Health service researchers frequently study length of hospital stay (LOS) as a health outcome. Generally originating from heavily skewed distributions, LOS data can be difficult to model with a single parametric model. Mixture models can be quite effective in dealing with such data. This example illustrates how to perform a Bayesian analysis of an exponential mixture model for LOS data. The experimental MCMC procedure is used for this analysis.
The LOS data analyzed in this example originate from geriatric patients in a psychiatric hospital in North East London in 1991 and were studied by Harrison and Millard ( 1991 ) and McClean and Millard ( 1993 ) . Each observation represents the LOS in days for an admitted patient.
data inputdata; input los @@; datalines; 1671 1300 722 586 552 525 364 359 321 302 272 248 226 216 208 182 141 141 132 120 117 115 114 113 104 103 101 99 96 94 93 92 88 84 83 81 79 74 70 63 62 62 61 56 55 53 53 51 51 50 49 36 33 33 33 29 28 26 24 19 16 16 15 ... more lines ... 35 28 20 19 9 5 2 2 2 22317 14006 11549 11006 8981 8402 7947 7266 6693 4408 4010 4003 1970 1857 1849 1833 1770 1769 1514 1217 956 924 611 386 280 93 ;
Before using this data in a mixture model setting, use the MEANS procedure to obtain summary statistics for the LOS data in table form. Figure 1 displays summary statistics for this data.
Analysis Variable : los | ||||
---|---|---|---|---|
N | Mean | Std Dev | Minimum | Maximum |
469 | 3712.36 | 5675.97 | 1.0000000 | 24028.00 |
Figure 2 displays a plot that illustrates the skewness of the data. The histogram of the data is overlaid with a scaled exponential density, and you can see that a single exponential density does not fit the lower values of LOS well.
Mixture models arise naturally when one mechanism generates data according to one model and another mechanism generates data according to a different model. In this data set, the variable that indicates which mechanism generated an observation is not recorded, and only the response variable is available. A mixture model is useful for analyzing this data set because it unveils the latent heterogeneity that arises from a latent categorical variable (Fruhwirth-Schnatter; 2006 ) .
Mixture models can be expressed as a weighted average of component densities. More formally, is said to arise from a finite mixture model with probability density function where
For all , is the component probability density function, and are the weights defined by the following constraints: and .
Congdon ( 2003 ) states that exponential mixture models with relatively small number of components are effective in modeling skewed LOS data. You can write a two-component exponential mixture model for LOS data with density as follows:
for patients .
There are three parameters in the density: , and . Researchers Harrison and Millard ( 1991 ) suggest the mean parameters, and , as the average LOS for the standard- and long-stay groups, respectively. In addition, represents an unknown fraction of patients in the standard-stay group with .
Suppose the following prior distributions are placed on the three parameters:
where indicates a prior distribution and is the density function for the inverse-gamma distribution. Priors of this type are often called diffuse priors . The prior expresses your lack of knowledge about the mixture proportion.
Label-switching is a common problem that arises in mixture models. It was described by as a result of the invariance of the mixture likelihood function under the relabeling of the mixture components. In an effort to remove label-switching, Harrison and Millard ( 1991 ) place an identifiability constraint that the average LOS of the standard-stay group is less than that of the long-stay group; that is, .
Using Bayes’ theorem, the likelihood function and prior distributions determine the posterior distribution of , and as follows:
PROC MCMC obtains samples from the desired posterior distribution, which is determined by the prior and likelihood specified. It does not require the form of the posterior distribution.
The following SAS statements use the prior distributions to fit the Bayesian exponential mixture model. The PROC MCMC statement invokes the procedure and specifies the input data set. The NMC= option specifies the number of posterior simulation iterations. The THIN=5 option specifies that one of every five samples is kept. The PROPCOV=QUANEW option uses the estimated inverse Hessian matrix as the initial proposal covariance matrix.
ods graphics on; proc mcmc data=inputdata seed=1010 nmc=50000 thin=5 propcov=quanew; ods output PostSummaries = post_summ; parms B 100 D 6000 pi 0.5; prior B D ~ igamma(3/10, scale = 10/3); prior pi ~ uniform(0,1); if (B < D) then llike = log(pi*pdf("expo", los, B) + (1-pi)*pdf("expo", los, D)); else llike = .; model general(llike); run; ods graphics off;
The ODS OUTPUT statement creates an output data set post_summ used later for a graphical analysis of the fit in Figure 8 . The PARMS statement puts all three parameters , , and in a single block and assigns initial values to each of them. The PRIOR statements specify priors for all the parameters. Note that and can be specified with one PRIOR statement because they have the same prior distribution.
The IF-ELSE statements enable different values of LOS to have different log-likelihood functions, depending on whether the order constraint placed on the mean parameters is satisfied. The MODEL statement specifies that llike is the log likelihood for each observation in the model and is simply a missing value when the order constraint is not met.
By turning ODS Graphics on, PROC MCMC produces graphs at the end of the procedure which enable you to visually examine the convergence of the chain. See Figure 3 . Inferences should not be made if the Markov chain has not converged.
Figure 3 displays convergence diagnostic graphs for parameters , , and . The trace plots indicate that the chains appear to have reached stationary distributions. The chain also has good mixing and is dense.
The autocorrelation plots indicate low autocorrelation and efficient sampling. Finally, the kernel density plots show the smooth, unimodal shape of posterior marginal distributions for each parameter. If label-switching had occured, you would see jumps in the trace plots or multimodal density plots.
Figure 4 contains the "Parameters" table which lists the names of the parameters, the blocking information, the sampling method used, the starting values, and the prior distributions.
Parameters | |||
---|---|---|---|
Parameter |
Sampling
Method |
Initial
Value |
Prior Distribution |
B | N-Metropolis | 100.0 | igamma(3/10, scale = 10/3) |
D | N-Metropolis | 6000.0 | igamma(3/10, scale = 10/3) |
pi | N-Metropolis | 0.5000 | uniform(0,1) |
The "Tuning History" table, shown in Figure 5 , displays how the tuning stage progresses for the multivariate random walk Metropolis algorithm used by PROC MCMC to generate samples from the posterior distribution. An important aspect of the algorithm is the calibration of the proposal distribution. The tuning of the Markov chain is broken into a number of phases. In each phase, PROC MCMC generates trial samples and automatically modifies the proposal distribution as a result of the acceptance rate.
The "Burn-In History" and the "Sampling History" tables show the burn-in and main phase sampling, respectively.
Tuning History | |||
---|---|---|---|
Phase | Block | Scale |
Acceptance
Rate |
1 | 1 | 2.3800 | 0.0280 |
2 | 1 | 1.0123 | 0.0700 |
3 | 1 | 0.5221 | 0.3420 |
Burn-In History | ||
---|---|---|
Block | Scale |
Acceptance
Rate |
1 | 0.5221 | 0.3980 |
Sampling History | ||
---|---|---|
Block | Scale |
Acceptance
Rate |
1 | 0.5221 | 0.3811 |
Figure 6 displays summary and interval statistics for each posterior distribution.
Posterior Summaries | ||||||
---|---|---|---|---|---|---|
Parameter | N | Mean |
Standard
Deviation |
Percentiles | ||
25% | 50% | 75% | ||||
B | 10000 | 619.1 | 74.2261 | 568.9 | 618.3 | 668.3 |
D | 10000 | 7752.6 | 721.2 | 7267.1 | 7710.4 | 8202.1 |
pi | 10000 | 0.5638 | 0.0403 | 0.5370 | 0.5655 | 0.5916 |
Posterior Intervals | |||||
---|---|---|---|---|---|
Parameter | Alpha | Equal-Tail Interval | HPD Interval | ||
B | 0.050 | 473.6 | 767.0 | 472.0 | 763.7 |
D | 0.050 | 6418.1 | 9288.4 | 6339.5 | 9195.0 |
pi | 0.050 | 0.4826 | 0.6389 | 0.4825 | 0.6387 |
Figure 7 reports a number of convergence diagnostics to assist in determining convergence. These are the Monte Carlo standard errors, the autocorrelations at selected lags, the Geweke diagnostics, and the effective sample sizes.
Monte Carlo Standard Errors | |||
---|---|---|---|
Parameter | MCSE |
Standard
Deviation |
MCSE/SD |
B | 3.9598 | 74.2261 | 0.0533 |
D | 38.2006 | 721.2 | 0.0530 |
pi | 0.00164 | 0.0403 | 0.0408 |
Posterior Autocorrelations | ||||
---|---|---|---|---|
Parameter | Lag 1 | Lag 5 | Lag 10 | Lag 50 |
B | 0.9150 | 0.6588 | 0.4459 | 0.0689 |
D | 0.9220 | 0.6766 | 0.4760 | 0.0354 |
pi | 0.5086 | 0.3438 | 0.2564 | 0.0267 |
Geweke Diagnostics | ||
---|---|---|
Parameter | z | Pr > |z| |
B | 0.1718 | 0.8636 |
D | -0.2669 | 0.7895 |
pi | -0.3073 | 0.7586 |
Effective Sample Sizes | |||
---|---|---|---|
Parameter | ESS |
Correlation
Time |
Efficiency |
B | 351.4 | 28.4595 | 0.0351 |
D | 356.4 | 28.0580 | 0.0356 |
pi | 601.9 | 16.6129 | 0.0602 |
Figure 6 displays the marginal posterior summaries. The larger group of standard-stay patients has an average LOS of days relative to the smaller group of long-stay patients whose stay averages days. The posterior average mixture proportions are and for the standard-stay and long-stay group, respectively.
Figure 8 illustrates the scaled mixture of exponential density and the gain in model fit from the two-component exponential mixture model.
The single-component exponential model had an average LOS of 3712 days. The mean parameters found when fitting an exponential mixture model to the standard-stay group and the long-stay group are 619 and 7752 days, respectively. The mixture distribution fits the data better than the exponential distribution, especially at the low values of LOS. Additional components could be fit and evaluated in similar methods or with information criteria.
Congdon, P. (2003), Applied Bayesian Modeling , John Wiley & Sons.
Fruhwirth-Schnatter, S. (2006), Finite Mixture and Markov Switching Models , Springer.
Harrison, G. and Millard, P. (1991), “Balancing Acute and Long-Term Care: The Mathematics of Throughput in Departments of Geriatric Medicine,” Meth. Infor. Medicine , 30, 221–228.
McClean, S. and Millard, P. (1993), “Patterns of Length of Stay after Admission in Geriatric Medicine: An Event History Approach,” The Statistician , 42(3), 263–274.