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Bayesian model-based clustering for longitudinal ordinal data

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Abstract

Traditional cluster analysis methods used in ordinal data, for instance k-means and hierarchical clustering, are mostly heuristic and lack statistical inference tools to compare among competing models. To address this we propose a latent transitional model, a finite mixture model that includes both observed and latent covariates and apply it for the first time to the case of longitudinal ordinal data. This model-based clustering model is an extension of the proportional odds model and includes a first-order transitional term, occasion effects and interactions which provide flexible ways to capture different time patterns by cluster as well as time-heterogeneous transitions. We estimate model parameters within a Bayesian setting using a Markov chain Monte Carlo scheme and block-wise Metropolis–Hastings sampling. We illustrate the model using 2001–2011 self-reported health status (SRHS) from the Household, Income and Labour Dynamics in Australia survey. SRHS is recorded as an ordinal variable with five levels: poor, fair, good, very good and excellent. Using the Widely Applicable Information Criterion for model comparison, we find evidence for six latent groups. Transitions in the original data and the estimated groups are visualized using heatmaps.

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Acknowledgements

The work is being supported by the Marsden Fund Grants 16-VUW-062 and E2987-3648 from the Royal Society of New Zealand. We would like to thank Professor Shirley Pledger from Victoria University of Wellington for many useful discussions. This paper uses unit record data unit record data from the Household, Income and Labour Dynamics in Australia (HILDA) Survey. The HILDA Project was initiated and is funded by the Australian Government Department of Social Services (DSS) and is managed by the Melbourne Institute of Applied Economic and Social Research (Melbourne Institute). The findings and views reported here, however, are those of the author and should not be attributed to either DSS or the Melbourne Institute. More information about the HILDA survey can be found at: https://www.melbourneinstitute.com/hilda/.

Author information

Authors and Affiliations

  1. Institute for Molecular Bioscience and Queensland Alliance for Agriculture and Food Innovation, The University of Queensland, Brisbane, QLD, 4072, Australia

    Roy Costilla

  2. School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand

    Ivy Liu, Richard Arnold & Daniel Fernández

  3. Institut de Recerca Sant Joan de Déu, Parc Sanitari Sant Joan de Déu, CIBERSAM, Barcelona, Spain

    Daniel Fernández

Authors
  1. Roy Costilla
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Correspondence to Roy Costilla.

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Appendices

Appendix A: Proposals

After choosing initial values for all model parameters (\(\mu \), \(\alpha \), \(\beta \), \(\gamma \), \(\pi \), \(\sigma ^2_{\mu }\), \(\sigma ^2_{\alpha }\), \(\sigma ^2_{\beta }\), and \(\sigma ^2_{\gamma }\)), we proceed to update them according to the following:

$$\begin{aligned}&\mu '_k \mid \mu _k, \mu _{k-1}, \mu _{k+1} {\sim } U[ \text { max}(\mu _k - \tau ,\mu _{k-1}), \text { min}(\mu _k + \tau ,\mu _{k+1}) ] \; k=2, \ldots , q-1,\\&\quad \mu _0=-\infty ,\mu _1=0 \, \mu _q=\infty \\&\alpha '_r \mid \alpha _r \overset{iid}{\sim }\text {Normal}(\alpha _r,\sigma ^2_{\alpha p}) \qquad r=1, \ldots R, \\&\beta '_{rk'} \mid \beta _{rk'} \overset{iid}{\sim }\text {Normal}(\beta _{rk'},\sigma ^2_{\beta p}) \qquad k'=1, \ldots q-1, \; \beta _{rq}=-\sum _{k'=1}^{q-1}\beta _{rk'}, \; \forall r=1, \ldots R, \\&\gamma '_j \mid \gamma _j \overset{iid}{\sim }\text {Normal}(\gamma _j,\sigma ^2_{\gamma p}) \qquad j=2, \ldots p-1, \; \gamma _{p}=-\sum _{j=2}^{p-1}\gamma _j \\&\text {logit}(w') \mid \text {logit}(w) \sim \text {Normal}(\text {logit}(w), \sigma ^2_{\pi p}) \qquad w=\pi _{r1}/(\pi _{r1}+\pi _{r2}) \; r1,r2 \in {1, \ldots ,R} \\&\pi '_{r1}=w'(\pi _{r1}+\pi _{r2}) \qquad \pi '_{r2}=(1-w')(\pi _{r1}+\pi _{r2}) \\&\text {log}(\sigma '^2_{\mu }) \mid \text {log}(\sigma ^2_{\mu }) \sim \text {Normal}(\text {log}(\sigma ^2_{\mu }), \sigma ^{2}_{\sigma \mu p}) \\&\text {log}(\sigma '^2_{\alpha }) \mid \text {log}(\sigma ^2_{\alpha }) \sim \text {Normal}(\text {log}(\sigma ^2_{\alpha }), \sigma ^{2}_{\sigma \alpha p}) \\&\text {log}(\sigma '^2_{\beta }) \mid \text {log}(\sigma ^2_{\beta }) \sim \text {Normal}(\text {log}(\sigma ^2_{\beta }), \sigma ^{2}_{\sigma \beta p}) \\&\text {log}(\sigma '^2_{\gamma }) \mid \text {log}(\sigma ^2_{\gamma }) \sim \text {Normal}(\text {log}(\sigma ^2_{\gamma }), \sigma ^{2}_{\sigma \gamma p}) \end{aligned}$$

with proposal step sizes scaled by: \(\tau =0.5\), \(\sigma ^2_{\alpha p}=0.1\), \(\sigma ^2_{\beta p}=0.1\), \(\sigma ^2_{\gamma p}=0.1\), \(\sigma ^2_{\pi p}=0.25\), \(\sigma ^{2}_{\sigma \mu p}=\text {log}(2)\), \(\sigma ^{2}_{\sigma \alpha p}=\text {log}(4)\), \(\sigma ^{2}_{\sigma \beta p}=\text {log}(1.5)\) and \(\sigma ^{2}_{\sigma \gamma p}=\text {log}(2)\).

Appendix B: Posterior summary statistics and convergence diagnostics, HILDA case study \(R=6\)

Par

Median

Mean

SE

Lower CI

Upper CI

PSRF

\(\mu _2\)

3.53

3.54

0.24

3.12

4.04

1.00

\(\mu _3\)

6.86

6.87

0.27

6.35

7.38

1.00

\(\mu _4\)

11.06

11.06

0.33

10.46

11.73

1.00

\(\sigma ^2_{\mu }\)

0.27

0.34

0.25

0.08

0.74

1.00

\(\alpha _1\)

2.76

2.76

0.49

1.74

3.67

1.00

\(\alpha _2\)

5.02

4.99

0.33

4.40

5.60

1.04

\(\alpha _3\)

5.27

5.27

0.24

4.77

5.74

1.00

\(\alpha _4\)

8.20

8.17

0.53

7.47

9.15

1.10

\(\alpha _5\)

9.87

9.84

0.73

8.39

11.23

1.07

\(\alpha _6\)

12.39

12.40

1.02

10.17

14.64

1.06

\(\sigma ^2_{\alpha }\)

28.60

31.96

15.05

10.72

59.19

1.00

\(\beta _{11}\)

\(-\) 0.15

\(-\) 0.17

0.42

\(-\) 0.97

0.68

1.03

\(\beta _{12}\)

\(-\) 0.31

\(-\) 0.31

0.36

\(-\) 1.01

0.37

1.00

\(\beta _{13}\)

0.23

0.24

0.34

\(-\) 0.40

0.93

1.01

\(\beta _{14}\)

0.17

0.19

0.44

\(-\) 0.72

1.06

1.00

\(\beta _{15}\)

0.06

0.05

0.91

\(-\) 1.86

1.78

1.02

\(\beta _{21}\)

\(-\) 0.53

\(-\) 1.92

2.74

\(-\) 7.06

0.58

1.00

\(\beta _{22}\)

\(-\) 1.03

\(-\) 1.50

1.06

\(-\) 3.68

\(-\) 0.35

1.00

\(\beta _{23}\)

0.32

0.17

0.51

\(-\) 1.05

0.89

1.00

\(\beta _{24}\)

1.44

2.16

1.45

0.90

5.06

1.00

\(\beta _{25}\)

\(-\) 0.26

1.09

2.72

\(-\) 1.45

6.08

1.00

\(\beta _{31}\)

\(-\) 6.06

\(-\) 4.63

2.78

\(-\) 7.12

0.50

1.00

\(\beta _{32}\)

\(-\) 3.08

\(-\) 2.67

1.04

\(-\) 3.92

\(-\) 0.69

1.00

\(\beta _{33}\)

\(-\) 0.45

\(-\) 0.42

0.61

\(-\) 1.53

0.66

1.02

\(\beta _{34}\)

4.36

3.63

1.53

0.85

5.19

1.00

\(\beta _{35}\)

5.51

4.09

2.64

\(-\) 0.85

6.34

1.00

\(\beta _{41}\)

\(-\) 0.19

\(-\) 0.29

0.81

\(-\) 1.19

0.73

1.18

\(\beta _{42}\)

\(-\) 0.19

\(-\) 0.22

0.50

\(-\) 0.98

0.58

1.13

\(\beta _{43}\)

\(-\) 0.29

\(-\) 0.30

0.29

\(-\) 0.85

0.27

1.03

\(\beta _{44}\)

\(-\) 0.10

\(-\) 0.03

0.60

\(-\) 0.62

0.41

1.25

\(\beta _{45}\)

0.78

0.84

0.78

\(-\) 0.27

1.88

1.17

\(\beta _{51}\)

0.17

0.17

0.44

\(-\) 0.70

1.00

1.01

\(\beta _{52}\)

0.29

0.31

0.46

\(-\) 0.68

1.22

1.03

\(\beta _{53}\)

0.26

0.27

0.40

\(-\) 0.55

1.04

1.04

\(\beta _{54}\)

0.23

0.24

0.36

\(-\) 0.46

0.94

1.00

\(\beta _{55}\)

\(-\) 1.03

\(-\) 0.99

0.78

\(-\) 2.44

0.70

1.10

\(\beta _{61}\)

\(-\) 0.06

\(-\) 0.05

0.44

\(-\) 0.94

0.87

1.00

\(\beta _{62}\)

\(-\) 0.06

\(-\) 0.06

0.46

\(-\) 1.00

0.84

1.01

\(\beta _{63}\)

\(-\) 0.15

\(-\) 0.16

0.45

\(-\) 1.07

0.66

1.04

\(\beta _{64}\)

0.04

0.06

0.39

\(-\) 0.77

0.86

1.00

\(\beta _{65}\)

0.24

0.22

0.69

\(-\) 1.17

1.57

1.04

\(\sigma ^2_{\beta 1}\)

0.28

0.33

0.20

0.09

0.73

1.02

\(\sigma ^2_{\beta 2}\)

0.77

3.54

5.67

0.12

15.43

1.00

\(\sigma ^2_{\beta 3}\)

8.29

8.56

6.86

0.17

20.43

1.00

\(\sigma ^2_{\beta 4}\)

0.25

0.45

1.51

0.09

0.67

1.31

\(\sigma ^2_{\beta 5}\)

0.29

0.35

0.23

0.09

0.77

1.00

\(\sigma ^2_{\beta 6}\)

0.27

0.33

0.24

0.08

0.72

1.00

\(\gamma _{2}\)

0.42

0.42

0.14

0.15

0.69

1.00

\(\gamma _{3}\)

0.17

0.17

0.13

\(-\) 0.09

0.42

1.00

\(\gamma _{4}\)

0.04

0.04

0.13

\(-\) 0.20

0.28

1.00

\(\gamma _{5}\)

\(-\) 0.08

\(-\) 0.08

0.13

\(-\) 0.33

0.17

1.00

\(\gamma _{6}\)

0.06

0.06

0.13

\(-\) 0.19

0.29

1.00

\(\gamma _{7}\)

0.06

0.06

0.13

\(-\) 0.18

0.32

1.00

\(\gamma _{8}\)

\(-\) 0.02

\(-\) 0.02

0.12

\(-\) 0.27

0.20

1.00

\(\gamma _{9}\)

0.06

0.06

0.13

\(-\) 0.19

0.31

1.00

\(\gamma _{10}\)

\(-\) 0.24

\(-\) 0.24

0.13

\(-\) 0.48

0.01

1.00

\(\gamma _{11}\)

\(-\) 0.46

\(-\) 0.47

0.13

\(-\) 0.71

\(-\) 0.21

1.00

\(\sigma ^2_{\gamma }\)

0.16

0.17

0.07

0.07

0.31

1.00

\(\pi _1\)

0.08

0.08

0.02

0.04

0.13

1.01

\(\pi _2\)

0.32

0.31

0.06

0.20

0.40

1.03

\(\pi _3\)

0.26

0.27

0.05

0.19

0.37

1.00

\(\pi _4\)

0.24

0.24

0.04

0.16

0.33

1.15

\(\pi _5\)

0.05

0.06

0.04

0.02

0.13

1.32

\(\pi _6\)

0.04

0.04

0.01

0.01

0.07

1.08

log-like

\(-\) 2121

\(-\) 2121

4.54

\(-\) 2130

\(-\) 2113

1.03

Appendix C: Traceplots and marginal posterior distributions, HILDA case study \(R=6\)

figure a

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Costilla, R., Liu, I., Arnold, R. et al. Bayesian model-based clustering for longitudinal ordinal data. Comput Stat 34, 1015–1038 (2019). https://doi.org/10.1007/s00180-019-00872-4

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