ESTIMATIONHYPOTHESIS TESTING FREQUENCY CLASSICAL BASIS Estimation Confidence Intervals

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ESTIMATION/HYPOTHESIS TESTING- FREQUENCY /
CLASSICAL BASISEstimation - Confidence Intervals

• From the statistical tables for a Standard Normal
  distribution, we note that Area Under
  From To Density Function 0.90 -1.64 1.64
  0.95 -1.96 1.96 0.99 -2.58 2.58
• From the central limit theorem, if and
  s2 are mean and variance of a random sample of
  size n, n gt 25 (from a large parent population),
  then we can say that 90 C.I. for parent mean m
• 
  where s replaces ?, ? unknown
• or

N (0,1)
?0.95?
0
-1.96
1.96
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Example-Attribute/Proportionate Sampling

• Attribute
• A random sample of size, n 25 has
  15 and s 2.
• Then a 95 confidence interval for m is
• i.e. 15 1.96 (2 / 5) so, C.I. is 14.22 to
  15.78
• Proportionate A random sample of size n 1000
  has p 0.40
• A 95 confidence interval for P is 0.40
  ?0.03 (i.e.) 0.37 to 0.43.

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Small Sampling Theory

• For reference purposes, useful to regard
  expression
• as the default formula for a confidence
  interval and to modify it to suit particular
  circumstances e.g. etc.
  
• Proportionate sampling, standard error (s.e.)
  term simplifies as follows
• (1-?) C.I. ? determines critical value 1.96,
  1.64 etc.
• If the sample size n lt 25, the Normal
  distribution must be replaced by Students t n -
  1 distribution.
• For sampling without replacement from a finite
  population, an fpc term must be used.

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Examples in brief

• A random sample of size n 10, drawn from a
  large parent population, has a mean of 12 and a
  standard deviation s 2. Then a 99 confidence
  interval for the parent mean is
• ie. 12 3.25 (2)/3
  that is an interval 9.83 to 14.17
• and 95 confidence limits for the parent mean is
  
• ie 12 2.262 (2)/3
  that is an interval 10.492 to 13.508.
• Note that for n 1000,
  for values of p between 0.3 and
  0.7.
• Refer to 3 swing or inherent error

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ESTIMATION - Rationale

• Estimator validity - good, bad, low, high
  confidence?
• Need measurement of statistical properties
  (variance, bias, distribution, confidence
  intervals)
• Bias
• where is the point estimate and ?
  the true parameter.
• Can be positive, negative or zero.
• Permits calculation of other properties,
  such as
• where this quantity and variance of
  estimator only the same if estimator is unbiased.
  Obtained by both analytical and bootstrap
  methods
• Similarly, for continuous variables
• or for b bootstrap replications,

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Estimation Rationale- contd.

• For any, even unbiased, estimator , still a
  difference between estimator and true parameter
  sampling error
• Hence the need for probability statements
  around
• with C.I. for estimator (T1 , T2),
  similarly to before and ? the confidence
  coefficient. If the estimator is unbiased in
  other words, ? is the probability that the true
  parameter falls into the interval.
• In general, confidence intervals can be
  determined using parametric and non-parametric
  approaches, where parametric construction needs a
  pivotal quantity variable which is a function
  of parameter and data, but whose distribution
  does not depend on the parameter.

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HYPOTHESIS TESTING - Rationale

• Starting Point of scientific research
• e.g. No Genetic Linkage between the genetic
  markers and genes
• when we design a linkage mapping
  experiment (Biological)
• H0 ? 0.5 (No Linkage)
  (2-locus linkage experiment)
• H1 ? ? 0.5 (two loci linked
  with specified R.F. 0.2)
• Critical Region
• Given a cumulative probability distribution
  of a test statistic, F(x) say, the critical
  region for the hypothesis test is the region of
  rejection in the distribution, i.e. the area
  under the probability curve where the observed
  test statistics value is unlikely to be observed
  if H0 true.
• 
  ? significance
  level

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HT Critical Regions and Symmetry

• For a symmetric 2-tailed hypothesis test
• 
  or
• distinction uni or bi-directional
  alternative hypotheses
• Non-Symmetric, 2-tailed
• For a0, b0, reduces to 1-tailed case

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HT-Critical Values and Significance

• Cut-off values Rejection and acceptance regions
  Critical Values, so hypothesis test can be
  interpreted as comparison between critical values
  and observed hypothesis test statistic, i.e.
• Significance Level p-value is the probability
  of observing a sample outcome if H0 true
• is cum. prob. that a statistic
  is less than observed test statistic for data
  under H0. For p-value less than or equal to ?, H0
  rejected at significance level ?

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HYPOTHESIS TESTING - Example
(1-sample)

• Suppose that it is claimed
• that the average survival time
• of patients with cancer at a specific site
• 60 months. A random sample of
• n 49 patents gives a mean of 55 with
• a standard deviation of 2. Is the sample
• finding consistent with the claim?
  rejection regions
• We regard the original claim as a null
  hypothesis (H0) which is tentatively accepted as
  true H0 m 60, with H1 m ?60
• If H0 true, test statistic
  as above

N(0,1)
0.95
1.96
-1.96
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Hypothesis Testing -POWER of the TEST

• Probability of False Positive and False Negative
  errors
• e.g. false positive if linkage between two
  genes declared, when really unlinked
• 
  Hypothesis Test Result
• Fact Accept
  H0 Reject H0
• H0 True
  1-? False positive
• 
  Type I error ?
• H0 False False
  negative Power of the Test
• Type
  II error? 1- ?
• Power of the Test or Statistical Power
  probability of rejecting H0 when correct to do
  so. (Related strictly to alternative hypothesis
  and ?)