1.9: A Closer Look at Screening Tools
- Page ID
- 147914
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)INTRODUCTION
In this collaboration, you'll be taking a deeper dive into sensitivity, specificity and related probabilities. Let's check in on a few things to get us started. In your group, discuss the following questions:
- Consider a deck of 52 playing cards. If you draw one card randomly, what is the probability that it is a queen?
- Now take the complete deck again and take out all of the cards except the face-cards (just leaving the kings, queens and jacks). Now randomly draw a card. What is the probability that the card is a queen this time?
- How do the numerators and denominators compare in each of these probabilities?
The first draw could be restated as: “Draw a card from the deck. What is the probability that it is a queen?”
The second draw would be restated as something like this: “If only the face cards are in the deck, then what is the probability of drawing a queen?”
The "if" part of this statement is adding a condition which changes the group from which you are drawing.
SPECIFIC OBJECTIVES
By the end of this collaboration, you should understand that
- conditional probability is the probability that an event will occur given that another event has already occurred.
- it is important to be careful when interpreting phrases about conditional probabilities.
- a two-way table is a useful tool to analyze conditional probabilities.
- some calculations related to a tool’s effectiveness cannot be computed directly from its sensitivity or specificity.
By the end of this collaboration, you should be able to
- correctly interpret phrases or statements referring to conditional probabilities.
- calculate different types of conditional probabilities with information in a two-way table (contingency table). These conditional probabilities include specificity and sensitivity.
- create a two-way table to model the effectiveness of a screening tool.
PROBLEM SITUATION: USING SCREENING TOOLS TO MAKE DECISIONS
In the previous collaboration (N.8), you learned how difficult it is to decide on a cut-off value when trying to identify people who may have cancer. There can be benefits to certain cut-off values, but also drawbacks. A screening tool’s sensitivity and specificity can help quantify how effective the screening tool is. In this collaboration, we will examine other factors that are important when analyzing the effectiveness of screening tools.
There are a variety of probabilities, or likelihoods, associated with analyzing the effectiveness of screening tools. Specificity and sensitivity are two examples, but there are many others. Describing these probabilities in everyday language can sometimes be very difficult. Doctors and scientists who study screening tools have to be careful about how they describe and distinguish between these probabilities.
When answering questions about a test’s effectiveness, it is important to read each question carefully to determine the right proportion.
For example, consider the following two questions:
Question A: If a screening tool is given to a person without cancer, what is the probability that the screening tool will suggest that cancer might be present?
Question B: If a screening tool suggests that cancer is present in a person, what is the probability that the individual does not have cancer?
(1) Compare Question A and Question B above. Are these two questions asking the same thing, phrased in two slightly different ways? Or, are these two questions asking for different information, and their answers could be different? Explain.
(2) Suppose you were being screened for a disease, such as cancer. Unfortunately, your doctor tells you that the screening tool produced a positive result. This suggests that cancer may be present.
Looking back at Question A and Question B, which question would you like your doctor to answer? Why?
The questions Question A and Question B in the Problem Situation are examples of conditional probabilities, which are questions that can be put into the following form:
If [something has happened], what is the probability that [something else will happen]?
For example, Jan rolls a six-sided die and does not show the result to her friend, but tells her friend she rolled an odd number. What is the probability of her friend guessing correctly that Jan rolled a 5? You can think of this problem as: If her friend knows an odd number was rolled, what is the probability that her friend will guess it? This is a conditional probability. (Note: The answer is ⅓, since there is one correct answer, 5, and three possibilities: 1, 3, and 5.)
As we discussed in Questions 1 and 2, great care must be taken to determine exactly what calculations should be performed to find a specific conditional probability. Questions that are asking for different conditional probabilities may appear quite similar at first glance. Determining the numerator and denominator for the conditional probability may require reading the problem carefully to guarantee understanding, and may require finding the sum of a row or column in a two-way table.
Suppose we are studying the effectiveness of Screening Tool X, and we decide to investigate the research questions A and B given earlier. First, we apply Screening Tool X to 1,200 people. For each patient, Screening Tool X produces either
- a positive result if the tool suggests the presence of cancer, or
- a negative result if it suggests the absence of cancer.
Remember, as we saw in Collaboration N.8, most screening tools are not guaranteed to be correct in either case, and it is possible for either a negative result or a positive result to be wrong.
For the purposes of our study, suppose that after being tested with Screening Tool X and given an initial diagnosis, everyone undergoes a biopsy to see if cancer was or was not actually present. Table 1 displays the results of this hypothetical experiment in a two-way table:
| Cancer is Present | Cancer is Not Present | Total | |
| Result of Screening Tool X is Positive | 53 | 56 | 109 |
| Result of Screening Tool X is Negative | 211 | 880 | 1091 |
| Total | 264 | 936 | 1200 |
(3) What is the sensitivity of this test? Remember, the sensitivity of the screening tool is the probability that the screening tool will be correct when cancer is actually present. In other words, what percentage of individuals with cancer received a positive result from the screening tool? Take a minute to try this on your own before sharing with your group. Write your answer as a percentage rounded to one decimal place.
(4) What is the specificity of this test? Remember, the specificity is the probability that the screening tool will be correct when cancer is not present. In other words, what percentage of individuals without cancer received a negative result from the screening tool? Write your answer as a percentage rounded to one decimal place.
(5) Recall Question A from the Problem Situation:
Question A: If a screening tool is given to a person without cancer, what is the probability that the screening tool will suggest that cancer might be present?
Use Table 1 above to answer Question A. Does your answer appear to be related in any way to either the specificity or sensitivity? If so, how? (Hint: Read through Question A carefully, thinking about a description of the numerator and denominator in that probability statement, and relating those descriptions to the problems you just completed on specificity and sensitivity.)
(6) Now think about Question B from the Problem Situation:
Question B: If a screening tool suggests that cancer is present in a person, what is the probability that the individual does not have cancer?
Use Table 1 above to answer Question B. Does this answer appear to be related in any way to either the specificity or sensitivity? If so, how? (Hint: Read through Question B carefully, thinking about a description of the numerator and denominator in that probability statement, and relating those descriptions to the problems you just completed on specificity and sensitivity.)
Now, suppose we study the effectiveness of Screening Tool X on a different population that is known to have a low cancer rate. We apply the screening tool to 1,200 people in this new population. Table 2 below displays the results of this hypothetical experiment:
| Cancer is Present | Cancer is Not Present | |
| Result of Screening Tool X is Positive | 3 | 70 |
| Result of Screening Tool X is Negative | 10 | 1117 |
(7) Determine the specificity and sensitivity of Screening Tool X in the new population. Write your answers rounded to the nearest percent.
(8) As you did for Questions 5 and 6, answer Question A and Question B using the results of Screening Tool X in this new population.
Question A: If a screening tool is given to a person without cancer, what is the probability that the screening tool will suggest that cancer might be present?
Question B: If a screening tool suggests that cancer is present in a person, what is the probability that the individual does not have cancer?
(a) Use Table 2 above to answer Question A. Does your answer appear to be related in any way to either the specificity or sensitivity? If so, how?
(b) Use Table 2 above to answer Question B. Does this answer appear to be related in any way to either the specificity or sensitivity? If so, how?
(c) Compare your answers to Questions 8(a) and 8(b) to your answers to Questions 5 and 6 above. Why do you think the results are similar or different?
MAKING CONNECTIONS
Record the important mathematical ideas from the discussion.