To all those who are suggesting standard nonparametric tests:

Do you understand that Massil has repeated measures for ~multiple~ dependent variables? The Kruskal-Wallis test is for comparing multiple groups on one dependent variable (or in Massil's case, one of the repeated measurements of one DV). Massil has told us he has repeated measures, but it's not clear whether he also has 2 or more independent groups. If there are no independent groups, then KW is altogether out of the picture. But even if there are 2 or more independent groups, KW does not provide the multivariate test Massil is asking for.

Friedman's test is a rank-based alternative to one-factor repeated measures ANOVA. As Massil noted, it applies to ONE dependent variable; but he has more than one DV, and is asking if there is a nonparametric test that handles repeated measures data for MULTIPLE DVs (a multivariate version of the Friedman test, if you will).

HTH.

MANOVA allows you to determine significant differences between groups. For a non paremétrico analysis, you can use Kruskal-Wallis test and post-test as the Mann-Whitney's U (each group with another). There is a further advantage, the relationship between variables need not be linear, you can work with certain relations determined through a contingency table and chi square.

Massil, what are the repeated measures variables? Can you give us some basic descriptive stats please (including sample size)? I ask these questions, because I think that sometimes people use nonparametric tests when a parametric test is actually more robust, given the nature of the data. E.g., take a look at Fagerland's nice article on the Mann-Whitney U test.

https://www.researchgate.net/publication/24045186_The_Wilcoxon-Mann-Whitney_test_under_scrutiny?ev=prf_pub

HTH.

Article The Wilcoxon-Mann-Whitney test under scrutiny

You can use Friedman Test. The Friedman test is the non-parametric alternative to the one-way ANOVA with repeated measures. It is used to test for differences between groups when the dependent variable being measured is ordinal. It can also be used for continuous data that has violated the assumptions necessary to run the one-way ANOVA with repeated measures (e.g., data that has marked deviations from normality). While Kruskal-Wallis test is non-parametric test for independent groups and It is equivalent to the F test in the ANOVA analysis. This URL will be helpful for you.

https://statistics.laerd.com/spss-tutorials/friedman-test-using-spss-statistics.php

To all those who are suggesting standard nonparametric tests:

Do you understand that Massil has repeated measures for ~multiple~ dependent variables? The Kruskal-Wallis test is for comparing multiple groups on one dependent variable (or in Massil's case, one of the repeated measurements of one DV). Massil has told us he has repeated measures, but it's not clear whether he also has 2 or more independent groups. If there are no independent groups, then KW is altogether out of the picture. But even if there are 2 or more independent groups, KW does not provide the multivariate test Massil is asking for.

Friedman's test is a rank-based alternative to one-factor repeated measures ANOVA. As Massil noted, it applies to ONE dependent variable; but he has more than one DV, and is asking if there is a nonparametric test that handles repeated measures data for MULTIPLE DVs (a multivariate version of the Friedman test, if you will).

HTH.

@ Paulo - I was not recommending the Kruskal-Wallis test. IMO, none of the standard nonparametric tests (KW or Friedman) does what Massil is asking for--i.e., none of them are multivariate (meaning they cannot handle multiple DVs). It may be that some kind of multivariate generalized linear model would work, depending on the nature of the dependent variables. That's why I asked for more info about the DVs.

Dear all,

Thank you for your answers.

As Bruce highlighted, I have 5 DV ( = Manova) with answers at 6 times, more specifically 6 experimental conditions in a radomized trial (= repeated mesures). Each DV represents one item. I know it s a bit weird but I m working on a second-hand database and my colleague usually treat results item by item. In this case, Cronbach analysis suggest that we can not compute a global score..That's why I'm stuck with a repeated measures MANOVA or its non parametric equivalent.

My sample is 44 participants and I only have one groupe (within subjects).

Scheirer-Ray-Hare Test

Why not simply undertake a Friedman's on each item rather than trying to do them simultaneously? I have never really understood why MANOVA is so popular!

@ Béatrice - As far as I can tell from Google searches, the Scheirer-Ray-Hare test is an extension of the Kruskal-Wallis test (e.g., http://listserv.uga.edu/cgi-bin/wa?A2=ind0802B&L=sas-l&F=&S=&P=41584). But as Massil has said, he has only one group, so KW (or extensions thereof) will not work.

@ Adrian - I also wonder why MANOVA is so popular. I think that in part, it's because (some) people think that MANOVA can/should be used as a preliminary test prior to a bunch of ANOVAs, and that it provides protection against Type I error in the same way that the omnibus F-test in Fisher's LSD procedure provides protection for the pair-wise contrasts. But the classic article by Huberty & Morris decries that use of MANOVA. The article is available here:

http://wweb.uta.edu/management/Dr.Casper/Fall10/BSAD6314/Coursematerial/Huberty%20&%20Morris%201989%20-%20MANOVA%20vs.%20ANOVA.pdf

Cheers,

Bruce

@Bruce, thank you. I was thinking naively to a Manova because of the covariance between my five DV. But if it is "over-rated", I may simply go with Friedman's Test on each itemas suggested by Adrian.

I'm going to read the article by huberty & Morris carefully. Massil

Dear Bruce

I know that Friedman Test or Kruskal-Wallis test not a Multivariate analysis. But I meant that applying Friedman test as an alternative test to the one-way ANOVA with repeated measures, applying it for each DV separately, and I pointed out that test Kruskal-Wallis test is non-parametric test for independent groups and It is equivalent to the F test in the ANOVA (as a note) and I did not mean that tests are represented multivariate.

Scheirer-Ray-Hare Test is an extention of Friedman

Choosing and using statistics: A biologist's guide by Calvin Dy (2011)

Blackwell

Hello Béatrice. I have been looking at Calvin Dytham's example of the Scheirer-Ray-Hare test shown here:

http://books.google.ca/books?id=ZvulFAhLIHYC&pg=PA175&lpg=PA175&dq=choosing+and+using+statistics+a+biologist%27s+guide+Scheirer-Ray-Hare+Test&source=bl&ots=kyiEC_Ncfm&sig=HrElYcyIt10k4QR99ybQ4acP-s0&hl=en&sa=X&ei=IS-kU6PwOYjl8AHtmIGoAw&ved=0CB4Q6AEwAA#v=onepage&q&f=false

He starts by; carrying out a two-way (independent groups) ANOVA on a rank-transformed dependent variable. He then computes a MS_total (or MS-Corrected-Total in SPSS), and uses it as the denominator for computing 3 test statistics. For each effect, the test statistic = SS_Effect / MS_Total. The p-values are equal to 1 - CDF.CHISQ with df = the numerator df for that term in the ANOVA table. (SPSS syntax for his example is given below.)

I will say that the introductory paragraph to that section in Dytham's book is a bit unclear as to whether this is an extension of KW or Friedman; but the example makes it pretty clear that it is an extension of KW.

Please don't take this post as an endorsement of the Scheirer-Ray-Hare test. I'd never heard of it before reading this thread, and have not consulted any sources other than Dytham's book (and not all of the pages are visible in Google Books).

Cheers,

Bruce

* Scheirer-Ray-Hare Test example from

* "Choosing and Using Statistics: A Biologist's Guide",

* by Calvin Dytham. The data are from p. 165.

NEW FILE.

DATASET CLOSE all.

DATA LIST list / Male LongDay (2f1) FoodIntake (f5.2).

BEGIN DATA

0 1 78.1

0 1 75.5

0 1 76.3

0 1 81.2

1 1 69.5

1 1 72.1

1 1 73.2

1 1 71.1

0 0 82.4

0 0 80.9

0 0 83.0

0 0 88.2

1 0 72.3

1 0 73.3

1 0 70.0

1 0 72.9

END DATA.

VARIABLE LABELS

Male "Sex"

LongDay "Length of day"

FoodIntake "Total 7-day food intake (g)"

.

VALUE LABELS

Male 0 'Female' 1 'Male' /

LongDay 0 "Short (8hr)" 1 "Long (16h)"

.

* On p. 167, Dytham shows resuls of a 2-way ANOVA on these data.

UNIANOVA FoodIntake BY Male LongDay

/EMMEANS=TABLES(Male)

/EMMEANS=TABLES(LongDay)

/EMMEANS=TABLES(Male*LongDay)

/DESIGN=Male LongDay Male*LongDay.

* Instructions for carrying out the Scheirer-Ray-Hare Test in SPSS

* are given on pp. 175-176. In a nutshell, they say to carry out

* the same two-way ANOVA shown above, but on the ranks for Food Intake.

* Dytham instructs readers to SORT the cases, and then manually enter

* ranks. But of course, the ranks can be obtained directly with the

* RANK command.

RANK VARIABLES=FoodIntake (A)

/RANK

/PRINT=YES

/TIES=MEAN.

SORT CASES by RFoodInt.

FORMATS RFoodInt (f5.1).

LIST.

* Two-way ANOVA on the ranks.

UNIANOVA RFoodInt BY Male LongDay

/EMMEANS=TABLES(Male)

/EMMEANS=TABLES(LongDay)

/EMMEANS=TABLES(Male*LongDay)

/DESIGN=Male LongDay Male*LongDay.

* Dytham says to compute a MS for the Corrected Total row,

* and then to compute 3 test statistics as SS_effect / MSCT.

DATA LIST list / Effect(a15).

BEGIN DATA

"Day length"

"Sex"

"Interaction"

END DATA.

IF ($casenum EQ 1) TestValue = 25 / (340/15).

IF ($casenum EQ 2) TestValue = 256 / (340/15).

IF ($casenum EQ 3) TestValue = 4 / (340/15).

COMPUTE CumChiSq = cdf.chisq(TestValue,1).

COMPUTE p = 1 - CumChiSq.

FORMATS TestValue (f5.3) /CumChiSq (f8.6) / p (f8.5).

LIST.

* I get slightly different values than those shown by Dytham on p. 180.

* It may be that he used the intermediate values shown in his table,

* thus introducting rounding error.

* Try again using intermediate values in the table.

IF ($casenum EQ 1) TestValue2 = 1.1.

IF ($casenum EQ 2) TestValue2 = 11.29.

IF ($casenum EQ 3) TestValue2 = .176.

COMPUTE CumChiSq2 = cdf.chisq(TestValue2,1).

COMPUTE p2 = 1 - CumChiSq2.

FORMATS TestValue2 (f5.3) /CumChiSq2 (f8.6) / p2 (f8.5).

LIST.

* Now my results match those reported by Dytham.

OUTPUT:

Effect TestValue CumChiSq p TestValue2 CumChiSq2 p2

Day length 1.103 .706378 .29362 1.100 .705734 .29427

Sex 11.29 .999222 .00078 11.29 .999221 .00078

Interaction .176 .325576 .67442 .176 .325166 .67483

Okay

Those with access to JSTOR can get the original article here:

http://www.jstor.org/stable/2529511

Note the title: "The Analysis of Ranked Data Derived from Completely Randomized Factorial Designs"

@Beatrice the Scheirer-Ray-Hare Test, the recommendation is to work with small samples and in no case sample covariance between the dependent variables.

@Massil

in this case, being a multivariate process is advisable to work with at least 20 replicates per treatment or factor.

you have 2 ways, one is proposed to make successive anova "less preferred" because it does not show, the interaction between the DP. The other is to generate the interaction model between the dependent variables, it is sometimes difficult to find the dependent variables are independent of each other.

What would lead to perform ANOVA with the variable MODEL

OK

Many thanks Carlos

Is it possible to use Sequential Equation Modelling for this?

Greetings all!

Are there any new developments on this question, which is of particular interest to me especially given my limited knowledge in statistics.

I'm trying out "Optimal Data Analysis" on a very similar problem

no IV,

multi DVs

[8DVs with 5 levels:

2 Levels, each for a pre post [i.e. repeat] measure;

and  1 level, post only;

with all measures being "noncommensurate" between DV levels,

although all on a 5-point Likert scale, therefore ORDINAL],

but would like a "classical"

model or method

as a means of comparison -

ie to answer the question

of pre vs. post differences

following an intervention (longitudinal, time-dependent study),

with regard to each DV and  level.

Thank you.

PS.

I found the following concerning a number of MANOVA assumption violations, which, unfortunately, do not cover my needs:

(http://psycnet.apa.org/journals/med/1/1/27/)

(http://www.entsoc.org/PDF/MUVE/6_NewMethod_MANOVA1_2.pdf)

provided by Dr Charles Zaiontz,

(http://www.real-statistics.com/)

In the first link (the article is generously provided by its author via Researchgate), Pr Finch states:

"In order for MANOVA to work appropriately, there

are assumptions which must be met, and which are very

similar to those required for the use of ANOVA: (1) observations must be independent, (2) the response variables are multivariate normal, and (3) the population covariance matrices are equal across the levels (P) of the

independent variable."

There is no mention of the variable having to be  either an  INTERVAL or a  RATIO measure.

In another discussion on ANOVA

(https://www.researchgate.net/post/Are_there_times_when_we_use_parametric_tests_ANOVA_for_ordinal_data_References_appreciated)

a link is provided  by Johan Larsson, who summarizes the work by Lantz as follows:

"The take-home message is that ANOVA seems relatively insensitive to violations of the assumption of equidistance given that the other assumptions of normality hold; otherwise, one should look to robust alternatives".

1. Lantz B. The impact of non-equidistance on Anova and alternative methods. Electron J Bus Res Methods [Internet]. 2014;12(1):16–26. Available from: http://www.scopus.com/inward/record.url?eid=2-s2.0-84912137275&partnerID=40&md5=e284285938f919aba5e5acbf52a90cfc

Can one therefore eventually state:

MANOVA does not necessarily require interval/ordinal scale values

or, even if it does, since it is similar to ANOVA,

MANOVA COULD be used

if equidistance and other assumptions of normality exist using ORDINAL scale values?

Dear massil, 

Abstract: "The conventional analysis of variance applied to designs in which each subject is measured repeatedly requires stringent assumptions regarding the variance-covariance (i. e., correlations among repeated measures) structure of the data. Violation of these assumptions results in too many rejections of the null hypothesis for the stated significance level. This paper considers several alternatives when heterogeneity of covariance exists, including nonparametric tests, randomization and matching procedures, Box and Greenhouse-Geisser corrections, and multivariate analysis. The presentation is from an applied rather than theoretical standpoint. Multivariate techniques that make no covariance assumptions and provide exact probability statements represent the most versatile solution."

Article Bias in the Analysis of Repeated-Measures Designs: Some Alte...