Supplementary Material for Analysis of Job Satisfaction: The Case of Japanese Private Companies

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1 Supplementary Material for Analysis of Job Satisfaction: The Case of Japanese Private Companies S1. Sampling Algorithms We assume that z i NX i β, Σ), i =1,,n, 1) where Σ is an m m positive definite covariance matrix. To complete the Bayesian model, we introduce the prior distributions of the parameters pβ, γ, Σ, z). On the basis of Bayes theorem, the joint posterior distribution can be written as pβ, γ, Σ, z y) pβ, γ, Σ, z)py β, γ, Σ, z) = pβ, γ, Σ)pz β, γ, Σ)py β, γ, Σ, z) [ n ] = pβ, γ, Σ) pz i β, γ, Σ)py i β, γ, Σ, z i ). Further, defining we have G ij = ] γ jc 1),γ jc if yij = c, c {1,,C j },j=1,,m G i = G i1 G im, i =1,,n, py i β, γ, Σ, z i )=1 zi G i), i =1,,n, where 1 ) is an indicator function. 1 Now, we specify the prior distributions as follows: m pβ, γ, Σ) =pβ)pγ)pσ) = pβ j )pγ j ) pσ), where j=1 β j Nβ j0, B j0 ), j =1,,m, Σ 1 Wκ 0, Q 1 0 ), 1 See Chib and Greenberg 1998, p.349). 1

2 and Wκ 0, Q 1 0 ) denotes a Wishart distribution with degrees of freedom κ 0 and scale matrix Q 1 0. Further, we introduce the prior distribution of γ j, pγ j )= pδ j γ j )) based on the following transformation for the cutoff points Chen and Dey, 2000, p.140): ) γjc γ jc 1) δ jc =log, c =2,,C j 2, 1 γ jc where δ j γ j )=δ j2,,δ jcj 2)) j =1,,m). follows: We specify pδ j γ j )) as δ j γ j ) Nδ j0, D j0 ), j =1,,m. Thus, the joint posterior distribution can be written as pβ, γ, Σ, z y) pβ, γ, Σ) { n [ 1 zi G i) Σ 1 2 exp 1 2 z i X i β) Σ 1 z i X i β)] }. 2) Using the MCMC sampling scheme, we can sample parameters β, γ, Σ, z) from the joint posterior distribution 2). S1.1. Sampling of β and Σ 1 For the sake of convenience of expression, we replace the jth factor of z i, X i, β, Σ and Σ 1 as the first factor, that is, ) ) ) zij x z i =, X z i = ij 0 βj, β = i j) 0 X i j) β j) σjj σ ) j) σ Σ =, Σ 1 jj σ j) ) = σ j) Σ j) σ j) Σ j). Then, we have the following full conditional distributions FCDs) of β j Σ 1. The FCD of β j is and β j N β j, B j ), j =1,,m, 3) where denotes that 3) is conditional on the other unspecified variables in the equation, and B j = B 1 j0 + σjj n x ij x ij β j = B j [B 1 j0 β j0 + σjj The FCD of Σ 1 is ) 1 n x ij z ij + n ) ] x ij σ j) z i j) X i j) β j). Σ 1 W κ, Q 1 ) 4) 2

3 where n κ = κ 0 + n, Q = Q0 + z i X i β)z i X i β). Applying a Gibbs sampling to the FCDs of 3) and 4), we can generate β j and Σ 1. S1.2. Sampling of z and γ Let z j) =z 1j,z 2j,,z nj ) denote the vector of the jth element z ij from z i i =1,,n). Further, let z j) denote the vector obtained by removing z j) from z, and let z i j) denote the vector of removing z ij from z i.wecanthen generate γ j and z j) from the joint conditional distribution pγ j, z j) β, Σ, z j), y) j =1,,m). The joint conditional distribution pγ j, z j) β, Σ, z j), y) can be written as pγ j, z j) β, Σ, z j), y) = pγ j β, Σ, z j), y)pz j) γ j, β, Σ, z j), y), j =1,,m. Similar to the sampling of β j, for the sake of convenience of expression, we replace the jth factor as the first factor. Since z i β, Σ, γ NX i β, Σ), from the property of the multivariate normal distribution, we have where z ij γ j, β, Σ, z j), y N μ ij, σ jj )1 zij G ij), i =1,,n; j =1,,m, 5) ) μ ij = x ij β j + σ j) Σ 1 j) z i j) X i j) β j) σ jj = σ jj σ j) Σ 1 j) σ j). The distribution of z ij is a truncated normal distribution. Since z 1j,z 2j,,z nj are independent, given γ j, β, Σ, wehave pγ j β, Σ, z j), y) pδ j γ j )) i:y ij=3 [ Φ γj3 μ ij i:y ij =C j 1 σ jj [ Φ i:y ij=2 [ Φ ) γj2 μ ij Φ 1 μij σ jj σ jj ) Φ γj2 μ ij )] σ jj γjcj 2) μ ij σ jj ) Φ μ )] ij σ jj )], where Φ ) is the distribution function of the standard normal distribution. Thus, the conditional distribution of δ j is C j 2 pδ j β, Σ, z j), y) pγ j β, Σ, z j), y) c=2 1 γjc 1) ) expδjc ) 1+expδjc ) ) 2. 6) 3

4 We use a multivariate t distribution, Mtδ j δ j, Σ δj,ν), as a proposal distribution for generating δ j,where δ j isthemodeof6), [ ] 1 Σ δj = 2 log pδ j ) δ j δ j δ j= δ j and ν is the degrees of freedom. The M-H algorithm for generating δ j is as follows: 1. Let δ t) j denote the value of δ j at the tth iteration. 2. At the t + 1)th iteration, sample δ p j from Mtδ j δ j, Σ δj,ν). 3. The transition probability from δ t) j to δ p j is { pδ p j α =min )Mtδt) j δ j, Σ } δj,ν) pδ t) j )Mtδp j δ j, Σ δj,ν), Generate u U0, 1), the uniform distribution on 0, 1), and take { δ p j if u<α δ t+1) j = δ t) j otherwise. We can obtain γ j from δ j using the equation γ jc = γ jc 1) +expδ jc ),c=2,,c j 2. 1+expδ jc ) S2. Partial Effects of Explanatory Variables We are more interested in the response probabilities Pry i1 = c) than in the coefficient parameters themselves. Dropping the suffix i in 1), we consider the population regression, that is, z NXβ, Σ), where X =diagx 1,,. x m), and divide z as z = z 1, z 1)) Suppose that z1 is a latent variable associated with a dependent variable of interest, y 1,andz 1) is a vector of latent variables corresponding to the other ordinal variables. Then, we have pz X, )=p z 1, z 1) X, ) = p z 1 z 1), X, ) p z 1) X, ). On the basis of the property of the multivariate normal distribution, we obtain where z 1 z 1), X, N μ 1, σ 11 ), 7) ) μ 1 = x 1 β 1 + σ 1) Σ 1 1) z 1) X 1) β 1) σ 11 = σ 11 σ 1) Σ 1 1) σ 1) 8) ) ) x X = 1 0 β1 σ11 σ ), β =, Σ = 1). O X 1) β 1) σ 1) Σ 1) 4

5 From 7), we have the following response probabilities of y 1 : Pry 1 = c) =Prγ 1c 1) <z 1 γ 1c ) γ1c 1) μ 1 =Pr < z 1 μ 1 γ ) 1c μ 1 σ11 σ11 σ11 Φ μ ) 1 c =1 σ11 ) ) γ1c μ 1 γ1c 1) μ 1 = Φ Φ c =2,,C 1 1 σ11 ) σ11 1 μ1 1 Φ c = C 1, σ11 9) where Φ ) is the distribution function of the standard normal distribution. If x is a binary explanatory variable, we can obtain the partial effect of x on the response probability Pry 1 = c) by calculating the average of Δ Pry 1 = c) = Pry 1 = c x =1) Pry 1 = c x = 0); that is, 1 n n ΔPry i1 = c) = 1 n n Pry i1 = c x i =1) 1 n S3. Correlation Matrices n Pry i1 = c x i =0). Following Hasegawa 2013), we calculate the simple version of the polychoric correlation matrix at each MCMC iteration based on the sample of Σ: R s = {r s ij} = D s ΣD s, 10) where D s = diag1/ σ 11,, 1/ σ mm )andσ jj is the jth diagonal element of Σ. Further, we can calculate the partial version of the polychoric correlation coefficients, defined as the lower triangular part without the diagonal elements of the following matrix: R p = {r p ij } = D rr 1 s D r, 11) where D r = diag1/ r 11,, 1/ r mm )andr jj is the jth diagonal element of R 1 s. 5

6 S4. Effects of gender and employment status differences on specific aspects related to job satisfaction Figures S4.1 to S4.9 show the effects of gender and employment satus differences on specific aspects related to the job satisfaction. Figure S4.1 : β 2,2, β 2,2 + β 2,4, β 2,3, β 2,3 + β 2,4, and economic indexes for motivation to work work1) β 2,2 β 2,2 + β 2,4 β 2,3 β 2,3 + β 2,4 6

7 Figure S4.2 : β 3,2, β 3,2 + β 3,4, β 3,3, β 3,3 + β 3,4, and economic indexes for adequately making use of your abilities and expertise work2) β 3,2 β 3,2 + β 3,4 β 3,3 β 3,3 + β 3,4 7

8 Figure S4.3 : β 4,2, β 4,2 + β 4,4, β 4,3, β 4,3 + β 4,4, and economic indexes for opportunities and support to enhance vocational skills and career work3) β 4,2 β 4,2 + β 4,4 β 4,3 β 4,3 + β 4,4 8

9 Figure S4.4 : β 5,2, β 5,2 + β 5,4, β 5,3, β 5,3 + β 5,4, and economic indexes for being given a certain amount of responsibility and discretion work4) β 5,2 β 5,2 + β 5,4 β 5,3 β 5,3 + β 5,4 9

10 Figure S4.5 : β 6,2, β 6,2 + β 6,4, β 6,3, β 6,3 + β 6,4, and economic indexes for wage and working conditions enough to run a household work5) β 6,2 β 6,2 + β 6,4 β 6,3 β 6,3 + β 6,4 10

11 Figure S4.6 : β 7,2, β 7,2 + β 7,4, β 7,3, β 7,3 + β 7,4, and economic indexes for adequate and satisfactory wage and working conditions, work6) β 7,2 β 7,2 + β 7,4 β 7,3 β 7,3 + β 7,4 11

12 Figure S4.7 : β 8,2, β 8,2 + β 8,4, β 8,3, β 8,3 + β 8,4, and economic indexes for feeling no excessive mental stress, work7) β 8,2 β 8,2 + β 8,4 β 8,3 β 8,3 + β 8,4 12

13 Figure S4.8 : β 9,2, β 9,2 + β 9,4, β 9,3, β 9,3 + β 9,4, and economic indexes for having good personal relationships in the workplace work8) β 9,2 β 9,2 + β 9,4 β 9,3 β 9,3 + β 9,4 13

14 Figure S4.9 : β 10,2, β 10,2 + β 10,4, β 10,3, β 10,3 + β 10,4, and economic indexes for work-life balance work9) β 10,2 β 10,2 + β 10,4 β 10,3 β 10,3 + β 10,4 References Chen M.-H. and Dey D. K. 2000) Bayesian analysis for correlated ordinal data models, In Dey D. K., Ghosh S. K. and Mallick B. K. Eds.), Generalized Linear Models: A Bayesian Perspective, New York: Marcel Dekker:

15 Chib S. and Greenberg E. 1998) Biometrika 852): Analysis of multivariate probit models, Hasegawa H. 2013) On polychoric and polyserial partial correlation coefficients: A Bayesian approach, Metron 712):