Intro to Statistical Inference

M. Drew LaMar
April 8, 2022

“To try to make a model of an atom by studying its spectrum is like trying to make a model of a grand piano by listening to the noise it makes when thrown downstairs.”

- Anonymous

Introduction to Quantitative Biology, Spring 2022

Class announcements

  • See Homework #7 assignment on Blackboard
  • Reading assignment: OpenIntro Stats, Chapters 1 and 2

Introduction to Statistical Inference

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Populations vs Samples

Definition: A parameter is a quantity describing a population, whereas an estimate or statistic is a related quantity calculated from a sample.

Parameter examples: Averages, proportions, measures of variation, and measures of relationship

What is statistics?

Statistics is a technology that describes and measures aspects of nature from samples.

Statistics lets us quantify the uncertainty of these measures.

Statistics makes it possible to determine the likely magnitude of measurements departure from the “truth”.

Statistics is about estimation, the process of inferring an unknown quantity of a target population using sample data.

What is statistics?

The two sides of the statistical coin:

  • Parameter estimation
  • Hypothesis testing
Definition: A statistical hypothesis is a specific claim regarding a population parameter.
Definition: Hypothesis testing uses data to evaluate evidence for or against statistical hypotheses.

What is statistics? Parameter estimation

The two sides of the statistical coin:

  • Parameter estimation
  • Hypothesis testing

Example: A trapping study measures the rate of fruit fall in forest clear-cuts.

What is statistics? Hypothesis testing

The two sides of the statistical coin:

  • Parameter estimation
  • Hypothesis testing

Example: A clinical trial is carried out to determine whether taking large doses of vitamin C benefits health of advanced cancer patients.

What is probability?

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Probability comes first!

…well, most of the time.

  • Many statistical techniques require assumptions about where your data is coming from (i.e. properties of the population)
  • In other words, an assumed probability model describes the population
  • Statistical techniques that are based on probability models are called parametric techniques, while those that are not are called non-parametric techniques.

Data as Information

For your question, there is desired and undesired information in your data.

Goals:

  • Get accurate information by reducing bias
  • Get precise information by reducing sampling error due to random variation (increase signal-to-noise ratio)
Definition: Bias is a systematic discrepancy between the estimates we would obtain, if we could sample a population again and again, and the true population characteristic.

Data as Information

For your question, there is desired and undesired information in your data.

Goals:

  • Get accurate information by reducing bias
  • Get precise information by reducing sampling error due to random variation (increase signal-to-noise ratio)
Definition: Sampling error is the difference between an estimate and the population parameter being estimated caused by chance.

Precision vs Accuracy

Data as Information

For your question, there is desired and undesired information in your data.

Goals:

  • Isolate desired information by reducing or controlling for confounding factors (i.e. undesired information)

“The aim … is to provide a clear and rigorous basis for determining when a causal ordering can be said to hold between two variables or groups of variables in a model…”

- H. Simon