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TWO 𝜈S OR MEDIANS: COMPARISONS

Business Statistics

Comparing two samples Comparing two unrelated samples Comparing the means of two unrelated samples Comparing the medians of two unrelated samples Old exam question Further study CONTENTS

It often happens that we want to compare two situations ▪ do I sell more when there is music in my shop? ▪ is the expensive machine more precise than the cheap

• ne?

▪ are adverisements on TV or internet equally profitable? ▪ do people buy more on Tuesdays than on Wednesday? ▪ in couples, who drinks more: the man or the woman? ▪ etc. COMPARING TWO SAMPLES

In all these questions we compare two populations ▪ Situation 1: two populations (or sub-populations) with similar variable

▪ sales in 105 days without music ▪ sales in 96 days with music

▪ Data matrix: two options COMPARING TWO SAMPLES

SPSS requires this data presentation

▪ Situation 2: one sample with paired observations

▪ drinks of the man in 78 couples ▪ drinks of the woman in the same 78 couples

▪ Data matrix: one option only ▪ Will be discussed in a later lecture COMPARING TWO SAMPLES

Situation 1 ▪ independent samples/unrelated samples ▪ introduce symbols for the two random variables ▪ e.g., using 𝑌1 en 𝑌2

▪ 𝑌1 with sample 𝑌1,1, 𝑌1,2, … , 𝑌1,𝑜1 and 𝑌2 with sample 𝑌2,1, 𝑌2,2, … , 𝑌2,𝑜2

▪ or using 𝑌 and 𝑍

▪ 𝑌: 𝑌1, 𝑌2, … , 𝑌𝑜𝑌 and 𝑍: 𝑍

1, 𝑍 2, … , 𝑍 𝑜𝑍

▪ sample sizes can be different COMPARING TWO UNRELATED SAMPLES

Or of course using “meaningful” indices: 𝑌𝐶 and 𝑌𝐻 for Belgium and Germany. Not 𝐶 and 𝐻, because we need to stress that it is “about” a variable 𝑌 (like sales)

We want to test hypothesis such as ▪ are the means equal?

▪ 𝐼0: 𝜈𝑌 = 𝜈𝑍 or 𝐼0: 𝜈1 = 𝜈2 or 𝐼0: 𝜈𝑌1 = 𝜈𝑌2 or ...

▪ are the variances equal?

▪ 𝐼0: 𝜏𝑌

2 = 𝜏𝑍 2 or etc.

▪ are the proportions equal

▪ 𝐼0: 𝜌𝑌 = 𝜌𝑍 or etc.

Also: ▪ inequalities, like 𝐼0: 𝜈𝑌 ≥ 𝜈𝑍 ▪ and non-zero differences, like 𝐼0: 𝜈𝑌 = 𝜈𝑍 + 85 COMPARING TWO UNRELATED SAMPLES

Context: ▪ sample 𝑌1: sales in 𝑜1 = 105 days without music ▪ sample 𝑌2: sales in 𝑜2 = 96 days with music General idea: ▪ ൠ 𝑌1~𝑒𝑗𝑡𝑢𝑠𝑗𝑐𝑣𝑢𝑗𝑝𝑜 𝜄1 𝑌2~𝑒𝑗𝑡𝑢𝑠𝑗𝑐𝑣𝑢𝑗𝑝𝑜 𝜄2 𝜄1 = 𝜄2? COMPARING TWO UNRELATED SAMPLES

Assumption (for now!): ▪ 𝑌~𝑂 𝜈𝑌; 𝜏𝑌

2

▪ 𝑍~𝑂 𝜈𝑍; 𝜏𝑍

2

▪ in words: both samples come from normally distributed populations with known variances Question ▪ are 𝜈𝑌 and 𝜈𝑍 different? ▪ can we test this, on the basis of the (limited!) evidence concerning ҧ 𝑦 and ത 𝑧? ▪ so, can we reject 𝐼0: 𝜈𝑌 = 𝜈𝑍? To decide ▪ use ത 𝑌 − ത 𝑍 ~𝑂 𝜈 ത

𝑌− ത 𝑍, 𝜏ത 𝑌− ത 𝑍 2

COMPARING THE MEANS OF TWO UNRELATED SAMPLES

So, the sampling distribution of the difference of means is normal

For one sample, we had ത 𝑌 − 𝜈 ത

𝑌

𝜏 ത

𝑌

~𝑂 0,1 As it turns out, for two samples, we have ത 𝑌 − ത 𝑍 − 𝜈 ത

𝑌 − 𝜈 ത 𝑍

𝜏 ത

𝑌− ത 𝑍

~𝑂 0,1

▪ 𝜈 ത

𝑌 − 𝜈 ത 𝑍 = 𝜈𝑌 − 𝜈𝑍 follows from the null hypothesis

▪ for instance 𝐼0: 𝜈𝑌 = 𝜈𝑍 or 𝐼0: 𝜈𝑌 − 𝜈𝑍 = 85 ▪ ҧ 𝑦 and ത 𝑧 are obtained from the data ▪ but what is 𝜏 ത

𝑌−ത 𝑍?

COMPARING THE MEANS OF TWO UNRELATED SAMPLES

For one sample, we had 𝜏ത

𝑌 2 = 𝜏𝑌 2

𝑜 As it turns out, for two independent samples, we have 𝜏ത

𝑌− ത 𝑍 2

= 𝜏ത

𝑌 2 + 𝜏ത 𝑍 2, so

𝜏 ത

𝑌−ത 𝑍 =

𝜏𝑌

2

𝑜𝑌 + 𝜏𝑍

2

𝑜𝑍

▪ recall that variances add up when 𝑌 and 𝑍 are independent ▪ e.g., 𝜏𝑌+𝑍

2

= 𝜏𝑌

2 + 𝜏𝑍 2 but also 𝜏𝑌−𝑍 2

= 𝜏𝑌

2 + 𝜏𝑍 2

COMPARING THE MEANS OF TWO UNRELATED SAMPLES

Example Context: ▪ do I sell more when there is music in my shop? Experiment ▪ on some days the music is turned on, on other days the music is turned off ▪ you keep track of the sales during each day Data: ▪ sample of sales on days with music (𝑦1, 𝑦2, … , 𝑦105) ▪ sample of sales on days without music (𝑧1, 𝑧2, … , 𝑧96) Five step procedure COMPARING THE MEANS OF TWO UNRELATED SAMPLES

▪ Step 1:

▪ 𝐼0: 𝜈𝑌 = 𝜈𝑍; 𝐼1: 𝜈𝑌 ≠ 𝜈𝑍; 𝛽 = 0.05

▪ Step 2:

▪ sample statistic: ത 𝑌 − ത 𝑍 ▪ reject for “too large” and “too small” values

▪ Step 3:

▪ null distribution

ത 𝑌−ത 𝑍 − 𝜈𝑌−𝜈𝑍 𝜏ഥ

𝑌−ഥ 𝑍

=

ത 𝑌−ത 𝑍 𝜏ഥ

𝑌−ഥ 𝑍

~𝑂 0,1 ▪ valid because ...

▪ Step 4:

▪ 𝑨𝑑𝑏𝑚𝑑 = ▪ 𝑨𝑑𝑠𝑗𝑢 =

▪ Step 5:

▪ reject or not reject because ...

COMPARING THE MEANS OF TWO UNRELATED SAMPLES

in a minute we will supply full details and a worked example ...

▪ But, wait ...

▪ ... isn’t it weird to assume that 𝜏𝑌

2 and 𝜏𝑍 2 are known, while 𝜈𝑌

and 𝜈𝑍 are not known?

▪ In reality the population variances will often be unknown as well!

▪ remember we had the same problem in the one-sample case? ▪ there we decided to estimate the value of 𝜏2 with the value of 𝑡2 ▪ and paid a price of using the wider 𝑢-distribution ▪ here we will do the same: estimate the two 𝜏2-values with two 𝑡2-values ▪ and pay the same price: use 𝑢-dsitribution instead of 𝑨-distribution

COMPARING THE MEANS OF TWO UNRELATED SAMPLES

For one sample, we had ത 𝑌 − 𝜈 ത

𝑌

𝑇 ത

𝑌

~𝑢df As it turns out, for two samples, we have ത 𝑌 − ത 𝑍 − 𝜈 ത

𝑌 − 𝜈 ത 𝑍

𝑇 ത

𝑌− ത 𝑍

~𝑢df

▪ 𝜈 ത

𝑌 − 𝜈 ത 𝑍 = 𝜈𝑌 − 𝜈𝑍 follows from the null hypothesis

▪ ҧ 𝑦 and ത 𝑧 are obtained from the data ▪ but what is 𝑡 ത

𝑌−ത 𝑍?

▪ and how to choose df?

COMPARING THE MEANS OF TWO UNRELATED SAMPLES

Two options for 𝑡 ത

𝑌− ത 𝑍:

▪ 1: estimating 𝜏𝑌

2 and 𝜏𝑍 2 from 𝑡𝑌 2 and 𝑡𝑍 2 respectively

▪ 2: assuming 𝜏𝑌

2 = 𝜏𝑍 2 = 𝜏2 and estimating 𝜏2 as the

weighted average of both sample variances Both options lead to a different value of df COMPARING THE MEANS OF TWO UNRELATED SAMPLES

Option 1: ▪ estimating 𝜏𝑌

2 and 𝜏𝑍 2 from 𝑡𝑌 2 and 𝑡𝑍 2 respectively (Welch’s test)

𝑡 ത

𝑌−ത 𝑍 =

𝑡𝑌

2

𝑜𝑌 + 𝑡𝑍

2

𝑜𝑍 ▪ testing with 𝑢-distribution with df = 𝑡𝑌

2

𝑜𝑌 + 𝑡𝑍

2

𝑜𝑍

2

𝑡𝑌

2

𝑜𝑌

2

𝑜𝑌 − 1 + 𝑡𝑍

2

𝑜𝑍

2

𝑜𝑍 − 1

COMPARING THE MEANS OF TWO UNRELATED SAMPLES

Compare to 𝜏 ത

𝑌−ത 𝑍 =

𝜏𝑌

2

𝑜𝑌 + 𝜏𝑍

2

𝑜𝑍 quick rule, but bad approximation: 𝑒𝑔 ≈ min 𝑜𝑌 − 1, 𝑜𝑍 − 1

Option 2: ▪ estimating the common 𝜏2 from both samples

▪ a “weighted mean” of 𝑡𝑌

2 and 𝑡𝑍 2, the pooled variance 𝑡P 2

𝑡P

2 = 𝑜𝑌 − 1 𝑡𝑌 2 + 𝑜𝑍 − 1 𝑡𝑍 2

𝑜𝑌 − 1 + 𝑜𝑍 − 1 ▪ and 𝑡 ത

𝑌− ത 𝑍 =

𝑡P

2

𝑜𝑌 + 𝑡P

2

𝑜𝑍 ▪ testing with 𝑢-distribution with df = 𝑜𝑌 − 1 + 𝑜𝑍 − 1 = 𝑜𝑌 + 𝑜𝑍 − 2

COMPARING THE MEANS OF TWO UNRELATED SAMPLES

Compare to 𝑡 ത

𝑌−ത 𝑍 =

𝑡𝑌

2

𝑜𝑌 + 𝑡𝑍

2

𝑜𝑍 You can read this as

df𝑌𝑡𝑌

2+df𝑍𝑡𝑍 2

df𝑌+df𝑍

You can read this as df𝑌 + df𝑍

COMPARING THE MEANS OF TWO UNRELATED SAMPLES

When to use:

• a. 𝑨 =

ҧ 𝑦− ത 𝑧

𝜏𝑌 2 𝑜𝑌+ 𝜏𝑍 2 𝑜𝑍

• b. 𝑢 =

ҧ 𝑦− ത 𝑧

𝑡𝑌 2 𝑜𝑌+ 𝑡𝑍 2 𝑜𝑍

• c. 𝑢 =

ҧ 𝑦− ത 𝑧

𝑡𝑄 2 𝑜𝑌+ 𝑡𝑄 2 𝑜𝑍

EXERCISE 1

Use of SPSS COMPARING THE MEANS OF TWO UNRELATED SAMPLES

a data set on Computer Anxiety Rating split by gender

Results split by gender Results of 𝑢-test COMPARING THE MEANS OF TWO UNRELATED SAMPLES

Zoom in COMPARING THE MEANS OF TWO UNRELATED SAMPLES

𝑢-test with pooled estimate of 𝜏𝑌

2 = 𝜏𝑍 2

𝑢-test with separate estimates of 𝜏𝑌

2 and 𝜏𝑍 2

value of the 𝑢-statistic (𝑢calc) degrees of freedom 𝑞-value (2-sided)

And one more thing ... COMPARING THE MEANS OF TWO UNRELATED SAMPLES

tests of the assumption of equal variance 𝐼0: 𝜏𝑌

2 = 𝜏𝑍 2 versus 𝐼1: 𝜏𝑌 2 ≠ 𝜏𝑍 2

𝑞-value for this test

For these two tests, we need both ത 𝑌 and ത 𝑍 to be normally distributed ▪ This means either of the following three requirements holds:

▪ 𝑌 is a normally distributed population ▪ 𝑌 has a symmetric distribution and 𝑜𝑌 ≥ 15 ▪ 𝑜𝑌 ≥ 30

▪ Also for 𝑍 one of these requirements must hold

▪ but not necessarily the same one as for 𝑌

COMPARING THE MEANS OF TWO UNRELATED SAMPLES

• a. Suppose 𝑌 and 𝑍 are two distributions. We sample with

𝑜𝑌 = 10 and 𝑜𝑍 = 20 and we want to test 𝐼0: 𝜈𝑌 = 𝜈𝑍. Do we need to make any assumption?

• b. Same as a, but now we know that 𝑌 and 𝑍 are

symmetrically distributed.

• c. Same as a, but now we know that 𝑌 and 𝑍 are normally

distributed EXERCISE 2

▪ Recall the non-parametric one-sample test for the median

▪ the Wilcoxon signed ranks test ▪ replacing the values by ranks and testing the sum of the positive ranks

▪ Can we also develop a non-parametric (rank-order) order test for two unrelated samples? ▪ Yes we can: Wilcoxon-Mann-Whitney test

▪ named after Frank Wilcoxon, Henry Mann, and Donald Whitney ▪ also named Wilcoxon (Mann-Whitney) test, Mann-Whitney test, etc. ▪ requirement: similar shape of distribution

COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES

▪ Computational steps of the Wilcoxon-Mann-Whitney test

▪ combine both samples (𝑌 and 𝑍) ▪ assign ranks to the combined sample ▪ ties get an average rank ▪ sum the ranks of both samples separately (𝑈

𝑌 and 𝑈𝑍)

▪ compare the test statistic 𝑈

𝑌 (or 𝑈𝑍) to a critical value from the

table

COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES

Example (same as before) ▪ Sample data are collected on the capacity rates (in %) for two factories

▪ factory A, the rates are 71, 82, 77, 94, 88 ▪ factory B, the rates are 85, 82, 92, 97

▪ Are the median operating rates for two factories the same (at a significance level 𝛽 = 0.05)? COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES

Example ▪ data A: 𝑦𝑗 (𝑜𝑌 = 5) ▪ data B: 𝑧𝑗 (𝑜𝑍 = 4) ▪ one case of ties (82) ▪ 𝑈𝑍 = 24.5 COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES

Capacity Rank

Factory A Factory B Factory A Factory B

71 1 77 2 82 3.5 82 3.5 85 5 88 6 92 7 94 8 97 9 Rank sums: 20.5 24.5 a tie: observations 3 and 4 are 82, so assign rank 3.5 to facilitate the discussion, we focus on the sample with the smallest sample size

Testing the Wilcoxon-Mann-Whitney 𝑈 statistic ▪ using a table of critical values

▪ included in tables at exam

▪ using a normal approximation

▪ valid for large samples when Wilcoxon-Mann-Whitney table of critical values is not sufficient

COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES

Table of critical values of Wilcoxon statistic ▪ for 𝑜𝑦 = 𝑜1 = 4 and 𝑜𝑧 = 𝑜2 = 5 at 𝛽 = 0.05:

▪ 𝑈lower = 11, 𝑈

upper = 29

▪ 𝑆crit = 0,11 ∪ [29,45]

COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES

Why 0 and 45? 0 because the sum

• f the ranks cannot be smaller than
• 0. 45 because the sum of the ranks

here cannot be larger than 45.

Conclusion from small sample Wilcoxon-Mann-Whitney test ▪ 𝑈𝑍 = 24.5 is between 𝑈lower = 11 and 𝑈

upper = 29

▪ Therefore, do not reject the null hypothesis (𝐼0: 𝑁𝑌 = 𝑁𝑍) at the 5% level ▪ There is not enough evidence to conclude that the medians are different COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES

Large sample approximation ▪ Under 𝐼0, it can be shown that

▪ 𝐹 𝑈𝑍 =

𝑜𝑍 𝑜𝑌+𝑜𝑍+1 2

▪ var 𝑈

𝑍 = 𝑜𝑌𝑜𝑍 𝑜𝑌+𝑜𝑍+1 12

▪ Further, when 𝑜𝑌 ≥ 10 or 𝑜𝑍 ≥ 10, we use a normal approximation:

▪ 𝑈𝑍~𝑂

𝑜𝑌 𝑜𝑌+𝑜𝑍+1 2

,

𝑜𝑌𝑜𝑍 𝑜𝑌+𝑜𝑍+1 12

▪ 𝑎 =

𝑈𝑍−𝑜𝑍 𝑜𝑌+𝑜𝑍+1

2 𝑜𝑌𝑜𝑍 𝑜𝑌+𝑜𝑍+1 12

~𝑂 0,1

COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES

Large sample approximation (continued) ▪ so you can compute 𝑨calc =

𝑈𝑍,calc − 𝑜𝑍 𝑜𝑌+𝑜𝑍+1

2 𝑜𝑌𝑜𝑍 𝑜𝑌+𝑜𝑍+1 12

▪ and compare it to 𝑨crit (e.g., ±1.96) COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES

Use of SPSS COMPARING THE MEDIANS OF TWO UNRELATED SAMPLES

𝑈 = 345 𝑨-score with normal approximation 𝑞-value (2-sided)

21 May 2015, Q2a OLD EXAM QUESTION

Doane & Seward 5/E 10.1-10.3, 16.4 extra document “Wilcoxon Mann-Whitney test” Tutorial exercises week 3 𝑨-test (known variance) t-test (pooled variance) t-test (separate variance estimates) Wilcoxon Mann-Whitney test FURTHER STUDY