LECTURE 1: introduction to statistics

Outline

1) What is statistics?

2) Statistics and the scientific method

3) Populations vs. samples

4) Causal inference

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)

\[Statistics = Information + Uncertainty\]

Why do we need statistics?

Common tasks

• Identify relationships between variables
  

• Estimate unknown parameters
  

• Test hypotheses
  

• Describe stochastic systems
  

• Make predictions that account for uncertainty

Statistics and the scientific method

Ways of learning

Inductive reasoning

• Often attributed to Francis Bacon

• 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

Statistics and the scientific method

Ways of learning (real world)

1. Pattern identification (i.e., exploratory studies)
   • Anecdotes
   • Correlations/visual analysis
   • Exploratory modeling (i.e., fishing)

Statistics and the scientific method

Ways of learning (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

Statistics and the scientific method

Ways of learning (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”

Statistics and the scientific method

Ways of learning (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

Statistics and the scientific method

Ways of learning (real world)

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

2. Hypothesis formation

3. Predictions

4. Data collection

5. Models and testing

   • Model = mathematical abstraction of hypothesis
   • Model used to “confront” hypothesis with data (via predictions)
   • Draw conclusions: Does data support hypothesis?

Statistics 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. Models²

Uncertainty

Populations vs samples

Hypothesis: New plant variety is more disease resistant than current variety³

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

• A collection of subjects of interest

• Often, a biologically meaningful unit

• Sometimes a process of interest

Question: What is the population in our example?

Population

Note:

1. These are the populations

2. There is variation within and among populations, but:

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

I say the populations because these distributions represent all individuals of each variety. Because we know (in this example) prevalence in every individual, there is no uncertainty about whether the hypothesis is true or not. The mean of each population tells us exactly how different prevalence is between the two varities. Of course, in reality, we could never measure every individual in each population so cannot know the true difference in prevalence

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

Sample

Each orange and blue tick mark under the distributions shows the prevalence value of a single observation in each sample. The histogram bars show the observed frequency of prevalence values in the samples.

Sample

Note:

1. These are samples

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

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

Summary statistics

Measures of central tendency

• Sample mean

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

• Median

• Mode

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

Sampling error

Every sample has a different mean (and standard deviation)⁴

Sampling = uncertainty

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

• But sampling is inherently stochastic

  • Sampling produces uncertainty

  • Unavoidable (but that’s ok!)

• 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

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

Causal inference

Causal inference

 

• Often, we want to know whether \(x\) influences \(y\)

  • In other words, if we change \(x\), will \(y\) also change (and by how much)?

• Statistical models measure associations between variables

  • In other words, when \(x\) changes, \(y\) also tends to change⁵

• Do associations tell us anything about cause and effect?

  • Correlation is not causation

Causal inference

• If correlation is not causation, what is it?

• \(Correlation = Causation + Confounding + Noise\)

• Confounding: Both \(x\) and \(y\) are influenced by a third variable \(z\)

• Separating causation, confounding, and noise is one of the central challenges of scientific inference

  • Statistics can help with this challenge but cannot tell us whether associations we observe are the result of causation, confounding, or noise

  • Manipulative experiments are generally considered the “gold standard” for causal inference because of randomization

  • But causal inference from observational studies is also possible

  • We will discuss all of these topics in more detail throughout the semester

Sampling error is the main source of noise and can make associations larger or smaller than they actually are. We will return to this point throughout the semester because of accounting for the risk of spurious associations due to sampling error is one of the main goals of statistical inference

Summary

• Statistical models allow us to quantify associations between variables

• Statistical models also allow us to makes inferences about associations in the face of uncertainty

• Uncertainty stems from multiple processes, the most fundamental of which is sampling

  • Inferences based on finite samples from a population will always be uncertain

  • Sampling error can make associations look bigger or smaller than they actually are

  • Sampling error is one of the reasons statistical associations may not be causal (or even real)

• We often want to know if one variable causes another but drawing causal inferences is challenging

  • As we will see, causal inference is primarily a scientific challenge, not a statistical one

Looking ahead

Next time: Introduction to linear models

Reading: Fieberg chp. 1.2-1.4

Footnotes

1. We’ll get to these!

2. We’ll get to these!

3. Note that this hypothesis is causal. Hypotheses should, in general, be causal since our goal is (usually) to learn about the mechanisms that underlie patterns we observe

4. We have now mentioned two types of uncertainty: The sample mean will never equal the population mean and no two samples will ever have exactly the same mean

5. Note the subtle difference here - in the previous sentence, we said “we change \(x\). Here, we say”when \(x\) changes. In one case, our definition is based on intervening in the system to specifically change the value of \(x\). If a change in \(y\) follows, we can (under certain circumstances) conclude \(x\) causes \(y\). In the other case, we observe a change in \(x\). Generally, we can’t conclude \(x\) causes \(y\) in that case. Why not?