Using tests to improve decisions: Cutting scores

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Using tests to improve decisionsCutting scores
base rates
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Review Conditional Probability

• Conditional probabilities arise when the
  probability of one thing A depends on the
  probability of something else B
• In such cases, we want to factor in the
  probability of B before we worry about A
• This amounts to focusing on the elements that are
  likely to be picked out by both A and B
• P(AB) P(A and B)/P(B)

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Three ways

• We can consider three ways to solve condition
  probability questions (all exactly equivalent)
• Common sense
• Probability tables
• Bayes Theorem

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a.) Common sense

• P(AB) P(A and B)/P(B)
• 5 males, one wears a dress 3 females, 4 wear
  dresses.
• What is the probability that you wear a dress,
  given that you are female?
• First We want to know how many people are both
  dress wearers and females P(A and B) 4
• Second We want to know what proportion of all
  woman are accounted for by the dress wearing
  females
• Dress wearing females / Females
• P(Female and dress-wearing)/P(Female)
• 4/5

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b.) Probability Tables

• P(AB) P(A and B)/P(B)
• What is the probability that you sometimes wear a
  dress, given that you are female?

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b.) Probability Tables

• P(AB) P(A and B)/P(B)
• What is the probability that you sometimes wear a
  dress, given that you are female?

JUST IGNORE ALL THE MALES!
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c.) Bayes Theorem

• P(AB) P(A and B)/P(B)
• What is the probability that you sometimes wear a
  dress, given that you are female?
• P(AB) P(BA) P(A) / P(B)
• Proof By definition, (1.) P(AB) P(A and B) /
  P(B)
• (2.) P(BA) P(A and B) / P(A)
• (3.) P(BA) P(A) P(A and B) Multiply (2.) by
  P(A)
• (4.) P(BA) P(A) P(AB) P(B) Substitute (1.)
  in (3.)
• (5.) P(BA) P(A) / P(B) P(AB) Divide by P(B)
• P(Dress-wearingFemale) P(FemaleDress-wearing)P
  (Female)/P(Dress-wearing)
• (4/5 5/10) / (5/10)
• 4/5

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Why use ! Bayes Theorem?

• Bayes Theorem is not intended to confuse, but to
  simplify you can use it to get the probability
  relation between any two cells in the 2x2 table
• It can also be generalized to more complex
  situations
• However, in this class we wont go outside of 2x2
  conditional probability tables so just draw a
  picture or think it through if you prefer!

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Cutting scores

• What is a cutting score or cutting line?
• How shall we evaluate how good any given test is?

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Cutting scores

• What is a cutting score or cutting line?
• In many tests we have criteria if a subject
  scores above score X, they are likely to be Y a
  genius, a moron, a good prospect, likely to die
  in six months
• X is a cutting score
• Note that this is a conditional probability
  P(diagnosistest result)
• Note also that in this case probability of X
  test result is not given by God we test
  designers are free to change the cutting score
  as we like
• In doing so, we can change P(diagnosistest
  result)

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Cutting scores

• As an example, think of the probability that a
  person is a genius (defined, lets say, as IQ gt
  130) given that they got an IQ score of 128, on
  the one hand, or 110, on the other.
• Assume the standard error for IQ is 10 points
• Then there is a fair chance that a person who got
  128 has an IQ above 130, but a very small (but
  non-zero) chance that that person who got 110 has
  an IQ above 130
• If we used 110 as a cutting score for genius,
  wed be wrong a lot P(diagnosistest result) is
  very low
• If we used 128 as a cutting score for genius,
  wed be wrong less often P(diagnosistest
  result) is higher

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Cutting scores

• What we want is some principled way of deciding
  what a good cutting score is for any particular
  purpose
• Clearly, our choice of cutting score will depend
  on that purpose
• When we are diagnosing a brain tumour, we want to
  be wrong almost never if the person does have a
  brain tumour AND we dont care too much if we
  make a false positive
• When we are trying to identify criminals, we
  might be more worried about minimizing false
  positives (we could ruin a life is we say someone
  is a criminal when they are not) and willing to
  pay the price by letting some real criminals go
  free (increase our false negative rate)

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False negative Incorrectly undiagnosed.
False positive Incorrectly diagnosed
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Low false negative rate
High false positive rate
Rewarding incompetence
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High false negative rate
Low false positive rate
Ignoring competence
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How shall we evaluate how good a test is?

• Three things need to be taken into account
• i.) The size of the correlation between test
  scores and criterion
• - The higher the correlation, the narrower the
  scatterplot (i.e. the ellipse) and the smaller
  the error rates

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How shall we evaluate how good a test is?

• Three things need to be taken into account
• ii.) The base rate
• iii.) The cutting score
• What is the relation between these two measures?

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The relation between base rate and cutting score

• Example from Meehl
• Group A 415 well-adjusted soldiers
• Group B 89 mal-adjusted soldiers
• A scale diagnosed 55 of Group B, and only 19 of
  Group A, so the authors advocated its use

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Example Assume N 10,000

• 500 are bad. 55 (275) are classified as bad
• 9500 are good. 81 (7695) are not classified as
  bad.
• (7695 275)/10000 79.97 are correctly
  classified.
• Why should this bother us?

We could have correctly classified 95 without
using a test!
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Lets use Bayes Theorem Is bad bad?
When we take base rates into account, an
identification of a person as bad actually has
only a 13 chance of being correct, not a 55
chance as claimed.
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Lets use Bayes Theorem Is not bad good?
When we take base rates into account, a failure
to identify a person as bad has 97 chance of
being correctbut remember that we were already
95 sure before we bothered to do the calculation!
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The relation between base rate and cutting score,
II

• A certain Rorschach configuration is seen in 8.1
  of schizophrenics, and 0 of non-schizophrenics
• The authors claim this is clinically useful Is
  it really?

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Lets do the math!
Although the sign is certain in this case, it is
so rare itself and applies to a group with such a
rare base rate that it is P(Rorschach) that is
worrying This information would be
diagnostically helpful in only 7 cases out of
10,000! it is clinically useless
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What can we do? Rule 1

• In order for a positive diagnostic assertion to
  be more likely true than false, the ratio of
  positive to negative base rates in the examined
  population must exceed the false positive to
  valid positive rate
• Base rate of positives False positive rate of
  test
• Base rate of negatives True positive rate of test

gt
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Example Rule 1

• Base rate of positives False positive rate of
  test
• Base rate of negatives True positive rate of test

A cutting score identifies 80 of brain-damaged
patients. 15 of nondamaged patients also exceed
that cut-off. What base rates can justify the use
of such a test? .15 (false positive) / .80 (true
positive) 0.19 The ratio of brain damaged to
non-brain damaged patients in the population
under consideration must be equal to or greater
than .19, or about 1 in 5.
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The easiest case Equal base rates (Rule 2)

• Iff base rates are equal, then the probability of
  a positive diagnosis is the ratio of the true
  positive rate to the sum of the true and false
  positive rates.
• Another way of saying this more simply is equal
  base rates render Bayes Theorem unnecessary.

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Example Equal base rates (Rule 2)

• Iff base rates are equal, then the probability of
  a positive diagnosis is the ratio of the true
  positive rate to the sum of the true and false
  positive rates.
• Two kinds of cancers occur equally often. A test
  diagnoses Type B with 68 accuracy, but is at
  chance for Type A. You get a positive test
  result. What is the probability you have Type B
  cancer?

For once life is simple. The probability is
68. 0.68 / (0.68 0.32) 0.68
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Example 2 Equal base rates (Rule 2)

• A test picks out 75 of people who will continue
  in school (true positives) but also 40 of those
  who will not (false positives). It is claimed
  that about half of all students in the population
  drop out of school. How far off can that claim be
  without the test being useless?
• The probability of a positive diagnosis with
  equal split is the ratio of the true positive
  rate to the sum of the true and false positive
  rates
• 0.75 / (0.75 0.40) 0.65
• So the test gets about 65 right. If less than
  35 of the students actually do drop out, the
  test will not do better than base rates.
• That is If it is a matter of fact that (say)
  only 10 of students drop out, then there is no
  use giving this test it cant beat the 90 odds
  you have of being correct before you bothered to
  give the test

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When can a test help? (Rule 3)

• A test result can only help if the base rate of
  the more numerous class (here, positive) is less
  than the ratio of the true negative rate to the
  sum of the true and false negative rate

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When can a test help? (Rule 3)

• A test result can only help if the base rate of
  the more numerous class (say, positive) is less
  than the ratio of the true negative rate to the
  sum of the true and false negative rate
• A test of maladjustment classifies 85 of
  maladjusted girls, but only mis-identifies 15 of
  adjusted girls. What base rates are needed to
  support these ratios? (Assume, reasonably, that
  there are more adjusted than unadjusted girls.)
• The ratio of the true negative rate to the sum of
  the true and false negative rate (0.85 true
  negative / (0.85 true negative 0.15 false
  negative) 0.85. The test can only help if less
  than 85 of girls are well-adjusted.

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What does this have to do with cutting lines?

• The proportion of people selected (diagnosed,
  chosen) from a sample is called the selection
  ratio
• When positive/negative base rates are not equal,
  there is a (fairly brutal) trade-off between the
  accuracy (error rate) of a diagnosis or
  prediction, and the size of the selection ratio

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The brutal trade-off

• If you want to be very sure you are right, you
  can speak of only a very small proportion of the
  sample (and you need a very large sample to get
  the cut-off points!)
• If you want to say something about everyone, then
  you must be prepared to be uncertain about your
  cut-off points, and wrong very often.
• In short you can be certain about a few people,
  or uncertain about a lot of people take your
  pick!

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False negative Incorrectly unselected
False positive Incorrectlyselected
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Low false Negative rate
High false positive rate
Rewarding incompetence
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High false negative rate
Low false positive rate
Ignoring competence
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Sensitivity Specificity

• The sensitivity of a test The probability of
  having a positive test result when the disease is
  present
• P(ResultDisease) True positive rate
• The specificity of a test The probability of
  having a negative test result when the disease is
  absent
• P(ResultDisease) True negative rate

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SENSITIVITY
False negative Incorrectly unselected
True positive Correctly selected
True negative Correctly unselected
False positive Incorrectly selected
SPECIFICITY
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What to do? 1

• 1.) Obviously, sometimes we can be satisfied with
  a small improvement on true negative base rates
  and with a large false positive rate
• As we have said, we dont mind mistaking 90 brain
  tumors in order not miss 20.
• 2.) Successive hurdles Take a chance, allow
  errors, and give the expensive, time-consuming,
  but accurate tests to those who are selected out
  from a first-pass of a less-expensive, less
  time-consuming, and more accurate test
• Repeat as necessary...

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What to do? 2

• 3.) Sometimes we can find sub-populations with
  less extreme base rates than in the
  world-at-large
• If our referrals are well-screened, we can have
  more confidence in base rates that are less
  onerous ( closer to being equal) than they would
  be in the world at large

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What to do? 3

• 4.) Sometimes so what? is the right thing to
  say.
• Since testing with any accuracy is so difficult
  to do well, we should not bother to give tests
  that dont lead to real changes in therapy or
  other treatment
• If you can identify good therapy candidates with
  70 accuracy, so what? Will you then ignore or
  refuse to treat those who dont make the cut?
• If not, dont waste time and effort giving the
  test

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What to do? 4

• Gather base rate information.