University:
Massachusetts Institute of Technology
Course:
15.501/516 | Accounting
Academic year:

2023
Views:

377

Pages:

48
Author:

Garrett Obrien

Inferential statistics Statistics has 3+ components Data description & analysis Probability calculations Statistical inference Inferential statistics Models .... Hypothesis testing Why test for differences? If we ﬁnd that females are rated as 8 on being nice and males are rated as 7 -- are they different? After all different things are different ! Why test these differences? Why test for differences? Two main reasons: Variance (measurement error, random error, other variables) Making inferences beyond the sample to the population at large Variance We are living in a stochastic world Hypotheses testing I How can we test that things are different? Set 2 hypotheses that cover the entire possible range of outcomes H0 & H1 H0- No difference Group 1 [=, , ] Group 2 H1-Difference Group 1 [, >, <] Group 2 Examples Hypotheses testing Ib H0 & H1-- Examples: Is a coin fair Gender and grades Healing with a new medication Ability to cheat Marriage over time For each please write H0 & H1 Hypothesis testing II Why test a hypothesis we don’t believe in? We do this because we can only show that something is wrong -- not that something is right What does it mean to reject H0? If H0 is correct, the probability of getting this result (or a more extreme result) is very low -thus we reject H0 and (for now) accept H1 2 types of errors Not conservative and liberal just balancing 2 types of error H0 is wrong Correct Type 2 error H0 is correct Type I error Reject H0 Correct Accept H0 2 types of errors H0 is H0 is wrong correct Type I Reject Correct error H0 Type 2 Accept Correct error H0 The meaning of p What does p means? What is the difference between p = 0.03, p = 0.001, & p = 0.11 What is the relationship between p and conﬁdence? What is the relationship between p and effect size? What is the relationship between p and number of subjects? The importance of effect size Always give effect size measures Mean difference Quartile differences etc. Summary Hypotheses testing H1 & H0 -- setting the hypothesis to something you don’t believe in The meaning of p 2 types of errors Effect size ! Statistical tests • • • • • T-test for 1 sample T-test for 2 samples ANOVA Linear Regression Non-parametric tests T test for 1 sample One sample t test One sample t test Difference from the comparison t= -M 2 (xi - ) n-1 Standard deviation n Conﬁdence What do you do with “t” Compare it to the “t table” When there is more data, the t distribution gets closer to normal Example step 1 Observation 1 2 3 4 5 6 7 Sum Aggressive 24 22 23 18 17 16 20 140 xi - µ 4 2 3 -2 -3 -4 0 0 (xi - µ)2 16 4 9 4 9 16 0 58 Example step 2 H0: average is 16 H1: average 16 = 2 (xi - ) t= n-1 -M n = 3.11 = 3.42 T test for 2 samples two samples t test Test for dependent samples t= (between di) - (expected di) sd of di n two samples t test Test for 2 independent samples Mean difference t= Hypothesized difference (1 - 2) - (M1 - M2) 2 2 n1 1 + n2 2 n1 + n2 -2 ( n1 + n2 n1 x n2 ) Example 1 Who eats more lollipops males of females? 7 females; 5 males followed for a month Females: = 27, 2 = 29.2 Males: = 19, 2 = 24.57 Is there a difference? Calculating ... t= (27 - 19) - (0) 5 x 24.57 + 7 x 29.2 5 + 7 -2 ( 5+ 7 5x 7 ) = 2.42 Example 2 Does the sun creates freckles? Each subject has one side of their body in the sun and one in the shade H0 sun side non-sun side H1 sun side > non-sun side Data Subject 1 2 3 4 5 6 7 8 9 Sum sun 6 12 3 4 7 9 4 0 4 shade 8 5 2 6 0 10 4 2 3 diff -2 7 1 -2 7 -1 0 -2 1 9 d-µ -3 6 0 -3 6 -2 -1 -3 0 0 (d - µ)2 9 36 0 9 36 4 1 9 0 104 Calculating ... = 104 8 = 3.606 (1) - (0) t = 3.606 9 = 0.831 t test summary t test as an example of inferential statistics Mean differences relative to variance ANOVA ANOVA I Analysis of variance This is the same logic of the t test but allowing for more tests So why not just do multiple t tests? 1 - doing many tests can cause errors 2 - there is beneﬁt in pooling observations across cells ANOVA II The story is the same ... Looking at the variance within cells relative to the variance across cells (as if there were no treatments) and asking how much does the distinction between cells helps reduce the variance ANOVA IIa ANOVA IIb ANOVA IIc ANOVA IIc A few examples I No variance within or between groups Case# 1 A 40 Group B 40 2 3 4 40 40 40 40 40 40 40 40 40 Mean 40 40 40 C 40 OA mean 40 A few examples II No variance within groups, but variance between groups Case# 1 A 38 Group B 40 2 3 4 38 38 38 40 40 40 36 36 36 Mean 38 40 36 C 36 OA mean 38 A few examples III Variance within groups, but not between groups Case# 1 A 36 Group B 46 2 3 4 40 44 40 37 37 40 38 43 43 Mean 40 40 40 C 36 OA mean 40 A few examples IV Variance within & between groups Case# 1 A 38 Group B 45 2 3 4 39 39 36 44 49 46 37 35 36 Mean 38 46 36 C 36 OA mean 40 The general formula SS = Sum of squares SS total = SS within + SS between SSt=SSw+SSb This means: Take each of the samples, subtract it from the appropriate mean, and square it Sum of squares Once we have the sum of squares we compare the SSb and SSw As the SSb relative to SSw gets larger the results are more likely to be signiﬁcant Degrees of freedom This is a base to think about the amount of independent observations we have, and thus the strength of the results In general we lose a degree of freedom when we use a mean... The formula F= SSb / dfb SSw / dfw = SSb / K-1 SSw / n-k Once you have the F value use the F table with the correct df ANOVAs One way & multiple way ANOVAs M M F E DV ~E F Summary I Hypothesis testing Because we can never prove anything and only disprove things we set a hypothesis (H0) as one we do not believe in. Once we reject H0 we are willing to accept H1 Summary II T test and ANOVA It is all about variance and mean differences How large is the mean difference relative to the variance! These tests give us the probability of the data given that H0 is correct