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06_StatisticalInference/01_Introduction/index.Rmd

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+---
+title       : Introduction to statistical inference
+subtitle    : Statistical inference
+author      : Brian Caffo, Jeff Leek, Roger Peng
+job         : Johns Hopkins Bloomberg School of Public Health
+logo        : bloomberg_shield.png
+framework   : io2012        # {io2012, html5slides, shower, dzslides, ...}
+highlighter : highlight.js  # {highlight.js, prettify, highlight}
+hitheme     : tomorrow      # 
+url:
+  lib: ../../librariesNew
+  assets: ../../assets
+widgets     : [mathjax]            # {mathjax, quiz, bootstrap}
+mode        : selfcontained # {standalone, draft}
+---
+
+## Statistical inference defined
+
+Statistical inference is the process of drawing formal conclusions from
+data. 
+
+In our class, we wil define formal statistical inference as settings where one wants to infer facts about a population using noisy
+statistical data where uncertainty must be accounted for.
+
+---
+
+## Motivating example: who's going to win the election?
+
+In every major election, pollsters would like to know, ahead of the
+actual election, who's going to win. Here, the target of
+estimation (the estimand) is clear, the percentage of people in 
+a particular group (city, state, county, country or other electoral
+grouping) who will vote for each candidate.
+
+We can not poll everyone. Even if we could, some polled 
+may change their vote by the time the election occurs.
+How do we collect a reasonable subset of data and quantify the
+uncertainty in the process to produce a good guess at who will win?
+
+---
+
+## Motivating example: is hormone replacement therapy effective? 
+
+A large clinical trial (the Women’s Health Initiative) published results in 2002 that contradicted prior evidence on the efficacy of hormone replacement therapy for post menopausal women and suggested a negative impact of HRT for several key health outcomes. **Based on a statistically based protocol, the study was stopped early due an excess number of negative events.**
+
+Here's there's two inferential problems. 
+
+1. Is HRT effective?
+2. How long should we continue the trial in the presence of contrary
+evidence?
+
+See WHI writing group paper JAMA 2002, Vol 288:321 - 333. for the paper and Steinkellner et al. Menopause 2012, Vol 19:616 621 for adiscussion of the long term impacts
+
+---
+
+## Motivating example 
+### Brain activation
+
+![fMRI salmon study](fig/fmri-salmon.jpg 'fMRI salmon study')
+
+http://www.wired.com/2009/09/fmrisalmon/
+
+
+---
+
+## Summary
+
+- These examples illustrate many of the difficulties of trying
+to use data to create general conclusions about a population.
+- Paramount among our concerns are:
+  - Is the sample representative of the population that we'd like to draw inferences about?
+  - Are there known and observed, known and unobserved or unknown and unobserved variables that contaminate our conclusions?
+  - Is there systematic bias created by missing data or the design or conduct of the study?
+  - What randomness exists in the data and how do we use or adjust for it? Here randomness can either be explicit via randomization
+or random sampling, or implicit as the aggregation of many complex uknown processes.
+  - Are we trying to estimate an underlying mechanistic model of phenomena under study?
+- Statistical inference requires navigating the set of assumptions and
+tools and subsequently thinking about how to draw conclusions from data.
+
+--- 
+## Example goals of inference
+
+1. Estimate and quantify the uncertainty of an estimate of 
+a population quantity (the proportion of people who will
+  vote for a candidate).
+2. Determine whether a population quantity 
+  is a benchmark value ("is the treatment effective?").
+3. Infer a mechanistic relationship when quantities are measured with
+  noise ("What is the slope for Hooke's law?")
+4. Determine the impact of a policy? ("If we reduce polution levels,
+  will asthma rates decline?")
+5. Talk about the probability that something occurs.
+
+---
+## Example tools of the trade 
+
+1. Randomization: concerned with balancing unobserved variables that may confound inferences of interest
+2. Random sampling: concerned with obtaining data that is representative 
+of the population of interest
+3. Sampling models: concerned with creating a model for the sampling
+process, the most common is so called "iid".
+4. Hypothesis testing: concerned with decision making in the presence of uncertainty
+5. Confidence intervals: concerned with quantifying uncertainty in 
+estimation
+6. Probability models: a formal connection between the data and a population of interest. Often probability models are assumed or are
+approximated.
+7. Study design: the process of designing an experiment to minimize biases and variability.
+8. Nonparametric bootstrapping: the process of using the data to,
+  with minimal probability model assumptions, create inferences.
+9. Permutation, randomization and exchangeability testing: the process 
+of using data permutations to perform inferences.
+
+---
+## Different thinking about probability leads to different styles of inference
+
+We won't spend too much time talking about this, but there are several different
+styles of inference. Two broad categories that get discussed a lot are:
+
+1. Frequency probability: is the long run proportion of
+ times an event occurs in independent, identically distributed 
+ repetitions.
+2. Frequency inference: uses frequency interpretations of probabilities
+to control error rates. Answers questions like "What should I decide
+given my data controlling the long run proportion of mistakes I make at
+a tolerable level."
+3. Bayesian probability: is the probability calculus of beliefs, given that beliefs follow certain rules.
+4. Bayesian inference: the use of Bayesian probability representation
+of beliefs to perform inference. Answers questions like "Given my subjective beliefs and the objective information from the data, what
+should I believe now?"
+
+Data scientists tend to fall within shades of gray of these and various other schools of inference. 
+
+---
+## In this class
+
+* In this class, we will primarily focus on basic sampling models, 
+basic probability models and frequency style analyses
+to create standard inferences. 
+* Being data scientists,  we will also consider some inferential strategies that  rely heavily on the observed data, such as permutation testing
+and bootstrapping.
+* As probability modeling will be our starting point, we first build
+up basic probability.
+
+---
+## Where to learn more on the topics not covered
+
+1. Explicit use of random sampling in inferences: look in references
+on "finite population statistics". Used heavily in polling and
+sample surveys.
+2. Explicit use of randomization in inferences: look in references
+on "causal inference" especially in clinical trials.
+3. Bayesian probability and Bayesian statistics: look for basic itroductory books (there are many).
+4. Missing data: well covered in biostatistics and econometric
+references; look for references to "multiple imputation", a popular tool for
+addressing missing data.
+5. Study design: consider looking in the subject matter area that
+  you are interested in; some examples with rich histories in design:
+  1. The epidemiological literature is very focused on using study design to investigate public health.
+  2. The classical development of study design in agriculture broadly covers design and design principles.
+  3. The industrial quality control literature covers design thoroughly.
+