Data Generating Process

2025-01-21

Data

  • Data

  • Measurements

  • Study Design

  • Sampling

  • Data Generating Process

  • Statistical Model

Observations

An observation is the unit which contains information to be obtained.

Also known as experimental unit.

Data

Data is information from a particular experimental unit. There can be more that one piece of information per experimental unit.

Data Structure

The structure of the data can be represented in various forms:

  • A list of long numbers

  • Table

Tabular Form

Measurements

  • Data

  • Measurements

  • Study Design

  • Sampling

  • Data Generating Process

  • Statistical Model

Measurements

Measurements are the processes where we represent an attribute of an experimental unit as either a number or category.

Quantitative Measurements

Quantitative measurements are data measurements that take a numeric form.

Qualitative Measurements

Qualitative measurements are data measurements that take a certain category.

Value

A value is a description (number or category) of a specific attribute of an experimental unit.

Variable

A variable is the descriptive attribute that we want to obtain from an experimental unit. In terms of a data set, the variable contains all the values of specific attribute in a sample.

Study Design

  • Data

  • Measurements

  • Study Design

  • Sampling

  • Data Generating Process

  • Statistical Model

Research Question

A research question is designed to create new knowledge of certain phenomenons observed in the world.

This is designed by conducting a research projects that systematically answers the question.

The study design is the procedure in which data is collected to answer the question, while reducing any potential bias while conducting the study.

Study Design

This is the procedure designed to answer a research question.

Entails procedures to collect data that answers a research question.

Dictates how the data will be analyzed.

Determines how the data is quanitified and what is the experimental unit.

Sampling

  • Data

  • Measurements

  • Study Design

  • Sampling

  • Data Generating Process

  • Statistical Model

Sampling

Sampling is the process of selecting units from a population interest to collect data from.

Image Provided by Simply Psychology.

Sampling

When Sampling you want to maintain these properties:

  • Representative sample

  • Large enough sample size

Sampling Example

We want to answer the question, are people happy?

Random Sampling

When sampling, we strive for random sampling

  • Each unit in the population of interest must have an equal probability of being selected for the study.

  • This ensures a representative population

Independent Sampling

In addition to random sampling, we strive to make sure each unit is independent from each other.

  • The probability of UNIT A being sampled will not affect the probability of UNIT B to be sampled.

Sampling Variation

  • Random samples may vary from the population of interest.

  • Due to randomness, samples many not look the or biased.

  • However, this is to be expected as because the sample will not be biased in one way or another.

  • Samples are then considered unbiased as long as experimental units were collected randomly.

Data Generating Process

  • Data

  • Measurements

  • Study Design

  • Sampling

  • Data Generating Process

  • Statistical Model

Data

The measurement collected from an experimental unit.

Data Distribution

When thinking about data, we know an attribute is allowed to vary. With this variation, some numbers are more likely to be observed than others.

Forest from the trees

Imamge Provided by Fine Art America.

Sample

When inspecting data:

  • Do not focus on one individual data point.

  • See how data points is in relation to other data points.

  • See what is common

  • See what is rare

Data Generation Process (DGP)

The data generation process is understanding how variation from the population is transferred to the data collected.

A population has a mechanism to produce data, understanding this mechanism is essential understanding the data.

DGP

A populations DGP can be defined with the following characteristics:

  • The potential outcomes that can be observed when measuring

  • Each potential outcome will have a probability of being observed

    • The probability must be between 0 and 1

  • Sum of all the probabilities of each outcome will add up to 1

Flipping a Coin

Flipping a coin results in either heads or tail. The probability for heads is 50%

The DGP of flipping a coin is the process of selecting an outcome, given the probability of both options are 50%.

Rolling a die

Measuring Body Temperature

Image Provided by Thermofisher Scientific.

Inference

How do we use DGP and sampling to understand the world?

We can use a sample to understand the DGP.

We can use the DGP to understand the sample.

Visualizing a DGP

Visualizing DGP

Example

Statistical Model

  • Data

  • Measurements

  • Study Design

  • Sampling

  • Data Generating Process

  • Statistical Model

Statistical Model

\[ Y \sim F(\theta) \]

  • \(Y\): A data value generated for DGP
  • \(F\): A generic mathematical model that describes the DGP
  • \(\theta\): Parameter controlling the shape and form of the DGP

Statistical Model

\[ Y = \mu + \varepsilon \]

  • \(\mu\): The average value we expect to see
  • \(\varepsilon \sim F(\cdot)\): Error Term

Statistical Model

\[ Y = \mu + \varepsilon \]

\(\mu\): Explained Part

  • This is what we expect to see if we took an educated guess
  • The educated guess can incorporate information that is related to \(Y\)

\(\varepsilon\): Unexplained Part

  • Randomness
  • Incorporates several types of variation
    • Measurement Error
    • Biological Error
    • Environmental Error