Statistical Inference for Data Science PDF

Statistical inference for data science A companion to the Coursera Statistical Inference Course Brian Caffo This book is for sale at http://leanpub.com/LittleInferenceBook This version was published on 2016-05-24 This is a Leanpub book. Leanpub empowers authors and publishers with the Lean Publi...

Statistical inference for data science A companion to the Coursera Statistical Inference Course Brian Caffo This book is for sale at http://leanpub.com/LittleInferenceBook This version was published on 2016-05-24 This is a Leanpub book. Leanpub empowers authors and publishers with the Lean Publishing process. Lean Publishing is the act of publishing an in-progress ebook using lightweight tools and many iterations to get reader feedback, pivot until you have the right book and build traction once you do. This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License Also By Brian Caffo Regression Models for Data Science in R Executive Data Science Advanced Linear Models for Data Science Developing Data Products in R To Kerri, Penelope and Scarlett Contents About this book........................................ 1 About the picture on the cover................................ 1 1. Introduction.......................................... 2 Before beginning........................................ 2 Statistical inference defined.................................. 2 Summary notes........................................ 3 The goals of inference..................................... 4 The tools of the trade..................................... 4 Different thinking about probability leads to different styles of inference......... 4 Exercises............................................ 5 2. Probability........................................... 6 Where to get a more thorough treatment of probability................... 6 Kolmogorov’s Three Rules................................... 7 Consequences of The Three Rules.............................. 7 Random variables....................................... 8 Probability mass functions.................................. 10 Probability density functions................................. 10 CDF and survival function.................................. 13 Quantiles............................................ 14 Exercises............................................ 15 3. Conditional probability................................... 16 Conditional probability, motivation.............................. 16 Conditional probability, definition.............................. 16 Bayes’ rule........................................... 17 Diagnostic Likelihood Ratios................................. 19 Independence......................................... 20 IID random variables..................................... 21 Exercises............................................ 22 4. Expected values........................................ 23 The population mean for discrete random variables..................... 23 The sample mean....................................... 23 CONTENTS Continuous random variables................................. 28 Simulation experiments.................................... 30 Summary notes........................................ 33 Exercises............................................ 33 5. Variation............................................ 34 The variance.......................................... 34 The sample variance...................................... 37 Simulation experiments.................................... 37 The standard error of the mean................................ 39 Data example......................................... 42 Summary notes........................................ 44 Exercises............................................ 44 6. Some common distributions................................. 46 The Bernoulli distribution................................... 46 Binomial trials......................................... 46 The normal distribution.................................... 47 The Poisson distribution.................................... 50 Exercises............................................ 52 7. Asymptopia.......................................... 54 Asymptotics.......................................... 54 Limits of random variables.................................. 54 The Central Limit Theorem.................................. 57 CLT simulation experiments................................. 57 Confidence intervals...................................... 60 Simulation of confidence intervals.............................. 62 Poisson interval........................................ 69 Summary notes........................................ 72 Exercises............................................ 73 8. t Confidence intervals.................................... 74 Small sample confidence intervals.............................. 74 Gosset’s t distribution..................................... 74 The data............................................ 76 Independent group t confidence intervals.......................... 78 Confidence interval...................................... 79 Mistakenly treating the sleep data as grouped........................ 79 Unequal variances....................................... 83 Summary notes........................................ 84 Exercises............................................ 84 9. Hypothesis testing...................................... 87 CONTENTS Hypothesis testing....................................... 87 Types of errors in hypothesis testing............................. 88 Discussion relative to court cases............................... 88 Building up a standard of evidence.............................. 88 General rules.......................................... 90 Two sided tests......................................... 91 T test in R........................................... 91 Connections with confidence intervals............................ 92 Two group intervals...................................... 92 Exact binomial test...................................... 93 Exercises............................................ 94 10. P-values............................................ 96 Introduction to P-values.................................... 96 What is a P-value?....................................... 96 The attained significance level................................ 97 Binomial P-value example................................... 97 Poisson example........................................ 98 Exercises............................................ 98 11. Power............................................. 100 Power.............................................. 100 Question............................................ 103 Notes.............................................. 103 T-test power.......................................... 103 Exercises............................................ 104 12. The bootstrap and resampling................................ 106 The bootstrap......................................... 106 The bootstrap principle.................................... 109 Group comparisons via permutation tests.......................... 113 Permutation tests....................................... 114 Variations on permutation testing.............................. 114 Permutation test B v C..................................... 114 Exercises............................................ 116 CONTENTS 1 About this book This book is written as a companion book to the Statistical Inference¹ Coursera class as part of the Data Science Specialization². However, if you do not take the class, the book mostly stands on its own. A useful component of the book is a series of YouTube videos that comprise the Coursera class. The book is intended to be a low cost introduction to the important field of statistical inference. The intended audience are students who are numerically and computationally literate, who would like to put those skills to use in Data Science or Statistics. The book is offered for free as a series of markdown documents on github and in more convenient forms (epub, mobi) on LeanPub and retail outlets. This book is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License³, which requires author attribution for derivative works, non-commercial use of derivative works and that changes are shared in the same way as the original work. About the picture on the cover The picture on the cover is a public domain image taken from Wikipedia’s article on Francis Galton’s quincunx. Francis Galton was an 19th century polymath who invented many of key concepts of statistics. The quincunx was an ingenious invention for illustrating the central limit theorem using a pinball setup. ¹https://www.coursera.org/course/statinference ²https://www.coursera.org/specialization/jhudatascience/1?utm_medium=courseDescripTop ³http://creativecommons.org/licenses/by-nc-sa/4.0/ 1. Introduction Before beginning This book is designed as a companion to the Statistical Inference¹ Coursera class as part of the Data Science Specialization², a ten course program offered by three faculty, Jeff Leek, Roger Peng and Brian Caffo, at the Johns Hopkins University Department of Biostatistics. The videos associated with this book can be watched in full here³, though the relevant links to specific videos are placed at the appropriate locations throughout. Before beginning, we assume that you have a working knowledge of the R programming language. If not, there is a wonderful Coursera class by Roger Peng, that can be found here⁴. The entirety of the book is on GitHub here⁵. Please submit pull requests if you find errata! In addition the course notes can be found also on GitHub here⁶. While most code is in the book, all of the code for every figure and analysis in the book is in the R markdown files files (.Rmd) for the respective lectures. Finally, we should mention swirl (statistics with interactive R programming). swirl is an intelligent tutoring system developed by Nick Carchedi, with contributions by Sean Kross and Bill and Gina Croft. It offers a way to learn R in R. Download swirl here⁷. There’s a swirl module for this course!⁸. Try it out, it’s probably the most effective way to learn. Statistical inference defined Watch this video before beginning.⁹ We’ll define statistical inference as the process of generating conclusions about a population from a noisy sample. Without statistical inference we’re simply living within our data. With statistical inference, we’re trying to generate new knowledge. Knowledge and parsimony, (using simplest reasonable models to explain complex phenomena), go hand in hand. Probability models will serve as our parsimonious description of the world. The use ¹https://www.coursera.org/course/statinference ²https://www.coursera.org/specialization/jhudatascience/1?utm_medium=courseDescripTop ³https://www.youtube.com/watch?v=WkOinijQmPU&list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ ⁴https://www.coursera.org/course/rprog ⁵https://github.com/bcaffo/LittleInferenceBook ⁶https://github.com/bcaffo/courses/tree/master/06_StatisticalInference ⁷http://swirlstats.com ⁸https://github.com/swirldev/swirl_courses#swirl-courses ⁹http://youtu.be/WkOinijQmPU?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ 2 Introduction 3 of probability models as the connection between our data and a populations represents the most effective way to obtain inference. 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, predicting the weather When a weatherman tells you the probability that it will rain tomorrow is 70%, they’re trying to use historical data to predict tomorrow’s weather - and to actually attach a probability to it. That probability refers to population. Motivating example, brain activation An example that’s very close to the research I do is trying to predict what areas of the brain activate when a person is put in the fMRI scanner. In that case, people are doing a task while in the scanner. For example, they might be tapping their finger. We’d like to compare when they are tapping their finger to when they are not tapping their finger and try to figure out what areas of the brain are associated with the finger tapping. Summary notes 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 unknown 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. Introduction 4 The goals of inference You should recognize the goals of inference. Here we list five examples of inferential goals. 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 pollution levels, will asthma rates decline?”) 5. Talk about the probability that something occurs. The tools of the trade Several tools are key to the use of statistical inference. We’ll only be able to cover a few in this class, but you should recognize them anyway. 1. Randomization: concerned with balancing unobserved variables that may confound infer- ences 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 permuta- tions 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: Introduction 5 1. Frequency probability: is the long run proportion of times an event occurs in independent, identically distributed repetitions. 2. Frequency style 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 style 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. Furthermore, there are so many shades of gray between the styles of inferences that it is hard to pin down most modern statisticians as either Bayesian or frequentist. In this class, we will primarily focus on basic sampling models, basic probability models and frequency style analyses to create standard inferences. This is the most popular style of inference by far. 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 as our first task. Exercises 1. The goal of statistical inference is to? Infer facts about a population from a sample. Infer facts about the sample from a population. Calculate sample quantities to understand your data. To torture Data Science students. 2. The goal of randomization of a treatment in a randomized trial is to? It doesn’t really do anything. To obtain a representative sample of subjects from the population of interest. Balance unobserved covariates that may contaminate the comparison between the treated and control groups. To add variation to our conclusions. 3. Probability is a? Population quantity that we can potentially estimate from data. A data quantity that does not require the idea of a population. 2. Probability Watch this video before beginning.¹ Probability forms the foundation for almost all treatments of statistical inference. In our treatment, probability is a law that assigns numbers to the long run occurrence of random phenomena after repeated unrelated realizations. Before we begin discussing probability, let’s dispense with some deep philosophical questions, such as “What is randomness?” and “What is the fundamental interpretation of probability?”. One could spend a lifetime studying these questions (and some have). For our purposes, randomness is any process occurring without apparent deterministic patterns. Thus we will treat many things as if they were random when, in fact they are completely deterministic. In my field, biostatistics, we often model disease outcomes as if they were random when they are the result of many mechanistic components whose aggregate behavior appears random. Probability for us will be the long run proportion of times some occurs in repeated unrelated realizations. So, think of the proportion of times that you get a head when flipping a coin. For the interested student, I would recommend the books and work by Ian Hacking to learn more about these deep philosophical issues. For us data scientists, the above definitions will work fine. Where to get a more thorough treatment of probability In this lecture, we will cover the fundamentals of probability at low enough of a level to have a basic understanding for the rest of the series. For a more complete treatment see the class Mathematical Biostatistics Boot Camp 1, which can be viewed on YouTube here². In addition, there’s the actual Coursera course³ that I run periodically (this is the first Coursera class that I ever taught). Also there are a set of notes on GitHub⁴. Finally, there’s a follow up class, uninspiringly named Mathematical Biostatistics Boot Camp 2, that is more devoted to biostatistical topics that has an associated YouTube playlist⁵, Coursera Class⁶ and GitHub notes⁷. ¹http://youtu.be/oTERv_vrmJM?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ ²Youtube:www.youtube.com/playlist?list=PLpl-gQkQivXhk6qSyiNj51qamjAtZISJ- ³Coursera:www.coursera.org/course/biostats ⁴http://github.com/bcaffo/Caffo-Coursera ⁵http://www.youtube.com/playlist?list=PLpl-gQkQivXhwOsKPQ4fbCBYOWjvdzrSM ⁶https://www.coursera.org/course/biostats2 ⁷https://github.com/bcaffo/MathematicsBiostatisticsBootCamp2 6 Probability 7 Kolmogorov’s Three Rules Watch this lecture before beginning.⁸ Given a random experiment (say rolling a die) a probability measure is a population quantity that summarizes the randomness. The brilliant discovery of the father of probability, the Russian mathematician Kolmogorov⁹, was that to satisfy our intuition about how probability should behave, only three rules were needed. Consider an experiment with a random outcome. Probability takes a possible outcome from an experiment and: 1. assigns it a number between 0 and 1 2. requires that the probability that something occurs is 1 3. required that the probability of the union of any two sets of outcomes that have nothing in common (mutually exclusive) is the sum of their respective probabilities. From these simple rules all of the familiar rules of probability can be developed. This all might seem a little odd at first and so we’ll build up our intuition with some simple examples based on coin flipping and die rolling. I would like to reiterate the important definition that we wrote out: mutually exclusive. Two events are mutually exclusive if they cannot both simultaneously occur. For example, we cannot simultaneously get a 1 and a 2 on a die. Rule 3 says that since the event of getting a 1 and 2 on a die are mutually exclusive, the probability of getting at least one (the union) is the sum of their probabilities. So if we know that the probability of getting a 1 is 1/6 and the probability of getting a 2 is 1/6, then the probability of getting a 1 or a 2 is 2/6, the sum of the two probabilities since they are mutually exclusive. Consequences of The Three Rules Let’s cover some consequences of our three simple rules. Take, for example, the probability that something occurs is 1 minus the probability of the opposite occurring. Let A be the event that we get a 1 or a 2 on a rolled die. Then Ac is the opposite, getting a 3, 4, 5 or 6. Since A and Ac cannot both simultaneously occur, they are mutually exclusive. So the probability that either A or Ac is P (A) + P (Ac ). Notice, that the probability that either occurs is the probability of getting a 1, 2, 3, 4, 5 or 6, or in other words, the probability that something occurs, which is 1 by rule number 2. So we have that 1 = P (A) + P (Ac ) or that P (A) = 1 − P (Ac ). We won’t go through this tedious exercise (since Kolmogorov already did it for us). Instead here’s a list of some of the consequences of Kolmogorov’s rules that are often useful. ⁸http://youtu.be/Shzt9uZ8BII?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ ⁹http://en.wikipedia.org/wiki/Andrey_Kolmogorov Probability 8 1. The probability that nothing occurs is 0 2. The probability that something occurs is 1 3. The probability of something is 1 minus the probability that the opposite occurs 4. The probability of at least one of two (or more) things that can not simultaneously occur (mutually exclusive) is the sum of their respective probabilities 5. For any two events the probability that at least one occurs is the sum of their probabilities minus their intersection. This last rules states that P (A ∪ B) = P (A) + P (B) − P (A ∩ B) shows what is the issue with adding probabilities that are not mutually exclusive. If we do this, we’ve added the probability that both occur in twice! (Watch the video where I draw a Venn diagram to illustrate this). Example of Implementing Probability Calculus The National Sleep Foundation (www.sleepfoundation.org¹⁰) reports that around 3% of the American population has sleep apnea. They also report that around 10% of the North American and European population has restless leg syndrome. Does this imply that 13% of people will have at least one sleep problems of these sorts? In other words, can we simply add these two probabilities? Answer: No, the events can simultaneously occur and so are not mutually exclusive. To elaborate let: A1 = {Person has sleep apnea} A2 = {Person has RLS} Then P (A1 ∪ A2 ) = P (A1 ) + P (A2 ) − P (A1 ∩ A2 ) = 0.13 − Probability of having both Given the scenario, it’s likely that some fraction of the population has both. This example serves as a reminder don’t add probabilities unless the events are mutually exclusive. We’ll have a similar rule for multiplying probabilities and independence. Random variables Watch this video before reading this section¹¹ ¹⁰http://www.sleepfoundation.org/ ¹¹http://youtu.be/Shzt9uZ8BII?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ Probability 9 Probability calculus is useful for understanding the rules that probabilities must follow. However, we need ways to model and think about probabilities for numeric outcomes of experiments (broadly defined). Densities and mass functions for random variables are the best starting point for this. You’ve already heard of a density since you’ve heard of the famous “bell curve”, or Gaussian density. In this section you’ll learn exactly what the bell curve is and how to work with it. Remember, everything we’re talking about up to at this point is a population quantity, not a statement about what occurs in our data. Think about the fact that 50% probability for head is a statement about the coin and how we’re flipping it, not a statement about the percentage of heads we obtained in a particular set of flips. This is an important distinction that we will emphasize over and over in this course. Statistical inference is about describing populations using data. Probability density functions are a way to mathematically characterize the population. In this course, we’ll assume that our sample is a random draw from the population. So our definition is that a random variable is a numerical outcome of an experiment. The random variables that we study will come in two varieties, discrete or continuous. Discrete random variables are random variables that take on only a countable number of possibilities. Mass functions will assign probabilities that they take specific values. Continuous random variable can conceptually take any value on the real line or some subset of the real line and we talk about the probability that they lie within some range. Densities will characterize these probabilities. Let’s consider some examples of measurements that could be considered random variables. First, familiar gambling experiments like the tossing of a coin and the rolling of a die produce random variables. For the coin, we typically code a tail as a 0 and a head as a 1. (For the die, the number facing up would be the random variable.) We will use these examples a lot to help us build intuition. However, they aren’t interesting in the sense of seeming very contrived. Nonetheless, the coin example is particularly useful since many of the experiments we consider will be modeled as if tossing a biased coin. Modeling any binary characteristic from a random sample of a population can be thought of as a coin toss, with the random sampling performing the roll of the toss and the population percentage of individuals with the characteristic is the probability of a head. Consider, for example, logging whether or not subjects were hypertensive in a random sample. Each subject’s outcome can be modeled as a coin toss. In a similar sense the die roll serves as our model for phenomena with more than one level, such as hair color or rating scales. Consider also the random variable of the number of web hits for a site each day. This variable is a count, but is largely unbounded (or at least we couldn’t put a specific reasonable upper limit). Random variables like this are often modeled with the so called Poisson distribution. Finally, consider some continuous random variables. Think of things like lengths or weights. It is mathematically convenient to model these as if they were continuous (even if measurements were truncated liberally). In fact, even discrete random variables with lots of levels are often treated as continuous for convenience. For all of these kinds of random variables, we need convenient mathematical functions to model the probabilities of collections of realizations. These functions, called mass functions and densities, take possible values of the random variables, and assign the associated probabilities. These entities Probability 10 describe the population of interest. So, consider the most famous density, the normal distribution. Saying that body mass indices follow a normal distribution is a statement about the population of interest. The goal is to use our data to figure out things about that normal distribution, where it’s centered, how spread out it is and even whether our assumption of normality is warranted! Probability mass functions A probability mass function evaluated at a value corresponds to the probability that a random variable takes that value. To be a valid pmf a function, p, must satisfy: 1. It must always be larger than or equal to 0. 2. The sum of the possible values that the random variable can take has to add up to one. Example Let X be the result of a coin flip where X = 0 represents tails and X = 1 represents heads. p(x) = (1/2)x (1/2)1−x for x = 0, 1. Suppose that we do not know whether or not the coin is fair; Let θ be the probability of a head expressed as a proportion (between 0 and 1). p(x) = θx (1 − θ)1−x for x = 0, 1 Probability density functions Watch this video before beginning.¹² A probability density function (pdf), is a function associated with a continuous random variable. Because of the peculiarities of treating measurements as having been recorded to infinite decimal expansions, we need a different set of rules. This leads us to the central dogma of probability density functions: Areas under PDFs correspond to probabilities for that random variable Therefore, when one says that intelligence quotients (IQ) in population follows a bell curve, they are saying that the probability of a randomly selected person from this population having an IQ between two values is given by the area under the bell curve. Not every function can be a valid probability density function. For example, if the function dips below zero, then we could have negative probabilities. If the function contains too much area underneath it, we could have probabilities larger than one. The following two rules tell us when a function is a valid probability density function. Specifically, to be a valid pdf, a function must satisfy 1. It must be larger than or equal to zero everywhere. 2. The total area under it must be one. ¹²http://youtu.be/mPe0Us4VYDM?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ Probability 11 Example Suppose that the proportion of help calls that get addressed in a random day by a help line is given by f (x) = 2x for 0 < x < 1. The R code for plotting this density is Code for plotting the density x pbeta(0.75, 2, 1) 0.5625 Notice the syntax pbeta. In R, a prefix of p returns probabilities, d returns the density, q returns the quantile and r returns generated random variables. (You’ll learn what each of these does in subsequent sections.) CDF and survival function Certain areas of PDFs and PMFs are so useful, we give them names. The cumulative distribution function (CDF) of a random variable, X, returns the probability that the random variable is less than or equal to the value x. Notice the (slightly annoying) convention that we use an upper case X to denote a random, unrealized, version of the random variable and a lowercase x to denote a specific number that we plug into. (This notation, as odd as it may seem, dates back to Fisher and isn’t going anywhere, so you might as well get used to it. Uppercase for unrealized random variables and lowercase as placeholders for numbers to plug into.) So we could write the following to describe the distribution function F : F (x) = P (X ≤ x) This definition applies regardless of whether the random variable is discrete or continuous. The survival function of a random variable X is defined as the probability that the random variable is greater than the value x. S(x) = P (X > x) Notice that S(x) = 1 − F (x), since the survival function evaluated at a particular value of x is calculating the probability of the opposite event (greater than as opposed to less than or equal to). The survival function is often preferred in biostatistical applications while the distribution function is more generally used (though both convey the same information.) Example What are the survival function and CDF from the density considered before? 1 1 F (x) = P (X ≤ x) = Base × Height = (x) × (2x) = x2 , 2 2 Probability 14 for 1 ≥ x ≥ 0. Notice that calculating the survival function is now trivial given that we’ve already calculated the distribution function. S(x) = 1 = F (x) = 1 − x2 Again, R has a function that calculates the distribution function for us in this case, pbeta. Let’s try calculating F (.4), F (.5) and F (.6) > pbeta(c(0.4, 0.5, 0.6), 2, 1) 0.16 0.25 0.36 Notice, of course, these are simply the numbers squared. By default the prefix p in front of a density in R gives the distribution function (pbeta, pnorm, pgamma). If you want the survival function values, you could always subtract by one, or give the argument lower.tail = FALSE as an argument to the function, which asks R to calculate the upper area instead of the lower. Quantiles You’ve heard of sample quantiles. If you were the 95th percentile on an exam, you know that 95% of people scored worse than you and 5% scored better. These are sample quantities. But you might have wondered, what are my sample quantiles estimating? In fact, they are estimating the population quantiles. Here we define these population analogs. The αth quantile of a distribution with distribution function F is the point xα so that F (xα ) = α So the 0.95 quantile of a distribution is the point so that 95% of the mass of the density lies below it. Or, in other words, the point so that the probability of getting a randomly sampled point below it is 0.95. This is analogous to the sample quantiles where the 0.95 sample quantile is the value so that 95% of the data lies below it. A percentile is simply a quantile with α expressed as a percent rather than a proportion. The (population) median is the 50th percentile. Remember that percentiles are not probabilities! Remember that quantiles have units. So the population median height is the height (in inches say) so that the probability that a randomly selected person from the population is shorter is 50%. The sample, or empirical, median would be the height so in a sample so that 50% of the people in the sample were shorter. Example What is the median of the distribution that we were working with before? We want to solve 0.5 = F (x) = x2 , resulting in the solution Probability 15 > sqrt(0.5) 0.7071 Therefore, 0.7071 of calls being answered on a random day is the median. Or, the probability that 70% or fewer calls get answered is 50%. R can approximate quantiles for you for common distributions with the prefix q in front of the distribution name > qbeta(0.5, 2, 1) 0.7071 Exercises 1. Can you add the probabilities of any two events to get the probability of at least one occurring? 2. I define a PMF, p so that for x = 0 and x = 1 we have p(0) = −0.1 and p(1) = 1.1. Is this a valid PMF? 3. What is the probability that 75% or fewer calls get answered in a randomly sampled day from the population distribution from this chapter? 4. The 97.5th percentile of a distribution is? 5. Consider influenza epidemics for two parent heterosexual families. Suppose that the proba- bility is 15% that at least one of the parents has contracted the disease. The probability that the father has contracted influenza is 10% while that the mother contracted the disease is 9%. What is the probability that both contracted influenza expressed as a whole number percentage? Watch a video solution to this problem.¹³ and see a written out solution.¹⁴ 6. A random variable, X, is uniform, a box from 0 to 1 of height 1. (So that it’s density is f (x) = 1 for 0 ≤ x ≤ 1.) What is it’s median expressed to two decimal places? Watch a video solution to this problem here¹⁵ and see written solutions here¹⁶. 7. If a continuous density that never touches the horizontal axis is symmetric about zero, can we say that its associated median is zero? Watch a worked out solution to this problem here¹⁷ and see the question and a typed up answer here¹⁸ ¹³http://youtu.be/CvnmoCuIN08?list=PLpl-gQkQivXhHOcVeU3bSJg78zaDYbP9L ¹⁴http://bcaffo.github.io/courses/06_StatisticalInference/homework/hw1.html#3 ¹⁵http://youtu.be/UXcarD-1xAM?list=PLpl-gQkQivXhHOcVeU3bSJg78zaDYbP9L ¹⁶http://bcaffo.github.io/courses/06_StatisticalInference/homework/hw1.html#4 ¹⁷http://youtu.be/sn48CGH_TXI?list=PLpl-gQkQivXhHOcVeU3bSJg78zaDYbP9L ¹⁸http://bcaffo.github.io/courses/06_StatisticalInference/homework/hw1.html#9 3. Conditional probability Conditional probability, motivation Watch this video before beginning.¹ Conditioning is a central subject in statistics. If we are given information about a random variable, it changes the probabilities associated with it. For example, the probability of getting a one when rolling a (standard) die is usually assumed to be one sixth. If you were given the extra information that the die roll was an odd number (hence 1, 3 or 5) then conditional on this new information, the probability of a one is now one third. This is the idea of conditioning, taking away the randomness that we know to have occurred. Consider another example, such as the result of a diagnostic imaging test for lung cancer. What’s the probability that a person has cancer given a positive test? How does that probability change under the knowledge that a patient has been a lifetime heavy smoker and both of their parents had lung cancer? Conditional on this new information, the probability has increased dramatically. Conditional probability, definition We can formalize the definition of conditional probability so that the mathematics matches our intuition. Let B be an event so that P (B) > 0. Then the conditional probability of an event A given that B has occurred is: P (A ∩ B) P (A | B) =. P (B) If A and B are unrelated in any way, or in other words independent, (discussed more later in the lecture), then P (A)P (B) P (A | B) = = P (A) P (B) That is, if the occurrence of B offers no information about the occurrence of A - the probability conditional on the information is the same as the probability without the information, we say that the two events are independent. ¹http://youtu.be/u6AH6qsSVA4?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ 16 Conditional probability 17 Example Consider our die roll example again. Here we have that B = {1, 3, 5} and A = {1} P (A ∩ B) P (A) 1/6 1 P (one given that roll is odd) = P (A | B) = = = = P (B) P (B) 3/6 3 Which exactly mirrors our intuition. Bayes’ rule Watch this video before beginning² Bayes’ rule is a famous result in statistics and probability. It forms the foundation for large branches of statistical thinking. Bayes’ rule allows us to reverse the conditioning set provided that we know some marginal probabilities. Why is this useful? Consider our lung cancer example again. It would be relatively easy for physicians to calculate the probability that the diagnostic method is positive for people with lung cancer and negative for people without. They could take several people who are already known to have the disease and apply the test and conversely take people known not to have the disease. However, for the collection of people with a positive test result, the reverse probability is more of interest, “given a positive test what is the probability of having the disease?”, and “given a given a negative test what is the probability of not having the disease?”. Bayes’ rule allows us to switch the conditioning event, provided a little bit of extra information. Formally Bayes’ rule is: P (A | B)P (B) P (B | A) =. P (A | B)P (B) + P (A | B c )P (B c ) Diagnostic tests Since diagnostic tests are a really good example of Bayes’ rule in practice, let’s go over them in greater detail. (In addition, understanding Bayes’ rule will be helpful for your own ability to understand medical tests that you see in your daily life). We require a few definitions first. Let + and − be the events that the result of a diagnostic test is positive or negative respectively Let D and Dc be the event that the subject of the test has or does not have the disease respectively The sensitivity is the probability that the test is positive given that the subject actually has the disease, P (+ | D) ²http://youtu.be/TfeaZ_26iQk?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ Conditional probability 18 The specificity is the probability that the test is negative given that the subject does not have the disease, P (− | Dc ) So, conceptually at least, the sensitivity and specificity are straightforward to estimate. Take people known to have and not have the disease and apply the diagnostic test to them. However, the reality of estimating these quantities is quite challenging. For example, are the people known to have the disease in its later stages, while the diagnostic will be used on people in the early stages where it’s harder to detect? Let’s put these subtleties to the side and assume that they are known well. The quantities that we’d like to know are the predictive values. The positive predictive value is the probability that the subject has the disease given that the test is positive, P (D | +) The negative predictive value is the probability that the subject does not have the disease given that the test is negative, P (Dc | −) Finally, we need one last thing, the prevalence of the disease - which is the marginal probability of disease, P (D). Let’s now try to figure out a PPV in a specific setting. Example A study comparing the efficacy of HIV tests, reports on an experiment which concluded that HIV antibody tests have a sensitivity of 99.7% and a specificity of 98.5% Suppose that a subject, from a population with a.1% prevalence of HIV, receives a positive test result. What is the positive predictive value? Mathematically, we want P (D | +) given the sensitivity, P (+ | D) =.997, the specificity, P (− | Dc ) =.985 and the prevalence P (D) =.001. P (+ | D)P (D) P (D | +) = P (+ | D)P (D) + P (+ | Dc )P (Dc ) P (+ | D)P (D) = P (+ | D)P (D) + {1 − P (− | Dc )}{1 − P (D)}.997 ×.001 =.997 ×.001 +.015 ×.999 =.062 In this population a positive test result only suggests a 6% probability that the subject has the disease, (the positive predictive value is 6% for this test). If you were wondering how it could be so low for this test, the low positive predictive value is due to low prevalence of disease and the somewhat modest specificity Suppose it was known that the subject was an intravenous drug user and routinely had intercourse with an HIV infected partner? Our prevalence would change dramatically, thus increasing the PPV. You might wonder if there’s a way to summarize the evidence without appealing to an often unknowable prevalence? Diagnostic likelihood ratios provide this for us. Conditional probability 19 Diagnostic Likelihood Ratios The diagnostic likelihood ratios summarize the evidence of disease given a positive or negative test. They are defined as: The diagnostic likelihood ratio of a positive test, labeled DLR+ , is P (+ | D)/P (+ | Dc ), which is the sensitivity/(1 − specif icity). The diagnostic likelihood ratio of a negative test, labeled DLR− , is P (− | D)/P (− | Dc ), which is the (1 − sensitivity)/specif icity. How do we interpret the DLRs? This is easiest when looking at so called odds ratios. Remember that if p is a probability, then p/(1 − p) is the odds. Consider now the odds in our setting: Using Bayes rule, we have P (+ | D)P (D) P (D | +) = P (+ | D)P (D) + P (+ | Dc )P (Dc ) and P (+ | Dc )P (Dc ) P (Dc | +) =. P (+ | D)P (D) + P (+ | Dc )P (Dc ) Therefore, dividing these two equations we have: P (D | +) P (+ | D) P (D) = × P (D | +) c P (+ | D ) P (Dc ) c In other words, the post test odds of disease is the pretest odds of disease times the DLR+. Similarly, DLR− relates the decrease in the odds of the disease after a negative test result to the odds of disease prior to the test. So, the DLRs are the factors by which you multiply your pretest odds to get your post test odds. Thus, if a test has a DLR+ of 6, regardless of the prevalence of disease, the post test odds is six times that of the pretest odds. HIV example revisited Let’s reconsider our HIV antibody test again. Suppose a subject has a positive HIV test DLR+ =.997/(1 −.985) = 66 Conditional probability 20 The result of the positive test is that the odds of disease is now 66 times the pretest odds. Or, equivalently, the hypothesis of disease is 66 times more supported by the data than the hypothesis of no disease Suppose instead that a subject has a negative test result DLR− = (1 −.997)/.985 =.003 Therefore, the post-test odds of disease is now 0.3% of the pretest odds given the negative test. Or, the hypothesis of disease is supported.003 times that of the hypothesis of absence of disease given the negative test result Independence Watch this video before beginning.³ Statistical independence of events is the idea that the events are unrelated. Consider successive coin flips. Knowledge of the result of the first coin flip tells us nothing about the second. We can formalize this into a definition. Two events A and B are independent if P (A ∩ B) = P (A)P (B) Equivalently if P (A | B) = P (A). Note that since A is independent of B we know that Ac is independent of B A is independent of B c Ac is independent of B c. While this definition works for sets, remember that random variables are really the things that we are interested in. Two random variables, X and Y are independent if for any two sets A and B P ([X ∈ A] ∩ [Y ∈ B]) = P (X ∈ A)P (Y ∈ B) We will almost never work with these definitions. Instead, the important principle is that probabil- ities of independent things multiply! This has numerous consequences, including the idea that we shouldn’t multiply non-independent probabilities. Example Let’s cover a very simple example: “What is the probability of getting two consecutive heads?”. Then we have that A is the event of getting a head on flip 1 P (A) = 0.5 B is the event of getting a head on flip 2 P (B) = 0.5 A ∩ B is the event of getting heads on flips 1 and 2. Then independence would tell us that: ³http://youtu.be/MY1EfrR1ZUs?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ Conditional probability 21 P (A ∩ B) = P (A)P (B) = 0.5 × 0.5 = 0.25 This is exactly what we would have intuited of course. But, it’s nice that the mathematics mirrors our intuition. In more complex settings, it’s easy to get tripped up. Consider the following famous (among statisticians at least) case study. Case Study Volume 309 of Science reports on a physician who was on trial for expert testimony in a criminal trial. Based on an estimated prevalence of sudden infant death syndrome (SIDS) of 1 out of 8,543, a physician testified that that the probability of a mother having two children with SIDS was (1/8, 543)2. The mother on trial was convicted of murder. Relevant to this discussion, the principal mistake was to assume that the events of having SIDs within a family are independent. That is, P (A1 ∩ A2 ) is not necessarily equal to P (A1 )P (A2 ). This is because biological processes that have a believed genetic or familiar environmental component, of course, tend to be dependent within families. Thus, we can’t just multiply the probabilities to obtain the result. There are many other interesting aspects to the case. For example, the idea of a low probability of an event representing evidence against a plaintiff. (Could we convict all lottery winners of fixing the lotter since the chance that they would win is so small.) IID random variables Now that we’ve introduced random variables and independence, we can introduce a central modeling assumption made in statistics. Specifically the idea of a random sample. Random variables are said to be independent and identically distributed (iid) if they are independent and all are drawn from the same population. The reason iid samples are so important is that they are a model for random samples. This is a default starting point for most statistical inferences. The idea of having a random sample is powerful for a variety of reasons. Consider that in some study designs, such as in election polling, great pains are made to make sure that the sample is randomly drawn from a population of interest. The idea is to expend a lot of effort on design to get robust inferences. In these settings assuming that the data is iid is both natural and warranted. In other settings, the study design is far more opaque, and statistical inferences are conducted under the assumption that the data arose from a random sample, since it serves as a useful benchmark. Most studies in the fields of epidemiology and economics fall under this category. Take, for example, studying how policies impact countries gross domestic product by looking at countries before and after enacting the policies. The countries are not a random sample from the set of countries. Instead, conclusions must be made under the assumption that the countries are a random sample and the interpretation of the strength of the inferences adapted in kind. Conditional probability 22 Exercises 1. I pull a card from a deck and do not show you the result. I say that the resulting card is a heart. What is the probability that it is the queen of hearts? 2. The odds associated with a probability, p, are defined as? 3. The probability of getting two sixes when rolling a pair of dice is? 4. The probability that a manuscript gets accepted to a journal is 12% (say). However, given that a revision is asked for, the probability that it gets accepted is 90%. Is it possible that the probability that a manuscript has a revision asked for is 20%? Watch a video of this problem getting solved⁴ and see the worked out solutions here⁵. 5. Suppose 5% of housing projects have issues with asbestos. The sensitivity of a test for asbestos is 93% and the specificity is 88%. What is the probability that a housing project has no asbestos given a negative test expressed as a percentage to the nearest percentage point? Watch a video solution here⁶ and see the worked out problem here⁷. ⁴http://youtu.be/E4kE4M1J15s?list=PLpl-gQkQivXhHOcVeU3bSJg78zaDYbP9L ⁵http://bcaffo.github.io/courses/06_StatisticalInference/homework/hw2.html#3 ⁶https://www.youtube.com/watch?v=rbI97tSvGvQ&list=PLpl-gQkQivXhHOcVeU3bSJg78zaDYbP9L&index=11 ⁷http://bcaffo.github.io/courses/06_StatisticalInference/homework/hw2.html#5 4. Expected values Watch this video before beginning.¹ Expected values characterize a distribution. The most useful expected value, the mean, characterizes the center of a density or mass function. Another expected value summary, the variance, charac- terizes how spread out a density is. Yet another expected value calculation is the skewness, which considers how much a density is pulled toward high or low values. Remember, in this lecture we are discussing population quantities. It is convenient (and of course by design) that the names for all of the sample analogs estimate the associated population quantity. So, for example, the sample or empirical mean estimates the population mean; the sample variance estimates the population variance and the sample skewness estimates the population skewness. The population mean for discrete random variables The expected value or (population) mean of a random variable is the center of its distribution. For discrete random variable X with PMF p(x), it is defined as follows: ∑ E[X] = xp(x). x where the sum is taken over the possible values of x. Where did they get this idea from? It’s taken from the physical idea of the center of mass. Specifically, E[X] represents the center of mass of a collection of locations and weights, {x, p(x)}. We can exploit this fact to quickly calculate population means for distributions where the center of mass is obvious. The sample mean It is important to contrast the population mean (the estimand) with the sample mean (the estimator). The sample mean estimates the population mean. Not coincidentally, since the population mean is the center of mass of the population distribution, the sample mean is the center of mass of the data. In fact, it’s exactly the same equation: ∑ n X̄ = xi p(xi ), i=1 where p(xi ) = 1/n. ¹http://youtu.be/zljxRbu6jyc?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ 23 Expected values 24 Example Find the center of mass of the bars Let’s go through an example of illustrating how the sample mean is the center of mass of observed data. Below we plot Galton’s fathers and sons data: Loading in and displaying the Galton data library(UsingR); data(galton); library(ggplot2); library(reshape2) longGalton 30 using a normal test (n was 100). Notice that we rejected the one sided test when α = 0.05, would we reject if α = 0.01, how about 0.001? The smallest value for alpha that you still reject the null hypothesis is called the attained significance level. This is mathematically equivalent, but philosophically a little different from, the P-value. Whereas the P-value is interpreted in the terms of how probabilistically extreme our test statistic is under the null, the attained significance level merely conveys what the smallest level of α that one could reject at. This equivalence makes P-values very convenient to convey. The reader of the results can perform the test at whatever α he or she choses. This is especially useful in multiple testing circumstances. Here’s the two rules for performing hypothesis tests with P-values. * If the P-value for a test is less than α you reject the null hypothesis * For two sided hypothesis test, double the smaller of the two one sided hypothesis test Pvalues Binomial P-value example Suppose a friend has 8 children, 7 of which are girls and none are twins. If each gender has an independent 50% probability for each birth, what’s the probability of getting 7 or more girls out of 8 births? P-values 98 This calculation is a P-value where the statistic is the number of girls and the null distribution is a fair coin flip for each gender. We want to test H0 : p = 0.5 versus Ha : p > 0.5, where p is the probability of having a girl for each birth. Recall here’s the calculation: Example of a Binomial P-value calculation in R. > pbinom(6, size = 8, prob = 0.5, lower.tail = FALSE) 0.03516 Since our P-value is less than 0.05 we would reject at a 5% error rate. Note, however, if we were doing a two sided test, we would have to double the P-value and thus would then fail to reject. Poisson example Watch this video before beginning.⁹ Suppose that a hospital has an infection rate of 10 infections per 100 person/days at risk (rate of 0.1) during the last monitoring period. Assume that an infection rate of 0.05 is an important benchmark. Given a Poisson model, could the observed rate being larger than 0.05 be attributed to chance? We want to test H0 : λ = 0.05 where λ is the rate of infections per person day so that 5 would be the rate per 100 days. Thus we want to know if 9 events per 100 person/days is unusual with respect to a Poisson distribution with a rate of 5 events per 100. Consider Ha : λ > 0.05. Poisson P-value calculation. > ppois(9, 5, lower.tail = FALSE) 0.03183 Again, since this P-value is less than 0.05 we reject the null hypothesis. The P-value would be 0.06 for two sided hypothesis (double) and so we would fail to reject in that case. Exercises 1. P-values are probabilities that are calculated assuming which hypothesis is true? the alternative the null 2. You get a P-value of 0.06. Would you reject for a type I error rate of 0.05? Yes you would reject the null ⁹http://youtu.be/Tcw2OVyEX3s?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ P-values 99 No you would not reject the null It depends on information not given 3. The proposed procedure for getting a two sided P-value for the exact binomial test considered here is what? Multiplying the one sided P-value by one half Doubling the larger of the two one sided P-values Doubling the smaller of the two one sided P-values No procedure exists 4. Consider again the mtcars dataset. Use a two group t-test to test the hypothesis that the 4 and 6 cyl cars have the same mpg. Use a two sided test with unequal variances. Give a P-value. Watch the video here¹⁰ and see the text here¹¹ 5. You believe the coin that you’re flipping is biased towards heads. You get 55 heads out of 100 flips. Give an exact P-value for the hypothesis that the coin is fair. Watch a video solution¹² and see the text¹³. 6. A web site was monitored for a year and it received 520 hits per day. In the first 30 days in the next year, the site received 15,800 hits. Assuming that web hits are Poisson. Give an exact one sided P-value to the hypothesis that web hits are up this year over last. Do you reject? Watch the video solutions¹⁴ and see the problem text¹⁵. 7. Suppose that in an AB test, one advertising scheme led to an average of 10 purchases per day for a sample of 100 days, while the other led to 11 purchases per day, also for a sample of 100 days. Assuming a common standard deviation of 4 purchases per day. Assuming that the groups are independent and that they days are iid, perform a Z test of equivalence. Give a P-value for the test? Watch a video solution¹⁶ and see the text.¹⁷ 8. Consider the mtcars data set. Give the p-value for a t-test comparing MPG for 6 and 8 cylinder cars assuming equal variance, as a proportion to 3 decimal places. Give the associated P-value for a z test. Give the common standard deviation estimate for MPG across cylinders to 3 decimal places. Would the t test reject at the two sided 0.05 level (0 for no 1 for yes)? Watch a video solution¹⁸ and see the text¹⁹. ¹⁰https://www.youtube.com/watch?v=Zo5TirzS9rU&list=PLpl-gQkQivXhHOcVeU3bSJg78zaDYbP9L&index=28 ¹¹http://bcaffo.github.io/courses/06_StatisticalInference/homework/hw4.html#4 ¹²https://www.youtube.com/watch?v=0sqOErsfhqo&list=PLpl-gQkQivXhHOcVeU3bSJg78zaDYbP9L&index=30 ¹³http://bcaffo.github.io/courses/06_StatisticalInference/homework/hw4.html#6 ¹⁴https://www.youtube.com/watch?v=cE_88-Q7TX0&index=31&list=PLpl-gQkQivXhHOcVeU3bSJg78zaDYbP9L ¹⁵http://bcaffo.github.io/courses/06_StatisticalInference/homework/hw4.html#7 ¹⁶https://www.youtube.com/watch?v=Or4ly4rOiaA&index=32&list=PLpl-gQkQivXhHOcVeU3bSJg78zaDYbP9L ¹⁷http://bcaffo.github.io/courses/06_StatisticalInference/homework/hw4.html#8 ¹⁸https://www.youtube.com/watch?v=m0B5p0w2wJI&list=PLpl-gQkQivXhHOcVeU3bSJg78zaDYbP9L&index=37 ¹⁹http://bcaffo.github.io/courses/06_StatisticalInference/homework/hw4.html#13 11. Power Power Watch this video before beginning.¹ and then watch this video as well.² Power is the probability of rejecting the null hypothesis when it is false. Ergo, power (as its name would suggest) is a good thing; you want more power. A type II error (a bad thing, as its name would suggest) is failing to reject the null hypothesis when it’s false; the probability of a type II error is usually called β. Note Power = 1 − β. Let’s go through an example of calculating power. Consider our previous example involving RDI. H0 : µ = 30 versus Ha : µ > 30. Then power is: ( ) X̄ − 30 P √ > t1−α,n−1 ; µ = µa. s/ n Note that this is a function that depends on the specific value of µa ! Further notice that as µa approaches 30 the power approaches α. Pushing this example further, we reject if X̄ − 30 Z= √ > z1−α σ/ n Or, equivalently, if σ X̄ > 30 + Z1−α √ n But, note that, under H0 : X̄ ∼ N (µ0 , σ 2 /n). However, under Ha : X̄ ∼ N (µa , σ 2 /n). So for this test we could calculate power with this R code: ¹http://youtu.be/-TsBOLiW4rQ?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ ²http://youtu.be/GRS2b1aedmk?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ 100 Power 101 Power calculation for the sleep example in R alpha = 0.05 z = qnorm(1 - alpha) pnorm(mu0 + z * sigma/sqrt(n), mean = mua, sd = sigma/sqrt(n), lower.tail = FALS\ E) Let’s plug in the specific numbers for our example where: µa = 32, µ0 = 30, n = 16, σ = 4. > mu0 = 30 > mua = 32 > sigma = 4 > n = 16 > z = qnorm(1 - alpha) > pnorm(mu0 + z * sigma/sqrt(n), mean = mu0, sd = sigma/sqrt(n), lower.tail = FA\ LSE) 0.05 > pnorm(mu0 + z * sigma/sqrt(n), mean = mua, sd = sigma/sqrt(n), lower.tail = FA\ LSE) 0.6388 When we plug in µ0 , the value under the null hypothesis, we get that the probability of rejection is 5%, as the test was designed. However, when we plug in a value of 32, we get 64%. Therefore, the probability of rejection is 64% when the true value of µ is 32. We could create a curve of the power as a function of µa , as seen below. We also varied the sample size to see how the curve depends on that. Power 102 Plot of power as µa varies. The code below shows how to use manipulate to investigate power as the various inputs change. Code for investigating power. library(manipulate) mu0 = 30 myplot µ0 , notice if power is 1 − β, then ( ) σ 1 − β = P X̄ > µ0 + z1−α √ ; µ = µa n where X̄ ∼ N (µa , σ 2 /n). The unknowns in the equation are: µa , σ, n, β and the knowns are: µ0 , α. Specify any 3 of the unknowns and you can solve for the remainder. Notes The calculation for Ha : µ < µ0 is similar For Ha : µ ̸= µ0 calculate the one sided power using α/2 (this is only approximately right, it excludes the probability of getting a large TS in the opposite direction of the truth) Power goes up as α gets larger Power of a one sided test is greater than the power of the associated two sided test Power goes up as µ1 gets further away from $\mu_0$ Power goes up as n goes up √ Power doesn’t need µa , σ and n, instead only n(µσa −µ0 ) – The quantity µa −µσ 0 is called the effect size, the difference in the means in standard deviation units. – Being unit free, it has some hope of interpretability across settings. T-test power Watch this before beginning.⁴ Consider calculating power for a Gosset’s t test for our example where we now assume that n = 16. The power is ( ) X̄ − µ0 P √ > t1−α,n−1 ; µ = µa. S/ n Calculating this requires the so-called non-central t distribution. However, fortunately for us, the R function power.t.test does this very well. Omit (exactly) any one of the arguments and it solves for it. Our t-test power again only relies on the effect size. Let’s do our example trying different options. ³http://youtu.be/3bWhP5MyuqI?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ ⁴http://youtu.be/1DiwutNpt5Y?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ Power 104 Example of using ‘power.t.test’ in R. # omitting the power and getting a power estimate > power.t.test(n = 16, delta = 2/4, sd = 1, type = "one.sample", alt = "one.side\ d")$power 0.604 # illustrating that it depends only on the effect size, delta/sd > power.t.test(n = 16, delta = 2, sd = 4, type = "one.sample", alt = "one.sided"\ )$power 0.604 # same thing again > power.t.test(n = 16, delta = 100, sd = 200, type = "one.sample", alt = "one.si\ ded")$power 0.604 # specifying the power and getting n > power.t.test(power = 0.8, delta = 2/4, sd = 1, type = "one.sample", alt = "one\.sided")$n 26.14 # again illustrating that the effect size is all that matters power.t.test(power = 0.8, delta = 2, sd = 4, type = "one.sample", alt = "one.sid\ ed")$n 26.14 # again illustrating that the effect size is all that matters > power.t.test(power = 0.8, delta = 100, sd = 200, type = "one.sample", alt = "o\ ne.sided")$n 26.14 Exercises 1. Power is a probability calculation assuming which is true: The null hypothesis The alternative hypothesis Both the null and alternative 2. As your sample size gets bigger, all else equal, what do you think would happen to power? It would get larger It would get smaller It would stay the same It cannot be determined from the information given 3. What happens to power as µa gets further from µ0 ? Power decreases Power increases Power 105 Power stays the same Power oscillates 4. In the context of calculating power, the effect size is? The null mean divided by the standard deviation The alternative mean divided by the standard error The difference between the null and alternative means divided by the standard deviation The standard error divided by the null mean 5. Recall this problem “Suppose that in an AB test, one advertising scheme led to an average of 10 purchases per day for a sample of 100 days, while the other led to 11 purchases per day, also for a sample of 100 days. Assuming a common standard deviation of 4 purchases per day.” Assuming that 10 purchases per day is a benchmark null value, that days are iid and that the standard deviation is 4 purchases for day. Suppose that you plan on sampling 100 days. What would be the power for a one sided 5% Z mean test that purchases per day have increased under the alternative of µ = 11 purchase per day? Watch a video solution⁵ and see the text⁶. 6. Researchers would like to conduct a study of healthy adults to detect a four year mean brain volume loss of.01 mm3. Assume that the standard deviation of four year volume loss in this population is.04 mm3. What is necessary sample size for the study for a 5% one sided test versus a null hypothesis of no volume loss to achieve 80% power? Watch the video solution⁷ and see the text⁸. ⁵https://www.youtube.com/watch?v=RiS6EFnPYY8&index=34&list=PLpl-gQkQivXhHOcVeU3bSJg78zaDYbP9L ⁶http://bcaffo.github.io/courses/06_StatisticalInference/homework/hw4.html#10 ⁷https://www.youtube.com/watch?v=lrXyJrtatzk&index=35&list=PLpl-gQkQivXhHOcVeU3bSJg78zaDYbP9L ⁸http://bcaffo.github.io/courses/06_StatisticalInference/homework/hw4.html#11 12. The bootstrap and resampling The bootstrap Watch this video before beginning.¹ The bootstrap is a tremendously useful tool for constructing confidence intervals and calculating standard errors for difficult statistics. For a classic example, how would one derive a confidence interval for the median? The bootstrap procedure follows from the so called bootstrap principle To illustrate the bootstrap principle, imagine a die roll. The image below shows the mass function of a die roll on the left. On the right we show the empirical distribution obtained by repeatedly averaging 50 independent die rolls. By this simulation, without any mathematics, we have a good idea of what the distribution of averages of 50 die rolls looks like. Image of true die roll distribution (left) and simulation of averages of 50 die rolls Now imagine a case where we didn’t know whether or not the die was fair. We have a sample of size 50 and we’d like to investigate the distribution of the average of 50 die rolls where we’re not allowed to roll the die anymore. This is more like a real data analysis, we only get one sample from the population. ¹http://youtu.be/0hNQx9nagq4?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ 106 The bootstrap and resampling 107 Image of empirical die roll distribution (left) and simulates of averages of 50 die rolls from this distribution The bootstrap principle is to use the empirical mass function of the data to perform the simulation, rather than the true distribution. That is, we simulate averages of 50 samples from the histogram that we observe. With enough data, the empirical distribution should be a good estimate of the true distribution and this should result in a good approximation of the sampling distribution. That’s the bootstrap principle: investigate the sampling distribution of a statistic by simulating repeated realizations from the observed distribution. If we could simulate from the true distribution, then we would know the exact sampling distribution of our statistic (if we ran our computer long enough.) However, since we only get to sample from that distribution once, we have to be content with using the empirical distribution. This is the clever idea of the bootstrap. Example Galton’s fathers and sons dataset Watch this video before beginning.² The code below creates resamples via draws of size n with replacement with the original data of the son’s heights from Galton’s data and plots a histogram of the median of each resampled dataset. ²http://youtu.be/yNTWcmbWvWg?list=PLpl-gQkQivXiBmGyzLrUjzsblmQsLtkzJ The bootstrap and resampling 108 Bootstrapping example library(UsingR) data(father.son) x