Risk Adjusted Xbar Chart

1
Risk Adjusted X-bar Chart
Based on Work of Eric Eisenstein and Charles
Bethea, The use of patient mix-adjusted control
charts to compare in hospital costs of care
Health Care Management Science, 2 (1999), 193-198

• Farrokh Alemi, Ph.D.

2
Why Chart Data?

• To discipline intuitions
• To communicate data in vivid graphical ways

Decision makers often attribute positive outcomes
to their own skills and negative outcomes to
others, while in reality both could be due to
chance
3
Purpose of Risk Adjustment
4
Data Needed

• Data collected over time
• Risk (expected outcomes) for each patient
• Outcomes for each patient measured as a
  continuous variable

The purpose is to improve not to get so lost in
measurement to loose sight of improvement.
5
What Is Risk?

• A patients condition or characteristics that
  affects the expected outcomes for the patient
• A severity index used to predict patient outcomes
• Clinicians consensus regarding expected outcomes
• Patients self rating of expected outcomes

6
Example Observed Expected Costs
7
Elements of a Control Chart

• X axis shows time
• Y axis shows average cost (or dependent variable
  of interest)
• Observed rates are plotted against time sequence
• Upper or lower control limit are drawn so that
  points 95 or 99 of data should fall within
  these limits

8
Steps in Creating X-bar Chart

• Check assumptions
• Calculate average costs and plot them
• Calculate average expected costs
• Calculate standard deviation of difference of
  observed and expected cost
• Calculate control limits and plot them
• Interpret findings
• Distribute chart

9
Step One Check Assumptions

• We are examining continuous variables measured on
  a ratio or interval scale, e.g. cost,
  satisfaction ratings, blood pressure, etc.
• Observations are independent from each other.
  This assumption is violated if current
  observations can accurately predict future
  values.
• More than 5 observations for each time period.

10
Check Normal Distribution

• Histogram the observed costs
• Eyeball test Is the shape bell shaped curve
  with most data in the middle and little data in
  both tails
• For more precise verification of assumption you
  can do statistical tests of Normal distribution

11
Check Equality of Variance

• Eyeball test Accept the assumption if ranges
  are within the same ball park (No range several
  multiple of the other ranges)
• For more precise test of the assumption you can
  do statistical test of equality of variances

12
Step 2 Calculate Average Cost

• Cij Cost of case i in time period j
• nj Number of cases in time period j
• Cj Average cost for time period j ? i1,
  nj Cij / nj

Plot of average costs
13
Plot of the Observed Rates

• A graph helps us see possible relationships.
  Maybe August was a low cost month.
• Wait, until you see control limits of what could
  have been expected.

14
Step 3 Average Expected Costs

• Eij Expected cost of case i in time period
  j
• Ej Expected cost for time period j
• Ej (?i1,,nj Eij ) / nj

15
Plot Expected Costs

• Plotting expected cost helps interpret the
  observed costs but does not settle the question
  of whether differences are due to chance.

16
Step 4 Standard Deviation of Differences

• Dij Difference of observed and expected cost of
  case i in time period j
• D Average difference of observed and expected
  cost for all cases in all time periods
• S Standard deviation of differences
• S ?i1,,nj j 1, m (Dij -D )2 / (n-1)0.5
• Sj Standard deviation of differences for time
  period j
• Sj S/(nj)0.5

See sample calculation
17
Standard Deviation of Difference
A. Calculate differences for each case
B. Calculate standard deviation of differences
C. Calculate standard deviation of differences in
each time period
18
Step 5 Calculate Limits

• UCLj Upper control limit for time period j
• LCLj Lower control limit for time period j
• UCLj Ej t Sj
• LCLj Ej - t Sj
• t Constant based on t-student distribution

19
T- Values
20
Control Limits for First Period

• UCL1 335.81 3.2 20.8
• LCL1 335.81 3.2 20.8

t-value
Negative limits are set to zero as negative
costs are not possible
21
Control Limits for All Time Periods
22
Control Chart
23
Step 6 Interpret Findings

• Two points are outside limits.
• In these months, costs were different from what
  could be expected from patients severity of
  illness.

24
Step 7 Distribute Control Chart

• Include in the information
• How was severity measured and expected costs
  anticipated?
• Why are assumptions met?
• What does the control chart look like?
• What is the interpretation of the findings?

25
Summary of Steps

• Check assumptions
• Calculate and plot observed cost
• Calculate expected cost
• Calculate standard deviation of differences
• Calculate and plot control limits
• Interpret findings
• Distribute control chart