Interaction in factorial designs when dependent variable is not normal

 

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*(Q) How can I study interaction in factorial designs if the dependent
variable is not normal? (Non parametric 2-way ANOVA)

*(A) There are several ways:

*References:
*1) Shirley EAC. "Applications of Ranking Methods to Multiple Comparison
   Procedures and Factorial Experiments". Appl Statist
	1987;36(2):205-213.
*2) Conover WJ and Iman RL. "Rank Transformations as a Bridge between
   Parametric and Nonparametric Statistics". The American Statistician
   1981;35:124-129.
* Code written by Marta Garcia-Granero, sent to rlevesque by email on 2002/02/05.

******************** EXAMPLE ****************************.
DATA LIST LIST /altit(f2.0) splenec(f2.0) fibrinog(f4.0).
BEGIN DATA
1	1	528
1	1	444
1	1	338
1	1	342
1	1	338
1	2	434
1	2	331
1	2	312
1	2	575
1	2	472
2	1	294
2	1	254
2	1	352
2	1	241
2	1	291
2	2	272
2	2	275
2	2	350
2	2	350
2	2	466
END DATA.
VARIABLE LABELS altit "Altitude".
VALUE LABELS altit
 1 "Sea level"
 2 "15.000 ft." .
VARIABLE LABELS splenec "Splenectomy".
VALUE LABELS splenec
 1 "No"
 2 "Yes" .

* SOLUTION #1: FOR BALANCED AND SQUARE DESIGNS ONLY
* (AS THIS ONE IS: n=5; 2x2 design)
* (in this situation, Chi square values are additive)
*********************************************************.

COMPUTE group = splenec+2*(altit=2) .
* (for a 3x3 design: COMPUTE group = splenec+3*(altit=2)+6*(altit=3)
* and so on...).
EXECUTE .
VARIABLE LABELS group "Combined group".
VALUE LABELS group
 1 "Sea lev/No"
 2 "Sea lev/Yes"
 3 "15.000 ft/No"
 4 "15.000 ft/Yes" .

NPAR TESTS
  /K-W=fibrinog BY group(1 4)
  /K-W=fibrinog BY altit(1 2)
  /K-W=fibrinog BY splenec(1 2).

* Now, read the 3 Chi-square values + df from output and compute
* Interaction Chi-square (and df) as difference: group - altit- splenec
* The values have to be copied to the two first COMPUTE statements
* (I still haven't found a solution for this).

DO IF $casenum=1.
COMPUTE chi=5.774-4.487-0.824.
COMPUTE df=3-1-1.
COMPUTE sig = 1-CDF.CHISQ(chi,df) .
END IF.
EXECUTE.
FORMAT chi (F8.3) df (F2.0) sig (F8.4).
REPORT FORMAT=LIST AUTOMATIC ALIGN(CENTER)
  /VARIABLES=chi df sig
  /MISSING=LIST
  /TITLE "Kruskal-Wallis test for interaction".


* For unbalanced and/or non-square designs (2x3, 3x4...), the above
* solution doesn't work (you can end up with a negative Chi-square!).

* SOLUTION #2 (GENERAL): 2-WAY ANOVA (WITH TYPE 3 SS) WITH THE RANKS
*****************************************************************************.
DATA LIST LIST /altit(f2.0) splenec(f2.0) fibrinog(f4.0).
BEGIN DATA
1	1	528
1	1	444
1	1	338
1	1	342
1	1	338
1	2	434
1	2	331
1	2	312
1	2	575
1	2	472
2	1	294
2	1	254
2	1	352
2	1	241
2	1	291
2	2	272
2	2	275
2	2	350
2	2	350
2	2	466
END DATA.
VARIABLE LABELS altit "Altitude".
VALUE LABELS altit
 1 "Sea level"
 2 "15.000 ft." .
VARIABLE LABELS splenec "Splenectomy".
VALUE LABELS splenec
 1 "No"
 2 "Yes" .


* First, rank the dependent variable (as Kruskal-Wallis test does).
RANK VARIABLES=fibrinog (A) /RANK INTO yranks /PRINT=NO /TIES=MEAN .

* Now, a factorial ANOVA with the ranks.
UNIANOVA  yranks BY altit splenec.

* According to Conover and Iman, the F values and significance are
* approximately valid, but, if you want to have more appropriate
* p-values you must calculate Kruskal-Wallis Chi squares with their
* significance (according to Shirley).

* The following MACRO does all the work (from ranking to Chi square
* significance, + two profile plots with medians, to interpret the
* interaction if significant).

********************
* MACRO DEFINITION *
********************.

DEFINE rank2w (facta=!CHAREND ('/') /factb=!CHAREND ('/')
/depvar=!CMDEND ).
GRAPH /LINE(MULTIPLE)MED(!depvar) BY !facta BY !factb.
GRAPH /LINE(MULTIPLE)MED(!depvar) BY !factb BY !facta.
RANK VARIABLES=!depvar (A) /RANK INTO yranks /PRINT=NO /TIES=MEAN .
UNIANOVA  yranks BY !facta !factb /OUTFILE=EFFECT ('report_.sav').
GET FILE='report_.sav' /KEEP= source_ ss df.
EXECUTE.
SELECT IF((source_='CorModel') or (source_='CorTotal') or (source_='S2')
or (source_='S3') or (source_='S4')).
EXECUTE .
COMPUTE pos=$casenum.
execute.
SORT CASES BY pos(D).
compute varri=ss/df.
if $casenum>1 varri=lag(varri,1).
execute.
DO IF $casenum>1.
COMPUTE chi=ss/varri.
COMPUTE sig = 1-CDF.CHISQ(chi,df) .
END IF.
execute.
VARIABLE LABELS sig "Sig.".
VARIABLE LABELS chi "Chi square".
FORMAT ss (F8.3) df (F4.0) chi (F8.3) sig (F9.4).
SORT CASES by pos(A).
REPORT FORMAT=LIST AUTOMATIC ALIGN(CENTER)
  /VARIABLES=source_(label) ss df chi sig
  /TITLE "2-way Kruskal-Wallis".
!ENDDEFINE.

* MACRO call *.
rank2w facta=altit /factb=splenec /depvar=fibrinog.
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