Apply Kruskal-Wallis test

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

Created this.

Name	Status	Author	Last Modified
Apply Kruskal-Wallis test	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Apply Kruskal-Wallis test - SES	draft	Chris Graham	10/12/2020 01:12
Apply Kruskal-Wallis test - Mock Test - SES2003	Needs to be tested	Chris Graham	25/02/2019 11:24

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Question statement

The following data arose in a comparison of the effects of alcohol on the time taken to complete a task. There were three groups of subjects; Group A had no alcohol, Group B had two units over 1 hour and Group C had 4 units over 1 hour. The responses are the times (in seconds) taken to complete a word-matching task.

Group A (0 units)	$\var{r1[0]}$	$\var{r1[1]}$	$\var{r1[2]}$	$\var{r1[3]}$	$\var{r1[4]}$	$\var{r1[5]}$
Group B (2 units)	$\var{r2[0]}$	$\var{r2[1]}$	$\var{r2[2]}$	$\var{r2[3]}$	$\var{r2[4]}$	$\var{r2[5]}$
Group C (4 units)	$\var{r3[0]}$	$\var{r3[1]}$	$\var{r3[2]}$	$\var{r3[3]}$	$\var{r3[4]}$	$\var{r3[5]}$

Apply the Kruskal-Wallis test to this data on reaction times under alcohol in order to test the null hypothesis that the alcohol consumption does not affect the mean time taken to complete the task.

Give any introductory information the student needs.

			Name	Type	Generated Value

rkt	list	List of 18 items
sd1	number	3.06
sd2	number	3.37
sd3	number	1.03
ran	list	List of 36 items
biglist	list	List of 18 items
vties	list	List of 3 items
mu3	number	28
rs	list	[ 5, 2, 3, 8 ]
s3	list	List of 19 items
s2	list	List of 19 items
s1	list	List of 19 items
m1	number	21.83
tol	number	0.001
m2	number	24.83
tiesplus	list	Nested 4×4 list
m3	number	26.33
sig1	number	3.4
sig3	number	3.4
sig2	number	3.4
corr	number	0.9742002064
ties	list	Nested 3×2 list
r1	list	[ 24, 20, 17, 21, 24, 25 ]
n3	integer	6
mu1	number	21
g	number	150
r2	list	[ 24, 24, 26, 19, 28, 28 ]
mu2	number	24.5
h	number	7.0555555556
k	integer	3
r3	list	[ 26, 27, 26, 25, 28, 26 ]
n	integer	18
kw	number	7.24
w	list	[ 0, 0, 1, 0, 0 ]
sr	list	[ 30.5, 61.5, 79 ]
n1	integer	6
n2	integer	6
dec	string	we have evidence against the n

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
ran[0..18]

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `list`

[ 6.5, 3, 1, 4, 6.5, 9.5, 6.5, 6.5, 12.5, 2, 17, 17, 12.5, 15, 12.5, 9.5, 17, 12.5 ]

← Depends on:

This variable doesn't seem to be used anywhere.

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Ask the student a question, and give any hints about how they should answer this part.

Kruskal-Wallis Test

First fill in this table with the appropriate values, all decimals to 1 decimal place. $R_1,\;R_2,\;R_3$ are the sums of the ranks in each row.

Group A (0 units)	$\var{r1[0]}$	$\var{r1[1]}$	$\var{r1[2]}$	$\var{r1[3]}$	$\var{r1[4]}$	$\var{r1[5]}$
Rank	Gap 0.	Gap 1.	Gap 2.	Gap 3.	Gap 4.	Gap 5.	$R_1=\;$Gap 6.
Group B (2 units)	$\var{r2[0]}$	$\var{r2[1]}$	$\var{r2[2]}$	$\var{r2[3]}$	$\var{r2[4]}$	$\var{r2[5]}$
Rank	Gap 7.	Gap 8.	Gap 9.	Gap 10.	Gap 11.	Gap 12.	$R_2=\;$Gap 13.
Group C (4 units)	$\var{r3[0]}$	$\var{r3[1]}$	$\var{r3[2]}$	$\var{r3[3]}$	$\var{r3[4]}$	$\var{r3[5]}$
Rank	Gap 14.	Gap 15.	Gap 16.	Gap 17.	Gap 18.	Gap 19.	$R_3=\;$Gap 20.

Advice

In order to find the ranks we order, in increasing order, all of the times for the tasks across all the three groups. We also work out the ranks for each time by including a row which simply numbers from $1$ to $\var{n}$, this we call the index of the numbers and the last row then takes equal values in the list and gives the averages of their indices, so that they all get the same rank. So you simply add up their corresponding indices in that group and divide by the number of equal entries. So if a number is not repeated then its rank is its index.

For this example we have:

{table([s1,s2,s3],[])}

We see that there are ties as follows:

{table(ties,[])}

We use this information later to find the correction factor.

Putting these ranks back into the original table gives, where $R_1,\;R_2$ and $R_3$ are the sums of the ranks in each row:

Group A (0 units)	$\var{r1[0]}$	$\var{r1[1]}$	$\var{r1[2]}$	$\var{r1[3]}$	$\var{r1[4]}$	$\var{r1[5]}$
Rank	{rkt[0]}	{rkt[1]}	{rkt[2]}	{rkt[3]}	{rkt[4]}	{rkt[5]}	$R_1=\;${sr[0]}
Group B (2 units)	$\var{r2[0]}$	$\var{r2[1]}$	$\var{r2[2]}$	$\var{r2[3]}$	$\var{r2[4]}$	$\var{r2[5]}$
Rank	{rkt[6]}	{rkt[7]}	{rkt[8]}	{rkt[9]}	{rkt[10]}	{rkt[11]}	$R_2=\;${sr[1]}
Group C (4 units)	$\var{r3[0]}$	$\var{r3[1]}$	$\var{r3[2]}$	$\var{r3[3]}$	$\var{r3[4]}$	$\var{r3[5]}$
Rank	{rkt[12]}	{rkt[13]}	{rkt[14]}	{rkt[15]}	{rkt[16]}	{rkt[17]}	$R_3=\;${sr[2]}

We now have enough information to start the calculation of the Kruskal-Wallis statistic.

We do this in three steps:

1. Calculate the statistic $H$, which assumes there are no ties.

2. Find the correction factor $C$ given by the ties in the data.

3. This gives the statistic $H^*=H/C$ we want, and we make a decision based on the Kruskal-Wallis table.

Step 1: Find $H$.

\[\begin{eqnarray*}H &=& \left[\frac{12}{N \times (N+1)} \times \left(\sum \frac{R_i^2}{n_i}\right)\right]-3(N+1)\\&=&\left\{\frac{12}{\var{n}\times\var{n+1}}\times\left(\frac{\var{sr[0]}^2}{6}+\frac{\var{sr[1]}^2}{6}+\frac{\var{sr[2]}^2}{6}\right)\right\}-3\times \var{n+1}\\&=&\var{h}\\&=&\var{precround(H,3)}\end{eqnarray*}\] to 3 decimal places.

Step 2: Find the Correction Factor $C$.

For each tie with $g$ equal data values we calculate $\displaystyle \frac{g^3-g}{N^3-N}$ and add these together over all ties to get $T$.

Then $C=1-T$.

So for our data we have:

{table(tiesplus,[' ','Number','Contribution','Value'])}

Hence $C=1-T = 1-\var{sum(vties)}=\var{1-sum(vties)}=\var{precround(corr,3)}$ to 3 decimal places.

Step 3: Find the Kruskal-Wallis test statistic and make a decision.

The statistic is given by $\displaystyle H^*=\frac{H}{C}=\frac{\var{precround(h,3)}}{\var{precround(corr,3)}}=\var{kw}$ to 2 decimal places.

Looking at the $\chi^2$ table our decision is that {dec}

$10\%$	$5\%$	$1\%$	$0.1\%$
$4.605$	$5.991$	$9.210$	$13.816$

Give a worked solution to the whole question.

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: `list`

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Apply Kruskal-Wallis test

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: list

Parts

Gap-fill

Generated value: `list`