236 ◾ Roque Perez-Velez
where the engineer ran 182 weekly replications with 1,274 data points. Figure27.4 depicts box
plots of daily census by day of week.
Sanderson
*
explains that “the ABC Analysis Technique is a widely used tool that manage-
ment uses to categorize materials and components into workable groups. e ABC Analysis is the
application of Pareto’s principle of analysis and segregation to the inventory investment.” Pareto’s
analysis is based on the premise that 80% of investment is concentrated in group A while the other
20% is distributed among groups B and C. is same technique is useful when analyzing data in
a healthcare setting.
*
G. A. Sanderson, “Inventory Control Records and Practices,” in Production and Inventory Control Handbook,
ed. J. H. Greene (New York: McGraw Hill Companies, Inc., 1997).
Sundays
Mondays
Tuesdays
Wednesdays
ursdays
Fridays
Saturdays
Births by Day of Week
17%
14%
17%
14%
13%
11%
14%
Figure 27.2 Pie chart: births by day of week.
Table27.3 Time per
Transaction (in minutes)
Minutes Transactions
1 472
2 346
3 85
4 36
5 7
6 4
7 1
8 1
9 0
10 0
Over 10 0
Statistical and Mathematical Analysis in a Healthcare Setting ◾ 237
500
450
400
350
300
250
200
150
100
50
0
Number of Transactions
Time (minutes)
Over
10
10987654321
Pneumatic Tube System’s Time per Transaction
Figure 27.3 Bar chart.
40
35
30
25
20
15
10
5
0
Number of Patients
Unit Weekly Census
SaturdayFridayursdayWednesdayTuesdayMondaySunday
Figure 27.4 Box plot graph.
238 ◾ Roque Perez-Velez
For example, an engineer wants to determine which radiological procedures are requested
most. Table27.4 shows 16 of the 43 most common radiological procedures, sorted in descending
order, and their respective cumulative percentage. e engineers used this large data set to create
the Pareto graph shown in Figure27.5.
Data mining is the process of handling, managing, and analyzing extremely large sets of data.
Due to the complexity of working with huge or extremely large data sets, such data sets must usu-
ally be sampled. Take, for example, a bed control manager wants to better prognosticate how bed
usage is aected by patients that are transferred into his institution. e manager may want to
look at daily transfer reports. It may become cumbersome and dicult to visualize daily transfers
on a graph. Perhaps the manager may sample weekly transfers and combine a box plot and linear
graphs to gain the desired eect.
By estimating the quartiles for weekly transfers and then plotting the box plots in a linear
graph, the manager can better prognosticate bed usage as seen in Figure27.6.
On this graph, each bicolor bar depicts the rst quartile, median, and third quartile for each
week. ese are plotted on a weekly basis in a linear pattern that shows a slight increase.
A word of caution: outliers. ese are noticeably unusual values. A data point is considered
an outlier if it is more than 1.5 times the interquartile range away from the nearest quartile. e
reader must detect outliers, but there are no general rules on how outliers should be handled once
Table27.4 Radiology Procedures
Procedure
Number Procedure Name Total Percentage
Cumulative
%
1060 HEAD W/O CNTRST 750 24.1% 24.1
1005 ABD W/CNTRST 492 15.8% 40.0
1225 PELVIS W/CNTRST 492 15.8% 55.8
2000 ABD 1 VIEW 231 7.4% 63.3
1055 CHEST W/CNTRST 185 6.0% 69.2
1000 ABD W/O CNTRST 114 3.7% 72.9
1045 CHEST W/O CNTRST 114 3.7% 76.6
1410 MULTI-PLANAR REFORMATIONS 107 3.4% 80.0
1056 CHEST W/CNTRST EXT 105 3.4% 83.4
1002 RENAL STONE W/O CNTRST 102 3.3% 86.7
1215 PELVIS W/O CNTRST 102 3.3% 90.0
1080 HEAD W/&W/O CNTRST 42 1.4% 91.3
1085 MXFACE 1 PJ W/O CNTRST 39 1.3% 92.6
1025 BIOPSY 30-60 MINUTES 30 1.0% 93.5
1015 ABD W&W/O CNTRST 24 0.8% 94.3
1255 C SPINE W/O CNTRST 21 0.7% 95.0
Statistical and Mathematical Analysis in a Healthcare Setting ◾ 239
50
45
40
35
30
25
20
15
10
5
0
36
38
40
42
44
46
48
50
52
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
2
4
6
Week Number
Transfers
Hospital-to-Hospital Transfers
Figure 27.6 Data mining: box plot and linear graph.
3000
2500
2000
1500
1000
500
0
100.0%
90.0%
80.0%
70.0%
60.0%
50.0%
40.0%
30.0%
20.0%
10.0%
0.0%
Procedures
Procedure Number
Pareto Chart
1060 1005 1225 2000 1055 1000 1045 1410 1056 Other
24.1%
750
492 492
231
185
114 114 107 105
516
40.0%
55.8%
63.3%
69.2%
72.9%
76.6%
80.0%
83.4%
Figure 27.5 Pareto graph.
240 ◾ Roque Perez-Velez
they have been detected. One thing is certain: we do not want to automatically reject them. We
want to understand if there is an acceptable explanation for why these values dier from the rest
of the data. Outliers often lead us to further study, analyze, and operationalize the data.
For example, a management engineer is studying the nurse triage process at the emergency
medicine department. Gathering observations from the electronic medical records (EMR) data-
base, the engineer found the data excerpt depicted in Table27.5.
Are there any outliers? If so, is there an acceptable explanation for these? Finally, what can be
done? Should the engineer keep or discard these outliers?
In order to answer these questions, we need to look at the data in context. Let’s look at the tri-
age process. e patient arrives at the triage area; the nurse proceeds to ask the patient a minimum
of six questions:
◾ Name
◾ Social Security number or medical record number
◾ Date of birth
◾ Sex
◾ Allergies (Yes/No)
◾ Chief complaint
e engineer must assess how long this can take. Can it take less than four minutes? How
about any longer than 8 minutes? Yes, there are 5 possible observations that may be outliers. Some
of these can be explained: for the high-end outlier, perhaps the nurse forgot to close out the triage
on the tracking system. As for the lower-end outliers, perhaps the patient did not have triage per-
formed and rather went straight to the next step. ese will need to be further analyzed.
Mathematical Analysis
Wainer
*
denes a Markov chain as “a discrete-time stochastic model described using a graph. One
important property of Markov chains is that they are memory-less; thus, no state has a cause–
eect relationship with the previous state. erefore, knowledge of previous states is irrelevant for
predicting the probability of the future states.” Tuery
†
mentions that “since 1962, Markov chains
have been used experimentally for modeling the behavior of customers on the basis of their previ-
ous behavior.” Markov chain Monte Carlo methods are widely used in the inference of parameters
*
G. A. Wainer, Discrete-Event Modeling and Simulation: A Practitioner’s Approach (Boca Raton, FL: CRC Press,
Taylor and Francis Group, 2009).
†
Tuery, Data Mining and Statistics for Decision Making.
Table27.5 Nursing Triage Times (in minutes)
Triage Time Triage Time Triage Time Triage Time
0.0 0.1 1.2 1.4
4.5 6.7 7.6 4.8
5.3 5.9 6.2 4.9
7.2 5.6 6.1 25.6
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