QTM 100 Lecture 2 Notes: Describing Variables and Study Design

QTM 100 Lecture 2 Notes: Describing Variables and Study Design

• Purpose of lecture: Introduce how to describe variables, distinguish variable types, and compare observational studies versus experiments; cover sampling design, bias, and key experimental design concepts.

Describing Variables

• A variable is any characteristic observed in a study.

• Categorical variables

  • Observations belong to one of a set of categories.

  • Typically contain descriptive words or phrases.

  • Examples: gender (male/female), types of shipping (standard, firstClass, upsGround, etc.).

• Quantitative variables

  • Observations take on numeric values.

• For any data set, rows = observations and columns = variables.

  • Example data set (Mario Kart on eBay):

  • Variables include: obs, nBids, cond, startPr, totalPr, shipSp, wheels.

  • A sample row: 1 20 new 0.99 51.55 standard 1

  • The data frame example ends with: 143 observations and 13 variables.

• Describing quantitative variables further

  • Quantitative variables can be discrete or continuous.

  • Discrete: finite number of possible values (e.g., counts like 0, 1, 2, …).

  • Continuous: a continuum of possible values (e.g., times, measurements).

  • Example illustrating continuous data: times for the 200m race at the Olympics (e.g., 1:54.27, 1:54.90, 1:56:22, etc.).

• Describing categorical variables further

  • A categorical variable with two categories is dichotomous (e.g., gender: male/female).

  • A categorical variable with more than two categories is simply multi-category; some have a natural ordering (ordinal), e.g., year in college: freshman, sophomore, junior, senior.

• Response vs explanatory variables (in data analysis)

  • Both can be categorical or quantitative.

  • There is an association between the response and an explanatory variable when a relationship exists.

  • If no association is present, the response and explanatory variables are independent.

• Quick conceptual notes

  • An association does not imply causation in observational studies.

  • The way data are collected (the design) influences what conclusions are valid.

Types of Studies

• Types of studies distinguish how data are collected and whether treatments are assigned.

• Observational studies

  • Researchers observe the response and the explanatory variable without assigning a treatment.

  • Non-experimental; potential confounding variables can affect results.

  • Association does not equal causation in general.

• Experiments

  • Subjects are assigned to experimental conditions (treatments) and then the response is observed.

  • Experimental conditions are called treatments.

  • A key advantage: stronger potential to infer causation due to randomization and control of variables.

Observational Studies: Types

• Prospective studies (cohort)

  • Identify a group and observe them over a period of time; characteristics recorded along the way.

• Retrospective studies (case-control)

  • Look back in time or use existing records.

• Cross-sectional studies

  • Collect information about individuals at a single point in time or over a very short period.

Conducting an Experiment vs Observational Study (examples)

• Research question example: does exercise improve energy level?

  • Observational approach: select people with/without exercise habits and measure energy level.

  • Experimental approach: randomly assign subjects to exercise vs no exercise and compare energy levels.

• General takeaway: experimental designs can better isolate causal effects than observational designs.

Sampling Design and Generalizability

• Goal of sampling: obtain a representative sample so that inferences generalize to the population.

• If the sample is not representative, results may be biased.

• Bias sources in observational studies

  • Sampling bias / coverage bias: the sampling frame does not represent the entire population; sampling may not be random.

  • Undercoverage: some groups are underrepresented.

  • Nonresponse bias: those who participate differ from those who do not participate; missing data may arise if respondents skip questions.

  • Response bias: participants provide inaccurate answers or are influenced by question wording.

• Practical example: survey bias illustrated via media/press coverage of a study; headlines might mislead about causation; emphasis on correct interpretation (association vs causation).

Sampling Methods (how to select individuals)

• Random sampling methods

  • Simple random sample: every individual has an equal chance of being selected; most representative and unbiased.

  • Stratified sample: population divided into strata; random sample drawn from within each stratum; useful for ensuring representation of specific groups.

  • Cluster sample: natural groups (clusters) are sampled; then individuals within chosen clusters are sampled; good when a reliable sampling frame is unavailable.

• Non-random sampling methods

  • Volunteer sample: participants volunteer to join.

  • Convenience sample: obtain subjects who are easy to recruit.

• Non-random designs are more prone to sampling bias/undercoverage.

• Quick summaries of methods:

  • Simple Random Sample: each individual equally likely to be sampled; generally most representative.

  • Cluster Sample: sample clusters first, then sample within clusters.

  • Stratified Sample: ensure representation across key subgroups by stratifying first.

Example Scenarios and Questions (practice concepts)

• Example: three-variable dataset related to hiring rates (Simpson’s Paradox) used to illustrate that the direction of an association can change after conditioning on a third variable.

• Observational headlines vs true results: caution in interpreting causal claims from observational data; must consider confounding variables and potential biases.

• Quick practice questions from the slides:

  • Question: In the helper vs hinder study, where 14 out of 16 (87.5%) infants prefer the helper toy, what type of variable was studied?

  • Answer: C. categorical, dichotomous (two categories: helper vs hinder).

  • Question: An experiment regarding the physiological cost of reproduction on male fruit flies includes multiple variables; how many quantitative variables does this dataset contain?

  • Answer (based on the slide): D. 3 (lifespan, thorax length, sleep).

Response vs Explanatory Variables and Data Relationships

• Response variable (outcome) can be categorical or quantitative.

• Explanatory variable (predictor) can be categorical or quantitative.

• Association between response and explanatory variables may exist even in the absence of a causal relationship in observational data.

• If a third variable explains both the explanatory and response variables, this is a confounding variable.

Experimental Design: Core Concepts

• Key concepts in experimental design

  • Control: compare treatment of interest to a control group.

  • Randomize: randomly assign subjects to treatment and control groups to balance characteristics.

  • Replicate: use a large enough sample size or replicate the study to ensure reliability.

  • Block: account for known or suspected variables that affect the response; group units into blocks and assign treatments within blocks.

• Random assignment benefits

  • Helps determine whether an intervention was effective by comparing treatment vs control.

  • Balances other characteristics across groups, reducing confounding.

  • Supports causal inference if well designed.

• Terminology

  • Placebo: fake treatment used as control.

  • Placebo effect: change observed due to belief in treatment.

  • Blinding: experimental units do not know which group they are in.

  • Double-blind: neither experimental units nor researchers know group assignments.

• Multifactor experiments

  • Explanatory variables may be multiple factors (treatment conditions).

  • Example: antidepressants and nicotine patches in a four-group, cross-classified design:
    1) antidepressant + nicotine patch
    2) antidepressant + placebo patch
    3) placebo antidepressant + nicotine patch
    4) placebo antidepressant + placebo patch

• Blocking vs factors

  • Factors: experimental conditions imposed on units (e.g., treatment vs control).

  • Blocking variables: characteristics of units that you want to control for.

  • Blocking in experiments is analogous to stratification in observational studies when sampling.

How to Analyze and Interpret Study Designs

• Experimental studies

  • Random assignment aims to limit confounding and support causal conclusions.

  • Control groups and blinding reduce bias and placebo effects.

• Observational studies

  • Can reveal associations and real-world behavior but are more susceptible to confounding; causation is harder to establish.

• Simpson’s Paradox (illustrative problem)

  • A relationship that appears in aggregated data can reverse when data are stratified by a confounding factor (e.g., gender, field of study, or other grouping).

Practical Examples and Recaps

• Example: data collection on Mario Kart eBay listings uses a data frame with both quantitative (nBids, startPr, totalPr) and categorical variables (cond, shipSp, wheels).

• Example: measuring IQ in children and spanking status illustrates how two variables (spanking and IQ) can be associated but may be confounded by age group (2-4 vs 5-9) or other variables; headlines can misrepresent association as causation.

• Example of study design choices:

  • To investigate eye color and emotional sensitivity, an experimental study would be inappropriate; stratified sampling or simple random sampling among eye color groups would be more appropriate to study association in an observational framework.

Quick Reference: Glossary Highlights

• Observational study: researchers observe without assigning treatments; potential confounding.

• Experimental study: researchers assign treatments and observe outcomes; stronger causal claims possible.

• Randomization: random assignment to groups to balance known and unknown factors.

• Blocking: grouping similar units and randomizing within blocks to control for blocking variables.

• Placebo and placebo effect: control treatment and belief-driven response.

• Blinding and double-blinding: reduce bias from participants and researchers.

• Confounding variable: a third variable that influences both explanatory and response variables, creating a spurious association.

• Prospective (cohort): follow a group over time.

• Retrospective (case-control): look back at past data.

• Cross-sectional: snapshot at a single point in time.

• Simpson’s Paradox: aggregated association can reverse within strata or subgroups.

Key Equations and Notation (illustrative)

• Proportion and percent example from a small study:

  • If 14 of 16 infants prefer the helper toy, the proportion is rac{14}{16} = 0.875 = 87.5\%.

• Example data frame notations (variables in Mario Kart dataset):

  • Observations: rows

  • Variables: columns such as obs,
    nBids,
    cond,
    startPr,
    totalPr,
    shipSp,
    wheels.

  • Example values: nBids = 20, ext{ cond} = 'new', ext{ startPr} = 0.99, ext{ totalPr} = 51.55, ext{ shipSp} = 'standard', ext{ wheels} = 1.

• Example terminologies in an experimental design context:

  • Treatments: the different conditions applied to experimental units.

  • Blocking variable example: gender, age group, baseline health status.

  • Cross-classified factors: factorial design with multiple factors and their levels.

If you’d like, I can tailor these notes to a specific topic focus (e.g., only sampling biases, or only experimental design), or expand any section with more examples or practice questions.