Calculating the Spearman rank correlation coefficient and p-value

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From users who are members of Newcastle University Sports Science :

Lauren Frances Desoysa

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Needs to be tested

6 years, 9 months ago

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Chris Graham commented 6 years, 5 months ago

Changed the context to sporting interests

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Lauren Frances Desoysa 6 years, 9 months ago

Gave some feedback: Needs to be tested

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Lauren Frances Desoysa 6 years, 9 months ago

Created this as a copy of Calculate the Spearman rank correlation coefficient and p-value.

Name	Status	Author	Last Modified
Calculate the Spearman rank correlation coefficient and p-value	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Calculating the Spearman rank correlation coefficient and p-value	Needs to be tested	Lauren Frances Desoysa	26/11/2018 11:50

Question statement

A researcher has developed a questionnaire, which has been completed by 8 married couples (A-H), to find out how interest in sports plays a role in interpersonal attraction. The questions sought to place each partner in the marriage on a 20 point scale in which low scores represent little interest in sports, and high scores represent very interested in sports.

The data were:

Couple	$\var{obj[0]}$	$\var{obj[1]}$	$\var{obj[2]}$	$\var{obj[3]}$	$\var{obj[4]}$	$\var{obj[5]}$	$\var{obj[6]}$	$\var{obj[7]}$
Partner $X$	$\var{r1[0]}$	$\var{r1[1]}$	$\var{r1[2]}$	$\var{r1[3]}$	$\var{r1[4]}$	$\var{r1[5]}$	$\var{r1[6]}$	$\var{r1[7]}$
Partner $Y$	$\var{r2[0]}$	$\var{r2[1]}$	$\var{r2[2]}$	$\var{r2[3]}$	$\var{r2[4]}$	$\var{r2[5]}$	$\var{r2[6]}$	$\var{r2[7]}$

In this exercise you will find Spearman's correlation coefficient for this data and test it's significance in order to test the following null hypothesis:

$H_0: \text{There is no association between the attitudes of wives and husbands.}$

Give any introductory information the student needs.

			Name	Type	Generated Value

r1	list	List of 8 items
r2	list	List of 8 items
n	integer	8
obj	list	List of 8 items

			Name	Type	Generated Value

spcoef	number	0.333
aspcoef	number	0.333
d	list	[ -1, -4, 3, 3, 2, -3, -2, 2 ]
ssd	number	56
vs	list	[ 0, 0, 0, 0, 1 ]

			Name	Type	Generated Value

ten	number	0.643
five	number	0.738
one	number	0.881
point2	number	0.952

			Name	Type	Generated Value

			rr1	list	[ 7, 2, 5, 8, 3, 4, 1, 6 ]
			rr2	list	[ 8, 6, 2, 5, 1, 7, 3, 4 ]

			Name	Type	Generated Value

ssq	list	[ 1372, 1164 ]
tsqovern	list	[ 1152, 924.5 ]
ss	list	[ 220, 239.5 ]
m	integer	20
t	list	[ 96, 86 ]

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
darr(n,m,[random(1..20)])

JME function reference

Description

The wife's data

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`

[ 18, 5, 14, 19, 9, 12, 4, 15 ]

← Depends on:

→ Used by:

Statement

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

To find the Spearman correlation coefficient for the original score data, you need to rank the data, the lowest rank has rank $1$ and the highest score has rank $8$.

Also calculate the differences in the ranks, i.e. for each couple calculate

\[ d_i = (\text{partner X score}) - (\text{partner Y score}).\]

Couple	$\var{obj[0]}$	$\var{obj[1]}$	$\var{obj[2]}$	$\var{obj[3]}$	$\var{obj[4]}$	$\var{obj[5]}$	$\var{obj[6]}$	$\var{obj[7]}$
Partner $X$	Gap 0.	Gap 1.	Gap 2.	Gap 3.	Gap 4.	Gap 5.	Gap 6.	Gap 7.
Partner $Y$	Gap 8.	Gap 9.	Gap 10.	Gap 11.	Gap 12.	Gap 13.	Gap 14.	Gap 15.
Differences $(d_i)$	Gap 16.	Gap 17.	Gap 18.	Gap 19.	Gap 20.	Gap 21.	Gap 22.	Gap 23.

Advice

(a)

To rank the couples scores, we allocate the lowest numerical score a value of $1$ and repeat this for numbers $1..8$.

To calculate the differences, we subract the partner $Y$'s rank from partner $X$'s, for each couple.

(b)

To calculate Spearman's correlation coefficient, we use the following the formula:

\[r_s=1 - \frac{6 \times D}{n(n^2-1)}\]

where $ D = \sum{d_i}^2.$

Firstly we calculate $D$,

\begin{align}
D = \sum{d_i}^2 &= (\var{d[0]})^2 +(\var{d[1]})^2+(\var{d[2]})^2+(\var{d[3]})^2+(\var{d[4]})^2+(\var{d[5]})^2+(\var{d[6]})^2+(\var{d[7]})^2\\
& = \var{ssd}.
\end{align}

\begin{align}
r_s &= 1 - \frac{ 6 \times \var{ssd}}{\var{n} \times (\var{n}^2-1)}\\
& = \var{spcoef}
\end{align}

(c)

The sample size is $ n=\var{n}$, so we look up the critical values with $ n =\var{n} $ and thus the degrees of freedom are $\var{n}.$

(d)

Using the statistical table for the critical values for the spearman correlation coefficient in your notes, with $ n = \var{n}$, we find the following values:

$10\%$	$5\%$	$1\%$	$0.2\%$
$\var{ten}$	$\var{five}$	$\var{one}$	$\var{point2}$

(e)

When $r_s < \var{ten}$, then $p > 0.1 $.

When $\var{ten}< r_s <\var{five} $, then $0.05< p < 0.1 $.

When $\var{five}< r_s <\var{one} $, then $0.01< p < 0.05$.

When $\var{one}< r_s <\var{point2} $, then $0.002< p < 0.01$.

When $r_s > \var{point2}$, then $p < 0.002 $.

(f)

When $p>0.1$, we have no evidence of an association between $x$ and $y$

When $0.05< p < 0.1$, we have weak evidence of an association between $x$ and $y$

When $0.01< p < 0.05$, there is evidence of an association between $x$ and $y$

When $0.002<p<0.01$, there is strong evidence of an association between $x$ and $y$

When $p\le 0.002$, there is very strong evidence of an association between $x$ and $y$

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

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This question is used in the following exam:

Correlation by Vicky Hall in Newcastle University Sports Science.

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Calculating the Spearman rank correlation coefficient and p-value

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Chris Graham commented 6 years, 5 months ago

Lauren Frances Desoysa 6 years, 9 months ago

Lauren Frances Desoysa 6 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: `list`

Parts

Gap-fill

(a)

(b)

(c)

(d)

(e)

(f)

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Error

Calculating the Spearman rank correlation coefficient and p-value

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Chris Graham commented 6 years, 5 months ago

Lauren Frances Desoysa 6 years, 9 months ago

Lauren Frances Desoysa 6 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Gap-fill

(a)

(b)

(c)

(d)

(e)

(f)

Generated value: `list`