Communicating Quantitative Information

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Communicating Quantitative Information

• Themes of course
• Testing
• Odds
• Out-of-wedlock babies, census
• Homework Postings

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Themes of course

• Meta level (over-arching) theme ask questions
• What is/are the definition(s)? To gather
  quantitative data, there needs to be an
  operational definition that may or may not be
  what you/the reader has in mind
• What is the denominator (the base)?
• absolute or relative
• What is the difference? the context? the
  relevant comparisons?
• time (temporal) or space MAY be significant
• data may be missing
• What is the distribution?
• The phenomenon may involve sets of numbers (test
  scores, infants who die before age 5, longevity,
  weight). The average (mean) is just one summary
  statistic. How are the values distributed range,
  variance,quintile, etc.
• What are the critical dimensions of the thing /
  phenomenon?
• How to characterize something. How does a diagram
  represent information?

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Recall

• "More women murdered"
• What is the definition of 'on the job'?
• For the comparison, the denominators/bases were
  of the men who die and of the women who die
• It probably was okay to talk relatively (without
  knowing absolute numbers) but needed to make the
  appropriate comparisons
• What was killing the men was missing data.

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gapminder.org

• More later
• Child mortality
• Shows what's the difference
• context over time
• comparison between countries
• distributions comparisons within countries

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Testing for X
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Quality of test

• Two values (may not be the same)
• Suppose test is
• 99 accurate in predicting condition X
  (returning positive) when subject has X.
• 98 accurate in predicting the absence of X when
  subject does NOT have X.

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Prevalence in population

• Suppose population is 300,000 and 2 have
  condition X
• How many have X?

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How many?

• 300000 .02 6000
• (How many is 1 answer is 3000. 2 is twice
  that.)

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Testing for X
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Testing for X
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Testing for X
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So.

• Of the 11820 subjects given positive results
  (positive here may not be good!)
• 5940 really have X
• 5880 false positiveit was a mistake!
• Procedure generally is to re-test all positives
  with same test or, mostly, a better (more
  expensive) test.

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Which statement was correct?

• For the situation just described The test is 98
  to 99 accurate. 2 of a population of 300000
  have the condition X. If you get a positive
  result
• there is 1 to 2 chance it is wrong.
• there is around a 50 chance it is wrong.

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Comments

• Even with what appears to be an accurate test,
  there are many people told the wrong thing
• Most especially, many False Positives. This is
  because the base for the potential false
  positives is nearly the whole population.
• There also are False Negatives, but much fewer.
• Actual tests are somewhat more accurate, but
  still not absolutely accurate.

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Testing

• A test maker may prefer more false positives than
  false negatives because a process could be put in
  place to use a second test on all the positives.
• However, false positives for conditions such as
  HIV make people very uncomfortable. BUT it could
  still be a good public health approach. (It may
  be that given this serious condition, the second
  test is performed on material from same person.)

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General issue

• Think of
• testing for drugs
• mammograms
• Cost of 'false positive'
• for individual
• shared health costs

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Posting opportunity

• Do research to find out
• Prevalence of HIV or something else in a typical
  population
• Accuracy of typical tests
• re-do calculation
• make posting

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Posting opportunity

• New test for breast cancer. Claim better than
  mammograms
• How better?

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Posting opportunity

• Procedures and accuracies of tests done for Tour
  de France
• Note the AIDS/HIV example assumed that most
  people were not infected. What is the assumption
  (now) about cyclists? (SIGH)

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Theme Questions

• What is definition of accuracy for test and how
  does it work?
• more complex (but understandable if you make the
  effort)
• What is the difference/who cares accuracy for
  general testing versus how someone should react
  to getting a notice.

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Single die (dice)

• Success is 1, 2, 3, or 4
• Failure is 5 or 6
• What is probability of success
• What is probability of failure?

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Answer

• p is 2/3
• 1-p is 1/3
• Use this example along with fair coin (p .5)
  when reading examples.

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Recall

• Coin toss probability of any particular string
  of length n is (1/2) to the nth power.
• H H H is 1/8 as is TTT, THT, HTH, etc.
• Probability of getting exactly 1 head in 3 throws
  is
• Add up all of these
• H TT
• THT
• TTH
• 3 1/8
• Probability of getting exactly 2 heads in 4
  throws is
• Sum of probability of getting HHTT, HTHT, HTTH,
  THHT, THTH, TTHH

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How many ways?

• Choosing K things out of N choices where it
  doesnt matter what order you choose.
• N N-1 (N-K1) / KK-11

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Generalize

• Permutation formula number of ways of selecting
  an ordered sequence of k things from set of 1 to
  N things is (N)(N-1)(N-K1) (K factors)
• Can also write this as N!/(N-K)!
• Combinations order does not matter, so group
  together / divide by number of different
  orderings of the K things
• N!/((N-K)!K!) can be written
  N!/(K!(N-K)!)
• This is called the Binomial Coefficent

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N 4 trials

• Throw the single die 4 times.
• How many successes?
• Throw the single die 4 times again?
• How many successes?

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Note

• We are constructing complex events out of simple
  event.
• Event is N trials, Big Success is having exactly
  K successes.

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Probability of

• Getting exactly k successes in n events
• Is
• The number of different ways of getting the k
  successes (which ones will be successes) times
  the probability of any one sequence.

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What is the probability of

• F S F F S F S F
• If p is the probability of a success and (1-p) of
  a failure, then the probability of this is
• (p)3 (1-p)5
• More generally,
• pk (1-p)(N-K)
• Note the coin case had p (1-p) .5

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Combine these together

• The probability of obtaining K 'successes' in N
  trials is
• binomial_coefficient(N,K) pk (1-p)n-k
• NOTE I use computer-style for multiplication ,
  not putting two factors together
• Note original format for binomial coefficient
  hard to type.

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Complete example

• Biased coin (.6 for heads)
• Sequence of 8 (N8), Probability of 3 heads is
• ((8 7 6)/(321))(.6)3(.4)5

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Out-of-wedlock births

• Fairly regular news story
• One or more populations have increases in
  out-of-wedlock births
• As previous situations
• Definitions matter
• Must make note of issues and terms such as
  percentages, rates, changing underlying
  population, absolute vs rates, rates of change,
  time interval under study

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Posting opportunities

• Story on chances of pregnancies of children of
  politicians in the news?
• Success (define) rates of places with abstinence
  only versus more general sex education
• Definitions of education
• Definition of success

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http//www.cdc.gov/nchs/pressroom/07newsreleases/t
eenbirth.htm

• Teenager births and unmarried births
• (Data are for U.S. in 2006)
• "Between 2005 and 2006, the birth rate for
  teenagers 15-19 years rose 3 percent, from 40.5
  live births per 1,000 females aged 15-19 years in
  2005 to 41.9 births per 1,000 in 2006. This
  follows a 14-year downward trend in which the
  teen birth rate fell by 34 percent from its
  recent peak of 61.8 births per 1,000 in 1991"

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Source?

• I chose the CDC source.
• Let's look at others
• What words do we put into google?
• What about scholar.google.com
• Posting opportunity find out more exactly than
  I'm saying what scholar.google.com is
• Note data is somewhat older

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Murphy Brown story

• TV character!!! choosing to have child as a
  single mom. Criticized by VP Dan Quayle
• News story Murphy Brown was accurate
  Out-of-wedlock births by professional,
  well-off, non-minority women increasing at
  greater rate than other groups.
• Be careful when comparing rates of change
• If 1 goes to 2, this is a doubling,100
  increase
• If 45 goes to 50, this is increase of 11.1

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Homework

• Postings