Binomial Distribution and Applications

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Binomial Distribution and Applications
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Binomial Probability Distribution

• A binomial random variable X is defined to the
  number of successes in n independent trials
  where the P(success) p is constant.
  Notation X BIN(n,p)
• In the definition above notice the following
  conditions need to be satisfied for a binomial
  experiment
• There is a fixed number of n trials carried out.
• The outcome of a given trial is either a
  success or failure.
• The probability of success (p) remains constant
  from trial to trial.
• The trials are independent, the outcome of a
  trial is not affected by the outcome of any other
  trial.

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Binomial Distribution

• If X BIN(n, p), then
• where

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Binomial Distribution

• If X BIN(n, p), then
• E.g. when n 3 and p .50 there are 8
  possible equally likely outcomes (e.g. flipping a
  coin)
• SSS SSF SFS FSS SFF FSF
  FFS FFF
• X3 X2 X2 X2 X1 X1 X1
  X0
• P(X3)1/8, P(X2)3/8, P(X1)3/8,
  P(X0)1/8
• Now lets use binomial probability formula
  instead

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Binomial Distribution

• If X BIN(n, p), then
• E.g. when n 3, p .50 find P(X 2)

SSF SFS FSS
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Example Treatment of Kidney Cancer

• Suppose we have n 40 patients who will be
  receiving an experimental therapy which is
  believed to be better than current treatments
  which historically have had a 5-year survival
  rate of 20, i.e. the probability of 5-year
  survival isp .20.
• Thus the number of patients out of 40 in our
  study surviving at least 5 years has a binomial
  distribution, i.e. X BIN(40,.20).

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Results and The Question

• Suppose that using the new treatment we find that
  16 out of the 40 patients survive at least 5
  years past diagnosis.
• Q Does this result suggest that the new therapy
  has a better 5-year survival rate than the
  current, i.e. is the probability that a patient
  survives at least 5 years greater than .20 or a
  20 chance when treated using the new therapy?

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What do we consider in answering the question of
interest?

• We essentially ask ourselves the following
• If we assume that new therapy is no better than
  the current what is the probability we would see
  these results by chance variation alone?
• More specifically what is the probability of
  seeing 16 or more successes out of 40 if the
  success rate of the new therapy is .20 or 20 as
  well?

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Connection to Binomial

• This is a binomial experiment situationThere
  are n 40 patients and we are counting the
  number of patients that survive 5 or more years.
  The individual patient outcomes are independent
  and IF WE ASSUME the new method is NOT better
  then the probability of success is p .20 or 20
  for all patients.
• So X of successes in the clinical trial is
  binomial with n 40 and p .20,
  i.e. X BIN(40,.20)

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Example Treatment of Kidney Cancer

• X BIN(40,.20), find the probability that
  exactly 16 patients survive at least 5 years.
• This requires some calculator gymnastics and some
  scratchwork!
• Also, keep in mind we need to find the
  probability of having 16 or more patients
  surviving at least 5 yrs.

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Example Treatment of Kidney Cancer

• So we actually need to find
• P(X gt 16) P(X 16) P(X 17) P(X
  40)
• .002936 YIPES!

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Example Treatment of Kidney Cancer

• X BIN(40,.20), find the probability that 16 or
  more patients survive at least 5 years.
• USE COMPUTER!
• Binomial Probability calculator in JMP

probabilities are computed automatically for
greater than or equal to and less than or equal
to x.
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Example Treatment of Kidney Cancer

• X BIN(40,.20), find the probability that 16 or
  more patients survive at least 5 years.
• USE COMPUTER!
• Binomial Probability calculator in JMP

P(X gt 16) .0029362
The chance that we would see 16 or more patients
out of 40 surviving at least 5 years if the new
method has the same chance of success as the
current methods (20) is VERY SMALL, .0029!!!!
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Conclusion

• Because it is high unlikely (p .0029) that we
  would see this many successes in a group 40
  patients if the new method had the same
  probability of success as the current method we
  have to make a choice, either
• we have obtained a very rare result by dumb luck.
  OR
• our assumption about the success rate of the new
  method is wrong and in actuality the new method
  has a better than 20 5-year survival rate making
  the observed result more plausible.

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Sign Test

• The sign test can be used in place of the paired
  t-test when we have evidence that the paired
  differences are NOT normally distributed.
• It can be used when the response is ordinal.
• Best used when the response is difficult to
  quantify and only improvement can be measured,
  i.e. subject got better, got worse, or no change.
• Magnitude of the paired difference is lost when
  using this test.

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Example Sign Test

• A study evaluated hepatic arterial infusion of
  floxuridine and cisplatin for the treatment of
  liver metastases of colorectral cancer.
• Performance scores for 29 patients were recorded
  before and after infusion.
• Is there evidence that patients had a better
  performance score after infusion?

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Example Sign Test
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Sign Test

• The sign test looks at the number of () and (-)
  differences amongst the nonzero paired
  differences.
• A preponderance of s or s can indicate that
  some type of change has occurred.
• If in reality there is no change as a result of
  infusion we expect s and s to be equally
  likely to occur, i.e. P() P(-) .50 and the
  number of each observed follows a binomial
  distribution.

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Example Sign Test

• Given these results do we have evidence that
  performance scores of patients generally improves
  following infusion?
• Need to look at how likely the observed results
  are to be produced by chance variation alone.

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Example Sign Test
17 nonzeros differences, 11 s 6 s
-


-

-





-

-

-

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Example Sign Test

• If there is truly no change in performance as a
  result of infusion the number of s has a
  binomial distribution with n 17 and p P()
  .50.
• We have observed 11 s amongst the 17 non-zero
  performance differences.
• How likely are we to see 11 or more s out 17?
• P(X gt 11) .166 for a binomial n 17, p
  .50
• There is 16.6 chance we would see this many
  improvements by dumb luck alone, therefore we are
  not convinced that infusion leads to improvement
  (Remember less than .05 or a 5 chance is what
  we are looking for statistical significance)

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Example 2 Sign Test

• Resting Energy Expenditure (REE) for Patient with
  Cystic Fibrosis
• A researcher believes that patients with cystic
  fibrosis (CF) expend greater energy during
  resting than those without CF. To obtain a fair
  comparison she matches 13 patients with CF to 13
  patients without CF on the basis of age, sex,
  height, and weight. She then measured there REE
  for each pair of subjects and compared the
  results.

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Example 2 Sign Test
There are 11 s 2 s out of n 13
paired differences.
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Example 2 Sign Test
The probability of seeing this many s is small.
We conclude that when comparing individuals with
cystic fibrosis to healthy individuals of the
same gender and size that in general those with
CF have larger resting energy expenditure (REE)
(p .0112).