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BAMS 580B Lecture 2 Part 1 – LTC Planning

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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?

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LBH Planning Case

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Simulation Based Planning and Survival

Analysis

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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

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Model Overview

Tradeoff – excess capacity vs. long waits

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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.

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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

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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 LOS Adjust LOS

Survival curves

Adjustment factors from

Excel

Clients loaded before simulation starts

Clients created as simulation progresses

Model operation and statistic collection

Pop’n estimatesand rates

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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.

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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).

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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?

AgeLOS 10

10 ,

ID ResidentGende

rBirth Date

Admission

Discharge Status

1 **** **** Male 03-13-30 09-24-84 11-13-99DECEASED

2 **** **** Female 01-31-43 05-21-92 10-31-08Active

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NCSS Output

-2000.0

0.0

2000.0

4000.0

6000.0

-4.0 -2.0 0.0 2.0 4.0

Normal Probability Plot of Residuals of LOS

Expected Normals

Re

sid

ua

ls o

f L

OS

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NCSS Output

0.0

2000.0

4000.0

6000.0

8000.0

50.0 65.0 80.0 95.0 110.0

LOS vs Age

Age

LO

S

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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?

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Sample Data and Censoring

Nov-0

3

Jan-

04

Mar

-04

May

-04

Jul-0

4

Sep-0

4

Nov-0

4

Jan-

05

Mar

-05

May

-05

Jul-0

5

Sep-0

5

Nov-0

5

Jan-

06

Mar

-06

May

-06

Jul-0

6

Sep-0

6

Nov-0

6

Jan-

07

Mar

-07

Calendar Time

C

lient

s

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Kaplan-Meier Curves

Age_ Gr oup=85+

0. 00

0. 25

0. 50

0. 75

1. 00

LOS_ year s

0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 3. 5 4. 0

STRATA: Gender =F Censor ed Gender =F Gender =M Censor ed Gender =M

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Why does this matter?

Length of Stay (years)

1.00

0.75

1.5 2.0 2.5 3.0 3.5 4.00.23 1.18

0.50

0.25

0.00

0.0 0.5 1.0

Median

Uncensored

CensoredProbability of Survival

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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

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Weibull Distribution PDF and CDF

Two parameters Shape: Scale:

t

t

etF

et

tf

1

1

)!1()(

)]1

1()2

1([dev std

)1

1(mean

0

1

22

xordtetx tx

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Various Weibull Distributions

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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!

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

SAS Output

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Interpretation of coefficients

39.76433.1*73.574)5.1(1*574.73 mean

0.67

1/1.50

1/SCALE

574.13

exp(6.35)

0.65) 0 0.28 exp(5.43

LHA061) Ggroup1 Agroup3 ept exp(Interc

For example, the estimated parameters for males in LHA061 who are 75-84 years old would be determined as follows:

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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

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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.

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Bisection Search

0

Ser

vice

Lev

el

100%

85%

# Beds

Upper Bound:

Lower Bound:

1000 0

# Beds to choose: 500

1000 500

750 500 750

625

750 625

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Simultaneous Search

0

Ser

vice

Lev

el

100%

85%

Year

2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

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Some Plans

2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

Year

Res

ourc

e S

ize Base case

LOS increased

LOS decreased

Arrival rate increased

LOS down, arrival rate up

Be

ds

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Comparison to Ratio Based Approachin two regions

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Comparison of Service Based Approach to Ratio Approach: two metrics

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Comparison of Simulation Approach to LBH Approach

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Comparison to other methods

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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