Biostat - Notes on Data Types, Description, and p-values

Description before inference

• The speaker emphasizes that the science part of data analysis begins with describing what you have before making any inference.
• You must describe the data you collected and examine what you observe prior to drawing conclusions or performing statistical tests.

Data types: categorical vs quantitative

• Categorical data
  • Definition: data that represent categories or groups rather than numeric quantities.
  • Often described with counts, percentages, and modes rather than means.
  • Subtypes: nominal (no inherent order) and ordinal (ordered categories).
• Quantitative data
  • Definition: numerical measurements that express quantity.
  • Subtypes: discrete (countable values) and continuous (any value within a range).
• In the transcript, examples are given to illustrate these types:
  • Categorical data examples (often responses are words):
  • Race
  • Gender
  • State of birth
  • Country of birth
  • Model of car
  • Quantitative data example:
  • Height (e.g., how tall you are)

Data collection and response formats from the transcript

• For categorical questions, respondents often provide word answers (text labels).
• For quantitative questions, responses are numeric (e.g., inches, centimeters, or other units).

P-values and statistical inference

• The speaker references the p-value as part of data analysis when a computer program crunches the data.
• Concept: a small p-value indicates evidence against the null hypothesis, making the observed data appear unlikely under the null.
• Formal definitions (to connect to the practical idea):
  • General idea: the p-value is the probability of obtaining data as extreme as observed, assuming the null hypothesis is true.
  • For a two-sided test with a test statistic T and observed value tobs: $p\text{-value} = P\left(|T| \ge |t</em>{\text{obs}}| \;\middle|\; H_0\right)$
  • In many standard tests (e.g., tests for proportions or means), the p-value is computed by comparing the observed statistic to a reference distribution under the null hypothesis.
• Practical conventions (explicit or implied):
  • A common significance level is $\alpha = 0.05$; if $p \le \alpha$, the result is considered statistically significant and the null hypothesis is rejected.
  • Note: a small p-value does not imply large practical importance; it depends on sample size and effect size.
  • Large samples can yield small p-values for trivial effects; conversely, meaningful effects in small samples might not reach significance.

Illustrative formulas for common tests

• Proportion test (example):
  • Test statistic (z-form) for testing $H<em>0: p = p</em>0$ with sample proportion $\hat{p}$ and sample size $n$:
    $z = \frac{\hat{p} - p<em>0}{\sqrt{\frac{p</em>0(1-p_0)}{n}}}$
  • Corresponding two-sided p-value:
    $p\text{-value} = P\left(|Z| \ge |z|\right)$ where $Z\sim N(0,1)$ under $H_0$.
• General two-sided p-value form (abstract):
  $p\text{-value} = P\left(|T| \ge |t<em>{\text{obs}}| \middle| H</em>0\right)$

How data type informs analysis options

• Descriptive summaries depend on data type:
  • Categorical: frequencies, relative frequencies, bar charts, contingency tables.
  • Quantitative: measures of central tendency (mean, median, mode) and dispersion (range, variance, standard deviation, IQR), histograms, box plots.
• Choice of statistical test depends on data type and distribution assumptions:
  • Categorical data lead to chi-square tests, Fisher exact test, logistic regression, etc.
  • Quantitative data lead to t-tests, ANOVA, regression, nonparametric alternatives when assumptions fail.

Ethical, philosophical, and practical implications

• Ethical considerations when collecting demographic data (race, gender, birthplace, etc.):
  • Privacy and confidentiality
  • Informed consent and data protection
  • Potential for bias or misuse of data in reporting or decision-making
• Practical implications:
  • Always describe data first to avoid overreaching conclusions
  • Be mindful of sample size, measurement units, and data quality when interpreting p-values
  • Report both statistical significance (p-values) and practical significance (effect sizes, context)

Connections to broader principles and real-world relevance

• Links to foundational principles:
  • Measurement and data quality affect all subsequent inference
  • Understanding the type of data guides appropriate descriptive statistics and inferential tests
  • Statistical evidence must be interpreted in the context of study design, sample size, and practical importance
• Real-world relevance:
  • Surveys routinely collect categorical data (demographics) and quantitative data (heights, incomes, scores)
  • Proper interpretation of p-values is crucial in research, policy, and science to avoid over- or underestimating effects

Quick reference: key takeaways

• Before inferring anything, describe the data you have.
• Categorical data: nominal/ordinal categories, answers like race, gender, location, car model.
• Quantitative data: numerical measurements, like height.
• Small p-values indicate evidence against the null hypothesis, but are influenced by sample size and may not imply practical importance.
• Use $p\text{-value} = P\left(|T| \ge |t<em>{\text{obs}}| \middle| H</em>0\right)$ for two-sided tests; for a proportion test, a typical form is $z = \frac{\hat{p} - p<em>0}{\sqrt{\frac{p</em>0(1-p_0)}{n}}}$ with the corresponding $p\text{-value}$ from $Z\sim N(0,1)$.
• Always connect statistical findings to real-world meaning and ethical considerations.