Hypothesis testing

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Lecture - 6 Review Until now about one-third!

• Background
• Related terms
• Exploratory data analysis data presentation
• Normal distribution/ frequency curve
• Data transformation
• Measure of central tendency
• Measure of dispersion/variability

Whats remaining?

• Binomial and Poisson distributions
• Hypothesis formulation and testing
• Experimental designs ANOVA and multiple
  comparisons
• Regression and correlation
• Covariance analysis
• Non-parametric tests
• Miscellaneous

2
Normal distribution _ revision

• Continuous variables
• Class intervals
• Intermediate steps/scales e.g.
• - weight (10 g, 10.4g, 10.45 g. etc.)
• - similarly - length, distance, percentage, etc.

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Normal distribution
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Binomial distribution and probability

• Discrete series/nominal scale
• Two mutually exclusive category/class and their
  frequency/occurrence
• For examples,
• - male and female
• - head and tail
• - dead or live
• - white and black

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Binomial distribution

• Frequency bar
• Total 60 prawns sampled

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Binomial distribution
(p q)2 p2 2pq q2
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Binomial distribution becomes normal if it is
expanded
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Binomial distribution

• Expansion of binomial distribution
• (p q)1 p q
• (p q)2 p2 2pq q2
• (p q)3 p3 3p2q 3pq2 q3
• (p q)4 p4 4p3q 6p2q2 4pq3 q4
• (p q)5 p5 5p4q 10p3q2 10p2q3 5pq4
  q5
• ---
• (p q)n ??
• P (X) pk.qn-k x n! / k! (n-k)!
• Note as it was first described by Bernoulli
  (1654-1705) binomial distribution is also
  referred after his name

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• Application Examples
• Testing the sex ratio (Male female)
• Effectiveness of hormonal sex-reversal
• Progeny testing - super-male (YY) male
  technologies
• Breeding trials with animals, plants etc.
• Sex-ratio of population of your study areas e.g.
  during war time no. of male decreased
  significantly in some countries
• Gender migration studies etc.etc.

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Applications Mendel's Law of inheritance
(Genetics)
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Poisson distribution

• First described by Simeon-Denis Poisson
  (1781-1840)
• Random occurrences of an event
• Equal chances in space or time
• They are independent of each other occurrence
  of one event does not affect the others
• Mean Variance (? ?2)
• It approaches binomial distribution when n is
  large and p is small

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Poisson distribution
Poisson distribution
Poisson (random) ?2 ?
Clustered ?2 gt ?
Uniform/normal ?2 lt ?
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Circular distribution

• Data are in circular and interval scale e.g.
  degrees, radians etc. and should be presented
  using pie charts
• Examples,
• Per cent (0-100)
• Direction E, S, N, W (measured by a Compass)
• Longitude (distance)
• Time 0-24 hours, Weekdays, months, etc.

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Hypothesis formulation and testing
What is a hypothesis?
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Scientific method _revision
Nature
Observation
New related hypothesis
Alternate hypothesis
Hypothesis
Prediction
Design/propose experiment
Law (universal)
Experiment/trial
Generate/collect data
Theory
Modify
Analyze interpret data
Compare/relate
Support hypothesis
Reject hypothesis
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• What is a hypothesis?
• Starting point of scientific inquiry
  (discovery/invention)
• an assumption made for the sake of argument
• a tentative and normally testable statement that
  proposes a possible explanation to some
  phenomenon or event
• A hypothesis is an embryo which needs to be
  tested and developed to theory and law

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Hypothesis gt theory gt law

• Hypothesis implies insufficient evidence to
  provide more than a tentative explanation e.g. a
  hypothesis explaining the extinction of the
  dinosaurs, origin of the universe etc.
• Theory implies a greater range of evidence and
  greater likelihood of truth e.g. the theory of
  evolution
• Law implies a statement of order and relation in
  nature that has been found to be invariable under
  the same conditions e.g. the law of gravitation,
  law of inheritance (Mendel) etc.

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Type of Hypothesis

• H0 - Null hypothesis no difference
• H1 - Alternate hypothesis another is better

• Examples use tentative word may
• H0 - Salt in soil may not affect plant growth
• H1 - Salt in soil may affect plant growth
• Hypothesis (H1)
• Bird flu virus may also infect cat and then human
• Feeding Vit C may increase survival of fish fry
• 100 mg of Vit C/kg diet is needed to increase fry
  survival

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Hypothesis testing

• Is an intelligent guess based on limited and
  untested information
• Although experimental conclusions may match
  predicted results, we should not guess without
  proper testing and analysis we must test and
  confirm it
• Should be tested against new information using
  appropriate tool(s)
• Can be expressed in mathematical language e.g.
• H0 H1 there is no difference accept H0
• H0 ? H1 there is a difference reject H0

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Testing processes tools

• Requirements
• Design trial/survey
• Collect data or information (raw)
• Descriptive statistics (central locations and
  dispersions)
• Significance level confidence limits
  (intervals)
• Appropriate statistical tools/packages

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Significance (?) level

• Probability (P) of occurring any event by random
  error or chance
• Social sciences research 10
• Biological research 5
• Medical or laboratory analysis 1
• Physics 99.99 and 0.01 (?)
• In biology, if calculated P is lower than
• 5 or 0.05 ? significant ()
• 1 or 0.01 ? highly significant ()

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• In fact, there is no fixed level of significance-
  researchers themselves determine what it is.
• For example
• Cure for AIDS may consider plt0.40 adequate (i.e.
  Pgt0.60 that a certain cocktail is effective
  against AIDS)
• For drug for common cold probability as high as
  0.0001 may be necessary to convince that the
  treatment does not cause side effects.
• The scientists objective is to understand the
  system and not to find mindless mechanistic
  statistical significance

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Biological/substantive significance

• Effect size or substantive importance/
  significance or meaningfulness
• Statistically difference but that may not be able
  to cause any effect in the nature or the
  situation where we use it. Therefore, the
  difference to be a difference must make a
  difference e.g.
• For example (weight of fish in g)
• Sample 1 100.1, 100.2, 100.3, 100.1, 100.2,
  100.3
• Sample 2 100.4, 100.5, 100.5, 100.6, 100.6,
  100.4
• Means 100.2 and 100.5 (statistically
  significant)
• Difference 0.3 g (no biological significance)

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Confidence Interval and limits
Confidence interval
Lower limit (L1)
Higher limit (L2)
?1 95
1.96?
-1.96?
Mean ? 1.96 SE or (SD/vn)
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Use of confidence Interval and limits
Upper CI Mean Lower CI
Mean ? 1.96 SE or (SD/vn)
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Confidence Intervals (CI)

• Sample mean estimates the true mean and standard
  error describes the variability of that
  estimation
• This variability can be conveniently expressed in
  terms of probabilities by calculating CI
• For example
• Sample (n) 25 fish from a pond of 10, 000 fish,
  suppose we got -
• Mean 350g SD 75g
• L1 ?
• L2 ?
• -

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• Here, SE 75.v25 15 g
• If Mean 1SE, we can be confident that there is
  68 chance (p) that true mean is 35015
• 335 is the lower limit (L1)
• 365 is the upper limit (L2)
• But we would like to be 95 confident.
• This is done by multiplying the SE by t-value
• From the table for the critical values of the
  t-distribution
• t-statistics (t 0.05, 24) (SE) for n-1 24
• 2.064 15 31 g
• L1 350 31g 381 and L2 350-31319

We can now say with 95 confidence the true mean
falls between these limits (319-381 g)
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• CI increases as we demand greater and greater
  confidence e.g. for 99
• CI Mean ? 2.797 15 350 ? 42 g
• Conversely, if we have 0.01 or less probability
  (p) value, we are over 99 sure that it does not
  fall within that limit or (significantly
  different)

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Hypothesis testing - An example

• H1 Hypothesis (alternate hypothesis)
• Homemade pellet feed may give better yield of
  tilapia than the expensive commercial feed
• H0 null hypothesis
• There might be no difference in tilapia yield
  between two diets

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The test

• Four tanks each fed two types of feed
• Nursing results after 1 month
• Homemade feed (G1)
• 50.310.1g/fish (Mean 1SE)
• Commercial feed (G2)
• 69.19.2g/fish
• Difference in mean values 69.1 -50.3 18.8g
  i.e. 37 bigger than the fish of G1, if it is
  statistically proved (of course this difference
  have biological meaning)
• Should the null hypothesis be accepted or
  rejected?

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Statistical errors in hypothesis testing

• It will not be correct if we reject the null
  hypothesis (there are no difference between the
  two means) or accept the null hypothesis (no
  difference between two means)
• Two types of errors one can make
• Type I error- claiming true significant
  difference when there is none
• Type II error- claiming no truly significant
  difference when there is significant difference

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Statistical error in hypothesis testing

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• Scientists generally prefer to decrease the
  possibility of making a Type 1 error -
• It is better to miss significant difference/
  relationship when there was, than to claim
  significant when there was none
• Selection of statistical tools has an important
  role in not committing these errors.

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Summary of statistical tools
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Statistical tables http//www.statsoft.com/textbo
ok/stathome.html
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New Year! Enjoy yourself!
Thank you!