# Fractional Factorial design for 3 factorial

I would like to be assisted on how to go about choosing the treatments to work with in a fractional factorial the following scenario. I have 3 factors ie 3 feeding intervals, 3 larval densities and 3 total feeds. This makes it a 3 factorial experiment.

How can I go about getting a fractional factorial design. The 27 possible observations are not possible considering that I would like to replicate twice, taking it to 54. I wanted help on how to use fractional factorial method to be able to select at least 16.

• This is very short. Can you please give some more details and background, what is the goal of the experiment, ... Commented Nov 20, 2017 at 10:48
• the experiment aims at getting the best feeding time, density and total feed to give to my larvae for best production. The values on table are; (1) feeding intervals - 1, 2, 3; 3 larval desities - 1.0 kg of larvae/unit area, 1.5 and 2, (c)total feed to give- 90kg, 150kg and 200kg. Commented Nov 20, 2017 at 10:54
• Can you please add additional information as an edit to the post? A full factorial has then 3*3*3=27 observations, is that impossible to run? How many runs can you afford? Commented Nov 20, 2017 at 11:02
• The 27 possible observations are not possible considering that I would like to replicate twice, taking it to 54. I wanted help on how to use fractional factorial method to be able to select atleast 16. Commented Nov 20, 2017 at 11:20
• I dont have time for a longer answer now: fractioning with 3-level factors is not as straightforward as with 2-level factors. I would go for D-optimal design. In R use package AlgDesign, or have a look at cran.r-project.org/web/views/ExperimentalDesign.html Commented Nov 20, 2017 at 11:26

You haven't really given enough information, but I will try to give some ideas. One method is to first construct the complete design with $$n=3^3$$, and then use some algorithm for optimal experimental design, say D-optimality, there are R implementations in the packages AlgDesign and OptimalDesign, on CRAN, see https://CRAN.R-project.org/view=ExperimentalDesign. But here I will exemplify another approach (maybe they can be combined), construction fractional factorial designs directly with algebraic methods , using the CRAN package planor (*). See https://r2012-bordeaux.sciencesconf.org/file/14501 for more details and examples!

library(planor)
# Note that planor is now archived, but it can still be
# found at CRAN under 'archived', and is still usable
ex1key  <-  planor.designkey(factors=c("block", LETTERS[1:3]),
nlevels = c(3, rep(3, 3)),
block = ~ block,
model= ~ block+ (A+B+C)^2,
estimate= ~ A + B + C ,
nunits=3*3^2, max.sol=1)
Preliminary step 1 : processing the model specifications
Preliminary step 2 : performing prime decompositions on the factors
*** Main step for prime p = 3 : key-matrix search
=> search for columns 2 to 4
first visit to column 2
first visit to column 3
---    col. 3 ( j = 2) 22 selected candidates
first visit to column 4
---    col. 4 ( j = 3) 18 selected candidates
The search is closed: max.sol = 1 solution(s) found

summary(ex1key)

********** Prime  3  design **********

--- Solution  1  for prime  3  ---

TREATMENT EFFECTS CONFOUNDED WITH THE MEAN
nil

BLOCK-and-TREATMENT EFFECTS CONFOUNDED WITH THE MEAN
1 = block^2  A^2  B

WEIGHT PROFILES
Treatment effects confounded with the mean: none
Treatment effects confounded with block effects: 2^1
Treatment pseudo-effects confounded with the mean: none
Treatment pseudo-effects confounded with block effects: 2^1

ex1Des
An object of class "planordesign"
Slot "design":
block A B C
1      1 1 1 1
2      1 1 1 2
3      1 1 1 3
4      1 2 2 1
5      1 2 2 2
6      1 2 2 3
7      1 3 3 1
8      1 3 3 2
9      1 3 3 3
10     2 1 2 1
11     2 1 2 2
12     2 1 2 3
13     2 2 3 1
14     2 2 3 2
15     2 2 3 3
16     2 3 1 1
17     2 3 1 2
18     2 3 1 3
19     3 1 3 1
20     3 1 3 2
21     3 1 3 3
22     3 2 1 1
23     3 2 1 2
24     3 2 1 3
25     3 3 2 1
26     3 3 2 2
27     3 3 2 3

Slot "factors":
An object of class "designfactors"
Slot "fact.info":
nlev block ordered model basic dummy
block    3  TRUE   FALSE  TRUE FALSE FALSE
A        3 FALSE   FALSE  TRUE FALSE FALSE
B        3 FALSE   FALSE  TRUE FALSE FALSE
C        3 FALSE   FALSE  TRUE FALSE FALSE

Slot "pseudo.info":
parent nlev block ordered model basic dummy
block      1    3  TRUE   FALSE  TRUE FALSE FALSE
A          2    3 FALSE   FALSE  TRUE FALSE FALSE
B          3    3 FALSE   FALSE  TRUE FALSE FALSE
C          4    3 FALSE   FALSE  TRUE FALSE FALSE

Slot "levels":
$block [1] 1 2 3$A
[1] 1 2 3

$B [1] 1 2 3$C
[1] 1 2 3

Slot "model":
[[1]]
[[1]]$Model ~block + (A + B + C)^2 [[1]]$Estimate
~A + B + C

Slot "designkey":
[[1]]
An object of class keymatrix

********** Prime  3  design **********

block A B C
*U*     1 0 1 0
*U*     0 1 1 0
*U*     0 0 0 1

Slot "nunits":
[1] 27

Slot "recursive":
[1] FALSE


(*) That package seems to be still under development and somewhat unfinished, and so is not the easiest package to use. Do not misunderstand, the package is a very ambitious project, and bound to be very useful. But at this moment it is not very userfriendly!