Assumptions for 2-way Anova - nonparametric equivalent?

[Staffan Lindberg]

Dear list!

 

I have as a column variable 2 treatment groups and as row variable 3 age groups.  Unfortunately my dependent variable is heavily skewed (reverse J-distribution). As far as I know there is no nonparametric equivalent to 2-way Anova. I have checked the distributions and dispersions in all 6 cells and they all show the same reverse J-distribution. The standard deviations also appears to be approximately the same.

 

Would I be terribly wrong if I applied a 2-way Anova  to this table? Must I run 2 non-parametric tests instead? I know it is a possibility to transform the dependent variable so that it will be approximately normal but this seems to involve a lot of trial and error.  Or is it known what transformation would be the correct one given an extreme reverse J-shaped distribution?

 

Maybe I should explain that by “reverse J-distribution” I mean that the hook on the J is not pointing to the left but to the right.

 

Thankful for any input on this

 

best

 

Staffan Lindberg

Sweden

[Marta Garcia-Granero]

Hi Staffan


Staffan Lindberg wrote:

>
>
> I have as a column variable 2 treatment groups and as row variable 3
> age groups.  Unfortunately my dependent variable is heavily skewed
> (reverse J-distribution). As far as I know there is no nonparametric
> equivalent to 2-way Anova. I have checked the distributions and
> dispersions in all 6 cells and they all show the same reverse
> J-distribution. The standard deviations also appears to be
> approximately the same.
>

Rank the data and run a two way ANOVA with them. According to Conover,
the p-values are asymptotically valid. Right now I'm leaving, but if you
can wait a few hours, I can send you some code (I have to find it first)
I wrote based on a paper by Shirley, that computes more accurate
p-values using a chi-square statistic that is like a factorial
Kruskal-Wallis test.

Profile (interaction) plots should be based on medians instead of means.

HTH,
Marta GG

--
For miscellaneous SPSS related statistical stuff, visit:
http://gjyp.nl/marta/

=====================
To manage your subscription to SPSSX-L, send a message to
[hidden email] (not to SPSSX-L), with no body text except the
command. To leave the list, send the command
SIGNOFF SPSSX-L
For a list of commands to manage subscriptions, send the command
INFO REFCARD

[Bruce Weaver]

Staffan Lindberg wrote

Dear list!



I have as a column variable 2 treatment groups and as row variable 3 age
groups.  Unfortunately my dependent variable is heavily skewed (reverse
J-distribution). As far as I know there is no nonparametric equivalent to
2-way Anova. I have checked the distributions and dispersions in all 6 cells
and they all show the same reverse J-distribution. The standard deviations
also appears to be approximately the same.



Would I be terribly wrong if I applied a 2-way Anova  to this table? Must I
run 2 non-parametric tests instead? I know it is a possibility to transform
the dependent variable so that it will be approximately normal but this
seems to involve a lot of trial and error.  Or is it known what
transformation would be the correct one given an extreme reverse J-shaped
distribution?



Maybe I should explain that by "reverse J-distribution" I mean that the hook
on the J is not pointing to the left but to the right.



Thankful for any input on this



best



Staffan Lindberg

Sweden

Here is some advice from Dave Howell's "Statistical Methods for Psychology" (1997, 4th Ed., p.  321):

"In general, if the populations can be assumed to be symmetric, or at least similar in shape (e.g., all negatively skewed), and if the largest variance is no more than four times the smallest, the analysis of variance is most likely to be valid.  It is important to note, however, that heterogeneity of variance and unequal sample sizes do not mix.  If you have reason to anticipate unequal variances, make every effort to keep your sample sizes as equal as possible."

Are the group/cell sizes equal, or nearly so?  How large are they?

Here's another interesting suggestion from W.J. Conover's "Practical Nonparametric Statistics" (1999, 3rd Ed., p. 419):

"The recommended procedure in experimental designs for which no nonparametric test exists is to use the usual analysis of variance on the data and then to use the same procedure on the rank transformed data.  If the two procedures give nearly identical results the assumptions underlying the usual analysis of variance are likely to be reasonable and the regular parametric analysis valid.  When the two procedures give substantially different results, the experimenter may want to take a closer look at the data and to look especially for outliers...or very nonsymmetric distributions."

[Art Kendall]

In addition to what Bruce recommended, take a look at the residuals.  Are they wildly non-normal?

Art

Staffan Lindberg wrote:

Dear list!

 

I have as a column variable 2 treatment groups and as row variable 3 age groups.  Unfortunately my dependent variable is heavily skewed (reverse J-distribution). As far as I know there is no nonparametric equivalent to 2-way Anova. I have checked the distributions and dispersions in all 6 cells and they all show the same reverse J-distribution. The standard deviations also appears to be approximately the same.

 

Would I be terribly wrong if I applied a 2-way Anova  to this table? Must I run 2 non-parametric tests instead? I know it is a possibility to transform the dependent variable so that it will be approximately normal but this seems to involve a lot of trial and error.  Or is it known what transformation would be the correct one given an extreme reverse J-shaped distribution?

 

Maybe I should explain that by “reverse J-distribution” I mean that the hook on the J is not pointing to the left but to the right.

 

Thankful for any input on this

 

best

 

Staffan Lindberg

Sweden

===================== To manage your subscription to SPSSX-L, send a message to [hidden email] (not to SPSSX-L), with no body text except the command. To leave the list, send the command SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD

[Marta Garcia-Granero]

I'm back (after a couple of beers with my husband to celebrate Friday,
and the end of a veeeery hectic week, Bologne reform is killing me).

Here's the code. Replace "depvar" by the name of the quantitative
variable, and "factor1" and "factor2" by the names of the grouping
variables. The MATRIX part doesn't need any changes at all.

Reference I used to develop the code: Shirley EAC (1987).  Application
of ranking methods to multiple comparison procedures and factorial
experiments. Applied Statistics 36:205-213.

I used to have a printed copy, but someone, ahem, "forgot" to return it,
I can't send you a copy, sorry.

HTH,
Marta GG

----------------------------

* Syntax. Warning: 'C:\Temp' folder must exist (replace by other path if
necessary) *.

* Replace names by actual ones *.
RANK
  VARIABLES=depvar  (A) /RANK INTO Ranked.
UNIANOVA
  Ranked  BY factor1 factor2
  /OUTFILE=EFFECT ('C:\TEMP\report_.sav')
  /DESIGN = factor1 factor2 factor1*factor2 .

* Leave this part untouched *.
MATRIX.
PRINT /TITLE='KRUSKAL-WALLIS FOR FACTORIAL DESIGNS'.
GET depname /VAR=depvar_ /FILE='c:\temp\report_.sav'.
GET tnames  /VAR=source_ /FILE='c:\temp\report_.sav'.
GET ss      /VAR=ss      /FILE='c:\temp\report_.sav'.
GET df      /VAR=df      /FILE='c:\temp\report_.sav'.
PRINT depname(1)
 /FORMAT='A8'
 /TITLE='Dependent variable:'.
COMPUTE k=NROW(ss).
COMPUTE varri=ss(k)/df(k).
COMPUTE chi2=ss/varri.
COMPUTE chi2sig=MAKE(k-3,1,0).
LOOP i=1 TO k-3.
- COMPUTE chi2sig(i)=1-CHICDF(chi2(i),df(i)).
END LOOP.
PRINT {ss(1:(k-3)),df(1:(k-3)),chi2(1:(k-3)),chi2sig(1:(k-3))}
 /FORMAT='F8.3'
 /RNAMES=tnames
 /CLABELS='SS','DF','Chi^2','Sig'
 /TITLE='Chi-square tests & asymptotic significance'.
PRINT varri
 /FORMAT='F8.3'
 /RLABEL='VAR(Ri)='
 /TITLE='Ranks Variance'.
END MATRIX.

DELETE VARIABLES Ranked.
> Thank you ever so much Marta!
>
> It never occurred to me to rank the data first. Of course I will wait for
> your code. I'm in no hurry. This is a problem I frequently encounter and a
> general solutions to this would be very much appreciated. I can't believe
> I'm the only person getting a headache over this so your solution would be
> important to other list members as well I imagine.

--
For miscellaneous SPSS related statistical stuff, visit:
http://gjyp.nl/marta/

=====================
To manage your subscription to SPSSX-L, send a message to
[hidden email] (not to SPSSX-L), with no body text except the
command. To leave the list, send the command
SIGNOFF SPSSX-L
For a list of commands to manage subscriptions, send the command
INFO REFCARD