LECTURE 1: introduction to statistics

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# LECTURE 1: introduction to statistics
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## FANR 6750 (Experimental design)
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.author[
### Fall 2024
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# outline

#### 1) What is statistics?

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#### 2) Statistics and the scientific method

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

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`$$Statistics = Information + Uncertainty$$`

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# why do we need statistics?

### Common tasks

- Identify relationships between variables

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- Estimate unknown parameters

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- Test hypotheses

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- Describe stochastic systems

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- Make predictions that account for uncertainty

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# stats and the scientific method

#### Ways of learnings

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**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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**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 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

2) **Hypotheses**

3) **Predictions**

4) **Data collection**1

5) **Models**1

.footnote[[1] We'll get to these!]

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# causal inference

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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

+ 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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# uncertainty

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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

### 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

#### 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

#### 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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- Median

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- 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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- Sample standard deviation

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

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- Range

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# sampling error

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

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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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# looking ahead

### **Next time**: Introduction to linear models

### **Reading**: [Fieberg chp. 1.2-1.4](https://statistics4ecologists-v1.netlify.app/linreg.html#data-example-sustainable-trophy-hunting-of-male-african-lions)