統計推斷week1

1.1統計推斷導論

統計推斷是從數據中得出正式結論的過程(frormal conclusions)

在此課中，我們對正式的統計推斷的定義是在考慮到不確定性的情況下，通過帶有噪音的數據，從大量的事實中推斷

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.

下面是兩個例子，一爲選舉，二爲更年期症狀（Menopausal symptoms） hormone激素

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.

Is HRT effective?
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: ECMO

In 1985 a group at a major neonatal intensive care center(新生兒重症中心） published the results of a trial comparing a standard treatment and a promising new extracorporeal membrane oxygenation treatment (ECMO，ECMO是體外膜肺氧合(extracorporeal membrane oxygenation)的英文簡稱，它是代表一個醫院，甚至一個地區、一個國家的危重症急救水平的一門技術) for newborn infants with severe respiratory failure（嚴重的呼吸）. Ethical considerations（倫理方面的考慮） lead to a statistical randomization scheme whereby（據此） one infant received the control therapy, thereby opening the study to sample-size based criticisms.

For a review and statistical discussion, see Royall Statistical Science 1991, Vol 6, No. 1, 52-88

Summary

These examples illustrate many of the difficulties of trying to use data to create general conclusions about a population.
Paramount(最重要的） 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?")

Example tools of the trade

Randomization: concerned with balancing unobserved variables that may confound inferences of interest
Random sampling: concerned with obtaining data that is representative of the population of interest
Sampling models: concerned with creating a model for the sampling process, the most common is so called "iid".
Hypothesis testing: concerned with decision making in the presence of uncertainty
Confidence intervals: concerned with quantifying uncertainty in estimation
Probability models: a formal connection between the data and a population of interest. Often probability models are assumed or are approximated.
Study design: the process of designing an experiment to minimize biases and variability.
Nonparametric bootstrapping: the process of using the data to, with minimal probability model assumptions, create inferences.
Permutation, randomization and exchangeability testing: the process of using data permutations to perform inferences

causal inference因果推斷

1.2概率基礎

Example

The National Sleep Foundation (www.sleepfoundation.org) reports that around 3% of the American population has sleep apnea(睡眠呼吸暫停). They also report that around 10% of the North American and European population has restless leg syndrome(不寧腿綜合症). Does this imply that 13% of people will have at least one sleep problems of these sorts?

Examples of variables that can be thought of as random variables

The $(0-1)$ outcome of the flip of a coin
The outcome from the roll of a die
The BMI of a subject four years after a baseline measurement
The hypertension status of a subject randomly drawn from a population

CDF and survival function

The cumulative distribution function (CDF) of a random variable X is defined as the function F(x) = P(X <= x)
This definition applies regardless of whether X is discrete or continuous.
The survival function of a random variable X is defined as S(x) = P(X > x)
Notice that S(x) = 1 - F(x)
For continuous random variables, the PDF is the derivative of the CDF

Quantiles

The a th quantile of a distribution with distribution function F is the point X a so that F(X a) = a
A percentile is simply a quantile with a expressed as a percent
The median is the $50^{th}$ percentile

1.3期望值

注意

因此，樣本均值的期望值是它試圖估計的總體值

當一個估計量的期望值是它試圖估計的，我們就把這種估計值稱爲無偏量

Remark

Therefore, the expected value of the sample mean is the population mean that it's trying to estimate
When the expected value of an estimator is what its trying to estimate, we say that the estimator is unbiased

小note:

設^θ(X1,X2,…,Xn)是θ的估計量，若E(^θ)=θ，對一切θ∈Θ，則稱^θ爲θ的無偏估計量，否則稱爲θ的有偏估計量。

無偏估計量的定義是：設（ξ∧）是ξ的一個估計量，若E（ξ∧）=ξ ，則稱ξ∧是ξ的無偏估計量

反正可能有點高估吧，實際可能沒那麼高

1.4獨立性

IID random variables

Random variables are said to be iid if they are independent and identically distributed
iid random variables are the default model for random samples
Many of the important theories of statistics are founded on assuming that variables are iid