The Stata module CUB for fitting mixture models for ordinal data - PowerPoint PPT Presentation

The statistical framework Empirical evidence The Stata module for CUB The Stata module CUB for fitting mixture models for ordinal data Christopher F. BAUM 1 , Giovanni CERULLI 2 , Francesca DI IORIO 3 , Domenico PICCOLO 3 , Rosaria SIMONE 3 1 Boston College 2 IRCrES-CNR, Roma 3 Universit` a degli Studi di Napoli Federico II November 15th, 2018 XV Convegno Italiano degli Utenti di STATA Bologna Baum, Cerulli, Di Iorio, Piccolo, Simone The Stata module CUB for fitting mixture models for ordinal data

The statistical framework Empirical evidence The Stata module for CUB Outline 1 The statistical framework 2 Empirical evidence 3 The Stata module for CUB Baum, Cerulli, Di Iorio, Piccolo, Simone The Stata module CUB for fitting mixture models for ordinal data

The statistical framework Empirical evidence The Stata module for CUB 1 The statistical framework 2 Empirical evidence 3 The Stata module for CUB Baum, Cerulli, Di Iorio, Piccolo, Simone The Stata module CUB for fitting mixture models for ordinal data

The statistical framework Empirical evidence The Stata module for CUB Ordinal data Human and relational variables such as happiness, job satisfaction, quality of life, consumers’ preferences , etc. are considered as the main responses in official sample surveys Ordinal variables: Associate positive integers to discrete choices Ranking : Rating : Numbers convey the Numbers ( ordinal scores ) convey the location/preference of the “object” level/evaluation in a given ordered list of a “perception” items, products, sports, applicants, sentences, teams, songs, . . . perception, opinion, taste, fear, worry, agreement . . . Baum, Cerulli, Di Iorio, Piccolo, Simone The Stata module CUB for fitting mixture models for ordinal data

The statistical framework Empirical evidence The Stata module for CUB Cumulative models Analyses of these data are generally performed in the context of Generalized Linear Models (McCullagh 1980; McCullagh and Nelder, 1989): ▸ Let R i be the ordinal score marked by the i -th respondent to an item of a questionnaire for i = 1 , . . . , n : 1 2 . . . r . . . m ▸ The discrete response is obtained by grouping the (continuous) latent variable i in classes by means of cut-points ( −∞ = α 0 < α 1 < ⋯ < α m = +∞ ): R ⋆ α r − 1 < R ⋆ i ≤ α r ⇐ ⇒ R i = r, r = 1 , . . . , m ▸ A systematic relationship is set between the cumulative function and selected subjects’ variables t i ( covariates ) via regression coefficients β = ( β 1 , . . . , β p ) : Pr ( R i ≤ r ∣ θ , t i ) = F θ ( α r − β T t i ) i = β T t i + ǫ i , ⇔ R ⋆ (Proportional odds model -POM) logit ( Pr ( R i ≤ r ∣ t i )) = α r − β T t i , ( logit ( p ) = log ( 1 − p ) , p ∈ ( 0 , 1 )) p In Stata: ologit, oprobit, oglm, ... Baum, Cerulli, Di Iorio, Piccolo, Simone The Stata module CUB for fitting mixture models for ordinal data

The statistical framework Empirical evidence The Stata module for CUB Rationale: cub models paradigm Psychologists assess that two main aspects are activated when people have to express their evaluation (agreement, worry, etc.) towards an item by selecting a category out of a list of m ordered alternatives (Tourangeau et al. (2000)): Perceptual aspects: the rater’s perception of the item content Decisional aspects : the rater’s use of the available scale cub models (Piccolo, 2003) assume that the data generating process is structured as the combination of: Feeling: generated by the sound perception of the respondent Uncertainty : generated by the intrinsic fuzziness of the final choice Baum, Cerulli, Di Iorio, Piccolo, Simone The Stata module CUB for fitting mixture models for ordinal data

The statistical framework Empirical evidence The Stata module for CUB Modelling Feeling Feeling is the result of a continuous (latent) variable that is discretized: depending on the framework, it is a direct measure of worry, satisfaction, preference, involvement, happiness, ... How likely is it that you would recommend this brand to a friend or colleague? (Net Promoter Score) 1 Does your family easily make ends meet? 2 3 . . . cub paradigm prescribes a shifted Binomial random variable for feeling: b r ( ξ ) = ( m − 1 r − 1 ) ξ m − r ( 1 − ξ ) r − 1 , r = 1 , . . . , m ▸ Pragmatic view . The shifted Binomial distribution m= 7 csi = 0.2 m= 7 csi = 0.5 m= 8 csi = 0.9 allows for modal values to be located everywhere 0.4 0.30 on the support { 1 , 2 , . . . , m } and on the basis of 0.4 0.25 0.3 a single parameter ( ξ ) , related in a simple way to 0.20 0.3 both mode and expectation ▸ Statistical view . When a respondent selects a 0.2 0.15 0.2 0.10 0.1 single value in a list of ordered categories, he/she 0.1 0.05 is comparing each score with all the others. The 0.0 0.0 Binomial distribution “counts” the number of 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 successes (number of times that the selected category is outclassed by the previous ones). Baum, Cerulli, Di Iorio, Piccolo, Simone The Stata module CUB for fitting mixture models for ordinal data

The statistical framework Empirical evidence The Stata module for CUB Modelling uncertainty 1 Limited set of information, Knowledge/Ignorance about the item 2 Personal interest/Engagement in activities related to item 0.30 0.30 0.30 0.20 0.20 0.20 3 Amount of time devoted to the 0.10 0.10 0.10 response 0.00 0.00 0.00 4 Range and wording of the scale 2 4 6 8 2 4 6 8 2 4 6 8 5 Tiredness or fatigue for a correct 0.30 0.30 0.30 comprehension of the wording 0.20 0.20 0.20 6 Willingness to joke and fake 0.10 0.10 0.10 7 Laziness/Apathy/Boredom 0.00 0.00 0.00 2 4 6 8 2 4 6 8 2 4 6 8 8 ... 0.30 0.30 0.30 The discrete Uniform random variable U 0.20 0.20 0.20 maximizes the entropy among all the 0.10 0.10 0.10 discrete distributions with finite support 0.00 0.00 0.00 2 4 6 8 2 4 6 8 2 4 6 8 Pr ( U = r ) = 1 r = 1 , 2 , . . . , m m , Baum, Cerulli, Di Iorio, Piccolo, Simone The Stata module CUB for fitting mixture models for ordinal data

The statistical framework Empirical evidence The Stata module for CUB cub models specification Let R i ∈ { 1 , 2 , . . . , m } the ordinal response given by the i -th subject characterized by variables t i ∈ T . If C i = ( r, t i ) denotes the information collected on the i -th subject, then the cub mixture is defined by: 1 A stochastic component : Pr ( R i = r ∣ C i , θ ) = π i [ b r ( ξ i )] + ( 1 − π i ) r = 1 , 2 , . . . , m ; i = 1 , 2 , . . . , n. [ 1 m ] , �ÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜ� �ÜÜÜÜÜÜ�ÜÜÜÜÜÜ� feeling uncertainty where θ = ( β T , γ T ) T 2 Two systematic components : if subject’s covariates x i and w i are chosen to explain π i and ξ i , respectively: logit ( π i ) = x i β = β 0 + β 1 x i 1 + ⋯ + β p x ip logit ( ξ i ) = w i γ = γ 0 + γ 1 w i 1 + ⋯ + γ q w iq If no covariate is included: π i = π ∈ ( 0 , 1 ] and ξ i = ξ ∈ [ 0 , 1 ] : logit ( π ) = β 0 ⇔ π = 1 1 + exp ( − β 0 ) logit ( ξ ) = γ 0 ⇔ ξ = 1 1 + exp ( − γ 0 ) Baum, Cerulli, Di Iorio, Piccolo, Simone The Stata module CUB for fitting mixture models for ordinal data

The statistical framework Empirical evidence The Stata module for CUB cub models visualization Model A Mode = 9 Model B Mode = 8 Model C Mode = 7 0.4 0.4 0.4 π = 0.9 ξ = 0.1 µ = 7.88 π = 0.3 ξ = 0.2 µ = 5.72 π = 0.7 ξ = 0.3 µ = 6.12 CUB models − m=9 0.2 0.2 0.2 1.0 A 0.0 0.0 0.0 2 4 6 8 2 4 6 8 2 4 6 8 B 0.8 C Model D Mode = 6 Model E Mode = 5 Model F Mode = 4 0.4 0.4 0.4 µ = 5.4 µ = 5 π = 0.4 ξ = 0.6 µ = 4.68 π = 0.5 ξ = 0.4 π = 0.2 ξ = 0.5 D 0.6 Feeling ( 1 − ξ ) 0.2 0.2 0.2 E 0.0 0.0 0.0 F 0.4 2 4 6 8 2 4 6 8 2 4 6 8 G H Model G Mode = 3 Model H Mode = 2 Model I Mode = 1 0.2 0.4 0.4 0.4 π = 0.6 ξ = 0.7 µ = 4.04 π = 0.8 ξ = 0.8 µ = 3.08 π = 0.1 ξ = 0.9 µ = 4.68 I 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 2 4 6 8 2 4 6 8 Uncertainty ( 1 −π ) Baum, Cerulli, Di Iorio, Piccolo, Simone The Stata module CUB for fitting mixture models for ordinal data

The statistical framework Empirical evidence The Stata module for CUB cub models and bimodality Bimodal observed distribution CUB distributions, given csi−covariate=0, 1 0.15 0.3 Prob(R|D=0) and Prob(R|D=1) Relative frequencies 0.10 0.2 0.05 0.1 0.00 0.0 1 2 3 4 5 6 7 8 9 2 4 6 8 ordinal Figure shows the simulated and estimated distributions (conditional to Di = 0 , 1 , respectively) of the shifted Binomial model ( m = 9 ): ⎧ j − 1 ) ξ 8 − j ( 1 − ξi ) j − 1 ; ⎪ ( 8 P r ( Ri = j ) = ⎪ ⎪ ⎪ ⎪ i ⎨ j = 1 , 2 , . . . , 9; i = 1 , 2 , . . . , n. ⎪ ⎪ ⎪ ⎪ ⎪ logit ( ξi ) = − 1 . 362 + 2 . 744 Di ; ⎩ Baum, Cerulli, Di Iorio, Piccolo, Simone The Stata module CUB for fitting mixture models for ordinal data