HYPOTHESIS TESTING PART II LEARNING GOALS get more intimate with p - - PowerPoint PPT Presentation

▶
hypothesis testing
SMART_READER_LITE
LIVE PREVIEW

HYPOTHESIS TESTING PART II LEARNING GOALS get more intimate with p - - PowerPoint PPT Presentation

Sep 20, 2023 501 likes •707 views

INTRODUCTION TO DATA ANALYSIS HYPOTHESIS TESTING PART II LEARNING GOALS get more intimate with p -values distribution under true H 0 relation to confidence intervals develop a basic sense of how clever math (e.g., Central Limit

slide-1
SLIDE 1

HYPOTHESIS TESTING

INTRODUCTION TO DATA ANALYSIS PART II

slide-2
SLIDE 2

LEARNING GOALS

▸ get more intimate with p-values ▸ distribution under true ▸ relation to confidence intervals ▸ develop a basic sense of how clever math (e.g., Central

Limit Theorem) helps approximate sampling distributions

▸

we don’t aim for perfect understanding of this math in this course!

▸ become able to interpret & apply some statistical tests ▸ Pearson’s

  • tests

▸ z-test ▸ one-sample t-test

H0 χ2

slide-3
SLIDE 3

p-value

revisit

slide-4
SLIDE 4

RECAP

▸ model

captures prior beliefs about data-generating process

▸ prior over latent parameters ▸ likelihood of data ▸ Bayesian posterior inference using

  • bserved data

▸ compare posterior beliefs to some

parameter value of interest

M Dobs

BAYESIAN PARAMETER ESTIMATION FREQUENTIST HYPOTHESIS TESTING ▸ model

captures a hypothetically assumed data-generating process

▸ fix parameter value of interest ▸ likelihood of data ▸ single out some aspect of the data

as most important (test statistic)

▸ look at distribution of test statistic

given the assumed model (sampling distribution)

▸ check likelihood of test statistic

applied to the observed data

M Dobs

slide-5
SLIDE 5

P-VALUE

p(Dobs) = P(T|H0 ⪰H0,a t(Dobs))

slide-6
SLIDE 6

RELATION OF P-VALUES AND CONFIDENCE INTERVALS

▸ assumptions: ▸ p-value and CI are constructed / approximated in the same way ▸ two-sided test with

and alternative

▸ correspondence result:

H0: θ = θ0 Ha: θ ≠ θ0

p(D) < α iff θ0 ∉ CI(D)

slide-7
SLIDE 7

approximating sampling distributions

slide-8
SLIDE 8

LAW OF LARGE NUMBERS

slide-9
SLIDE 9

CENTRAL LIMIT THEOREM

CLT gives us information about the distribution of estimated means, e.g., as when we estimate repeatedly in different (hypothetical experiments).

slide-10
SLIDE 10

Pearson’s

  • tests

χ2

slide-11
SLIDE 11

PEARSON

  • TESTS

χ2

▸ tests for categorical data (with more than two categories) ▸ two flavors: ▸ test of goodness of fit ▸ test of independence ▸ sampling distribution is a

  • distribution

χ2

slide-12
SLIDE 12
  • DISTRIBUTION

χ2

▸ standard normal random variables: ▸ derived RV: ▸ it follows (by construction) that:

X1, …Xn Y = X2

1 + … + X2 n

y ∼ χ2-distribution(n)

slide-13
SLIDE 13

PEARSON’S -TEST [GOODNESS OF FIT]

χ2

Is it conceivable that each category (= pair of music+subject choice) has been selected with the same flat probability of 0.25?

slide-14
SLIDE 14

FREQUENTIST MODEL FOR PEARSON’S -TEST [GOODNESS OF FIT]

χ2

⃗ n ∼ Multinomial( ⃗ p , N)

FACT: The sampling distribution of is approximately:

χ2

χ2 ∼ χ2-distribution(k − 1)

⃗ n N χ2 ⃗ p

χ2 =

k

∑

i=1

(ni − npi)2 npi

slide-15
SLIDE 15

PEARSON’S -TEST [GOODNESS OF FIT]

χ2

⃗ n N χ2 ⃗ p

χ2 ∼ χ2-distribution(k − 1)

χ2 =

k

∑

i=1

(ni − npi)2 npi

slide-16
SLIDE 16

PEARSON’S -TEST [GOODNESS OF FIT]

χ2

⃗ n N χ2 ⃗ p

χ2 ∼ χ2-distribution(k − 1)

χ2 =

k

∑

i=1

(ni − npi)2 npi

slide-17
SLIDE 17

PEARSON’S -TEST [GOODNESS OF FIT]

χ2

⃗ n N χ2 ⃗ p

χ2 ∼ χ2-distribution(k − 1)

χ2 =

k

∑

i=1

(ni − npi)2 npi

slide-18
SLIDE 18

PEARSON’S -TEST [GOODNESS OF FIT]

χ2

How to interpret / report the result:

What about the lecturer’s conjecture that (colorfully speaking) logic + metal = 🥱?