Two Classes Meet the Bell Curve

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Two Classes Meet the Bell Curve

• December 2004
• MUPGRET Workshop

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Math and Science

• Mathematics is an integral part of science.
• Used every day by bench scientists to perform
  experiments, interpret data, and make predictions.

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Statistics and Science

• Necessity for analyzing datasets
• Experiment must be well designed to be meaningful
• Ex. replications and controls
• Should know how youll analyze data before you
  start the experiment

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Data Analysis

• Data Come in Different Types
• Testing How Well Data Fit Hypotheses

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Data Types

• Yes or No (Qualitative Discontinuous)
• ---Ratios of Two or More Classes
• How Much? (Quantitative Continuous)
• ---Frequencies of Different Measurements
• But These Two Shade into Each Other
• ---Depending on Numbers Observed and on
  Measurement Discreteness

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Statistical Testing for Fit

• For Ratios, Chi-Square tests are often used
• For Frequencies, means, standard deviations, and
  linear regression are often used

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Chi-squared

• Tests if your ratios are statistically different
  from your expectation. Can be applied to any set
  of ratios.
• For example, do your data fit the 31 hypothesis?
• Chi-squared
• ?(observed-expected)2/expected

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Replications

• Give a better estimate of the true mean.
• Help to remove environmental variation from
  measurements.
• Reduce noise.
• Reduce effect of outliers in the dataset.

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Outliers
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Mean

• Average of a group of datapoints.
• Treatment mean
• Replicate mean
• Grand mean

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Standard Deviation

• The difference between the mean treatment value
  and the grand mean.
• Can think of it as the distance of the mean
  treatment value from the line of best fit.

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Linear regression

• Line of best fit.
• Algebraic equation.

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Genetic Models, Simple
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One Gene, Two Genes,
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Four Genes,
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Six Genes, Twelve Genes
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Genetic Models, Complex
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Genetic Models, Whats This?
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Continuous Distributions

• Test if your distributions are statistically
  different from hypothetical distributions.
• For example, do your measured data fit with
  chance, or are they biased?
• Mean, Standard Deviation

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The Bell Curve
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Testing Selection Advance
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High Heritability!
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Lower Heritability!
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Probability

• Tests the likelihood that something will or will
  not occur.
• Used extensively in everyday life.
• Las Vegas type gaming
• Lotto
• Insurance amortization
• Decisions regarding medical treatment

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Everyday examples

• Rolling the dice
• 1 in 6 chance that you will roll a one with a
  single die.
• (1/6)2 1/36 chance you will roll snake eyes.
• Playing cards
• 4 in 52 chance (1/13) of drawing an ace at random
  from a deck.
• Whats the chance of a full house?

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Biology examples

• Punnett square
• Nucleotide frequencies along a gene are used to
  examine evolutionary forces.
• Mutation rates
• Testing limits and sample sizes for transgenics.
• DNA forensics

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Mendels Results
Parent Cross F1 Phenotype F2 data
Round x wrinkled Round 5474 1850
Yellow x green Yellow 6022 2001
Purple x white Purple 705 224
Inflated x constricted pod Inflated 882 299
Green x yellow pod Green 428 152
Axial x terminal flower Axial 651 207
Long x short stem Long 787 277
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Important Observations

• F1 progeny are heterozygous but express only one
  phenotype, the dominant one.
• In the F2 generation plants with both phenotypes
  are observed?some plants have recovered the
  recessive phenotype.
• In the F2 generation there are approximately
  three times as many of one phenotype as the
  other.

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3 1 Ratio

• The 3 1 ratio is the key to interpreting
  Mendels data and the foundation for the the
  principle of segregation.

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Punnett Square
A (½) a (½)
A (½) AA (½ x ½ ¼) Aa (½ x ½ ¼)
a(½) Aa (½ x ½ ¼) aa (½ x ½ ¼)
Male
Female
¼ AA ½ Aa ¼ aa
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A Molecular View
Parents
F1
F2 Progeny
WW ww Ww ¼WW ¼Ww ¼wW ¼ww
1 2 1 Genotype 3 1 Phenotype
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Alleles

• People have thousands of genes.
• Each gene has one to many alleles.
• Each allele has a different DNA sequence.
• Some DNA differences are small, some large.
• Some allelic differences result in different
  phenotypes, e.g., brown vs. blue eyes.
• Frequencies of alleles vary.

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Molecularly Differing Alleles
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Using and Predicting

• How often is a given allele from a heterozygous
  parent transmitted to offspring?
• How often is an allele in a population,
  occurring at a frequency of 0.1, found in a
  sample of individuals of size n?
• How large a sample of individuals from a
  population is needed to be 95 sure of including
  at least one individual with an allele that is
  present at frequency p?