Notes taken from Mathematical Biostatistics Boot Camp I

Coursera: Lecture 1

Institution of Origin: John Hopkins Bloomberg School of Public Health
Student: Sherri Verdugo

Definition: Biostatistics

Biostatistics is a theory and methodolgy for the acquisition and use of quantitative evidence in biomedical research. Biostatisticians develop innovative deisngs and analytic methods targeted at increasing available information, improving the relevance and validity of statistical analyses, making best use of available information and communicating relevant uncertainties. Johns Hopkins Department of Biostatistics, 2007 self study.

Examples of clinical trials

1) Negative Health Outcomes & Hormone Replacement Therapy (HRT)

Research in 2002 contradicted Hormone Replacement Therapy for post menopausal women and indicated a negative impact of HRT for key health outcomes. The result from the statistical outcome is that the study was stopped early. WH1 writing group paper JAMA 2002, Vol 288: 331- 333 for the paper and Steinkelner et al. Menopause 2012, Vol 19:616 - 621 for a recent discussion of the long term impacts.

2) Extra-corporeal Membrane Oxygenation Treatment (ECMO)

Research in 1985 at a major neonatal intensive care center published results of a clinical trial comparing standard treatment and ECMO in newborn infants suffering from severe respiratory failure. Ethical considerations resulted in a statistic randomization scheme: control and treatment groups. In the end the study was subjected to sample-size critiques.

Summary:

• Biostatistics plays a central role in public health in both the importance of design, analysis, and interpretation of statistical data.

• Prevailing philosophy at John Hopkins Public Health:

  • tight coupling of the statistical methods with ethical and scientific goals
  • emphasizing scientific interpretation of statistical evidence to impact policy
  • acknowledging and assumptions and evaluating the robustness of conclusions to them.

Basics of biostatistics: Experiments and Outcomes

Experiments:

• Measurements are collected from a sample population
• Measurements come from a laboratory experiment or other repositories
• Results of clinical trials
• Public health data
• Results can be simulated by a computer for an experiment
• Hospital records can be sampled retrospectively

Mathematical Concepts

• The sample space, \( \Omega \)
  • Collection of possible outcomes of an experiment
  • Example 6 sided die roll \( \Omega \) = {1, 2, 3, 4, 5, 6}
• Event, E is a subset of \( \Omega \)
  • Die Roll is odd: E = {1, 3, 5}
• Elementary or Simple Event is a subset of the Event, E
  • Die roll is three, \( \omega \) = 3
• Empty Set or Null Event, \( \emptyset \)
  • For a six sided die, an example of \( \emptyset \) would be: \( \emptyset \) = 7
  • In this example, there is no element of \( \emptyset \) for which the property holds

Normal set operations with specific interpretations

1) \( \omega \) \( \in \) E implies that E occurs when \( \omega \) occurs
2) \( \omega \) \( \notin \) implies that E does not occur when \( \omega \) occurs
3) E \( \subset \) F implies that the occurrence of E implies the occurrence of F
4) E \( \cap \) F implies that both E and F occur
5) E \( \cup \) F implies that at least one of E or F occur
6) E \( \cap \) F = \( \emptyset \) means that E and F are mutually exclusive and cannot both occur. Important for hypothesis testing.
7) E^c or \( \overline{E} \) is the event that E does NOT occur

DeMorgan's Laws

• \( (A \cap B)^c \) = \( (A^c \cup B^c) \)
• \( (A \cup B)^c \) = \( (A^c \cap B^c) \)
• \( (A^c)^c \) = A
• \( (A \cup B) \cap C \) = \( (A \cap C) \cup (B \cap C) \)

Useful applications of DeMorgan's Laws

• hypothesis testing
• attribution of all known or theorized variables to a systematic model (a.k.a. “Mathematical Function”)
• using probability to quantify uncertainty in conclusions
• evaluate sensitivity of conclusions and assumptions of the model

Key questions for analyzing data

• what is actually being modeled as random?
• where does the “randomness” come from?
• where did the systematic model components come from?
• how did the observational units come to be in the study?
• is there importance to any of the missing data points?
• do the results generalize beyond the study in question?
• were important variables unaccounted for in the model?
• how drastically different would the inferences change depending on the answers to the previous listed questions?

Conclusion and discussion

Probability is the foundation and core of statistics and biostatistics. It is a systematic method that allows an investigator to set up a system of tests using hypotheses (null and alternative –> both mutually exclusive and exhaustive) to evaluate a situation.

Future tutorials will include how to use R Studio to create a seamless workflow for all stages of data analysis. These are key notes taken from the course and serve as a timeless reminder of the rules that allow us to investigate situations and outcomes systematically and unbiased.

Cheers, Sherri Verdugo, M.S.