LSD and Tukey yardsticks, and two-way ANOVA - SES2003

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Tom Lowe

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Has some problems

4 years, 4 months ago

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Tom Lowe 4 years, 4 months ago

Gave some feedback: Has some problems

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Tom Lowe commented 4 years, 4 months ago

Could use some adaptive marking in parts e) and f), as the LSD and Tukey yardsticks depend on the value found for the sqrt(RMS) in part a), and which differences are found to be significant depends on the means for the groups found in part d).

Also there is no advice for parts a), b), c) and d), and the advice for part e) doesn't explain where most of the terms in the calculation of the yardsticks come from (ie the critical values or the sample size of one group).

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Chris Graham 6 years, 2 months ago

Created this as a copy of LSD and Tukey yardsticks, and two-way ANOVA.

Name	Status	Author	Last Modified
LSD and Tukey yardsticks, and two-way ANOVA	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:51
LSD and Tukey yardsticks, and two-way ANOVA - SES2003	Has some problems	Chris Graham	03/12/2020 20:41
LSD and Tukey yardsticks, and two-way ANOVA - Mock - SES2003	Needs to be tested	Chris Graham	25/02/2019 11:00

Question statement

To test the effectiveness of pain-relief creams on back pain, five volunteers (A-E) suffering with back pain tried each of four creams (W-Z). The degree of pain relief was measured on a 0-50 scale (higher figures indicate higher levels of pain relief). The results are given below with some totals:

	A	B	C	D	E	Totals
W	{r[0][0]}	{r[1][0]}	{r[2][0]}	{r[3][0]}	{r[4][0]}	{cols[0]}
X	{r[0][1]}	{r[1][1]}	{r[2][1]}	{r[3][1]}	{r[4][1]}	{cols[1]}
Y	{r[0][2]}	{r[1][2]}	{r[2][2]}	{r[3][2]}	{r[4][2]}	{cols[2]}
Z	{r[0][3]}	{r[1][3]}	{r[2][3]}	{r[3][3]}	{r[4][3]}	{cols[3]}
Totals	{t[0]}	{t[1]}	{t[2]}	{t[3]}	{t[4]}	{tot}

Test the null-hypothesis using two-way ANOVA that the creams are equally effective.

Write down the sample mean for each cream together with the LSD and Tukey yardsticks so that you can see if there is any significant difference between the sample means given by these yardsticks..

Give any introductory information the student needs.

			Name	Type	Generated Value

lsd	number	6.05
tukey	number	8.25
stderror	number	1.96
bbss	number	31.50
cols	list	[ 165, 177, 158, 200 ]
vrbb	number	0.41
w6	list	[ 1, 0, 0 ]
vr	number	3.53
w4	list	[ 0, 0, 1 ]
w3	list	[ 0, 1, 0 ]
w2	list	[ 0, 0, 1 ]
w1	list	[ 0, 0, 1 ]
dfr	integer	12
rt	list	Nested 4×5 list
rs	number	19.24
t90	number	13.36
tot	number	700
sqrms	number	4.39
sig	number	4.2
tol	number	0.01
btss	number	203.60
tss	number	466.00
dfbt	integer	3
yn	list	Nested 6×3 list
msbt	number	67.87
ssq	number	24966
v1	list	[ 0, 0, 1, 0, 0 ]
dfbb	integer	4
t99	number	20.09
t95	number	15.51
msbb	number	7.88
rss	number	230.9
me	list	[ 33, 35.4, 31.6, 40 ]
r1	list	Nested 4×5 list
m	integer	4
w5	list	[ 0, 0, 1 ]
n	integer	5
mu	list	[ 33, 35, 30, 41, 40 ]
r	list	Nested 5×4 list
t	list	[ 139, 143, 136, 134, 148 ]
w	list	Nested 6×3 list
v	list	[ 0, 0, 1, 0, 0 ]
pvalue	number	0.049

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
precround(2.179*sqrms*sqrt(2/5),2)

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: `number`

6.05

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Number entry
  Add
Part b) Choose one from a list
Part c) Choose one from a list
Part d) Choose one from a list
Part e) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
3. Gap Gap 2. Number entry
  Add
4. Gap Gap 3. Number entry
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5. Gap Gap 4. Number entry
  Add
6. Gap Gap 5. Number entry
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Part f) Match choices with answers

Gap-fill

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

Here is the ANOVA table corresponding to this data:

Source	df	SS	MS	VR
Between Treatments	$\var{m-1}$	$\var{btss}$	$\var{msbt}$	$\var{vr}$
Between Blocks	$\var{n-1}$	$\var{bbss}$	$\var{msbb}$	$\var{vrbb}$
Residual	$\var{dfr}$	$\var{rss}$	$\var{rs}$	-
Total	$\var{m*n-1}$	$\var{tss}$	-	-

Input $\sqrt{RMS}$ here: Unnamed gap to 2 decimal places.

This will be used later to calculate the yardsticks.

Advice

Using the Yardsticks

The mean values for each cream are:

	$\overline{x}_i$
W	$\var{me[0]}$
X	$\var{me[1]}$
Y	$\var{me[2]}$
Z	$\var{me[3]}$

The differences between the mean values for the creams are:

Between $W$ and $X=\;|\var{me[0]}-\var{me[1]}|=\var{abs(me[0]-me[1])}$

Between $W$ and $Y=\;|\var{me[0]}-\var{me[2]}|=\var{abs(me[0]-me[2])}$

Between $W$ and $Z=\;|\var{me[0]}-\var{me[3]}|=\var{abs(me[0]-me[3])}$

Between $X$ and $Y=\;|\var{me[1]}-\var{me[2]}|=\var{abs(me[1]-me[2])}$

Between $X$ and $Z=\;|\var{me[1]}-\var{me[3]}|=\var{abs(me[1]-me[3])}$

Between $Y$ and $Z=\;|\var{me[2]}-\var{me[3]}|=\var{abs(me[2]-me[3])}$

We compare these differences with the LSD and Tukey yardsticks:

LSD yardstick = $2.179\times\var{sqrms}\times\sqrt{2/\var{n}}=\var{lsd}$ to 2 decimal places, where $\var{sqrms}$ is the value of $\sqrt{RMS}$ found above.

Tukey yardstick = $4.2\times\var{sqrms}\times\sqrt{1/\var{n}}=\var{tukey}$ to 2 decimal places.

If the difference of the means:

is greater than the Tukey yardstick we say that there is evidence of a definite significant difference.

less than the Tukey yardstick but greater than the LSD yardstick we say that there is evidence of a possible significant difference.

is less than the LSD yardstick then we say that there is no evidence of a significant difference.

Hence we have the following for the creams:

Pairs of creams	Definite Significant Difference	Possible Significant Difference	No Significant Difference
Means of W and X	{yn[0][0]}	{yn[0][1]}	{yn[0][2]}
Means of W and Y	{yn[1][0]}	{yn[1][1]}	{yn[1][2]}
Means of W and Z	{yn[2][0]}	{yn[2][1]}	{yn[2][2]}
Means of X and Y	{yn[3][0]}	{yn[3][1]}	{yn[3][2]}
Means of X and Z	{yn[4][0]}	{yn[4][1]}	{yn[4][2]}
Means of Y and Z	{yn[5][0]}	{yn[5][1]}	{yn[5][2]}

Give a worked solution to the whole question.

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

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LSD and Tukey yardsticks, and two-way ANOVA - SES2003

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Tom Lowe 4 years, 4 months ago

Tom Lowe commented 4 years, 4 months ago

Chris Graham 6 years, 2 months ago

Error in variable testing condition

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Generated value: `number`

Parts

Gap-fill

Using the Yardsticks

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LSD and Tukey yardsticks, and two-way ANOVA - SES2003

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Tom Lowe 4 years, 4 months ago

Tom Lowe commented 4 years, 4 months ago

Chris Graham 6 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Using the Yardsticks

Generated value: `number`