Trinity College Dublin
Diploma in Statistics
Computer Laboratory 2
(1) Measurement System Analysis and
(2) Analysis of a Split Plot experiment
Invitations to consider the results of Minitab analysis and their statistical and substantive
interpretations are printed in italics. Take some time for this; consult your neighbour or tutor.
Enter your responses in a Word document, as draft contributions to a report on the
experiment and its analysis.
Learning Objectives
On completion of this laboratory, students should be able to
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use the General Linear Model command to calculate an analysis of variance for a balanced
replicated fully crossed random effects layout,
report on patterns of variations in one way and two way tables of means and corresponding
factor plots,
explain the Minitab "Expected Mean Squares" output, relate it to components of variance and
the analysis of variance,
explain basic quantities in measurement system capability, repeatability and reproducibility
and estimate them from a relevant analysis of variance,
explain the make of a split plot experiment,
explain the make up of the analysis of variance table associated with a split plot experiment,
provide a comprehensive report on the analysis of results of a split plot experiment with tables
and graphs, analysis of variance, comparisons of treatments with control and pairwise
comparisons of treatments.
Exercise 1; Measurement System Analysis
Basic requirements of a measurement system are that it be accurate, or unbiased, and precise. A
measurement system is accurate if repeated measurements are centred in the right place and
precise if the spread of the repeated measurements is small. These definitions recognise that
measurement processes are subject to variation. The purpose of measurement system analysis is to
classify and quantify the extent of such inaccuracy and imprecision.
It is important to recognise that measurement variation may be influenced by a range of factors,
including
the instrument itself,
the user of the instrument,
materials used as part of the measurement process,
the laboratory in which the measurement is made,
the environment,
and others. This implies that we should assess the precision of the instrument itself, repeatability,
and also how that precision varies in varying circumstances, reproducibility.
It is also important to recognise that the objects being measured are themselves subject to
manufacturing variation. This imposes two requirements on a measurement process, that it is
capable of distinguishing differing objects and that it is valid across the range of variation of the
objects.
In a situation where measurements are made on manufactured parts where the standard deviation of
manufacturing variation is P, using an instrument whose error standard deviation is E,
measurement system capability may be expressed as
 T (P )
E
,
where
T(P) =
 2P   2E ,
referred to as Total Parts standard deviation.
Repeatability may be expressed simply as
E.
Where several operators make the measurements and variation between them has standard
deviation O, reproducibility of the measurement system may be expressed in terms of
2
O2  OP
 2E ,
referred to as Total Reproducibility standard deviation, which reflects the overall variation of the
measurement process including instrument and operator variation, or as
2
O2  OP
which reflects the additional variation associated with the use of different operators, additional to
repeatability variation.
In practice, different definitions derived from these are recommended by different standards
organisations; see Measurement Notes on the course web page for more details and some
references.
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Application
When fixing floor covering made of fibrous materials to a floor using suitable glue, a key determinant
of the strength of the bond is the typical angle of incidence of fibres to the floor. Varying approaches
to the measurement of such angles are available. In selecting a suitable approach, knowledge of the
precision of such methods is important. In a study designed to quantify the precision of one such
method, each of four analysts made eight measurements of the fibre angle of each of five sections
cut from a piece of floor covering that had been fixed to a sheet of base material. The results follow
overleaf and may be copied from Fibre angles.xls on the course website or mstuart's get folder.
Use the General Linear Model option from the ANOVA section of the Stat menu to calculate a full
analysis of variance, with Analyst and Section as random effects. To facilitate study of main effects
and interactions, choose to display least squares means corresponding to the term Section*Analyst
and factor plots corresponding to the terms Section Analyst Section* Analyst. Also, choose to
display expected mean squares and variance components.
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Section
Analyst 1
Analyst 2
Analyst 3
Analyst 4
1
10, 20, 20, 23
20, 20, 20, 15
20, 25, 17, 22
23, 15, 23, 20
20, 19, 15, 16
20, 19, 12, 14
10, 10, 10,
5, 5, 5,
2
15, 17, 20, 20
10, 15, 15, 15
15, 13, 5, 10
8, 8, 10, 12
15, 20, 14, 16
13, 20, 15, 15
10, 10, 10, 10
10, 15, 15, 10
3
23, 20, 22, 20
25, 22, 20, 23
20, 23, 20, 20
23, 23, 22, 20
15, 20, 22, 18
15, 20, 16, 20
10, 10. 10, 15
15, 10, 10, 10
4
15, 16, 22, 15
15, 15, 22, 17
20, 22, 18, 23
23, 23, 24, 20
13, 13, 15, 20
11, 20, 13, 15
5, 10, 10, 10
10, 10, 10, 10
5
20, 20, 22, 20
27, 17, 20, 15
18, 20, 18, 23
20, 20, 18, 15
10, 14, 17, 12
11, 10, 15, 10
5, 10, 10, 10
10, 10, 10, 10
5
5
Make the table directly
Copy the output on least squares means into a new worksheet and make a two way table; use Stat /
Tables / Descriptive Statistics, clear summaries for "Categorical Variables" and choose to display
Means of "Associated Variable" Angle.
Copy your factor plots and two way table into a Word document, for coordinated examination.
Report on patterns of variation in Analyst means, Section means and Section* Analyst
means, as seen in factor plots, with numerical highlights from the table.
Interpret the "Expected Mean Squares" output in terms of A, S and AS.
Explain the basis for calculating the F-ratios in the analysis of variance table.
Interpret the analysis of variance table, with reference to your earlier report, editing your
report as appropriate.
Calculate Repeatability standard deviation, E and Total Reproducibility standard deviation,
2A  2AS  E2 . Comment on their relative magnitude.
Calculate Total Parts standard deviation, T(P) =
2S  E2 . Comment on its magnitude
relarive to Repeatability standard deviation, E.
Exercise 2; Split Plot analysis
A classic split plot experiment was carried out on the Cambridge University Farm in 1931. The
original plan was to investigate the effects of two novel cultivation treatments of grass-land pasture,
by comparison with no treatment. The novel treatments were use of a grassland "Rejuvenator" (R)
and use of a conventional harrow (H). These treatments were compared with no treatment, taken to
be a control. The experiment was carried out on a field of old pasture, laid out in plots consisting of
strips 4 yards wide and 45 yards long. The treatments were applied to a block of 3 adjacent plots, in
a randomly chosen order. The experiment was replicated in a total of 6 independently randomised
blocks, placed side by side.
In addition to the randomised blocks layout for investigating treatment effects, each 45 yard plot was
subdivided into 4 subplots, each of which was treated with one of four fertilisers, Farmyard manure
(F), Straw (S), Artificial fertiliser, (A) and no treatment, acting as Control (C). The fertilisers were also
applied in random order to the subplots within each whole plot, independently.
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Yield was recorded in pounds (lbs) of green produce from a single cut of each plot made on June 31,
1931. The data follow and are available in GrassLand.xls on the course website or mstuart's get
folder. The main plots correspond to the columns in the table, the subplots to the entries within each
column.
The organisation of the data in subplots within whole plots, with the independent randomisations
within whole plots, means that the comparison of treatments C, H and R is assessed by reference to
the variation at whole plot level, effectively by way of a randomised blocks analysis of treatments C,
H and R in blocks 1 to 6., while comparison of fertilisers, including their interactions with treatments
(and, possibly, blocks) is assessed by reference to the variation at subplot level.
A
C
F
S
C
266
165
198
184
H
213
127
180
127
R
208
155
200
150
C
210
150
247
188
H
222
167
203
167
R
266
163
228
157
C
220
155
190
140
H
184
118
168
128
R
184
153
174
141
C
216
159
225
174
H
178
125
149
107
R
207
135
162
113
C
202
147
184
154
H
175
118
175
112
R
184
98
144
113
C
169
132
164
116
H
142
104
145
89
R
151
69
116
101
The analysis will be reported in two stages, with the randomised blocks layout of cultivation
treatments analysed initially, followed by the full split plot analysis. Reports will be in terms of tables
of means and corresponding plots with significance tests and confidence intervals, as we did for the
measurement system analysis.
To set up the randomised blocks analysis, first calculate the whole plot yields. To do this, copy the
data to Minitab, then use Data / Unstack1 on the Yields column using subscripts in the Fertiliser
column, then add the unstacked columns to each other, using the Calculator in the Calc command.
Since each whole plot yield is the sum of the corresponding subplot yields, this results in a column
with the whole plot yields.
Next, we need the appropriate Block and Treatment codes in two further columns. The first three
entries in the whole plot yield column are the whole plot yields of C, H and R, respectively, in Block 1.
Succesive sets of three whole plot yields are the same for successive blocks. Thus, we need a
column of Block numbers, 1 to 6, in sets of 3, and a column of succesive sets of C, H, R. Do get the
block numbers column, use the Make Patterned Data command in the Calc menu. To get the
treatment codes column, enter C, H and R in the first three cells and copy toi the remaining cells
(down to Row 18).
Next, use the General Linear Model command to do the analysis of variance, with model including
the new block and treatment variables as main effects; recall that the Block*Treatmentc interaction
serves as the error term in a simple ransomised block experiment. Include factor plots. Use the
Tables command to make a two way table, including marginal means (see Options).
Provide a comprehensive report on the randomised blocks analysis of treatment effects in
blocks, with table and graphs, analysis of variance, comparisons of treatments with control
and pairwise comparisons of treatments.
1
To understand the Unstack command, first look at the Stack command; it "stacks" several columns into one
big stack of numbers, that is, one big column. Unstack reverses this process, taking a single column and
putting its numbers into several columns. The "subscript" column tells you which numbers go in which column.
In this case, we unstack the Yield column using the "subscripts" in the Fertiliser column. Thus, all the numbers
in the Yield column corresponding to Fertiliser code A go in one unstacked column, all those corresponding to
code F go in another unstacked column, etc. The result is that we get four columns corresponding to the four
fertilisers, A, C, F and S. In fact, they form the transpose of the table above.
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Next, calculate the split plot analysis of the original data; first list all relevant main effects and
interactions to help decide on the Model terms to enter. Enter Block as a Random Effect, to force
Minitab to calculate appropriate error terms.
Compare the randomised blocks analysis with the top half of the split plots analysis.
What is the connection?
Why?
Provide a comprehensive report on the split plots analysis, with tables and graphs, analysis of
variance, comparisons of treatments with control and pairwise comparisons of treatments.
This ends the formal laboratory.
Review the Learning Objectives. Have they been achieved?
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