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 ]
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.
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 |
| 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)