Ordinal Models

1
Ordinal Models
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Ordinal Models

• Estimating gender-specific LLCA with repeated
  ordinal data
• Examining the effect of time invariant covariates
  on class membership
• The effect of class membership on a later outcome

3
The data

• 5 repeated measures of bedwetting
• 4½, 5½, 6½ ,7½ 9½ yrs
• 3-level ordinal
• Dry
• Infrequent wetting (lt 2 nights/week)
• Frequent wetting (2 nights/week)

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4 time point ordinal LLCA (Boys)

• title 4 time point LLCA of ordinal bedwetting
• data file is 'C\Work\bedwet_dsm4_llca\spss\llc
  a_dsm4.txt'
• listwise is ON
• variable
• names ID sex
• nwet_kk2 nwet_kk3 nwet_kk4 nwet_kk5
• nwet_km2 nwet_km3 nwet_km4 nwet_km5
• nwet_kp2 nwet_kp3 nwet_kp4 nwet_kp5
• nwet_kr2 nwet_kr3 nwet_kr4 nwet_kr5
• nwet_ku2 nwet_ku3 nwet_ku4 nwet_ku5
• categorical nwet_kk3 nwet_km3 nwet_kp3
  nwet_kr3 nwet_ku3
• usevariables nwet_kk3 nwet_km3 nwet_kp3
  nwet_kr3 nwet_ku3
• missing are nwet_kk3 nwet_km3 nwet_kp3
  nwet_kr3 nwet_ku3 (999)
• classes c (4)

5
RESULTS IN PROBABILITY SCALE

• Latent Class 1
• NWET_KK3
• Category 1 0.190 0.030
  6.430 0.000
• Category 2 0.672 0.033
  20.134 0.000
• Category 3 0.138 0.026
  5.409 0.000
• NWET_KM3
• Category 1 0.224 0.038
  5.929 0.000
• Category 2 0.727 0.036
  20.254 0.000
• Category 3 0.048 0.019
  2.533 0.011
• NWET_KP3
• Category 1 0.160 0.045
  3.540 0.000
• Category 2 0.823 0.044
  18.613 0.000
• Category 3 0.017 0.011
  1.473 0.141
• NWET_KR3
• Category 1 0.075 0.064
  1.178 0.239
• Category 2 0.903 0.061
  14.686 0.000
• Category 3 0.022 0.011
  1.929 0.054
• NWET_KU3

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RESULTS IN PROBABILITY SCALE

• Latent Class 1
• NWET_KK3
• Category 1 0.190 0.030
  6.430 0.000
• Category 2 0.672 0.033
  20.134 0.000
• Category 3 0.138 0.026
  5.409 0.000
• NWET_KM3
• Category 1 0.224 0.038
  5.929 0.000
• Category 2 0.727 0.036
  20.254 0.000
• Category 3 0.048 0.019
  2.533 0.011
• NWET_KP3
• Category 1 0.160 0.045
  3.540 0.000
• Category 2 0.823 0.044
  18.613 0.000
• Category 3 0.017 0.011
  1.473 0.141
• NWET_KR3
• Category 1 0.075 0.064
  1.178 0.239
• Category 2 0.903 0.061
  14.686 0.000
• Category 3 0.022 0.011
  1.929 0.054
• NWET_KU3

Dry
Infrequent wetting
Frequent wetting
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Alternative 1 three dimensions

• A 3D plot

or something made out of plasticine
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Alternative 2 two figures
Infrequent bedwetting
Frequent bedwetting
9
Alternative 3 two figures
Any bedwetting
Frequent bedwetting
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Alternative 3 two figures
Any bedwetting
Frequent bedwetting
(1) A persistent wetting group who mostly wet to
a frequent level (2) A persistent wetting group
who mostly wet to an infrequent level (3) A
delayed group comprising mainly infrequent
wetters (4) Normative group
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Fit statistics - Boys
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5-class model (boys)
Any bedwetting
Frequent bedwetting
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5-class model (boys)

• Normative (63.8)
• Mild risk of infrequent wetting at start which
  soon disappears
• Delayed-infrequent (18.2)
• Delayed attainment of nighttime bladder control
  but rarely attains frequent levels
• Persistent-infrequent (11.4)
• Persistent throughout period but rarely attains
  frequent levels
• Persistent-frequent (4.0)
• Persistently and frequently until late into
  period. Appears to be turning into lower
  frequency wetting however over 80 are still
  wetting to some degree at 9.5yr
• Delayed-frequent (2.7)
• Frequent wetting until half-way through time
  period, reducing to a lower level of wetting
  which appears to be clearing up by 9.5yr

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Fit statistics Girls Oh!
15
Fit statistics Girls Oh!
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6-class model (girls)
Any bedwetting
Frequent bedwetting
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6-class model (girls)

• Normative (78.6)
• Delayed-infrequent (11.7)
• Persistent-infrequent (4.6)
• Persistent-frequent (1.6)
• Delayed-frequent (1.3)
• Relapse (2.0)
• Initial period of dryness followed by a return to
  infrequent wetting

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Incorporating covariates

• 2-stage method
• Export class probabilities to another package
  Stata
• Model class membership as a multinomial model
  with probability weighting
• Using classes derived from repeated BW measures
  with partially missing data (gloss over)

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Multinomial models (boys)

• label values class class_label
• label define class_label ///
• 1 "Pers INF 1" ///
• 2 "DelayFRQ 2" ///
• 3 "Normal 3" ///
• 4 "Pers FRQ 4" ///
• 5 "DelayINF 5", add
• tab class
• foreach var of varlist bedwet_m bedwet_p
  toilet
• tab var' if class1
• xi mlogit class var' iw boy_weights, rrr
• test var'

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Typical output

• Multinomial logistic regression
  Number of obs 5004
• 
  LR chi2(4) 85.09
• 
  Prob gt chi2 0.0000
• Log likelihood -5256.9295
  Pseudo R2 0.0080
• --------------------------------------------------
  ----------------------------
• class RRR Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• Pers INF 1
• bedwet_m 2.456404 .3459902 6.38
  0.000 1.863829 3.237378
• -------------------------------------------------
  ----------------------------
• DelayFRQ 2
• bedwet_m 3.019243 .6958002 4.79
  0.000 1.921915 4.743095
• -------------------------------------------------
  ----------------------------
• Pers FRQ 4
• bedwet_m 3.795014 .6783016 7.46
  0.000 2.673461 5.387073
• -------------------------------------------------
  ----------------------------
• DelayINF 5
• bedwet_m 1.540813 .2046864 3.25
  0.001 1.18761 1.999062

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Selection of covariates (boys)
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What if we had used modal-class?

• The further the posterior probabilities for class
  assignment are from 1 (i.e. the lower the
  entropy) the poorer the estimates from a model
  using the modal class
• In this example (partial missing data)
• entropy 0.788
• 1 2 3 4 5
• 1 0.800 0.011 0.017 0.020 0.153
• 2 0.063 0.804 0.000 0.069 0.064
• 3 0.008 0.001 0.923 0.000 0.068
• 4 0.049 0.104 0.004 0.813 0.030
• 5 0.110 0.009 0.170 0.006 0.706

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Estimates using mod class
Bias depends on class and also covariate
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A later outcome

• Boys data
• Outcomes
• Key Stage 3 at 13-14 yrs
• Achieved level 5 or greater in English/Sci/Maths
• English failed 1210 (27.9)
• Science failed 895 (20.6)
• Maths failed 845 (19.5)

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2 stage procedure - Stata

• label values class class_label
• label define class_label ///
• 1 "Pers INF 1" ///
• 2 "DelayFRQ 2" ///
• 3 "Normal 3" ///
• 4 "Pers FRQ 4" ///
• 5 "DelayINF 5", add
• recode class 30
• foreach var of varlist maths science english
• tab var' if class1
• xi logit var' i.class iw b_par_p, or
• test _Iclass_2 _Iclass_3 _Iclass_4 _Iclass_5

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KS3 - English

• Logistic regression
  Number of obs 4341
• 
  LR chi2(4) 7.73
• 
  Prob gt chi2 0.1018
• Log likelihood -2564.536
  Pseudo R2 0.0015
• --------------------------------------------------
  ----------------------------
• k3_lev5e Odds Ratio Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• Pers INF .9896705 .1131557 -0.09
  0.928 .7909827 1.238267
• Delay FRQ .8946308 .1964416 -0.51
  0.612 .5817527 1.375781
• Pers FRQ 1.474409 .2323757 2.46
  0.014 1.082589 2.008041
• Delay INF .9135896 .0852754 -0.97
  0.333 .76085 1.096991
• --------------------------------------------------
  ----------------------------
• ( 1) _Iclass_1 0
• ( 2) _Iclass_2 0
• ( 3) _Iclass_4 0
• ( 4) _Iclass_5 0

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KS3 - Maths

• Logistic regression
  Number of obs 4341
• 
  LR chi2(4) 11.29
• 
  Prob gt chi2 0.0235
• Log likelihood -2133.6412
  Pseudo R2 0.0026
• --------------------------------------------------
  ----------------------------
• k3_lev5m Odds Ratio Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• Pers INF .9460608 .1241854 -0.42
  0.673 .7314512 1.223637
• Delay FRQ .9601245 .2358923 -0.17
  0.868 .5931937 1.554027
• Pers FRQ 1.688991 .2836716 3.12
  0.002 1.215249 2.347413
• Delay INF .8976914 .095965 -1.01
  0.313 .7280008 1.106935
• --------------------------------------------------
  ----------------------------
• ( 1) _Iclass_1 0
• ( 2) _Iclass_2 0
• ( 3) _Iclass_4 0
• ( 4) _Iclass_5 0

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KS3 - Science

• Logistic regression
  Number of obs 4341
• 
  LR chi2(4) 8.94
• 
  Prob gt chi2 0.0626
• Log likelihood -2203.9946
  Pseudo R2 0.0020
• --------------------------------------------------
  ----------------------------
• k3_lev5s Odds Ratio Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• Pers INF .8913295 .1159146 -0.88
  0.376 .6907838 1.150097
• Delay FRQ 1.067237 .2482602 0.28
  0.780 .6764799 1.683709
• Pers FRQ 1.525329 .2566845 2.51
  0.012 1.096786 2.121314
• Delay INF .8888344 .092835 -1.13
  0.259 .7242967 1.09075
• --------------------------------------------------
  ----------------------------
• ( 1) _Iclass_1 0
• ( 2) _Iclass_2 0
• ( 3) _Iclass_4 0
• ( 4) _Iclass_5 0

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Summary

• Fitting ordinal models is similar to binary data
  however results (trajectories) are harder to
  interpret graphically
• Resulting classes can be used either as outcomes
  or categorical predictors using weighted
  regression in Stata
• Using variables derived from modal class
  assignment can often introduce very biased
  estimates