Statistics, Visualization and More Using R (298.916) Block IV: - PowerPoint PPT Presentation

An introductory example Binom.test One sample t -test Two sample t -test Statistics, Visualization and More Using ”R” (298.916) Block IV: (Elementary) hypothesis testing Ass.-Prof. Dr. Wolfgang Trutschnig Research group for Stochastics/Statistics Department for Mathematics University Salzburg www.trutschnig.net Salzburg, April 2017 Wolfgang Trutschnig Statistics, Visualization and More Using ”R” (298.916)

An introductory example Binom.test One sample t -test Two sample t -test Plan for Block IV: ◮ An introductory example for the Binomial Distribution. ◮ Error of first and second kind; power. ◮ The p -value. ◮ Tests for the binomial (alternative) distribution. ◮ t -tests. ◮ Exercises. Wolfgang Trutschnig Statistics, Visualization and More Using ”R” (298.916)

An introductory example Binom.test One sample t -test Two sample t -test Alternative and binomial distribution ◮ The alternative distribution is fully determined by one single parameter p ∈ [0 , 1]. ◮ If X has an alternative distribution we will write X ∼ A ( p ) in the sequel. ◮ If X ∼ A ( p ) then X can only assume two values: 0 and 1; more precisely P ( X = 1) = p , P ( X = 0) = 1 − p . ◮ We all know examples of variables with alternative distribution: ◮ If we denote the result of flipping a coin by 0 (tails) and 1 (heads), then p = 1 2 and X ∼ A ( 1 2 ). ◮ If X denotes the result of rolling a dice and we write 1 if the result is either 5 or 6 and 0 otherwise then p = 1 3 and X ∼ A ( 1 3 ). Wolfgang Trutschnig Statistics, Visualization and More Using ”R” (298.916)

An introductory example Binom.test One sample t -test Two sample t -test Alternative and binomial distribution ◮ In practice, we do not know the parameter p and have to estimate it based on a sample x 1 , x 2 , . . . , x n . Example (Election forecasts simplified) ◮ Suppose that one week before the election 100 (randomly drawn) people are asked which of the two candidates ’0’ and ’1’ they will vote for. ◮ 42 answer ’0’ and 58 answer ’1’. ◮ How would you estimate p = P ( X = 1)? ◮ Natural choice is x 100 = 0 . 58 =: ˆ p 100 . Wolfgang Trutschnig Statistics, Visualization and More Using ”R” (298.916)

An introductory example Binom.test One sample t -test Two sample t -test Alternative and binomial distribution ◮ Suppose that Z ∼ A ( p ) and we repeat the ’experiment’ n times. ◮ Let X denote the number of 1s observed in the n trials. ◮ We will write X ∼ Bin ( n , p ) and say that X has binomial distribution (with parameters n and p ). ◮ X can attain all integer values between 0 and n . ◮ With a little bit of mathematics we get the following well-known formula: � n � p k (1 − p ) n − k , P ( X = k ) = k ∈ { 0 , 1 , . . . , n } . k Example (Election again) ◮ Suppose that in the election exactly 50% voted for candidate ’0’ and candidate ’1’ each. ◮ We ask 100 randomly selected voters, which candidate they voted for. ◮ What is the probability that 42 answer ’0’ and 58 answer ’1’? 0 . 5 58 0 . 5 42 ≈ 0 . 022. � 100 ◮ P ( X = 58) = � 58 Wolfgang Trutschnig Statistics, Visualization and More Using ”R” (298.916)

An introductory example Binom.test One sample t -test Two sample t -test Toy example hypothesis testing Example (Toy example hypothesis testing) ◮ Suppose that somebody rolls a dice (that you can not see). ◮ You only know that the dice either has (i) a ’1’ on four sides and a ’0’ on the other two sides or (ii) a ’1’ on two sides and a ’0’ on the other four sides. ◮ If we let X denote the result of rolling this dice once, then we either have 4 6 = 2 P ( X = 0) = 2 6 = 1 ( i ) p := P ( X = 1) = 3 = 1 − p and 3 or 2 6 = 1 P ( X = 0) = 4 6 = 2 ( ii ) p := P ( X = 1) = 3 = 1 − p . and 3 ◮ In other words, X ∼ A ( p ) and we know that p ∈ Θ = { 2 3 , 1 3 } . ◮ We will call H 0 : p = 2 3 the null hypothesis and H 1 : p = 1 3 the alternative hypothesis (for whatever reason). Wolfgang Trutschnig Statistics, Visualization and More Using ”R” (298.916)

An introductory example Binom.test One sample t -test Two sample t -test Toy example hypothesis testing Example (Toy example hypothesis testing, cont.) ◮ For the moment we focus on H 0 : p = 2 3 . ◮ Suppose that the dice is rolled twice and the result is denoted by ( X 1 , X 2 ). ◮ Possibility 1: ( X 1 , X 2 ) = (1 , 1). Would you stick to H 0 or reject H 0 (i.e. change to H 1 ), and why? ◮ Possibility 2: ( X 1 , X 2 ) = (1 , 0). Would you stick to H 0 or reject H 0 , and why? ◮ Possibility 3: ( X 1 , X 2 ) = (0 , 1). Would you stick to H 0 or reject H 0 , and why? ◮ Possibility 4: ( X 1 , X 2 ) = (0 , 0). Would you stick to H 0 or reject H 0 , and why? ◮ Which criterion is your decision based upon? ◮ For a given observation we check under which of the two hypotheses the observation has higher probability. Wolfgang Trutschnig Statistics, Visualization and More Using ”R” (298.916)

An introductory example Binom.test One sample t -test Two sample t -test Toy example hypothesis testing Example (Toy example hypothesis testing, cont.) ◮ If H 0 is correct then we have 4 P H 0 ( X 1 = 1 , X 2 = 0) = 2 P H 0 ( X 1 = 1 , X 2 = 1) = 9 , 9 2 P H 0 ( X 1 = 0 , X 2 = 0) = 1 P H 0 ( X 1 = 0 , X 2 = 1) = 9 , 9 . ◮ If H 1 is correct then we have 1 P H 1 ( X 1 = 1 , X 2 = 0) = 2 P H 1 ( X 1 = 1 , X 2 = 1) = 9 , 9 2 P H 1 ( X 1 = 0 , X 2 = 0) = 4 P H 1 ( X 1 = 0 , X 2 = 1) = 9 , 9 . ◮ In case of (1 , 1) we do not reject H 0 . ◮ In case of (1 , 0) and in case of (0 , 1) we do not reject H 0 (the observation is equally probable under both hypotheses, so by changing from H 0 to H 1 we don’t gain anything). ◮ In case of (0 , 0) we reject H 0 . Wolfgang Trutschnig Statistics, Visualization and More Using ”R” (298.916)

An introductory example Binom.test One sample t -test Two sample t -test Toy example hypothesis testing Example (Toy example hypothesis testing, cont.) ◮ We intuitively reject H 0 if - under the assumption that H 0 is true - the observation we made is very unlikely (in the sense of having low probability). ◮ In our toy setting we can make two different mistakes: ◮ Type I error : We reject H 0 although it is correct. ◮ Type II error : We do not reject (accept) H 0 although it is wrong. ◮ Let us calculate the probability of a type I and the probability of a type II error in our toy setting: ◮ @type I error α : α := P H 0 ( reject H 0 ) = P H 0 ( X 1 = 0 , X 2 = 0) = 1 9 ◮ We have a chance of more than 11% to make a type I error. Wolfgang Trutschnig Statistics, Visualization and More Using ”R” (298.916)

An introductory example Binom.test One sample t -test Two sample t -test Toy example hypothesis testing Example (Toy example hypothesis testing, cont.) ◮ @type II error β : β := P H 1 ( accept H 0 ) = P H 1 ( X 1 = 1 , X 2 = 1) + P H 1 ( X 1 = 1 , X 2 = 0) + P H 1 ( X 1 = 0 , X 2 = 1) 1 − P H 1 ( X 1 = 0 , X 2 = 0) = 5 = 9 ◮ We have chance of more than 55% to make a type II error. ◮ Could we improve our decision criterion to reduce the type I and the type II error? ◮ Is there a perfect decision rule such that α = β = 0? ◮ If we want α = 0 then we can NEVER reject H 0 , so we get β = 1. ◮ If we want β = 0 then we always have to reject H 0 , so we get α = 1. ◮ α and β are antagonists. ◮ Which one is more important? Think of a criminal trial... Wolfgang Trutschnig Statistics, Visualization and More Using ”R” (298.916)

An introductory example Binom.test One sample t -test Two sample t -test Toy example hypothesis testing Hypothesis testing vs. criminal trials ◮ Consider a criminal trial. ◮ Based on evidence the jury (or the judge) has to decide whether the defendant is guilty or not. ◮ Suppose that H 0 = { innocent } and that H 1 = { guilty } . ◮ Right at the start the jury (or the judge) accepts H 0 and assumes that the defendant is innocent. ◮ Only if enough evidence is brought in, H 0 will be rejected and the defendant will be declared guilty. ◮ The afore-mentioned type I error α corresponds to the situation that the defendant will be declared guilty although he is innocent. ◮ The afore-mentioned type II error β corresponds to the situation that the defendant will be declared innocent although he is guilty. Wolfgang Trutschnig Statistics, Visualization and More Using ”R” (298.916)

An introductory example Binom.test One sample t -test Two sample t -test Toy example hypothesis testing ◮ Which error has worse consequences for the defendant? ◮ Obviously the type I error. ◮ In the Anglo-Saxon jurisdiction system there there is the term ’Beyond reasonable doubt’ underlining this fact. ◮ In other words: We want to keep the type I error α (very) small. ◮ The same applies to hypothesis testing: α should be small, standard significance levels are α = 0 . 05 and α = 0 . 01 (one error out of twenty or one out of hundred). ◮ As soon as α is fixed it is the statisticians’ job to develop optimal tests, i.e. decision rules (criteria) with a probability of (at most) α for a type I error and, at the same time, minimal type II error β . Wolfgang Trutschnig Statistics, Visualization and More Using ”R” (298.916)