog Multinomial logistic regression Number of obs = 337 LR chi2(12) = Prob > chi2 = Log likelihood = Pseudo R2 = occ Coef. Std. Err. z P> z [95% Conf. Interval] Menial (base outcome) BlueCol 1.white ed exper _cons Craft 1.white ed exper _cons Nominal LHS \ 21

22 WhiteCol 1.white ed exper _cons Prof 1.white ed exper _cons Nominal LHS \ 22

23 Questions (for you) What does the coefficient for ed under Prof represent? Using this minimal set of coefficients, how would we calculate: the effect of education on the log-odds of having a Professional occupation compared to a Blue Collar occupation? the effect of white on the log-odds of having a White-Collar occupation compared to having a Professional occupation? the effect of experience on the log-odds of having a Blue Collar occupation compared to having a Craft occupation? Hint: β P W = β P M β W M Nominal LHS \ 23

24 Interpretation In even a simple MNLM there are a lot of parameters Too often, the MNLM is estimated, the parameters are listed, and statistical significance is noted, while the magnitudes and even directions of the effects are ignored We will consider: Factor change in the odds (odds ratio) Predicted probabilities Nominal LHS \ 24

25 Factor change coefficients For a model with three independent variables, ( x, ) β β x β x β x Ω mn x2 = e e e e 0, m n 1, m n 1 2, m n 2 3, m n 3 A change of one unit in x can be measured by the ratio of the odds: 2 ( x, 1) ( x, ) β0, m n β1, m nx1 β2, m nx2 β2, m n β3, m nx3 Ω mn x2+ e e e e e = = e β0, m n β1, m nx1 β2, m nx2 β3, m nx3 Ω x e e e e mn 2 β 2, m n Nominal LHS \ 25

26 Interpretation For a unit change in x, the odds are expected to change by a factor of k exp( ), holding all other variables constant β kmn, For a standard deviation change in x, the odds are expected to change k by a factor of exp( β s ), holding kmn, k Nominal LHS \ 26

27 Computing factor change. listcoef, help mlogit (N=337): Factor change in the odds of occ Variable: 1.white (sd=0.276) b z P> z e^b e^bstdx Menial vs BlueCol Menial vs Craft Menial vs WhiteCol Menial vs Prof BlueCol vs Menial BlueCol vs Craft BlueCol vs WhiteCol BlueCol vs Prof Craft vs Menial Craft vs BlueCol Craft vs WhiteCol Craft vs Prof WhiteCol vs Menial WhiteCol vs BlueCol WhiteCol vs Craft WhiteCol vs Prof Prof vs Menial Prof vs BlueCol Prof vs Craft Prof vs WhiteCol Nominal LHS \ 27

28 Variable: ed (sd=2.946) b z P> z e^b e^bstdx Menial vs BlueCol Menial vs Craft Menial vs WhiteCol Menial vs Prof BlueCol vs Menial BlueCol vs Craft BlueCol vs WhiteCol BlueCol vs Prof Craft vs Menial Craft vs BlueCol Craft vs WhiteCol Craft vs Prof WhiteCol vs Menial WhiteCol vs BlueCol WhiteCol vs Craft WhiteCol vs Prof Prof vs Menial Prof vs BlueCol Prof vs Craft Prof vs WhiteCol Nominal LHS \ 28

29 Variable: exper (sd=13.959) b z P> z e^b e^bstdx Menial vs BlueCol Menial vs Craft Menial vs WhiteCol Menial vs Prof BlueCol vs Menial BlueCol vs Craft BlueCol vs WhiteCol BlueCol vs Prof Craft vs Menial Craft vs BlueCol Craft vs WhiteCol Craft vs Prof WhiteCol vs Menial WhiteCol vs BlueCol WhiteCol vs Craft WhiteCol vs Prof Prof vs Menial Prof vs BlueCol Prof vs Craft Prof vs WhiteCol b = raw coefficient z = z-score for test of b=0 P> z = p-value for z-test e^b = exp(b) = factor change in odds for unit increase in X e^bstdx = exp(b*sd of X) = change in odds for SD increase in X Nominal LHS \ 29

30 Question (for you) Were your coefficient calculations correct? For any pair of contrasts (e.g.,! β k,m n & β k,n m ): How are the b coefficients related? How are the e^b coefficients related? Nominal LHS \ 30

31 Computing all contrasts at a given p value and for one x. listcoef white, pval(.05) mlogit (N=337): Factor Change in the Odds of occ when P> z < 0.05 Variable: white (sd= ) Odds comparing Alternative 1 to Alternative 2 b z P> z e^b e^bstdx Menial -Prof Craft -Prof Prof -Menial Prof -Craft Question (for you) Is the variable white a significant predictor of occupational class? Nominal LHS \ 31

32 Predicted probabilities As before predict, mtable, mchange, mgen, and margins + mlincom can be used Nominal LHS \ 32

33 Something new A discrete change plot The steps: Run mlogit, for example: mlogit occ exper ed i.white Run mchange, for example: mchange, atmeans Nominal LHS \ 33

34 . quietly mlogit occ white ed exper, base(1). mchange mlogit: Changes in Pr(y) Number of obs = 337 Expression: Pr(occ), predict(outcome()) Menial BlueCol Craft WhiteCol Prof white 1 vs p-value ed +1 cntr p-value SD cntr p-value Marginal p-value exper +1 cntr p-value SD cntr p-value Marginal p-value Average predictions Menial BlueCol Craft WhiteCol Prof Pr(y base) Nominal LHS \ 34

35 Discrete Change Plot white 1 vs 0 C M W B P ed SD change B C M W P exper SD change B M WP C Marginal Effect on Outcome Probability Job: 1=Menial 2=BlCol 3=Craft 4=WhCol 5=Prof Nominal LHS \ 35

36 . mchangeplot 1.white ed exper, /// > note(job: M=Menial B=BlCol C=Craft W=WhCol P=Prof) Nominal LHS \ 36

37 Adding CI to the DC. margins, at(white=(0 1)) atmeans post Adjusted predictions Number of obs = 337 Model VCE : OIM Expression : Pr(occ==Menial), predict() 1._at : white = 0 ed = (mean) exper = (mean) 2._at : white = 1 ed = (mean) exper = (mean) Delta-method Margin Std. Err. z P> z [95% Conf. Interval] _at mlincom (2-1) lincom pvalue ll ul Nominal LHS \ 37

38 Getting the Odds Ratio out of the doghouse Discrete change does not indicate the dynamics among the dependent outcomes. For example, a decrease in education increases the probability of both blue collar and craft jobs, but, how does it affect the odds of a person choosing a craft job relative to a blue-collar job? To answer these questions, consider the factor change in the odds Nominal LHS \ 38

39 Nominal LHS \ 39

40 β B A ( β ) exp B A p-value x x x x Factor Change Scale Relative to Category A x1 B A x2 BA x3 A B x4 A B Logit Coefficient Scale Relative to Category A Nominal LHS \ 40

41 Factor Change Scale Relative to Category A x1 B A x2 BA x3 A B x4 A B Logit Coefficient Scale Relative to Category A Nominal LHS \ 41

42 Consider a hypothetical model with three outcomes: Logit Coefficient for Comparison x 1 x 2 x 3 B A β B A exp( β B A ) p C A β C A exp( β C A ) p C B β C B exp( β C B ) p Nominal LHS \ 42

43 Nominal LHS \ 43

44 Nominal LHS \ 44

45 . listcoef, help mlogit (N=337): Factor Change in the Odds of occ Variable: white (sd= ) Odds comparing Alternative 1 to Alternative 2 b z P> z e^b e^bstdx BlueCol -Craft BlueCol -WhiteCol BlueCol -Prof BlueCol -Menial Craft -BlueCol Craft -WhiteCol Craft -Prof Craft -Menial WhiteCol-BlueCol WhiteCol-Craft WhiteCol-Prof WhiteCol-Menial Prof -BlueCol Prof -Craft Prof -WhiteCol Prof -Menial Menial -BlueCol Menial -Craft Menial -WhiteCol Menial -Prof Nominal LHS \ 45

46 Variable: ed (sd= ) Odds comparing Alternative 1 to Alternative 2 b z P> z e^b e^bstdx BlueCol -Craft BlueCol -WhiteCol BlueCol -Prof BlueCol -Menial Craft -BlueCol Craft -WhiteCol Craft -Prof Craft -Menial WhiteCol-BlueCol WhiteCol-Craft WhiteCol-Prof WhiteCol-Menial Prof -BlueCol Prof -Craft Prof -WhiteCol Prof -Menial Menial -BlueCol Menial -Craft Menial -WhiteCol Menial -Prof Nominal LHS \ 46

47 Variable: exper (sd= ) Odds comparing Alternative 1 to Alternative 2 b z P> z e^b e^bstdx BlueCol -Craft BlueCol -WhiteCol BlueCol -Prof BlueCol -Menial Craft -BlueCol Craft -WhiteCol Craft -Prof Craft -Menial WhiteCol-BlueCol WhiteCol-Craft WhiteCol-Prof WhiteCol-Menial Prof -BlueCol Prof -Craft Prof -WhiteCol Prof -Menial Menial -BlueCol Menial -Craft Menial -WhiteCol Menial -Prof Nominal LHS \ 47

48 Odds Ratio Plot Nominal LHS \ 48

49 What do you see? Question (for you) Note the different ordering of categories for the different variables Would the OLM allow for this different ordering? Why or why not? Nominal LHS \ 49

50 Why predicted probabilities remain important While the factor change in the odds is constant across the levels of all variables, the discrete changes get larger or smaller at different values of the variables. E.g., if the odds increase by a factor of ten but the current odds are 1 in 10,000, then the substantive impact is small. Nominal LHS \ 50

51 Putting it all together Incorporate information about the discrete change in the probability by making the height of the letter in the odds ratio plot proportional to the square root of the DC. Odds Ratio Scale Relative to Category Prof white 1 vs 0 M_ C_ B W P ed SD increase B_ M_ C_ W_ P exper B_ P M_ W SD increase C Logit Coefficient Scale Relative to Category Prof Job: M=Menial B=BlColl C=Craft W=WhColl P=Prof Nominal LHS \ 51

52 Stata Code. //OR plot. mlogitplot, amount(sd) symbols(m B C W P) mcolor(rainbow) /// note(job: M=Menial B=BlColl C=Craft W=WhColl P=Prof) /// min(-3) max(0.5) gap(.5). graph export mnlm-02-orplot.emf, replace. //DC and OR combined mlogitplot, amount(sd) symbols(m B C W P) mcolor(rainbow) /// note(job: M=Menial B=BlColl C=Craft W=WhColl P=Prof) /// min(-3) max(0.5) gap(.5) meffect. graph export mnlm-03-dcorplot.emf, replace Nominal LHS \ 52

53 Testing that a Variable Has No Effect The hypothesis that x does not affect the dependent variable can be k written as: H : β = β = β = β = 0 0 kbm, kc, M kw, M kpm, Nominal LHS \ 53

54 LR test using lrtest. quietly mlogit occ i.white ed exper, base(1) nolog. estimates store base. quietly mlogit occ i.white exper, base(1) nolog. estimates store noed. lrtest base noed Likelihood-ratio test LR chi2(4) = (Assumption: noed nested in base) Prob > chi2 = Wald test using test. quietly mlogit occ i.white ed exper,base(1). test ed ( 1) [Menial]o.ed = 0 ( 2) [BlueCol]ed = 0 ( 3) [Craft]ed = 0 ( 4) [WhiteCol]ed = 0 ( 5) [Prof]ed = 0 Constraint 1 dropped chi2( 4) = Prob > chi2 = Nominal LHS \ 54

55 Either, using mlogtest. quietly mlogit occ white ed exper,base(1). mlogtest ed, lr wald **** Likelihood-ratio tests for independent variables (N=337) Ho: All coefficients associated with given variable(s) are 0. chi2 df P>chi ed **** Wald tests for independent variables (N=337) Ho: All coefficients associated with given variable(s) are 0. chi2 df P>chi ed Nominal LHS \ 55

56 Testing that outcome categories can be combined The hypothesis that P and W are indistinguishable is H : β = β = β = 0 0 1, PW 2, PW 3, PW Nominal LHS \ 56

57 A Wald test using mlogtest. mlogtest, combine **** Wald tests for combining outcome categories Ho: All coefficients except intercepts associated with given pair of outcomes are 0 (i.e., categories can be collapsed). Categories tested chi2 df P>chi Menial- BlueCol Menial- Craft Menial-WhiteCol Menial- Prof BlueCol- Craft BlueCol-WhiteCol BlueCol- Prof Craft-WhiteCol Craft- Prof WhiteCol- Prof Nominal LHS \ 57

58 A LR test using mlogtest. mlogtest, lrcom **** LR tests for combining outcome categories Ho: All coefficients except intercepts associated with given pair of outcomes are 0 (i.e., categories can be collapsed). Categories tested chi2 df P>chi Menial- BlueCol Menial- Craft Menial-WhiteCol Menial- Prof BlueCol- Craft BlueCol-WhiteCol BlueCol- Prof Craft-WhiteCol Craft- Prof WhiteCol- Prof Nominal LHS \ 58

59 Question (for you) Do you notice any logical inconsistencies? Nominal LHS \ 59

60 Specification Searches Given the complexities in interpreting the MNLM, it is tempting to search for a more parsimonious model constructed by excluding variables or combining outcome categories. Tests for combining categories and that all coefficients for a variable are zero can guide a specification search, but great care is required to avoid over-fitting or misfitting the model. Nominal LHS \ 60

61 Independence of Irrelevant Alternatives For a model with outcome categories M, N, and L In the MNLM, the odds of M compared to N do not depend on L ( x, 1) ( x, ) β0, m n β1, m nx1 β2, m nx2 β2, m n β3, m nx3 Ω mn x2+ e e e e e = = e β0, m n β1, m nx1 β2, m nx2 β3, m nx3 Ω x e e e e mn 2 β 2, m n In other words, outcome L is irrelevant to the comparison of M to N This property is called the independence of irrelevant alternatives (IIA) Nominal LHS \ 61

62 McFadden's Classic example of IIA A person has two choices: Pr( car) = 1 /2 and Pr( red bus) = 1 /2 Odds of taking the car versus the red bus are ( ) ( ) = = Pr car 1/2 1 Pr red bus 1 / 2 Nominal LHS \ 62

63 A new bus company opens with identical service to the red bus. IIA requires: Pr( car) = 1 / 3; Pr( red bus) = 1 / 3; Pr( blue bus) = 1 / 3 So that the original odds can be maintained: ( ) ( ) = 1 = Pr car 1/3 Pr red bus 1 / 3 Nominal LHS \ 63

64 But what makes sense is: Pr( car) = 1 /2; Pr( red bus) = 1 / 4; Pr( blue bus) = 1 / 4 But this violates IIA: ( ) ( ) = = Pr car 1/2 2 Pr red bus 1 / 4 Nominal LHS \ 64

65 This implies that MNLM should only be used in cases where the outcome categories can plausibly be assumed to be distinct Red bus and Blue bus can be viewed as "perfect substitutes" Care in specifying the model to involve distinct outcomes that are not substitutes for one another seems to be reasonable advice But, many reviewers like to see formal tests of IIA Nominal LHS \ 65

66 Formal tests of IIA Hausman-type test Comparison of two estimators of the same parameter One estimator is consistent and efficient if the null hypothesis is true The second estimator is consistent but inefficient. Question (for you) What would be a consistent but inefficient estimator? Nominal LHS \ 66

67 . set seed 112. mlogtest, iia **** Hausman tests of IIA assumption (N=337) Ho: Odds(Outcome-J vs Outcome-K) are independent of other alternatives. Omitted chi2 df P>chi2 evidence BlueCol for Ho Craft WhiteCol Prof Note: If chi2<0, the estimated model does not meet asymptotic assumptions of the test. **** suest-based Hausman tests of IIA assumption (N=337) Ho: Odds(Outcome-J vs Outcome-K) are independent of other alternatives. Omitted chi2 df P>chi2 evidence BlueCol for Ho Craft against Ho WhiteCol for Ho Prof for Ho Nominal LHS \ 67

68 **** Small-Hsiao tests of IIA assumption (N=337) Ho: Odds(Outcome-J vs Outcome-K) are independent of other alternatives. Omitted lnl(full) lnl(omit) chi2 df P>chi2 evidence BlueCol against Ho Craft against Ho WhiteCol for Ho Prof against Ho Nominal LHS \ 68

69 . set seed mlogtest, iia **** Hausman tests of IIA assumption (N=337) Ho: Odds(Outcome-J vs Outcome-K) are independent of other alternatives. Omitted chi2 df P>chi2 evidence BlueCol for Ho Craft WhiteCol Prof Note: If chi2<0, the estimated model does not meet asymptotic assumptions of the test. **** suest-based Hausman tests of IIA assumption (N=337) Ho: Odds(Outcome-J vs Outcome-K) are independent of other alternatives. Omitted chi2 df P>chi2 evidence BlueCol for Ho Craft against Ho WhiteCol for Ho Prof for Ho Nominal LHS \ 69

70 **** Small-Hsiao tests of IIA assumption (N=337) Ho: Odds(Outcome-J vs Outcome-K) are independent of other alternatives. Omitted lnl(full) lnl(omit) chi2 df P>chi2 evidence BlueCol for Ho Craft for Ho WhiteCol for Ho Prof for Ho Nominal LHS \ 70

71 Case-specific vs. alternative-specific models Sometimes we want to model nominal outcomes as a function of decision-maker characteristics (e.g., education, experience, age as predictors of occupational class). These predictors are referred to as case-specific. Other times we want to model nominal outcomes as a function of alternative-specific characteristics (e.g., income, hours worked, number of years required for each occupational class). These predictors are referred to as alternative-specific. Nominal LHS \ 71

72 Conditional logit model The conditional logit model (CLM) uses alternative-specific data to model multiple nominal categories as alternatives to one another. Transportation alternatives: car, bus, bike, taxi Alternative-specific variables measure aspects of each different alternative. How long does it take to get to class with each alternative? How much does it cost to get to class with each alternative? The CLM estimates a single parameter for each variable that translates the value (or cost) of that alternative into a probability of choosing that alternative. Nominal LHS \ 72

73 CLM example Continuing the transportation example, imagine that we had three different transportation options. For one independent variable (time to class) and four observations, our data would look like this: Nominal LHS \ 73

74 MNLM vs. CLM In MNLM, coefficients for a variable differ for each outcome. Values for a variable are the same for a given variable (e.g., we have only one measure for each observation). In CLM, coefficients for a variable are the same for each outcome. Values for a variable differ for each outcome within the same observation. Nominal LHS \ 74

75 The transportation example: CLM vs. MNLM MNLM The effect of time differs for each mode of transport. The amount of time is the same for each mode of transport. CLM The effect of time is the same for each mode of transport. The amount of time differs for each mode of transport. Nominal LHS \ 75

76 . usecda cda_travel4. asclogit choice time, alt(mode) case(id) nolog Alternative-specific conditional logit Number of obs = 456 Case variable: id Number of cases = 152 Alternative variable: mode Alts per case: min = 3 avg = 3.0 max = 3 Wald chi2(1) = Log likelihood = Prob > chi2 = choice Coef. Std. Err. z P> z [95% Conf. Interval] mode time Train (base alternative) Bus _cons Car _cons Nominal LHS \ 76

77 Another example: Occupational attainment MNLM: Race, education and experience affect the odds of individuals have different occupations. For a given individual, the values of the regressors are the same for all outcomes (i.e., the value of race doesn t vary depending on which occupation we are examining). CLM The regressors are the costs and benefits of each occupation. For each observation, the present value of full-time employment is computed for each occupation. The effect of the present value is the same across all occupations, but the present value of holding that occupation differs by occupation. o For example, the present value of a professional occupation will exceed the present value of a menial occupation, thus making a professional occupation more likely, all else being equal. Nominal LHS \ 77

78 CLM and MNLM models reflect different aspects of the processes by which individuals choose or attain occupations, or choose models of transportation. Selection of the appropriate model should be driven by specification of the process you are interested in modeling. Nominal LHS \ 78

79 Also, note that both CLM & MNLM require IIA as a fundamental assumption (i.e., that the odds of being in any one category/choosing any one outcome) do not depend on the other outcomes. Nominal LHS \ 79

80 Other models for nominal outcomes Multinomial probit: Can allow for alternative- and case-specific predictors in the same model; IIA assumption can be relaxed. Requires modeling of variance structure (e.g., unstructured, exchangeable, etc.). Requires simulated ML estimates (or MCMC). o In Stata: mprobit; asmprobit o In R: mnp package (mlogit perhaps?) Nominal LHS \ 80

81 Nested logit: Allows for nested structure of outcomes (i.e., errors of certain outcomes to be correlated conditional on group membership). Can include alternative- and case-specific predictors; IIA assumption relaxed. Uses full ML. o In Stata: nlogit o In R: mlogit; mnp packages Nominal LHS \ 81

82 End MNLM Nominal LHS \ 82