R help - Jun 2010 - Nested ANOVA with covariate using Type III sums of squares

Hello,

I have been trying to get an ANOVA table for a linear model containing a
single nested factor, two fixed factors and a covariate:

 carbonmean<-lm(C.Mean~ Mean.richness + Diversity + Zoop + Diversity/Phyto +
Zoop*Diversity/Phyto)



where, *Mean.richness* is a covariate*, Zoop* is a categorical variable (the
species), *Diversity* is a categorical variable (Low or High), and
*Phyto*(community composition) is also categorical but is nested
within the level
of *Diversity*. Quinn & Keough's statistics text recommends using Type
III
SS for a nested ANOVA with a covariate.

I get the following output using the Type I SS ANOVA:



Analysis of Variance Table
Response: C.Mean
                                                Df        Sum Sq           Mean
Sq          F value            Pr(>F)
Mean.richness                        1          56385326        56385326
23.5855           3.239e-05 ***
Diversity                                 1          14476593        14476593
      6.0554             0.019634 *
Zoop                                        1          13002135
13002135
      5.4387             0.026365 *
Diversity:Phyto                      6          126089387      21014898
8.7904             1.257e-05 ***
Diversity:Zoop                       1          263036
263036
0.1100              0.742347
Diversity:Zoop:Phyto             6          61710145        10285024
     4.3021
           0.002879 **
Residuals                                31        74110911
2390675

I have tried using both the drop1() command and the Anova() command in the
car package.

When I use the Anova command I get the following error message:
>Anova(carbonmean,type="III")
“Error in linear.hypothesis.lm(mod, hyp.matrix, summary.model = sumry,: One
or more terms aliased in model.”



I am not sure why this is aliased. There are no missing cells, and the cells
are balanced (aside from for the covariate). Each Phyto by Zoop cross is
replicated 3 times, and there are four Phyto levels within each level of
Diversity. When I remove the nested factor (Phyto), I am able to get the
Type III SS output.



Then when I use drop1(carbonmean,.~.,Test=”F”) I get the following output:
> drop1(carbonmean,.~.,Test="F")
Single term deletions



Model:

C.Mean ~ Mean.richness + Diversity + Zoop + Diversity/Phyto + Zoop *
Diversity/Phyto

                                                Df        Sum of Sq
RSS                 AIC

<none>                                                74110911       718

Mean.richness                        1          49790403        123901314
741

Diversity                                 0         0
74110911        718

Zoop                                        0         0
74110911        718

Diversity:Phyto                      6          118553466      192664376
752

Diversity:Zoop                       0          -1.49e-08        74110911
718

Diversity:Zoop:Phyto             6          61710145        135821055
735



There are zero degrees of freedom for Diversity, Zoop and their interaction,
and zero sums of sq for Diversity and Zoop. This cannot be correct, however
when I do the model simplification by dropping terms from the models
manually and comparing them using anova(), I get virtually the same results.



I would appreciate any suggestions for things to try or pointers as to what
I may be doing incorrectly.



Thank you.

Anita Narwani.

	[[alternative HTML version deleted]]

That's what one would expect with type III sum of squares. You have Phyto
twice in your model, but only as a nested factor. To compare the full model
with a model without diversity of zoop, you have either the combination
diversity/phyto, zoop/phyto or phyto twice in the formula. That's aliasing.

Depending on how you stand on type III sum of squares, you could call that a
"bug". Personally, I'd just not use them.

https://stat.ethz.ch/pipermail/r-help/2001-October/015984.html

Cheers
Joris


On Thu, Jun 3, 2010 at 2:13 AM, Anita Narwani
<anitanarwani@gmail.com>wrote:
> Hello,
>
> I have been trying to get an ANOVA table for a linear model containing a
> single nested factor, two fixed factors and a covariate:
>
>  carbonmean<-lm(C.Mean~ Mean.richness + Diversity + Zoop +
Diversity/Phyto
> +
> Zoop*Diversity/Phyto)
>
>
>
> where, *Mean.richness* is a covariate*, Zoop* is a categorical variable
> (the
> species), *Diversity* is a categorical variable (Low or High), and
> *Phyto*(community composition) is also categorical but is nested
> within the level
> of *Diversity*. Quinn & Keough's statistics text recommends using
Type III
> SS for a nested ANOVA with a covariate.
>
> I get the following output using the Type I SS ANOVA:
>
>
>
> Analysis of Variance Table
> Response: C.Mean
>                                                Df        Sum Sq
> Mean
> Sq          F value            Pr(>F)
> Mean.richness                        1          56385326        56385326
> 23.5855           3.239e-05 ***
> Diversity                                 1          14476593
>  14476593
>      6.0554             0.019634 *
> Zoop                                        1          13002135
> 13002135
>      5.4387             0.026365 *
> Diversity:Phyto                      6          126089387      21014898
> 8.7904             1.257e-05 ***
> Diversity:Zoop                       1          263036
> 263036
> 0.1100              0.742347
> Diversity:Zoop:Phyto             6          61710145        10285024
>     4.3021
>           0.002879 **
> Residuals                                31        74110911
> 2390675
>
> I have tried using both the drop1() command and the Anova() command in the
> car package.
>
> When I use the Anova command I get the following error message:
>
> >Anova(carbonmean,type="III")
>
> “Error in linear.hypothesis.lm(mod, hyp.matrix, summary.model = sumry,: One
> or more terms aliased in model.”
>
>
>
> I am not sure why this is aliased. There are no missing cells, and the
> cells
> are balanced (aside from for the covariate). Each Phyto by Zoop cross is
> replicated 3 times, and there are four Phyto levels within each level of
> Diversity. When I remove the nested factor (Phyto), I am able to get the
> Type III SS output.
>
>
>
> Then when I use drop1(carbonmean,.~.,Test=”F”) I get the following output:
>
> > drop1(carbonmean,.~.,Test="F")
>
> Single term deletions
>
>
>
> Model:
>
> C.Mean ~ Mean.richness + Diversity + Zoop + Diversity/Phyto + Zoop *
> Diversity/Phyto
>
>                                                Df        Sum of Sq
> RSS                 AIC
>
> <none>                                                74110911      
718
>
> Mean.richness                        1          49790403        123901314
> 741
>
> Diversity                                 0         0
> 74110911        718
>
> Zoop                                        0         0
> 74110911        718
>
> Diversity:Phyto                      6          118553466      192664376
> 752
>
> Diversity:Zoop                       0          -1.49e-08        74110911
> 718
>
> Diversity:Zoop:Phyto             6          61710145        135821055
> 735
>
>
>
> There are zero degrees of freedom for Diversity, Zoop and their
> interaction,
> and zero sums of sq for Diversity and Zoop. This cannot be correct, however
> when I do the model simplification by dropping terms from the models
> manually and comparing them using anova(), I get virtually the same
> results.
>
>
>
> I would appreciate any suggestions for things to try or pointers as to what
> I may be doing incorrectly.
>
>
>
> Thank you.
>
> Anita Narwani.
>
>        [[alternative HTML version deleted]]
>
>
> ______________________________________________
> R-help@r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
> http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>
>

-- 
Joris Meys
Statistical Consultant

Ghent University
Faculty of Bioscience Engineering
Department of Applied mathematics, biometrics and process control

Coupure Links 653
B-9000 Gent

tel : +32 9 264 59 87
Joris.Meys@Ugent.be
-------------------------------
Disclaimer : http://helpdesk.ugent.be/e-maildisclaimer.php

	[[alternative HTML version deleted]]

Thanks for your response Joris.

I was aware of the potential for aliasing, although I thought that this was
only a problem when you have missing cell means. It was interesting to read
the vehement argument regarding the Type III sums of squares, and although I
knew that there were different positions on the topic, I had no idea how
divisive it was. Nevertheless, Type III SS are generally recommended by
statistical texts in ecology for my type of experimental design.
Interestingly, despite the aliasing, SPSS has no problems calculating Type
III SS for this data set. This is simply because I am entering a co-variate,
which causes non-orthogonality. I would be happier using R and the default
Type I SS, which are the same as the Type III SS anyway when I omit the
co-variate of Mean.richness, except that these results are very sensitive to
the order in which I add the variables into the model when I do enter the
co-variate. I understand that the order is very important relates back to
the scientific hypothesis, but I am equally interested in the main effects
of Zoop, Diversity, and the nested effect of Phyto, so entering either of
these variables before the other does not make sense from an ecological
perspective, and because the results do change, the order cannot be ignored
from a statistical perspective.
Finally, I have tried using the Type II SS and received similar warnings.

Do you have a recommendations?
Anita.

	[[alternative HTML version deleted]]

I see where my confusion comes from. I counted 4 levels of Phyto, but
you have 8, being 4 in every level of Diversity. There's your
aliasing.
> table(Diversity,Phyto)
???????? Phyto
Diversity M1 M2 M3 M4 P1 P2 P3 P4
??????? H? 0? 0? 0? 0? 6? 6? 6? 6
??????? L? 6? 6? 6? 6? 0? 0? 0? 0

There's no need to code them differently for every level of Diversity.
If you don't, all is fine :
> Phyto <- gsub("M","P",as.character(Phyto))
> Phyto <- as.factor(Phyto)
>
> test <- lm(C.Mean~? Mean.richness + Diversity + Zoop + Diversity/Phyto +
+ Zoop*Diversity/Phyto)
>
> Anova(test,type="III")
Anova Table (Type III tests)

Response: C.Mean
??????????????????????? Sum Sq Df F value??? Pr(>F)
(Intercept)?????????? 23935609? 1 10.0121 0.0034729 **
Mean.richness???????? 49790385? 1 20.8269 7.471e-05 ***
Diversity???????????? 35807205? 1 14.9779 0.0005234 ***
Zoop????????????????? 10794614? 1? 4.5153 0.0416688 *
Diversity:Phyto????? 118553464? 6? 8.2650 2.184e-05 ***
Diversity:Zoop????????? 261789? 1? 0.1095 0.7429356
Diversity:Zoop:Phyto? 61710162? 6? 4.3021 0.0028790 **
Residuals???????????? 74110938 31
---
Signif. codes:? 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ?
1
>
You can check with summary(test) that the model is fitted correctly.

On Fri, Jun 4, 2010 at 12:48 AM, Anita Narwani <anitanarwani at gmail.com>
wrote:
>
> You have everything right except that there are only 2 zooplankton species
(C & D, which stand for Ceriodaphnia and Daphnia).
>