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Lecture_03.md

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Lecture #3

Probability

probability - the chance that something happens

  • example of crossing street

    not hit: IIIII IIIII IIIII

    hit: I

$$ \frac{\mbox{(hit)}}{\mbox{(hit + not hit)}} = \frac{\mbox{(# of outcomes)}}{\mbox{(# of trials)}} $$

$$ 0 \le p \le 1 $$

Variation

$$ mean = \bar{x} = \frac{\displaystyle{\sum_{i=1}^{n}x_i}}{n} $$

$$ variance = \sigma^2_x = \frac{\displaystyle{\sum_{i=1}^{n}(x_i - \bar{x})^2}}{(n - 1)} $$

IMAGE 3.1 Normal distribution

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measurement variation biological variation
measurement error,
measurement method,
measurement conditions		 space, time, species,
age, sex, genotype

randomness - a mixture of measurement error and sources of variation too complex to measure and/or not of primary interest

null hypothesis -( “no effect” H0) observed differences among groups reflect measurement error and/or other sources of unspecified variation

Hypothesis Testing

alternative hypothesis - NOT H0

statistical hypothesis - a test of whether the pattern in the data is better explained by H0 (null hypothesis) or NOT H0 (alternative hypothesis)

IMAGE 3.2 Null and alternative hypothesis for sex differences

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Statistical P-Values

statistical p - probability of obtaining the observed results (or something more extreme) if the null hypothesis (H0) were true (p(data|H0)

  • if p is “large” (p = 0.87), no evidence to reject H0
  • if p is “small” (p = 0.02), sufficient evidence to reject H0

INCORRECT definition of statistical p - the probability that H0 is true

Graphing of Variables

discrete variable - groupings (sex,age,genotype)

continuous variable - continuous numeric scale (mass, abundance)

Discrete Predictor Variables (t-test, ANOVA)

IMAGE 3.3 schematic of bar chart

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IMAGE 3.4 null versus alternative bar chart

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Continuous Predictor Variablers (Linear Regression)

IMAGE 3.5 schematic of linear regression

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IMAGE 3.6 null versus alternative regression model

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Testing Hypotheses

HBiol = biological hypothesis (= mechanism)

H0 = null hypothesis (= pattern)

IMAGE 3.7 flow chart for logic tree of HBiol and H0

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Sample Inference Problem

  • HBiol In high-alpine lakes, increased algal biomass increases fish species diversity (“bottom-up control”)

IMAGE 3.8 Schematic for outcome of fish experiment

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Determinants of Statistical P-Value

  • larger sample size ($N$) $\rightarrow$ small p
  • larger differences among group means $(\bar{G_1} - \bar{G_2})$ $\rightarrow$ small p
  • larger within-group variance $\sigma^2$ $\rightarrow$ large p

Type I and Type II Statistical Errors

  • Reality = H0 true or H0 false

  • Conclusion = not reject H0 or reject H0

  • stomach ache in the night H0 not true:appendicitis; H0 true: no appendicitis

H0 True H0 FALSE
not reject H0 correct decision
(go home)		 Type II statistical error
(go home)
reject H0 Type I statistical error
(operate)		 correct decision
(operate)

Type I Statistical Error - incorrectly rejecting a null hypothesis that is true (error of falsity)

Type II Statistical Error - incorrectly accepting a null hypothesis that is false (error of ignorance)

  • which kind of error is more serious?
  • in clinical applications, usually assume Type II error is more important
  • in standard science applications, priority is placed on Type I error
    • errors of ignorance are less serious than errors of falsity
    • Type I error is measured readily in mainstream statistics, but Type II error is much harder (Bayesian analysis)

alternative definition for statistical p - probability of making a Type I error by rejecting H0