To everyone
Thank you to everyone who responded to my original query on Type 1 & 3 SS in ANOVA. In trying to describe the background, I think I might have confused some people. Therefore I would like to clarify a few things.
The data is supposed to be liking scores (on a 1 to 7 scale) for 9 variants of a confectionery from 200 consumers per product. Unfortunately no data was collected for 1 product as below due to a misunderstanding i.e. the N/A cell is empty making this an incomplete design.
@8wks @16wks @24wks
Std 5.38 5.39 5.20
AltI 5.40 5.43 5.15
AltP 5.50 5.35 N/A
The Mean Square Error is 1.5 on 1649 df.
The 2 factors shown are Product Type (Std, AltP, AltI) and product Age (8, 16, 24 weeks). Ideally there should have been a fourth product type in the design but this was overlooked by the original project team.
There is in fact a third factor which is type of consumer (using a very simple clustering method as follows)
0 : They have not bought the product and it is not a favourite.
1 : They have bought the product but it is not a favourite.
2 : They have bought the product and it is a favourite.
Originally I analysed the data using XLSTAT (an excel add-in) which was only capable of Type 1 & 3 SS. Following feedback from some of you, I reanalysed using STATISTICA which is also capable of Type 2 SS. The F statistics for each type of SS and type of model are given below.
Type 3 SS Type 2 SS
Effect df F3 F2 F1 F3 F2 F1
Product 2 0.97 1.11 0.12 0.11 0.11 0.12
Age 2 0.77 0.93 4.16 4.12 4.13 4.16
Consumr 2 19.82 19.25 22.35 22.15 22.22 22.35
PxA 3 0.24 0.57 0.57 0.57
PxC 4 1.09 1.19 1.19 1.19
AxC 4 0.56 0.53 0.53 0.53
PxAxC 6 0.16 0.16
MS Error is 1.5 on 1649df
Where
F3 - Interactions up to degree 3 i.e. full model
F2 - Interactions up to degree 2
F1 - No interactions i.e. main effects model.
These values show that with no significant interactions are present the main effects model is the correct one for drawing conclusions. In this case both Type 2 and 3 SS will give the same results. But I had based my original query on the Type 3 SS full model whilst ignoring the interactions and the Consumer term. I had not appreciated the fact that the significance of the Age factor would change so dramatically once the interaction terms were removed. However in the Type 2 (and Type 1 models) removal of the interaction terms does not change the significance of the main effects much.
I am left with the impression that Type 2 SS is the most appropriate choice. Type 1's are dependent on the order in which the 3 factors are presented and in this study there is no natural order to the factors. Type 3's are not order dependent but there is a big risk on being misled as to the significance of the main effects when interactions are included in the models as has happened here. Type 2's seem to get around both of these issues.
Three other comments I would like to make.
First, I tried each one of the 6 possible orders of the 3 main effects when doing the Type 1 SS models and discovered that the SS for each effect in Type 3 SS appeared to be the minimum SS obtained for that effect from the 6 potential Type 1 SS models. Is this how Type 3 SS is supposed to work?
Second, someone has suggested that Type 4 SS would be suitable. What is the difference between this and the other types.
Lastly, in general will an incomplete design as happened here exacerbate the difference in results obtained from Type 1, 2, 3 or 4 SS?
Overall I have learnt a lot from your feedback so far but I would greatly appreciate any more feedback on what I have written here.
Regards
Nigel Marriott
Senior Statistician
R&D, Masterfoods Europe
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