LECTURE 1: introduction to statistics
FANR 6750 (Experimental design)

Fall 2024
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outline

1) What is statistics?

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outline

1) What is statistics?

2) Statistics and the scientific method

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outline

1) What is statistics?

2) Statistics and the scientific method

3) Populations vs. samples

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what is statistics?

The study of the collection, analysis, interpretation, presentation, and organization of data (Dodge 2006)

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what is statistics?

The study of the collection, analysis, interpretation, presentation, and organization of data (Dodge 2006)

The science of learning from data in the face of uncertainty (various)

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what is statistics?

The study of the collection, analysis, interpretation, presentation, and organization of data (Dodge 2006)

The science of learning from data in the face of uncertainty (various)

$Statistics = Information + Uncertainty$

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why do we need statistics?Common tasks4 / 25

why do we need statistics?Common tasksIdentify relationships between variables

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why do we need statistics?Common tasksIdentify relationships between variables

Estimate unknown parameters

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why do we need statistics?Common tasksIdentify relationships between variables

Estimate unknown parameters

Test hypotheses

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why do we need statistics?Common tasksIdentify relationships between variables

Estimate unknown parameters

Test hypotheses

Describe stochastic systems

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why do we need statistics?Common tasksIdentify relationships between variables

Estimate unknown parameters

Test hypotheses

Describe stochastic systems

Make predictions that account for uncertainty  
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stats and the scientific methodWays of learnings5 / 25

stats and the scientific method

Ways of learnings

Inductive reasoning

Often attributed to Francis Bacon (and others)
Consistent observations -> general principle
Problem: "confirmatory" observations can't disprove theory
Example: I've only seen birds that fly :: all birds can fly

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stats and the scientific method

Ways of learnings

Inductive reasoning

Often attributed to Francis Bacon (and others)
Consistent observations -> general principle
Problem: "confirmatory" observations can't disprove theory
Example: I've only seen birds that fly :: all birds can fly

Deductive reasoning

Formalized by Karl Popper
Theory -> predictions -> observations
Based on falsification
Example: All birds can fly :: penguins are birds :: penguins can fly

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stats and the scientific methodWays of learnings (real world)6 / 25

stats and the scientific method

Ways of learnings (real world)

1) Pattern identification (i.e., exploratory studies)

Anecdotes
Correlations/visual analysis
Exploratory modeling (i.e., fishing)

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stats and the scientific method

Ways of learnings (real world)

1) Pattern identification (i.e., exploratory studies)

2) Hypothesis formation

Formed from patterns
Should focus on mechanisms ("because", "controls", "adapted to")
Should be falsifiable
Ideally > 1 alternatives

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stats and the scientific method

Ways of learnings (real world)

1) Pattern identification (i.e., exploratory studies)

2) Hypothesis formation

3) Predictions

If the hypothesis is true, what do you expect to see?
Focus on things we can measure
More = better
"associated", "correlated", "greater/less than"

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stats and the scientific method

Ways of learnings (real world)

1) Pattern identification (i.e., exploratory studies)

2) Hypothesis formation

3) Predictions

4) Data collection

Can be observational but ideally manipulative experiment
Sampling must be designed to answer question

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stats and the scientific method

Ways of learnings (real world)

1) Pattern identification (i.e., exploratory studies)

2) Hypothesis formation

3) Predictions

4) Data collection

5) Models and testing

Model is mathematical abstraction of hypothesis
Model used to "confront" hypothesis with data (via predictions)
Draw conclusions: Does data support hypothesis?

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stats and the scientific method

Example

1) Pattern: Trees at higher elevations are shorter than at low elevations

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stats and the scientific method

Example

1) Pattern: Trees at higher elevations are shorter than at low elevations

2) Hypotheses

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stats and the scientific method

Example

1) Pattern: Trees at higher elevations are shorter than at low elevations

2) Hypotheses

3) Predictions

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stats and the scientific method

Example

1) Pattern: Trees at higher elevations are shorter than at low elevations

2) Hypotheses

3) Predictions

4) Data collection¹

5) Models¹

[1] We'll get to these!

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causal inference12 / 25

causal inferenceOften, we want to know whether x influences y13 / 25

causal inferenceOften, we want to know whether x influences yIn other words, if we change x, will y change also (and by how much)? 
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causal inference

Often, we want to know whether $x$ influences $y$

In other words, if we change $x$ , will $y$ change also (and by how much)?
Harder than it seems! Why?

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causal inference

Often, we want to know whether $x$ influences $y$

In other words, if we change $x$ , will $y$ change also (and by how much)?
Harder than it seems! Why?
Generally restricted to manipulative experiments

13 / 25

causal inference

Often, we want to know whether $x$ influences $y$

In other words, if we change $x$ , will $y$ change also (and by how much)?
Harder than it seems! Why?
Generally restricted to manipulative experiments
- Well-designed experiments ensure that "treatment assignment is independent of the potential outcomes" (Gelman et al. 2021)

13 / 25

causal inference

Often, we want to know whether $x$ influences $y$

In other words, if we change $x$ , will $y$ change also (and by how much)?
Harder than it seems! Why?
Generally restricted to manipulative experiments
- Well-designed experiments ensure that "treatment assignment is independent of the potential outcomes" (Gelman et al. 2021)

Increasing interest in causal inference from observational studies
- This is something we may discuss later in the semester

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uncertainty14 / 25

populations vs samples

Hypothesis: New plant variety is more disease resistant than current variety

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populations vs samples

Hypothesis: New plant variety is more disease resistant than current variety

Prediction: Disease prevalence is lower in new variety than in current variety

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populations vs samples

Hypothesis: New plant variety is more disease resistant than current variety

Prediction: Disease prevalence is lower in new variety than in current variety

How can we determine whether the hypothesis is true?

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populations vs samples

Population

A collection of subjects of interest
Often, a biologically meaningful unit
Sometimes a process of interest

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populations vs samples

Population

A collection of subjects of interest
Often, a biologically meaningful unit
Sometimes a process of interest

Question: What is the population in our example?

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population

Note:

1) This is the population

2) There is variation within and among groups, but:

3) The hypothesis is correct (mean prevalence is lower in new vs. current variety)

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populations vs samples

Population

A collection of subjects of interest
Often, a biologically meaningful unit
Sometimes a process of interest

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populations vs samples

Population

A collection of subjects of interest
Often, a biologically meaningful unit
Sometimes a process of interest

Sample

A finite subset of the population of interest, i.e. the data we collect
Samples allow us to draw inferences about the population
Good samples are:
- Random
- Representative
- Sufficiently large

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sample

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sample

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sample

Note:

1) This is a sample

2) The sample means are our best estimates of the population means

3) But the sample means will never equal the population means (uncertainty!)

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summary statistics

Measures of central tendency

Sample mean

$\large \bar{y} = \frac{\sum_{i=1}^n y_i}{n}$

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summary statistics

Measures of central tendency

Sample mean

$\large \bar{y} = \frac{\sum_{i=1}^n y_i}{n}$

Median

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summary statistics

Measures of central tendency

Sample mean

$\large \bar{y} = \frac{\sum_{i=1}^n y_i}{n}$

Median

Mode

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summary statistics

Measures of dispersion

Sample variance

$\large s^2 = \frac{\sum_{i=1}^n (y_i - \bar{y})^2}{n-1}$

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summary statistics

Measures of dispersion

Sample variance

$\large s^2 = \frac{\sum_{i=1}^n (y_i - \bar{y})^2}{n-1}$

Sample standard deviation

$\large s = \sqrt{s^2}$

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summary statistics

Measures of dispersion

Sample variance

$\large s^2 = \frac{\sum_{i=1}^n (y_i - \bar{y})^2}{n-1}$

Sample standard deviation

$\large s = \sqrt{s^2}$

Range

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sampling error

Every sample has a different mean (and standard deviation) - more uncertainty!

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sampling = uncertaintyBecause populations (usually) cannot be measured, sampling is essential24 / 25

sampling = uncertainty

Because populations (usually) cannot be measured, sampling is essential

But sampling is inherently stochastic

sampling produces uncertainty
unavoidable (but that's ok!)

Doubt is not a pleasant condition, but certainty is absurd -- Voltaire

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sampling = uncertainty

Because populations (usually) cannot be measured, sampling is essential

But sampling is inherently stochastic

sampling produces uncertainty
unavoidable (but that's ok!)

Doubt is not a pleasant condition, but certainty is absurd -- Voltaire

Statistics is what allows us to learn about the population using samples in the face of uncertainty

the primary goal of this class is for you to understand how to make robust inferences that account for uncertainty (and the limitations of those inferences)
we will return to this basic concept (sampling error) many times this semester

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LECTURE 1: introduction to statistics

FANR 6750 (Experimental design)

Fall 2024

outline

1) What is statistics?

outline

1) What is statistics?

2) Statistics and the scientific method

outline

1) What is statistics?

2) Statistics and the scientific method

3) Populations vs. samples

what is statistics?

what is statistics?

what is statistics?

why do we need statistics?

Common tasks

why do we need statistics?

Common tasks

why do we need statistics?

Common tasks

why do we need statistics?

Common tasks

why do we need statistics?

Common tasks

why do we need statistics?

Common tasks

stats and the scientific method

Ways of learnings

stats and the scientific method

Ways of learnings

stats and the scientific method

Ways of learnings

stats and the scientific method

Ways of learnings (real world)

stats and the scientific method

Ways of learnings (real world)

stats and the scientific method

Ways of learnings (real world)

stats and the scientific method

Ways of learnings (real world)

stats and the scientific method

Ways of learnings (real world)

stats and the scientific method

Ways of learnings (real world)

stats and the scientific method

Example

stats and the scientific method

Example

stats and the scientific method

Example

stats and the scientific method

Example

causal inference

causal inference

Often, we want to know whether xx influences yy

causal inference

Often, we want to know whether xx influences yy

causal inference

Often, we want to know whether xx influences yy

causal inference

Often, we want to know whether xx influences yy

causal inference

Often, we want to know whether xx influences yy

causal inference

Often, we want to know whether xx influences yy

uncertainty

populations vs samples

Hypothesis: New plant variety is more disease resistant than current variety

populations vs samples

Hypothesis: New plant variety is more disease resistant than current variety

Prediction: Disease prevalence is lower in new variety than in current variety

populations vs samples

Hypothesis: New plant variety is more disease resistant than current variety

Prediction: Disease prevalence is lower in new variety than in current variety

How can we determine whether the hypothesis is true?

populations vs samples

Population

populations vs samples

Population

Often, we want to know whether $x$ influences $y$

Often, we want to know whether $x$ influences $y$

Often, we want to know whether $x$ influences $y$

Often, we want to know whether $x$ influences $y$

Often, we want to know whether $x$ influences $y$

Often, we want to know whether $x$ influences $y$