1. 1 BAMS 580B Lecture 2 Part 1 – LTC Planning

2. 2 Topics  LTC Capacity Planning  Objectives  Approaches LBH Deterministic Model – Parameter Estimation Simulation Model – Concept – Data – Optimization  Comparisons  Queuing Models and Capacity Planning  What they are  Why use them?

3. 3 LBH Planning Case

4. 4 Simulation Based Planning and Survival Analysis

5. 5 Overview Overview  Goal: Develop a model to support long term care capacity planning decisions  Model must forecast the annual bed requirements 2020 Regional level Facility level  Must allow sensitivity and “What if?” analysis  This is a fundamental planning problem faced by all health system planners  Standard approach – Ratio based planning  Ratios of population 75 and older  Usually between 75-90 beds per 1000 aged 75 or older  Our approach – Service criteria based planning  Methods -simulation model, survival analysis, goal seeking  Determine capacity levels to meet a standard For example 85% of clients wait less than 30 days for admission

6. 6 Model Overview Tradeoff – excess capacity vs. long waits

7. 7 Model Inputs  Demographics from BC Stats projections  Arrival rate by age and gender in each LHA  Historical length of stay by age and gender  In 2003 a significant change was made to admissions criteria for complex care that allowed only clients of higher acuity into care  This causes complications in models because we need different LOS models for pre-2003 clients.

8. 8 Simulation Logic  Preload clients at start of planning horizon  Sample appropriate remaining lifetime distributions  Generate a case from the appropriate inter-arrival time distribution  Allocate age and gender proportionally  Generate LOS from appropriate distribution  Adjust LOS if desired  Enter case into queue  When case exits queue:  Record time in queue  Record if service criterion has been met  Occupy “bed” for determined LOS  Leave  At the end of each year of simulation time:  Calculate the percentage of people served within the criteria and record

9. 9 Simulation Logic Schematic Clients enter queue and then enter care Clients exit care Create pre- load clients and waitlist clients Choose LOS Create new clients Choose LOSAdjust LOS Survival curves Adjustment factors from Excel Clients loaded before simulation starts Clients created as simulation progresses Model operation and statistic collection Pop’n estimates and rates

10. 10 Arrival Rates  Usually expressed as a rate per 1000 in a particular age and gender group  Relevant data may not be available!  In LBH setting, it is difficult to determine true arrival rate since arrivals are triggered by departures and so pure arrival process is not visible.  At VIHA we could only obtain a snapshot of the arrival list at a date.  We can do the best we can and then use sensitivity analysis to measure impact of arrival rate assumptions on capacity.

11. 11 Analyzing Length of Stay  A key driver in capacity planning  Data is censored; many clients remain in the system at the end of the data period  Ignoring censored clients seriously biases the estimates for LOS  Censored cases tend to be those with long lengths of stay  Survival analysis takes into account clients still in the system when fitting LOS distributions  A statistical technique for estimating LOS distributions accounting for censored data.  We will need whole distribution to generate LOS in simulation model.  Fit parametric models stratified by region with age group and gender as covariates (Weibull).

12. 12  To examine the relationship between LOS and the age at admission  : random error with normal distribution  : regression coefficients, to be estimated from the data  Data: All discharges from LB Home for the Aged – 1978 to 2008 Why not linear regression? IDResidentGenderBirth DateAdmissionDischargeStatus 1**** Male03-13-3009-24-8411-13-99DECEASED 2**** Female01-31-4305-21-9210-31-08Active

13. 13 NCSS Output

14. 14 NCSS Output

15. 15 Why Survival Analysis  Linear regression is problematic because data is skewed and censored  Survival analysis takes into account clients still in the system when fitting LOS distributions  Parametric models provide the “whole distribution” so that we can generate LOS in the simulation model  We use models with age group, gender and region as covariates (or strata)  Questions Which models? Interpretation?

16. 16 Sample Data and Censoring Nov-03 Jan-04 Mar-04 May-04 Jul-04 Sep-04 Nov-04 Jan-05 Mar-05 May-05 Jul-05 Sep-05 Nov-05 Jan-06 Mar-06 May-06 Jul-06 Sep-06 Nov-06 Jan-07 Mar-07 Calendar Time Clients

17. 17 Kaplan-Meier Curves

18. 18 Why does this matter? Length of Stay (years) 1.00 0.75 1.5 2.0 2.5 3.03.54.0 0.23 1.18 0.50 0.25 0.00 0.0 0.5 1.0 Median Uncensored Censored Probability of Survival

19. 19 Survival Distributions  In order to simulate LOS, a distribution is required  Several distributions are commonly used in survival analysis:  Weibull  Exponential – a special case of Weibull  Gompertz, log-normal, log-logistic  Weibull is most common & was used for our simulations  Two parameters required:  Shape, α  Scale, β

20. 20 Weibull Distribution  PDF and CDF  Two parameters  Shape:  Scale:

21. 21 Various Weibull Distributions

22. 22 Fitting Parameters  Finding a suitable model involves regression  Ordinary regression problematic Length of stay times are not normally distributed Data has large percentage of right censoring  Models are fit by maximizing the likelihood function  When censoring exists this becomes the product of the likelihood for each type of data (censored & uncensored)  Requires analyst involvement!

23. 23 Type III Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq Agroup 4 33.9101 <.0001 Ggroup 1 156.4401 <.0001 LHA 11 66.7901 <.0001 Analysis of Parameter Estimates Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept 1 5.4206 0.3097 4.8136 6.0275 306.42 <.0001 Agroup 0 1 0.0530 0.1819 -0.3035 0.4096 0.08 0.7706 Agroup 1 1 0.1909 0.1351 -0.0739 0.4558 2.00 0.1576 Agroup 2 1 0.2362 0.0837 0.0721 0.4002 7.96 0.0048 Agroup 3 1 0.2822 0.0503 0.1837 0.3807 31.50 <.0001 Agroup 4 0 0.0000..... Ggroup 0 1 0.5936 0.0475 0.5006 0.6866 156.44 <.0001 Ggroup 1 0 0.0000..... LHA 061 1 0.6501 0.3090 0.0444 1.2558 4.43 0.0354 LHA 062 1 0.7035 0.3283 0.0601 1.3469 4.59 0.0321 LHA 063 1 0.9557 0.3161 0.3362 1.5752 9.14 0.0025 LHA 064 1 0.2955 0.3415 -0.3738 0.9648 0.75 0.3868 LHA 065 1 0.2329 0.3194 -0.3930 0.8588 0.53 0.4658 LHA 067 1 0.2346 0.3262 -0.4047 0.8740 0.52 0.4720 LHA 068 1 0.6631 0.3137 0.0483 1.2780 4.47 0.0345 LHA 069 1 0.6593 0.3165 0.0391 1.2795 4.34 0.0372 LHA 070 1 0.6176 0.3271 -0.0234 1.2587 3.57 0.0590 LHA 071 1 0.5475 0.3174 -0.0746 1.1697 2.98 0.0846 LHA 072 1 0.3302 0.3281 -0.3128 0.9733 1.01 0.3141 LHA 085 0 0.0000..... Scale 1 1.4992 0.0193 1.4618 1.5375 Weibull Shape 1 0.6670 0.0086 0.6504 0.6841 SAS Output

24. 24 Interpretation of coefficients  For example, the estimated parameters for males in LHA061 who are 75-84 years old would be determined as follows:

25. 25 More on coefficient interpretation  A female of the same age and in the same location as a male will have a mean time in long term care that is exp(0.59) = 1.80 times greater than that of a male 25

26. 26 Using Simulation to Determine Capacities  A simulation optimization approach is adopted  Capacities are determined by iteratively running the simulation and adjusting resource levels  Stopping conditions are determined by the service criteria  The service criteria we used was that 85% of clients are placed within 30 days.

27. 27 Bisection Search 0 Service Level 100% 85% # Beds Upper Bound: Lower Bound: 1000 0 # Beds to choose: 500 1000 500 750 500 750 625 750 625

28. 28 Search Simultaneous Search 0 Service Level 100% 85% Year 2009 201020112012 2013 201420152016 2017 201820192020

29. 29 Some Plans 200920102011201220132014201520162017201820192020 Year Resource Size Base case LOS increased LOS decreased Arrival rate increased LOS down, arrival rate up Beds

30. 30 Comparison to Ratio Based Approach in two regions

31. 31 Comparison of Service Based Approach to Ratio Approach: two metrics

32. 32 Comparison of Simulation Approach to LBH Approach

33. 33 Comparison to other methods

34. 34 Some Observations  These are important and costly decisions  In depth analysis is required  Ratio based plans and service base plans differ  Improved ratios do not give reliable service levels  We recommend using simulation optimization to determine “how many beds”.  Managers should not relax acuity standards if there is excess capacity  Will extend LOS and invalidate planning assumptions  Capacity is usually added in discrete blocks which necessitates some further analyses