Orthogonal Arrays with R. Example of Orthogonal Arrays and… | by George Pipis

Whenever we are able to control the way the data is serving we should take advantage. For example in a poll by applying sampling techniques or in medical statistics by splitting the participants into groups and treatments and so on.

We will give an example of the Orthogonal Arrays which is a part of the family of Experimental Designs.

Let’s assume that Joe sometimes suffers from stomach ache during the night. His gastroenterologist suspects that his diet is responsible for these occasional symptoms. Let’s also assume that Joe’s diet includes:

So all the possible combinations are 4 x 2 x 4 x 4 x 2 x 2 = 512. The Doctor would like to detect which food(s) may cause him this discomfort and he is planning to apply the Orthogonal Arrays. Assuming that there is no interaction in the meals, he asks Joe to follow the following diet.

Question: Which are all the possible Orthogonal Arrays from this case?

Answer: Notice that we have 3 factors of 4 levels and 3 factors of 2 levels. Using the library DoE.base we can get the list of them.

library(DoE.base)## the orthogonal arrays with 3 4-level
## factors and 3 2-level show.oas(factors = list(nlevels=c(4,2),number=c(3,3)))

And we get:

5  resolution IV or more  arrays found
name nruns lineage
10   L64.2.8.4.3    64        
12   L64.2.6.4.4    64        
23 L128.2.20.4.3   128        
26 L192.2.36.4.3   192        
29 L256.2.52.4.3   256        
990  orthogonal  arrays found, 
the first  10  are listed
name nruns                           lineage
17      L16.2.6.4.3    16                   4~5;:(4~1!2~3;)
18      L16.2.3.4.4    16                   4~5;:(4~1!2~3;)
53     L32.2.22.4.3    32 4~8;8~1;:(8~1!2~4;4~1;)(4~1!2~3;)
55     L32.2.19.4.4    32 4~8;8~1;:(8~1!2~4;4~1;)(4~1!2~3;)
57     L32.2.16.4.5    32 4~8;8~1;:(8~1!2~4;4~1;)(4~1!2~3;)
59 L32.2.15.4.3.8.1    32               4~8;8~1;:(4~1!2~3;)
60     L32.2.13.4.6    32 4~8;8~1;:(8~1!2~4;4~1;)(4~1!2~3;)
61 L32.2.12.4.4.8.1    32               4~8;8~1;:(4~1!2~3;)
62     L32.2.10.4.7    32 4~8;8~1;:(8~1!2~4;4~1;)(4~1!2~3;)
63  L32.2.9.4.5.8.1    32               4~8;8~1;:(4~1!2~3;)

From the R output, we can see that 8 runs is the minimum number of runs that we can get from this experiment. The ID code of this experiment is L16.2.6.4.3 which tells you that you can also use 6 2-level factors and 3 4-level factors.

Question: What is the recommended diet for Joe?

Answer: The doctor could ask Joe to follow the diet below for the next 16 days. Notice that 16 was the number of minimum runs that we got from that particular experimental design.

OA<-oa.design(nruns=16, factor.names=list(Breakfast=c("Sandwich","Pancakes","Omelette", "Yogurt+Honey+Nuts"), Beverage=c("Coffee","Orange Juice"),Lunch=c("Pork","Fish", "Chicken", "Salad"), Dinner=c("Pasta","Rice", "Milk+Cereals", "Pizza"),Dessert=c("Ice-Cream","Nothing"), Drink=c("Tea", "Wine")))OA

And we get:

Breakfast     Beverage   Lunch         Dinner   Dessert Drink
1  Yogurt+Honey+Nuts       Coffee    Fish Milk+Cereals Ice-Cream   Tea
2  Yogurt+Honey+Nuts Orange Juice   Salad        Pizza   Nothing   Tea
3           Omelette Orange Juice Chicken Milk+Cereals Ice-Cream  Wine
4           Pancakes       Coffee    Fish         Rice   Nothing  Wine
5           Omelette Orange Juice    Fish        Pasta   Nothing   Tea
6           Pancakes       Coffee Chicken        Pizza Ice-Cream   Tea
7  Yogurt+Honey+Nuts Orange Juice    Pork         Rice Ice-Cream  Wine
8           Pancakes Orange Juice   Salad        Pasta Ice-Cream  Wine
9           Sandwich       Coffee    Pork        Pasta Ice-Cream   Tea
10          Sandwich       Coffee   Salad Milk+Cereals   Nothing  Wine
11 Yogurt+Honey+Nuts       Coffee Chicken        Pasta   Nothing  Wine
12          Sandwich Orange Juice    Fish        Pizza Ice-Cream  Wine
13          Omelette       Coffee   Salad         Rice Ice-Cream   Tea
14          Omelette       Coffee    Pork        Pizza   Nothing  Wine
15          Sandwich Orange Juice Chicken         Rice   Nothing   Tea
16          Pancakes Orange Juice    Pork Milk+Cereals   Nothing   Tea

Every row in the table above represents one day.

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aggregate(Lunch~Breakfast+Dessert, OA, length)

Breakfast   Dessert Lunch
1          Sandwich Ice-Cream     2
2          Pancakes Ice-Cream     2
3          Omelette Ice-Cream     2
4 Yogurt+Honey+Nuts Ice-Cream     2
5          Sandwich   Nothing     2
6          Pancakes   Nothing     2
7          Omelette   Nothing     2
8 Yogurt+Honey+Nuts   Nothing     2