To Allstatters
Here is a summary of the responses I received. In general, I took on board the following points.
1. This discussion only arises because my design ended up being unbalanced. Had it been balanced, all effects would be orthogonal to each other and standard ANOVA analysis and interpretation would have been possible.
2. Type II is the most appropriate SS for my problem and possibly for most problems.
3. Type I SS is only appropriate if there is a natural order to the effects being fitted. I decided there wasn't such an order in my case, though some argued otherwise. See the bottom of this note for a description of the effects.
4. Type III is inappropriate for most types of problems. They do not appear to handle models involving interactions very well (as can be seen in my case in the table at the bottom of this note).
5. However, if there are no interactions in the model then Type II and Type III give the same results.
6. The comments from John Nelder and Peter Lane who cowrote a paper on this subject are reproduced in this note. In essence they say Type III is discredited and should not used for linear models
7. Type IV SS is apparently recommended by SPSS (though I have not tried this). However, Nelder & Lane criticisms of Type III also apply to Type IV.
The comments of Nelder & Lane were
Nelder - "Two points:
(1) I hope you are using generalised models for the analysis and not transforming the data.
(2) I have shown that type-III SS have no part to play in making inferences from GLMs (which includes normal models) See my paper in Statistics and Computing called Back to Basics ...
I would ask you to look in detail at my paper in Computing and Statistics , which shows the irrelevance of Type III and IV sums of squares to the making of inferences from linear models (and GLMs generally). It is unfortunate that SAS has propagated these false methods, which are widely used. What you need is to fit sequential models (The so-called type II SS) in different orders and go on from there. (A better solution is to use a decent package like Genstat which does these things properly)."
Peter Lane - "It's good to see another example of Type III SS producing silly results. I regret I don't have time to try to understand and explain the difference between Type I and III for your dataset specifically, as I have far too much work waiting, but I felt moved to reply to encourage you to use neither Type I or Type III SS. I think Type II is what you want, even though some software does not always allow you to get it very easily. The Type II SS for Age is the SS associated with Age, after eliminating the SS associated with Product, and ignoring the SS associated with the interaction. This is different from the Type I SS for Age if you happen to fit the factors in the order Age, Product rather than Product, Age. If you have fitted in the order Product, Age, then Type I = Type II, and you are done. The Type II is different from the Type III because the Type III SS also eliminates the effect of the interaction -- which in my view, and that of many others, is a totally !
ridiculous thing to do; see Nelder & Lane, 1995, "The computer analysis of factorial experiments: in memoriam Frank Yates", American Statistician 49, 382-385. If you fit the model with no interaction term, then you also have Type III = Type II, so you can get it that way as well."
For reasons of space, I have only given excerpts of other comments from some respondents as they mainly repeated each other. If I have not included your comments, then can I please say thank you for them as I did find every comment helpful. Please let me know if you think I have missed a pertinent point in my summary above.
Allan White - "If you have an unbalanced experimental design, various of the effects are not orthogonal, i.e. they can be thought of as overlapping. In these circumstances, for at least some of the effects, there is more than one way of attributing the sum of squares for an effect, depending on whether the SS is adjusted for any other effects or not. The Type I SS does not adjust for any other effect. By contrast, the Type III SS adjusts for the presence of every other term in the model."
Ross Darnell - "The F-value (from a Type III ANOVA) for Age depends on what other terms are in the model. A type III SS is sometimes described as a "adjusting for all other terms" effect which in the case of a main effect in the presence of an interaction is impossible to interpret"
If I have not included your comments, then can I please say thank you for them as I did find every comment helpful. Please let me know if you think I have missed a pertinent point in my summary above.
An overview of the problem and results obtained is appended at the bottom of this note.
Regards
Nigel Marriott
Senior Statistician - R&D
Masterfoods Europe
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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.
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