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Design Examples

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Factor Structure Diagram

A factor structure diagram shows all the factors in a design, where each factor is linked by a line to the maximal factors that are coarser than itself. E.g. if A > B and B > C then the diagram would look like this:

A --> B --> C

i.e. we do not draw the arrow from A to C.

The number of non-empty levels is then written as a subscript for each factor, e.g..

A _~a --> B _~b --> C _~c

The degrees of freedom can them be calculated, and are written in as superscripts. In our rather artificial example it would look like this

A _~a^{~a - ~b} --> B _~b^{~b - ~c} --> C _~c^~c

[ remember that _ df(A) _ = _ |A| - df(B) - df(C) _ = _ ~a - (~b - ~c) - ~c _ = _ ~a - ~b ]

Two Way Analysis of Variance

We have ~n observations and two factors, R (rows) and C (columns) with ~r rows and ~c columns respectively, their cross product R # C, the identity factor I, and the null factor O.

_ I _~n^{~n - ~r~c} _ zDgrmRight{,} _ R # C _{~r~c}^{(~r - 1)(~c - 1)} _ zDgrmRight{,} _ R _~r^{~r - 1}
    zDgrmDown{,}   zDgrmDown{,}
    _ C _~c^{~c - 1} _ zDgrmRight{,} _ 0 _1^1 _

Table of Variances

Factor |F| d_F SS_F SSD_F
O 1 1 SS_O SS_O
C ~c ~c - 1 SS_C SS_C - SS_O
R ~r ~r - 1 SS_R SS_R - SS_O
R # C ~r~c ~r~c - ~r + 1 - ~c + 1 - 1
= ~r~c - ~r - ~c + 1
= (~r - 1)(~c - 1)
SS_{R # C} SS_{R # C} - SS_R - SS_C + SS_O
I ~n ~n - ~r~c + ~r + ~c - 1 - ~r + 1 - ~c + 1 - 1
= ~n - ~r~c
SS_I SS_I - SS_{R # C}

Note: In tables of actual observations the columns for |F| and SS_F are not usually shown.

Latin Square

A latin square design involves several factors, one factor being a "treatement" factor, the others factors being "blocking" factors. For example to test the effects of three different types of fertilizer (T) they are tested in three different fields (F), each split up into three plots, each with a different crop (C = wheat, barley, or oats). The treatments are distributed so that each combination of field-crop is treated with each of the different fertilizers,e.g. If the fields are labeled 1, 2, and 3, and the

  Field
Crop 1 2 3
Wheat A B C
Barley C A B
Oats B C A

Note that each treatment is applied only once in each field, and only once to each crop.

_ I _{~k^2} ^{~k^2 - 3~k + 2} _ zDgrmRight{,} _ F _~k^{~k - 1} _ zDgrmRight{,} _
zDgrmDown{,} _ _ _ zDgrmDown{,}
_ zDgrmRight{,} _ C _~k^{~k - 1} _ zDgrmRight{,} _
zDgrmDown{,} _ _ _ zDgrmDown{,}
_ zDgrmRight{,} _ T _~k^{~k - 1} _ zDgrmRight{,} _ 0 _1^1 _

Table of Variances

Factor |F| d_F SS_F SSD_F
O 1 1 SS_O SS_O
F ~k ~k - 1 SS_F SS_F - SS_O
C ~k ~k - 1 SS_C SS_C - SS_O
T ~k ~k - 1 SS_T SS_T - SS_O
I ~k^2 ~k^2 - ~3k + 2 SS_I SS_I - SS_T - SS_C - SS_F + 2SS_O

Split Plot

Plot Treatment on
sub-plots
i ii
1 A B C A B C
2 C A B A B C
3 A B C B C A
4 B C A A B C
5 C A B C B A

Note that the naming of the subplots (i and ii) is for illustrative purposes, it is implicit in the design, the essential being that each treatment occurs once in each sub-plot of a plot.

_ I _{~m~p~t}^{~{mpt} - ~m~p + ~p - ~p~t} _ zDgrmRight{,} _ S _{~m~p}^{~m~p - ~p} _
zDgrmDown{,}   zDgrmDown{,}
_ P # T _{~p~t}^{~p~t - ~p - ~t + 1} _ zDgrmRight{,} _ P _~p^{~p - 1} _
zDgrmDown{,}   zDgrmDown{,}
_ T _~t^{~t - 1} _ zDgrmRight{,} _ 0 _1^1 _

Table of Variances

Factor |F| d_F SS_F SSD_F
O 1 1 SS_O SS_O
T ~t ~t - 1 SS_T SS_T - SS_O
P ~p ~p - 1 SS_P SS_P - SS_O
P # T ~p~t (~p - 1)(~t - 1) = ~p~t - ~p - ~t + 1 SS_{P # T} SS_{P # T} - SS_T - SS_P + SS_O
S ~m~p ~m~p - ~p SS_S SS_S - SS_P
I ~{mpt} ~{mpt} - ~m~p + ~p - ~p~t + ~p + ~t - 1 &minus. ~p + 1 - ~t + 1 - 1
= ~{mpt} - ~m~p + ~p - ~p~t
SS_I SS_I - SS_S + SS_P - SS_{P # T}

In the example above there are three treatments (~t = 3), five plots (~p = 5) and two subplots for each plot (~m = 2)