Introductory Biostatistics

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Introductory Biostatistics

• Runhua Shi MD, MPH, PhD
• Associate Professor of Medicine and Feist-Weiller
  Cancer Center
• rshi_at_lsuhsc.edu
• 813-1434
• Office FWCC B-444

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Review
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Arithmetic Mean The arithmetic mean is the sum
of all the observations divided by the number of
observations.
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Measures of Spread/Dispersion
Sample Variance A measure of the difference of
each value from the mean value. Sample
Standard Deviation A measure of dispersion
expressed in terms of the original units.
.
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Median
The sample median is n is odd the (n1)/2 th
largest observation n is even the average of the
n/2 and n/21 th largest observation Example
Arrange values in ordered array, and choose the
middle value(s). 23,23,24,28,30,40,43,44,48
n9, The median is (91)/2(10/2)5th
median30 23,23,24,28,30,38,40,43,44,48 n10,
the median is the average of 10/25th and
10/216th obs (3038)/234
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Geometric mean
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Chapter 3. Concepts of Basic Probability

• Definition of Probability
• Conditional Probability
• Screening Tests-Bayes Rule
• ROC curves
• Prevalence and Incidence

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Sensitivity the probability of a positive test
given you have the diseases sensitivity
P(TD) a/(ac) Specificity the probability
of a negative test given you dont have the
diseases specificity P(T_ D_) d/(bd)
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sensitivity P(TD) a/(ac) specificity P(T_
D_) d/(bd)
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What is PV in our example? PVP(DT)a/(ab
) 132/1115 0.11838 or 11.8 Interpretation
11.8 of the women who screened positive actually
had cancer. What is PV- in our example?
PV_P(D_T_) d/(cd) 63650/63695 0.99929 or
99.9 Interpretation almost all of the women who
screened negative really were disease free.
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Random Variable A variable whose values arise as
a result of chance factors and cannot be exactly
predicted in advance. Discrete Random Variable
A random variable that is characterized by gaps
or interruptions in the values that it can
assume. (yes, no positive, negative racial
categories) Continuous Random Variable A random
variable that does not possess interruptions or
gaps. (age, concentration, BP)
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Chapter 4 Discrete Probability

• Definition of Random Variables.
• Mass function for a discrete random variable
• The expected value of a discrete random variable
• The variance of a discrete random variable
• Binomial Distribution
• Poisson Distribution

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Random Variable

• A random variable (RV) is numeric function that
  assigns probabilities to different events in a
  sample space.
• A RV for which there exists a discrete set of
  values with specified probabilities is a discrete
  RV (/ - of a test).
• A RV whose possible values cannot be enumerated
  is continuous RV (cumulative of exposure-smoking,
  in a life time)

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Arithmetic Mean The arithmetic mean is the sum
of all the observations divided by the number of
observations.
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Measures of Spread/Dispersion
Sample Variance A measure of the difference of
each value from the mean value. Sample
Standard Deviation A measure of dispersion
expressed in terms of the original units.
.
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A discrete RV

• Expected value
• Variance

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Cumulative distribution function of a Discrete RV

• The cumulative distribution function (cdf) of a
  random variable X is denoted by F(X) and, for a
  specific values x of X, is denoted by P(Xx) or
  by F(x)---step function

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Permutation

• The number of permutation of n things taken k at
  a time is

A, B, C AB, AC, BC BA, CA, CB
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Combination

• The number of combination of n things taken k at
  a time is

A, B, C AB, AC, BC BA, CA, CB
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Properties of The Binomial Distribution

• A sample of n independent trials, each of which
  have only two possible outcomes, which are
  denoted as success or failure.
• Furthermore, the probability of a success at each
  trial is assumed to be some constant p, and hence
  the probability of a failure at each trial is
• 1-pq.
• Example Flip a coin for 100 times, at each time
  (flip), 2 possible outcomes are head or tail, the
  probability of head is ½, the probability of tail
  is 1- ½ ½ , here n100, p ½ , q ½

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The Binomial Distribution

• The distribution of the number of successes
  (k0,1,2,..n) in n statistically independent
  trials, where the probability of success on each
  trial is p, is known as the binomial distribution
  and has a probability-mass function (pdf) given by

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example

• What is the probability of obtaining 2 boys out
  of 5 children if the probability of a boy is 0.51
  at each birth and the sexes of successes children
  are considered independent random variable.
• N5, k2, p0.51, q1-p0.49
• P(X2)0.306

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example

• What is the probability of obtaining 2 boys out
  of 5 children if the probability of a boy is 0.51
  at each birth and the sexes of successes children
  are considered independent random variable.
• N5, k0,1,2, p0.51, q1-p0.49
• P(Xlt2)P(X0)P(X1)P(X2)
  0.3060.1470.028 0.481
• P(Xlt2)0.481

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Get probability from excel

• In excel BINOMDIST (k, n, p, FALSE)
• False if for probability mass function
• True if for cumulative distribution function

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The Binomial Parameters (expected value and
variance) The binomial distribution has 2
parameters, n and p. The binomial distribution is
a group of distributions, with each possible
value of n and p designating a different member
of the group. Measures of central tendency
(mean) and dispersion ( Variance, SD) for the
binomial distribution are
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• The Poisson Distribution-rare event
• The probability of k events occurring in a time
  period t for a Poisson random variable with
  parameter ? is
• e is approximately 2.71828
• Represents expected number of events per unit
  timerate ? Represents expected number of events
  over a time period t, ?? t,
• For rare event, cancer count by age group and race

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example

• If A100 cm2 and ?0.02 colonies per cm2,
  calculate the probability distribution of the
  number of bacterial colonies.
• Assuming that the probability of finding 1 colony
  in an area size of ?A at any point on an agar
  plate is ??A for some ? and that the number of
  bacterial colonies found at 2 different points of
  the plate are independent random variables, then
  the probability of finding k bacterial colonies
  in an area of size A is

?? A 0.021002
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Expected value and variance of Poisson
Distribution For a Poisson distribution with
parameter ?, the mean and variance are both equal
? . Poisson approximation to the binomial
distribution The binomial distribution with
large n and small p can be accurately
approximated by a Poisson distribution with
parameter ?np When n is very large, its hard to
compute the nCk and (1-p)(n-k)
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example

• Suppose we are interested in the genetic
  susceptibility to breast cancer. we find that 4
  out of 1000 women aged 40-49 whose mothers have
  had breast cancer also develop BCa over the next
  10 year of life. We would expect from large
  population studies that 1 in 1000 women of this
  age group will develop a new case of the disease
  over this period of time.
• How unusual is this event?

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• Exact binomial distribution,
• N1000, k4, p0.001
• P(X4)1-P(X3)0.0189
• Using Poisson distribution, ?1000(0.001)1,
• K0,1,2,3
• P(X4)1-P(X3)0.0190
• This event is indeed unusual and suggests a
  genetic susceptibility to BCa among daughters of
  women who have had BCa. Plt0.05

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An drug company is designed test the potency of a
bird flu vaccine on 2000 children, previous study
have shown that one dose of the bird flu vaccine
has side effect (i.e. fever) of 0.2 of time
within the first 48 hours. What is the
probability of that 0 child will experience side
effect in the first 48 hours. Using Poisson
distribution, ?2000(0.002)4 e2.718 k0
The probability of that 0 child will experience
side effect in the first 48 hours is 1.8.
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Binomial or Poison distribution

• If the event is rare such as rate is lt 2 and the
  sample size is large- Poison
• If the event is not rare such as the rate is
  greater than 2 and the sample size is not very
  large such as n lt100. -Binomial

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Get probability from excel

• In excel POISSON(x,m,FALSE)
• False if for probability mass function
• True if for cumulative distribution function

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Review

• The probability of obtain k out of n events where
  the individual event with the success probability
  of p. -Binomial Distribution
• The probability of finding k bacterial colonies
  in an area of size A is where ??A (A can also be
  time t)

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Review

• Definition of Random Variables.
• Mass function for a discrete random variable
• The expected value of a discrete random variable
• The variance of a discrete random variable
• Binomial Distribution
• Poisson Distribution (??t t can be time or area)

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Home work-1

• An experiment is designed test the potency of a
  drug on 20 rats, previous study have shown that a
  10-mg dose of the drug is lethal 5 of time
  within the first 4 hours.
• What is the probability of that 0 rat will die in
  the first 4 hours.
• What is the probability of that 1 rat will die in
  the first 4 hours.
• What is the probability of that 2 rat will die in
  the first 4 hours.
• What is the probability of that 3 or more rats
  will die in the first 4 hours.

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Home work-2

• An experiment is designed test the potency of a
  drug on 20 rats, previous study have shown that a
  20-mg dose of the drug is lethal 10 of time
  within the first 8 hours.
• What is the probability of that 0 rat will die in
  the first 8 hours.
• What is the probability of that 1 rat will die in
  the first 8 hours.
• What is the probability of that 2 rats will die
  in the first 8 hours.
• What is the probability of that 3 or more rats
  will die in the first 8 hours.

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Home work-3

• An drug company is designed test the potency of a
  bird flu vaccine on 2000 children, previous study
  have shown that one dose of the bird flu vaccine
  has side effect (i.e. fever) of 0.2 of time
  within the first 48 hours.
• What is the probability of that 0 child will
  experience side effect in the first 48 hours.
• What is the probability of that 1 child will
  experience side effect in the first 48 hours.
• What is the probability of that 2 children will
  experience side effect in the first 48 hours.
• What is the probability of that 3 or more
  children will experience side effect in the first
  48 hours