Context of the problem:
A nurse needs to collect blood specimens from a patient who needs to
have his blood tested for 9 different blood tests in the course of 3
weeks. The nurse needs to collect 2 tubes in the first week, 3 tubes
in the second week, and 4 tubes in the third week. There are 11
possible test tubes to choose from depending on the test that needs to
be performed. These tubes are identified by the color tops to indicate
the coagulant they carry. When collecting blood from the patient, test
tubes must be collected in a specific order to insure accurate lab
results. The adopted "order of draw" is as follows:
1) Light Blue top tube
2) Red top tube
3) Light Green top tube
4) Green top tube
5) Lavender top tube
6) Pink top tube
7) White top tube
8) Gray top tube
9) Yellow top tube
10) Light Blue top tube
11) Draw Royal Blue
A diagram of the ?order of draw? can be seen here:
http://www.medicine.uiowa.edu/path_handbook/Appendix/new_tubes/tube_tops.html
Problem:
On each week, the nurse needs to pick test tubes and select, based on
the order of draw above, the order in which tubes will be filled. What
is the probability that tubes will be selected in the right order in
week 1, week 2 and week 3? And what is the probability of selecting
the wrong tube in week 1, week 2 and week 3?
ps. let me know if you need more details to solve this problem. |
Request for Question Clarification by
mathtalk-ga
on
24 Jun 2005 05:49 PDT
Hi, braunow-ga:
Let's limit the discussion temporarily to the first week to see if we
can clear up some of the premises. Recall that the nurse will be
required to collect two tubes in the first week.
The "order of draw" presents "11 possible test tubes" but in your list
numbers 1 and 10 are described as identical (they are depicted
distinctly at the link you provide).
Focusing now on the simplified question of the two tubes of blood to
be drawn in Week 1, we would think first of a naive answer that the
chance of getting the tubes in the right or wrong order is one-half
for each.
But this is doubtless an unrealistic result. It assumes that the
nurse correctly selects the two correct tubes, that they are different
from one another, and that the nurse then randomly draws blood in one
or the other.
However it helps to identify some issues for defining the problem:
- What if anything is known about the tubes needed for the Week 1
tests? Is there for example a probability that the two tests might
required two of the same tube?
- What allowance if any must be made for the possibility that the
tubes selected for the draw are different from the ones actually
needed?
- Finally, and perhaps the central focus of your Question, what
model should be assumed for the order of the draw by the nurse
phlebotomist? A random shuffling of the tubes would seem overly
pessimistic, and a realistic model would perhaps take account of the
level of training/experience by the nurse as well as the physical
distinctions among the tubes. For example, a certain number of people
have red/green color blindness, so that transposing the order of draw
given only these two tubes (2 and 4) might be more likely for some
percentage of cases than transposing red and white (2 and 7).
Formulating a mathematical model for your problem requires some
thought to be given to setting these parameters. For example, if the
two tube combination of red and green is the most likely requirement
for Week 1, and this is at the same time the most likely pair to be
transposed, it will have a crucial bearing on the calculations (as
opposed to a random shuffling of randomly selected tubes).
regards, mathtalk-ga
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Clarification of Question by
braunow-ga
on
24 Jun 2005 15:15 PDT
Mathtalk-ga,
Thanks for the clarifications and for catching the discrepancy on the
order of draw. Let me start this message by making a correction to the
"order of draw" so that we can agree with a final ?order of draw?
version for this problem. Please, adopt the following "order of draw":
1. Light yellow top tube
2. Light blue top tube
3. Yellow top tube
4. Red top tube
5. Green top tube
6. Light green top tube
7. Lavender top tube
8. Pink top tube
9. Grey top tube
As described by Becton, Dickinson and Company (BD) at:
http://www.bd.com/vacutainer/pdfs/plus_plastic_tubes_instructions_orderofdraw_VS5734.pdf
NOTICE that this order has only "9 possible test tubes" and not 11 as
previously described.
Now to your points:
- What if anything is known about the tubes needed for the Week 1
tests? On any given week, the nurse knows what tubes will be needed
for the test. Usually, the nurse receives a paper request with the
patient information (name, required tests, etc).
- Is there for example a probability that the two tests might require
two of the same tube? If tubes are the same, there is no need to put
them in any order because they have the same coagulant inside and will
not contaminate each other during the procedure. For more information
on how cross-contamination between tubes actually happens, please
refer to the comments provided below (3rd paragraph). So, I think it
is OK to assume that the test tubes requested will always be different
every week and there will be no repeats.
- What allowance if any must be made for the possibility that the
tubes selected for the draw are different from the ones actually
needed? That's not the problem since the nurse knows what test tubes
need to be used. What she may NOT KNOW is the sequence in which they
need to be filled. That's when she needs to consult the "order of
draw" chart. Most blood rooms will have a chart of the ?order of draw?
displaying on the wall. This is really the crocks of this problem
because we don't know whether or not she will follow that order
appropriately and I am hopping we can create a model to calculate the
probability of making mistakes while consulting the order and actually
performing the blood draw in the right sequencing.
To help clarify even more, let?s imagine a common scenario for week 1:
1) The patient comes in.
2) The nurse receives the request - two tests need to be done:
"Platelet Count" (Lavender top tube) and "ABO/Rh Testing" (Ret top
tube).
3) The nurse chooses the test tubes from the shelf.
4) The nurse consults the "order of draw" chart.
5) The nurse chooses the sequence in which the tubes must be drawn.
6) The nurse performs the venipuncture.
- Finally, and perhaps the central focus of your Question, what model
should be assumed for the order of the draw by the nurse phlebotomist?
Yes, it makes a lot of sense taking into account the level of
training/experience. Should we then assume different ?error rates? for
each experience level nurses have? The physical distinction among the
tubes is simply the color of the top. The notion of "Color Vision
Deficiency" will surely impact - I never thought about that, thanks
for bringing that up. I looked into the percentages of people with
that deficiency and found that it occurs in about 8% - 12% of males of
European origin and about one-half of 1% of females. In the blood
room, it is common to see a 50/50 split of male and female.
Let me know if you need more clarification.
|
Request for Question Clarification by
mathtalk-ga
on
26 Jun 2005 17:17 PDT
Hi, braunow-ga:
It sounds as if you are willing to assume:
a) all different tubes, not only in a given week, but across all three weeks
b) the "the nurse knows what test tubes need to be used", so zero mistakes there
c) there is a chart in the "blood room", but some likelihood of
mistakes (while consulting the order of draw shown there) must be
imputed
So far we don't have much to go on in terms of actually assigning a
probability for a mistake. We could assume that the nurse, although
aware of the chart, proceeds to randomly order the draw each week.
Since any order of the given tubes except the one specified by the
chart will then entail an error, the computation would be
straightforward (but obviously pessimistic, since in the best case,
Week 1 with its two tubes, we already have a 50-50 chance of error).
I'm afraid that mathematics cannot take us further without some data,
possibly experimental in nature, that proposes what the nurse is
likely to do correctly and what incorrectly, e.g. the lavender and
grey tubes are three times as likely to be confused as the red and
green ones, but the red and green pair are ten times as likely to be
ordered for a random patient.
If this were only a math homework assignment, I'd be more confident
about the intended interpretation. But a real world application like
this deserves careful analysis.
regards, mathtalk-ga
|
Clarification of Question by
braunow-ga
on
28 Jun 2005 20:56 PDT
Hi Mathtalk-ga,
To your points:
a) "all different tubes, not only in a given week, but across all three weeks"
Yes, every week the nurse is asked to do a new test with tubes that
are different than the week before. Considering the pace in which they
draw blood, the nurse most likely won't know what tubes were required
in the past week. She needs to start from zero and verify the types of
tests required for the week.
b) "the "the nurse knows what test tubes need to be used", so zero mistakes there".
The fact that the nurse knows what test tubes need to be used does not
remove the space for error when it comes to define the sequence in
which they should be drawn. You are right that this may be about doing
a real life observation to detect when or not the selection is done
correctly. However I also hoping we would be able to calculate the
propagation of human errors by taking into account the elements of
human performance such as task complexity, working/short term memory,
vigilance, time pressure, information availability,
training/experience, etc.
c) "there is a chart in the "blood room", but some likelihood of
mistakes (while consulting the order of draw shown there) must be
imputed"
Considering the access to the blood room is limited and pursuing an
experimental study would involve many resources not available to me at
this time, I would like to explore 2 scenarios for calculating the
chances of errors:
1. considering random shuffling of randomly selected tubes when trying
to follow the order of draw for weeks 1, 2 and 3 (as much as this
sound very pessimistic as you pointed out, at least we would have a
worse case scenario).
2. considering the human performance elements into the probability of
making errors. This may be more complex to solve. Please, let me know
if you have ideas and/or solutions for any of these scenarios.
Thanks and kind regards!
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