7.3.0. Overview
As we have seen in Chapter 6, ttest is the appropriate procedure for testing whether two samples belong to the same population (where the null hypothesis tested is expressed as “H_{0}: μ_{1} = μ_{2}”). When we need to test whether three (or more) samples belong to the same population (the null hypothesis “H_{0}: μ_{1} = μ_{2} = μ_{3}”), it may look as though performing a series of ttests between all possible pairs of samples (the three null hypotheses “H_{0}: μ_{1} = μ_{2}”, “H_{0}: μ_{1} = μ_{3}” and “H_{0}: μ_{2} = μ_{3}”), would solve the problem. Unfortunately this is not the case, since each ttest is associated with a confidence level (say 0.95) and when three are performed in a row, the final confidence level would drop to 0.95 x 0.95 x 0.95 = 0.86. Or in other words, the chance of rejecting the null hypothesis when it is in fact true (Type I error) would increase to 14%. As the number of samples to test increases, the chance of introducing an error would also increase. Analysis of Variance (ANOVA) was designed to overcome this problem by R A Fisher in the 1920s.
The data for a simple ANOVA problem consists of a number of measurements taken from a number of different groups. An example would be the weights of a sample of five people from four different regions of the country. The criterion used in grouping (country) is called a factor and each group (North, South, East West) a level of the factor. If an ANOVA problem has only one factor, then it is called a oneway ANOVA. There may, however, be another factor defined on the same set of measurements, such as sex (a factor with two levels), which makes the problem a twoway ANOVA. In this case, we can compare the means of groups defined by each factor separately (the main effects) and also compare the means of groups defined by the combinations of the two factor levels (interactions), males in the North, females in the East, etc. In theory, there are no limits to the number of factors and the number of interactions that can be defined in an ANOVA design. ANOVA and GLM procedures assume that each factor has a maximum of 2000 levels, though this number may be increased by entering and editing the following line in Documents\Unistat10\Unistat10.ini file under the [Options] group:
MaxFactorLevel=2000
Also see 3.2.12. Long String Table.
Let k be the number of groups and n_{i} the number of observations in group i for i = 1,…, k. in a oneway ANOVA problem. Let us also define the total number of observations as:
the mean of group i as:
and the grand mean as:
If we express the deviation of an observation from the grand mean as the sum of its deviation from its own group mean (withingroup) plus the deviation of its group mean from the grand mean (betweengroup) we have:
If we then take the squares of both sides, sum over i and j and rearrange, we obtain:
In other words:
Total Ssq = WithinGroups Ssq + BetweenGroups Ssq
where Ssq stands for sum of squares.
Our aim is to test the null hypothesis that “all means are equal”. Therefore, the entity we are interested in is the BetweenGroups Ssq. Since the WithinGroups Ssq term represents the rest of variation in data, we can also call it the Error Term. We construct the ANOVA table as follows:

Sum of Squares (Ssq) 
Degrees of Freedom 
Mean Squares (MSQ) 
FStatistic 
Probability 
Factor 
BetweenGroups Ssq 
k – 1 
BetweenGroups Ssq / (k – 1) 
BetweenGroups MSQ / WithinGroups MSQ 
Pvalue for F_{(k1)(Nk)} 
Error 
WithinGroups Ssq 
N – k 
WithinGroups Ssq / (N – k) 


Total 
Total Ssq 
N – 1 
Total Ssq / (N – 1) 


The Fstatistic for the Factor is the test statistic. The associated onetail probability from the Fdistribution is calculated with (k – 1) and (N – k) degrees of freedom and it is reported in the last column of the ANOVA table. If this pvalue is less than or equal to a given confidence level (usually 5%), then we reject the null hypothesis “H_{0}: μ_{1} = μ_{2} =…= μ_{k}”, or in other words, we conclude that the k samples tested do not belong to the same population.
Note that all we can conclude using ANOVA is whether the k population means are all equal or not. If they are not equal, then ANOVA does not tell us which ones are different. In order to find out which pairs or groups of population means are different, one of the Multiple Comparisons tests should be used. If the ANOVA design is more complicated than oneway, then the General Linear Model procedure may also be used to find out the significantly different groups (see 7.3.2.3. GLM Output Options).
7.3.0.1. ANOVA and GLM Data Format
In order to analyse ANOVA and GLM designs in a general purpose statistical program, the data should be organised in an accurate and logical way. The approach adopted in almost all serious statistical packages involves stacking all measurement data (the explanatory variable) in a single column, and expressing the various group memberships of these observations in separate corresponding categorical data columns (factors). The user will often face the problem of having to convert a published data table into this format.
For instance, consider the Randomised Block Design ANOVA example given in Table 54 p. 140 by Montgomery, D. C. (1991). Measurements on Hardness are given in the following format:

Coupon (Block) 

Type of Tip 
1 
2 
3 
4 
1 
2 
1 
1 
5 
2 
1 
2 
3 
4 
3 
3 
1 
0 
2 
4 
2 
1 
5 
7 
This should be entered into UNISTAT as follows:
Hardness 
Tip 
Coupon 
2 
1 
1 
1 
1 
2 
1 
1 
3 
5 
1 
4 
1 
2 
1 
2 
2 
2 
3 
2 
3 
4 
2 
4 
3 
3 
1 
1 
3 
2 
0 
3 
3 
2 
3 
4 
2 
4 
1 
1 
4 
2 
5 
4 
3 
7 
4 
4 
If the data has already been entered into a spreadsheet in the form of a table, you do not have to retype it in the above format manually. UNISTAT’s own spreadsheet Data Processor provides a number of functions that will help you to do the transformation automatically. The Data → Stack Columns procedure can be used to stack the hardness measurements in a column and create the Tip column automatically (see 3.3.9. Stack Columns).
You may also use the function Level() which is designed to generate factor columns containing regular (balanced) levels automatically (see 3.4.2.5. Statistical Functions). To do this, first create the data column and then enter the function Level(4);B in a blank column (to create the Tip column) and Level(4) into the next blank column (to create the Coupon column).
For the analysis, select Tip and Coupon as [Factor]s and Hardness as [Dependent].
Let us also consider a more complex example known as GraecoLatin Square Design given in Table 520 p. 168 by Montgomery, D. C. (1991).
Batches of Raw 
Operators 

Material 
1 
2 
3 
4 
5 
1 
Aα = 1 
Bc = 5 
Ce = 6 
Db = 1 
Ed = 1 
2 
Bb = 8 
Cd = 1 
Dα = 5 
Eχ = 2 
Ae = 11 
3 
Cc = 7 
De = 13 
Eb = 1 
Ad = 2 
Bα = 4 
4 
Dd = 1 
Eα = 6 
Ac = 1 
Be = 2 
Cb = 3 
5 
Ee = 3 
Ab = 5 
Bd = 5 
Cα = 4 
Dc = 6 
This table is entered into UNISTAT as follows:
Operator 
Batch 
Formulation 
Test assemblies 
Coded Data 
1 
1 
A 
a 
1 
2 
1 
B 
c 
5 
3 
1 
C 
e 
6 
4 
1 
D 
b 
1 
5 
1 
E 
d 
1 
1 
2 
B 
b 
8 
2 
2 
C 
d 
1 
3 
2 
D 
a 
5 
4 
2 
E 
c 
2 
5 
2 
A 
e 
11 
1 
3 
C 
c 
7 
2 
3 
D 
e 
13 
3 
3 
E 
b 
1 
4 
3 
A 
d 
2 
5 
3 
B 
a 
4 
1 
4 
D 
d 
1 
2 
4 
E 
a 
6 
3 
4 
A 
c 
1 
4 
4 
B 
e 
2 
5 
4 
C 
b 
3 
1 
5 
E 
e 
3 
2 
5 
A 
b 
5 
3 
5 
B 
d 
5 
4 
5 
C 
a 
4 
5 
5 
D 
c 
6 
where Formulation, Batch, Operator and Test assemblies are the factors and Coded Data is the dependent variable.
7.3.0.2. ANOVA Designs
It is possible to test a large number of experimental designs using UNISTAT’s Analysis of Variance and GLM procedures. In this section we shall describe seven major designs and demonstrate how we can solve these problems using UNISTAT with the help of published examples.
7.3.0.2.1. Randomised Block Design
In many ANOVA problems, it is desirable to control the variability from known nuisance factors (blocks). We want to remove this variability from the error sum of squares to increase the power of the test. For example if we wish to determine the effectiveness of different fertilisers on a particular crop, we might try each fertiliser on the crop in a number of different fields. But the soil in each field might not be of the same quality and this would add variability to the results. As a result, the experimental error will reflect both the random error and the variability between fields. A better design would be the randomised block design, where each fertiliser is tested in each field. A randomised block design is said to be complete if all the treatments are used in all the blocks.

Block 1 

Block 2 

Block b 
Treatment 1 
y_{11} 

y_{12} 
… 
y_{1b} 
Treatment 2 
y_{21} 

y_{22} 
… 
y_{2b} 

. 

. 

. 

. 

. 

. 
Treatment a 
y_{a1} 

y_{a2} 
… 
y_{ab} 
These designs can be constructed in UNISTAT using one of ANOVA or GLM procedures.
Example
Table 54 on p. 140 from Montgomery, D. C. (1991). The table format given in the book can be transformed into the factor format by using UNISTAT’s Data → Stack Columns procedure and the Level() function (see 3.4.2.5. Statistical Functions). All data should be stacked in a single column Hardness and two factor columns Tip and Coupon created to keep track of the group memberships. Therefore, the resulting data matrix should have 16 rows and 3 columns.
Open ANOVA, select Statistics 1 → ANOVA and GLM → Analysis of Variance, and select Tip (C2) and Coupon (C3) as [Factor]s and Hardness (C1) as [Dependent]. Then select Classic Experimental Approach and no interaction terms to obtain the following ANOVA table:
Analysis of Variance
Approach: Classic Experimental
Dependent variable: Hardness
Due To 
Sum of Squares 
DoF 
Mean Square 
Fstat 
Probability 
Main Effects 
121.000 
6 
20.167 
22.688 
0.0001 
Tip 
38.500 
3 
12.833 
14.438 
0.0009 
Coupon 
82.500 
3 
27.500 
30.938 
0.0000 
Explained 
121.000 
6 
20.167 
22.688 
0.0001 
Error 
8.000 
9 
0.889 


Total 
129.000 
15 
8.600 


The result is the Fstatistic and its probability value for Tip. Since the probability 0.0009 is much smaller than 5%, we reject the null hypothesis and conclude that the means differ significantly. In other words, the conclusion is that the type of tip affects the hardness values. The Fstatistic and its probability for Coupon is not strictly meaningful, but informally we can see that the Coupon were an important source of variation in the resulting hardness, and power of the analysis has been increased by using a Randomised Block Design.
7.3.0.2.2. Repeated Measures Design
It is often the case that repeated measurements are taken on the same subject. This typically happens when the subjects are people and measurements are taken from the same people at different times, following a particular treatment.
If the subject receives different treatments and the treatments are administered in a random order, then it is possible to regard the experiment as having a Randomised Block Design with the subject as a blocking factor. If the subject receives different treatments in a specified order, then it is possible to regard the experiment as a Crossover Design. If each subject receives the same treatment a number of times, this should be considered as a repeated measures design.
In a repeated measures design, the total sum of squares can be partitioned into the Between Subjects sum of squares and the Within Subjects sum of squares. It is assumed that these terms are statistically independent. This means that the residual sum of squares can also be partitioned into Error Between Subjects and Error Within Subjects. These terms can be used to find the various factor Fratios to increase the power of the analysis.
7.3.0.2.2.1. Repeated Measures over all Factors
This is the case when each subject only receives one level of each factor. The twofactor experiment of this kind is shown schematically below.


c_{1} 
c_{2} 
L 
c_{r} 

b_{1} 
X_{111} 
X_{112} 
L 
X_{11r} 
a_{1} 
M 
M 
M 

M 

b_{q} 
X_{1q1} 
X_{1q2} 
L 
X_{1qr} 
M 
M 





b_{1} 
X_{p11} 
X_{p12} 
L 
X_{p1r} 
a_{p} 
M 
M 
M 

M 

b_{q} 
X_{pq1} 
X_{pq2} 
L 
X_{pqr} 
Each row represents a group of subjects and each subject is measured on r occasions. This means that the factors A and B sum of squares are contained in the Between Subjects sum of squares. Only the Trial (factor C) sum of squares is contained in the Within Subjects sum of squares.
This can be done using the Analysis of Variance (ANOVA) procedure and selecting the Repeated Measures over all Factors option or using the General Linear Model (GLM). In order to use the GLM procedure, an additional factor column must be created in the spreadsheet to give information about the repeated measure (the Trial factor). In ANOVA with Repeated Measures over all Factors, this additional factor is created internally by the program, assuming a repeated measure across all the factors.
Example 1: Using ANOVA with Repeated Measures over all Factors
Table 7.43 on p. 341 from Winer, B. J. (1970). The table format given in the book can be transformed into the factor format by using UNISTAT’s Data → Stack Columns procedure and the Level() function (see 3.4.2.5. Statistical Functions). All data should be stacked in a single column Score and three factor columns created Subject, Anxiety and Tension.
Open ANOVA and select Statistics 1 → ANOVA and GLM → Analysis of Variance. Select Score (C24) as [Dependent], Subject (C23) as [Repeated], Anxiety (C21) and Tension (C22) as [Factor]s. From the next two dialogues select the Repeated Measures over all Factors and Classic Experimental Approach options and include all interactions at the last dialogue.
Analysis of Variance
Design: Repeated Measures over all Factors
Approach: Classic Experimental
Dependent Variable: Score
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Between Subjects 
181.000 
11 
16.455 


Anxiety 
10.083 
1 
10.083 
0.978 
0.3517 
Tension 
8.333 
1 
8.333 
0.808 
0.3949 
Anxiety x Tension 
80.083 
1 
80.083 
7.766 
0.0237 
Error Between 
82.500 
8 
10.313 


Within Subjects 
1077.000 
36 
29.917 


Trial 
991.500 
3 
330.500 
152.051 
0.0000 
Anxiety x Trial 
8.417 
3 
2.806 
1.291 
0.3003 
Tension x Trial 
12.167 
3 
4.056 
1.866 
0.1624 
Anxiety x Tension x Trial 
12.750 
3 
4.250 
1.955 
0.1477 
Error Within 
52.167 
24 
2.174 


Total 
1258.000 
47 
26.76596 


Example 2: Using General Linear Model
To use the GLM procedure with this example, an extra factor column Trial must be created. This is defined as the number of times a measurement is made on a particular subject. So the second measurement taken with subject i would result in a 2 in the Trial column, the third measurement with subject j would results in a 3 in the Trial column, etc.
Anxiety 
Tension 
Subject 
Trial 
1 
1 
1 
1 
1 
2 
1 
2 
2 
1 
1 
3 
2 
2 
1 
4 
1 
1 
2 
1 
1 
2 
2 
2 
2 
1 
2 
3 
2 
2 
2 
4 
So the two factors, Anxiety and Tension play no part in the calculation of the Trial column.
Open ANOVA and select Statistics 1 → ANOVA and GLM → General Linear Model. Select Score (C24) as [Dependent], Subject (C23) as [Repeated]. Select the following terms as factors, and at the following dialogue select the FStatistic denominators as shown.
C21 Anxiety 
Error Between C23 Subject 
C22 Tension 
Error Between C23 Subject 
C21 Anxiety x C22 Tension 
Error Between C23 Subject 
C25 Trial 
Error Within C23 Subject 
C21 Anxiety x C25 Trial 
Error Within C23 Subject 
C22 Tension x C25 Trial 
Error Within C23 Subject 
C21 Anxiety x C22 Tension x C25 Trial 
Error Within C23 Subject 
From the Output Options Dialogue select only the ANOVA option to obtain the following results.
General Linear Model
ANOVA
Dependent Variable: Score
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 

Prob 
Constant 
4800.000 
1 
4800.000 
35.644 

0.0000 
Between Subjects 
181.000 
11 
16.455 



Anxiety 
10.083 
1 
10.083 
0.978 
a 
0.3517 
Tension 
8.333 
1 
8.333 
0.808 
a 
0.3949 
Anxiety x Tension 
80.083 
1 
80.083 
7.766 
a 
0.0237 
Error Between 
82.500 
8 
10.313 



Within Subjects 
1077.000 
36 
29.917 



Trial 
991.500 
3 
330.500 
152.051 
b 
0.0000 
Anxiety x Trial 
8.417 
3 
2.806 
1.291 
b 
0.3003 
Tension x Trial 
12.167 
3 
4.056 
1.866 
b 
0.1624 
Anxiety x Tension x Trial 
12.750 
3 
4.250 
1.955 
b 
0.1477 
Error Within 
52.167 
24 
2.174 



Explained 
1123.333 
15 
74.889 
17.795 

0.0000 
Error 
134.667 
32 
4.208 



Total 
1258.000 
47 
26.766 



Rsquared = 
0.8930 
Adjusted Rsquared = 
0.8428 
a FStatistic: Error Between
b FStatistic: Error Within
In this particular example, ANOVA with the Repeated Measures over all Factors option and GLM produce exactly the same results. This would not always be the case since GLM always adopts the Regression Approach and with ANOVA you can select different approaches. However with a balanced design (as above) all three approaches will give the same result.
7.3.0.2.2.2. Repeated Measures over some Factors
This is the case when each subject receives only one level of some factors but all levels of other factors. The threefactor experiment of this kind is shown schematically below.


b_{1} 

L 

b_{q} 


c_{1} 
L 
c_{r} 
L 
c_{1} 
L 
c_{r} 
a_{1} 
X_{111} 
L 
X_{11r} 
L 
X_{1q1} 

X_{1qr} 
a_{2} 
X_{211} 
L 
X_{21r} 
L 
X_{2q1} 

X_{2qr} 
M 
M 

M 

M 

M 
a_{p} 
X_{p11} 
L 
X_{p1r} 
L 
X_{pq1} 

X_{pqr} 
Each row represents a group of subjects and each subject only receives one level of factor A. However each subject receives all levels of factors B and C. This means that only the factor A sum of squares is contained in the Between Subjects sum of squares. The factor B and C sum of squares is contained in the Within Subjects sum of squares.
Example
Table 7.33 on p. 324 from Winer, B. J. (1970). The table format given in the book can be transformed into the factor format by using UNISTAT’s Data → Stack Columns procedure and the Level() function (see 3.4.2.5. Statistical Functions). All data should be stacked in a single column Score and five factor columns created Subject, Noise, Period, Dial and SubjWGrps.
The artificial factor SubjWGrps has been set up to partition the Error Within Subjects, and these partitions are used for the Fstatistic ratios in this example. The factor SubjWGrps is equivalent to Subject(Noise) nest term. However since terms of the form FactorA x FactorB(FactorC) cannot be specified, SubjWGrps needs to be built explicitly. The table below shows how this is done. Each combination of Subject and Noise results in a different level. However, when each level of Noise is met for the first time this is pooled into level 0.
Subject 
Noise 
Count 
SubjWGrps 
1 
High 
1 
0 
2 
High 
2 
2 
3 
High 
3 
3 
4 
Low 
4 
0 
5 
Low 
5 
5 
6 
Low 
6 
6 
The SubjWGrps column calculated here has nothing to do with the Trial column calculated in the previous example (see 7.3.0.2.2.1. Repeated Measures over all Factors). In fact it can informally be considered as the opposite of the Trial column from the previous example.
Open ANOVA and select Statistics 1 → ANOVA and GLM → General Linear Model. Select Score (C26) as [Dependent] and Subject (C27) as [Repeated]. Select the following terms as factors, and on the following dialogue select the Fstatistic denominators as shown.
C30 Noise 
Error Between C27 Subject 
C29 Period 
C29 Period x C31 SubjWGrps 
C29 Period x C30 Noise 
C29 Period x C31 SubjWGrps 
C29 Period x C31 SubjWGrps 
Error Term 
C28 Dial 
C28 Dial x C31 SubjWGrps 
C28 Dial x C30 Noise 
C28 Dial x C31 SubjWGrps 
C28 Dial x C31 SubjWGrps 
Error Term 
C28 Dial x C29 Period 
C28 Dial x C29 Period x C31 SubjWGrps 
C28 Dial x C29 Period x C30 Noise 
C28 Dial x C29 Period x C31 SubjWGrps 
C28 Dial x C29 Period x C31 SubjWGrps 
Error Term 
The following results are obtained:
General Linear Model
ANOVA
Dependent Variable: Score
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 

Prob 
Constant 
105868.167 
1 
105868.167 
35.782 

0.0000 
Between Subjects 
2959.278 
5 
591.856 



Noise 
468.167 
1 
468.167 
0.751 
a 
0.4348 
Error Between 
2491.111 
4 
622.778 



Within Subjects 
6965.556 
48 
145.116 



Period 
3722.333 
2 
1861.167 
63.389 
b 
0.0000 
Period x Noise 
333.000 
2 
166.500 
5.671 
b 
0.0293 
Dial 
2370.333 
2 
1185.167 
89.823 
c 
0.0000 
Dial x Noise 
50.333 
2 
25.167 
1.907 
c 
0.2102 
Dial x Period 
10.667 
4 
2.667 
0.336 
d 
0.8499 
Dial x Period x Noise 
11.333 
4 
2.833 
0.357 
d 
0.8357 
Error Within 
467.556 
32 
14.611 



Period x SubjWGrps 
234.889 
8 
29.361 



Dial x SubjWGrps 
105.556 
8 
13.194 



Dial x Period x SubjWGrps 
127.111 
16 
7.944 



Explained 
6966.167 
17 
409.775 
4.986 

0.0000 
Error 
2958.667 
36 
82.185 



Total 
9924.833 
53 
187.261 



a FStatistic: Error Between
b FStatistic: Period x SubjWGrps
c FStatistic: Dial x SubjWGrps
d FStatistic: Dial x Period x SubjWGrps
The Between Subjects sum of squares and the Within Subjects sum of squares partition the Total sum of squares. The Error Between and the Error Within partition the Total Error sum of squares. And the Period x SubjWGrps, Dial x SubjWGrps and Dial x Period x SubjWGrps errors partition the Error Within.
So, in the example output above, Period x SubjWGrps, Dial x SubjWGrps and Dial x Period x SubjWGrps are the terms that are not included in the model. The factor Noise is Between Subjects. The terms Period, Period x Noise, Dial, Dial x Noise, Dial x Period, and Dial x Period x Noise are Within Subjects. All the Between Subjects and Within Subjects terms are included in the model.
7.3.0.2.3. Latin Square Design
Consider an experiment to compare k treatments in which there are two nuisance factors (blocks) each at k levels (this is more common than it sounds). A complete factorial design with one observation at each level would need kӠobservations, but a Latin Square needs only kSquared observations. Consider the following design with k = 5. The treatments are A, B, C, D and E, the two other sources of variation are represented by the rows and columns of the table.

Column 

Row 
1 
2 
3 
4 
5 
1 
A 
B 
C 
D 
E 
2 
E 
A 
B 
C 
D 
3 
D 
E 
A 
B 
C 
4 
C 
D 
E 
A 
B 
5 
B 
C 
D 
E 
A 
Only kSquared (= 25) observations are made, since at each combination of a row and a column only one of the five treatments is used. Each treatment occurs in each row and column precisely once. It is assumed that there are no interactions between the three factors.
These designs are constructed in UNISTAT using the ANOVA procedure. Treatment, columns and rows are selected as factors and all interaction terms are omitted. The main result is the Fstatistic on the treatment.
Example
Table 511 on p. 159 from Montgomery, D. C. (1991). The table format given in the book can be transformed into the factor format by using UNISTAT’s Data → Stack Columns procedure and the Level() function (see 3.4.2.5. Statistical Functions). All data should be stacked in a single column Coded Data and three factor columns Operator, Batch and Formulation created to keep track of the group memberships.
Open ANOVA and select Statistics 1 → ANOVA and GLM → Analysis of Variance, Operator (C4), Batch (C5) and Formulation (C6) as [Factor]s and Coded Data (C7) as [Dependent]. Then select Classic Experimental Approach and omit all interaction terms to obtain the following ANOVA table:
Analysis of Variance
Approach: Classic Experimental
Dependent Variable: Coded Data
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Main Effects 
548.000 
12 
45.667 
4.281 
0.0089 
Operator 
150.000 
4 
37.500 
3.516 
0.0404 
Batch 
68.000 
4 
17.000 
1.594 
0.2391 
Formulation 
330.000 
4 
82.500 
7.734 
0.0025 
Explained 
548.000 
12 
45.667 
4.281 
0.0089 
Error 
128.000 
12 
10.667 


Total 
676.000 
24 
28.167 


The result is the Formulation Fstatistic and its tail probability. The Batch and Operator variables are nuisance factors (blocks) which are removed from the error sum of squares to increase the power of the test. The Fstatistic and probability for Batch and Operator are not strictly meaningful, but informally we can see that the power of the test has been increased by including them in the design.
7.3.0.2.4. GraecoLatin Square Design
Consider a Latin square design with another factor added. Denote the levels of this extra factor by Greek letters. If each Latin letter appears once and only once with each Greek letter then the design is called a GraecoLatin square.

Column 

Row 
1 
2 
3 
4 
1 
Aα 
Bb 
Cc 
Dd 
2 
Bd 
Ac 
Db 
Cα 
3 
Cb 
Dα 
Ad 
Bc 
4 
Dc 
Cd 
Bα 
Ab 
A GraecoLatin square design exists for all k ≥ 3 except for k = 6. The GraecoLatin square design allows investigation of four factors (rows, columns, Latin letters and Greek letters), each at k levels with only kSquared observations.
These designs are constructed in UNISTAT using ANOVA. Selecting the Latin letters, Greek letters, columns and rows as factors. All interaction terms are omitted. The main result is the Fstatistic on the Latin letters.
Example
Table 520 on p. 168 from Montgomery, D. C. (1991). The table format given in the book can be transformed into the factor format by using UNISTAT’s Data → Stack Columns procedure and the Level() function (see 3.4.2.5. Statistical Functions). All data should be stacked in a single column Coded Data and four factor columns Operator, Batch, Formulation and Test created to keep track of the group memberships.
Open ANOVA, select Statistics 1 → ANOVA and GLM → Analysis of Variance, select Operator (C4), Batch (C5), Formulation (C6) and Test assemblies (C8) as [Factor]s and Coded Data (C7) as [Dependent]. Then select Classic Experimental Approach and no interaction terms (the default) to obtain the following ANOVA table:
Analysis of Variance
Approach: Classic Experimental
Dependent Variable: Coded Data
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Main Effects 
610.000 
16 
38.125 
4.621 
0.0171 
Operator 
150.000 
4 
37.500 
4.545 
0.0329 
Batch 
68.000 
4 
17.000 
2.061 
0.1783 
Formulation 
330.000 
4 
82.500 
10.000 
0.0033 
Test assemblies 
62.000 
4 
15.500 
1.879 
0.2076 
Explained 
610.000 
16 
38.125 
4.621 
0.0171 
Error 
66.000 
8 
8.250 


Total 
676.000 
24 
28.167 


The result is the Formulation Fstatistic and its probability. The Batch, Operator and Test assemblies variables are nuisance factors (blocks) which are removed from the error sum of squares to increase the power of the test. The Fstatistic and its probability for Batch, Operator and Test assemblies are not strictly meaningful, but informally we can see that the power of the test has been increased by including them in the design.
7.3.0.2.5. SplitPlot Design
In some experimental designs, one of the factors may be a subunit of another factor. For example a field may be divided into main plots and these main plots split into sub plots. If one factor is allocated to the main plots and another factor to their sub plots, then we have a SplitPlot design. The sub plots factor is compared against the variation between sub plots, the main plots factor is compared against the variation between the main plots. The variation within the main plots (between the sub plots) is likely to be less than the variation between the main plots. So the sub plots factor is tested with more power. The main plots factor is said to be confounded with blocks.
These designs are also called splitunit designs, in which case the terms main units and sub units are used instead of main plots and sub plots. Some examples of main plots and sub plots are as follows:
Main Plot 
Sub Plot 
Days 
Hours within day 
Subject 
Occasion with the subject 
Field 
Area within the field 
These designs are constructed in UNISTAT using ANOVA with the Repeated Measures over some Factors option. Select the main plot as the first factor and the sub plot as the first repeated measure.
Example
Example 9.5 on p. 266 from Armitage & Berry (2002). The table format given in the book can be transformed into the factor format by using UNISTAT’s Data → Stack Columns procedure and the Level() function (see 3.4.2.5. Statistical Functions). All data should be stacked in a single column Swabs and three factor columns Crowding, Status and Family created to keep track of the group memberships.
Open ANOVA, select Statistics 1 → ANOVA and GLM → Analysis of Variance and select Crowding (C9) and Status (C10) as [Factor]s, Family (C11) as [Repeated] and Swabs (C12) as [Dependent]. From the next two dialogues select the Repeated Measures over some Factors and Classic Experimental Approach options. At the interaction terms dialogue check the only interaction term Crowding x Status to obtain the following ANOVA table:
Analysis of Variance
Design: Repeated Measures over some Factors
Approach: Classic Experimental
Dependent Variable: Swabs
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Main Effects 
2004.156 
6 
334.026 
13.214 
0.0000 
Crowding 
470.489 
2 
235.244 
5.223 
0.0190 
Error (Family) 
675.600 
15 
45.040 


Status 
1533.667 
4 
383.417 
15.167 
0.0000 
2 Way Interactions 
72.400 
8 
9.050 
0.358 
0.9384 
Crowding x Status 
72.400 
8 
9.050 
0.358 
0.9384 
Explained 
2076.556 
14 
148.325 
5.868 
0.0000 
Error 
1516.733 
60 
25.279 


Total 
4268.889 
89 
47.965 


Crowding is compared against the between family variation instead of the overall error term. This increases the power of the test on the Crowding effect. The interaction between Crowding and Status is not significant, so we might consider removing it from the model.
7.3.0.2.6. Nested Design
Consider an experiment where the levels of one factor (child) are different depending on the level of another factor (parent). For example the parent factor may be Country and the child factor Region. The north of England is not related to the north of Germany and thus Region is a nested factor of Country.
Nested designs resemble factorial designs with certain cells missing. This is because one factor is nested under another so that not all combinations of the two factors are observed.
These designs are constructed in UNISTAT using ANOVA selecting the parent as a factor and the child as the corresponding repeated measure. Then the Nested Factors option is selected from the next dialogue.
Example
Table 133 on p. 443 from Montgomery, D. C. (1991). The table format given in the book can be transformed into the factor format by using UNISTAT’s Data → Stack Columns procedure and the Level() function (see 3.4.2.5. Statistical Functions). All data should be stacked in a single column Purity and two factor columns Supplier and Batches created to keep track of the group memberships.
Open ANOVA and select Statistics 1 → ANOVA and GLM → Analysis of Variance. Select Supplier (C13) as [Factor], Batches (C14) as [Repeated] and Purity (C15) as [Dependent]. Then select Nested Factors and Classic Experimental Approach from the next two dialogues to obtain the following results:
Analysis of Variance
Design: Nested Factors
Approach: Classic Experimental
Dependent Variable: Purity
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Main Effects 
84.972 
11 
7.725 
2.927 
0.0135 
Supplier 
15.056 
2 
7.528 
2.853 
0.0774 
Batches(Supplier) 
69.917 
9 
7.769 
2.944 
0.0167 
Explained 
84.972 
11 
7.725 
2.927 
0.0135 
Error 
63.333 
24 
2.639 


Total 
148.306 
35 
4.237 


We conclude that the difference between Batches is a source of variation.
7.3.0.2.7. Crossover Design
Crossover designs occur when subjects are reused, typically in time. For instance, in an experiment designed to compare the effects of drugs A and B, half the sample (chosen at random) take drug A and the remaining half take drug B at the start of the experiment. Sometime later the first sample now take drug B and the second sample take drug A. It is important that effect of the first drug taken does not carryover and affect the performance of the second drug taken. If this does happen it is called the carryover effect.
In UNISTAT the crossover design is analysed in two steps. The first step tests whether the carryover effect is significant. If the carryover effect is not significant then a standard ANOVA can be used on the remaining factors. If the carryover effect is significant then analysis should be restricted to the first trial, and in future experiments a larger time period left between the trials.
The significance of the carryover effect is tested using a SplitPlot design (see 7.3.0.2.5. SplitPlot Design) of the treatment order against the subjects. To do this a factor column needs to be created which represents the order in which the treatments were given. The easiest way to do this is to have a string column with characters representing each treatment in the order they were given, say, for 3 treatments A, B and C. The column would contain ABC, ACB, BAC, BCA, CAB and CBA as required. When this sequence factor column is defined, it will be selected as the first factor and the subjects as a repeated measure. The sequence should not be significant. If it is not, continue to analyse the full data. If it is significant, then it may only be possible to use the results from the first trial.
Example
Table 11.5 on p. 380 from Bolton, S. (1990). The table format given in the book can be transformed into the factor format by using UNISTAT’s Data → Stack Columns procedure and the Level() function (see 3.4.2.5. Statistical Functions). All data should be stacked in a single column AUC and four factor columns Period, Subject, Sequence and Treatment created to keep track of the group memberships.
The first step is to test for any carryover (Sequence) effects. This is a SplitPlot design (see 7.3.0.2.5. SplitPlot Design), with Sequence against Subject. Open ANOVA and select Statistics 1 → ANOVA and GLM → Analysis of Variance. Select Sequence (C18) as [Factor], Subject (C17) as [Repeated] and AUC (C20) as [Dependent]. Then select the Repeated Measures over some Factors and Classic Experimental Approach options to obtain the following ANOVA table:
Analysis of Variance
Design: Repeated Measures over some Factors
Approach: Classic Experimental
Dependent Variable: AUC
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Main Effects 
4620.375 
1 
4620.375 
1.590 
0.2313 
Sequence 
4620.375 
1 
4620.375 
1.187 
0.3016 
Error(Subject) 
38940.083 
10 
3894.008 


Explained 
4620.375 
1 
4620.375 
1.590 
0.2313 
Error 
34870.500 
12 
2905.875 


Total 
78430.958 
23 
3410.042 


The critical value is the Fstatistic and its probability for Sequence. This shows that the sequence is not significant, so there are no significant crossover effects in the data. We can then proceed to analyse the full data set. This is done by selecting ANOVA and Subject (C17), Period (C16) and Treatment (C19) as [Factor]s and AUC (C20) as [Dependent]. Select Classic Experimental Approach and no interaction terms:
Analysis of Variance
Approach: Classic Experimental
Dependent Variable: AUC
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Main Effects 
67760.875 
13 
5212.375 
4.885 
0.0083 
Subject 
43560.458 
11 
3960.042 
3.711 
0.0240 
Period 
13490.042 
1 
13490.042 
12.643 
0.0052 
Treatment 
10710.375 
1 
10710.375 
10.038 
0.0100 
Explained 
67760.875 
13 
5212.375 
4.885 
0.0083 
Error 
10670.083 
10 
1067.008 


Total 
78430.958 
23 
3410.042 


This shows that Subject, Period and Treatment are all significant.