Discussion:
Equivalent of MIXED ANOVA FOR NON PARAMETRIC STATISTICS
(too old to reply)
l***@gmail.com
2014-11-17 11:51:53 UTC
Permalink
HI,
I hope someone could help.
My experiment:
within subject (time) with 3 level
between subject (condition) with 2 levels
In each condition 10 subjects (total 20)
I have different DP variables but I want to analyse these one by one.

Unfortunately my data are not normally distributed (as expected)and even with the correction I cannot achieve this assumption. The idea to use a mixed anova is not possible. For sure I could use MANN-WHITNEY test.

I was wondering if there is a different way to analyse the data or a sort of nonparametric GLM.

Thank you very much

L
Bruce Weaver
2014-11-17 23:09:05 UTC
Permalink
Post by l***@gmail.com
HI,
I hope someone could help.
within subject (time) with 3 level
between subject (condition) with 2 levels
In each condition 10 subjects (total 20)
I have different DP variables but I want to analyse these one by one.
Unfortunately my data are not normally distributed (as expected)and even with the correction I cannot achieve this assumption.
1. As George Box reminded us, normal distributions and straight lines do
not exist in nature. They are models that can be useful under certain
conditions.

2. The normality assumption for ANOVA is normality of the errors (i.e.,
normality within groups, not of the DV overall). How are you assessing
normality? If you perform the mixed design ANOVA you would like to do
and save the residuals, how are those residuals distributed?

3. When they are used to test hypotheses about location, rank-based
tests are very sensitive to heterogeneity of variances and small
differences in skewness. Google for articles by Morten Fagerland (in
the journal Statistics in Medicine, IIRC--something about the
Wilcoxon-Mann-Whitney test under scrutiny in the title). So using a
rank-based procedure may introduce more problems than it solves,
depending on how similar the population shapes are.
Post by l***@gmail.com
The idea to use a mixed anova is not possible. For sure I could use MANN-WHITNEY test.
I was wondering if there is a different way to analyse the data or a sort of nonparametric GLM.
Conover (author of the book on nonparametric statistics) discusses the
use of the usual parametric test on rank-transformed data. But as I
recall, that does not work well when the parametric model includes
interaction terms. So you would only be able to look at the main effects.

If after all this you still think you want to use a rank-based method,
see this note:

http://www.angelfire.com/wv/bwhomedir/notes/alternative_to_anova.txt

HTH.
Post by l***@gmail.com
Thank you very much
L
--
Bruce Weaver
***@lakeheadu.ca
http://sites.google.com/a/lakeheadu.ca/bweaver/Home
"When all else fails, RTFM."
Rich Ulrich
2014-11-18 06:39:45 UTC
Permalink
Post by l***@gmail.com
HI,
I hope someone could help.
within subject (time) with 3 level
between subject (condition) with 2 levels
In each condition 10 subjects (total 20)
I have different DP variables but I want to analyse these one by one.
Unfortunately my data are not normally distributed (as expected)and even with the correction I cannot achieve this assumption. The idea to use a mixed anova is not possible. For sure I could use MANN-WHITNEY test.
I was wondering if there is a different way to analyse the data or a sort of nonparametric GLM.
I like Bruce's comments; he hits a number of good points.

I am going to say some more about normality and non-normality.
Beginners often either over-rate the importance of normality, or
under-rate it, or both (in varying circumstances). If the residuals
do not have outliers, the testing is probably okay. Making decent
models with more than one d.f. can be a tougher matter, and that
is where rank-transformations fail. Using a further transformation
of the ranks (logit or probit) might rescue that, but that step does
make assumptions that you might need to justify for an audience.

- Do you have any effects apparent when you plot your data?
- Is this just hypothetical froth, or do you have something that
you might show? Does an analysis of ranks show it?

The syntax is potentially ambiguous, but I think you say that you
expected "not normally distributed". Well, is a simple transformation
possible? Is one reasonable? - What are you measuring? The best
clue for what transformation is reasonable is the knowledge of where
the numbers come from.

I don't know what you mean when you say, "even with the correction,
I cannot achieve this assumption." Huh? You are jumping some step.
Is this a reference to a conventional transformation?

Data collected across time are often correlated well enough that
it is useful to look at the lagged scatterplots to figure what the
underlying metric looks like. N of 20 would not tell you much about
really minor distortions, but, then, those would not be much concern.
What you want to figure is what transformation would make the
average changes across lags be homogeneous for the low scorers
and the high scorers.

Trying my internet ESP:
What comes to my mind as a potential cause of "non-normal"
expectations is that you have a bunch of scores at zero
(or some other number). If this is inherently "off the scale"
of what the other numbers represent -- such that you cannot
(say) arbitrarily rescore all values so that equal-increments do
hold what you would consider to be equal increments of the
underlying meaning -- then there may be effectively two (or
more) d.f. necessary for the modeling in concept, even if you
end up creating a test that uses a weighted average of the
two tests. I think that this is effectively what over-dispersed,
zero-inflated models do. Otherwise, zeroes are easier to handle
for a predictor than for an outcome variable. I won't bother
creating examples that will have nothing to do with your problem.
--
Rich Ulrich
y***@hotmail.com
2020-05-13 00:40:28 UTC
Permalink
Post by l***@gmail.com
HI,
I hope someone could help.
within subject (time) with 3 level
between subject (condition) with 2 levels
In each condition 10 subjects (total 20)
I have different DP variables but I want to analyse these one by one.
Unfortunately my data are not normally distributed (as expected)and even with the correction I cannot achieve this assumption. The idea to use a mixed anova is not possible. For sure I could use MANN-WHITNEY test.
I was wondering if there is a different way to analyse the data or a sort of nonparametric GLM.
Thank you very much
L
Hey I was wondering how did you solve your problem at the end?
In fact, I have a very similar problem right now where I have a design in which it has both between subject and within subject component, and the distribution are not at all normal. I was struggling in deciding which non-parametric test to use.

How did you tackle the stat at the end in your research? because if we tease apart the groups and within subject level, we can't detect interaction effect.

best
J
Rich Ulrich
2020-05-13 19:28:36 UTC
Permalink
Post by y***@hotmail.com
Post by l***@gmail.com
HI,
I hope someone could help.
within subject (time) with 3 level
between subject (condition) with 2 levels
In each condition 10 subjects (total 20)
I have different DP variables but I want to analyse these one by one.
Unfortunately my data are not normally distributed (as expected)and even with the correction I cannot achieve this assumption. The idea to use a mixed anova is not possible. For sure I could use MANN-WHITNEY test.
I was wondering if there is a different way to analyse the data or a sort of nonparametric GLM.
Thank you very much
L
Hey I was wondering how did you solve your problem at the end?
In fact, I have a very similar problem right now where I have a design in which it has both between subject and within subject component, and the distribution are not at all normal. I was struggling in deciding which non-parametric test to use.
How did you tackle the stat at the end in your research? because if we tease apart the groups and within subject level, we can't detect interaction effect.
best
J
To the Questioner: Why is your distribution "not at all normal"?
More importantly, do equal point-differences describe "equal
intervals" of whatever is important in outcome? (If not, why
not, and can't you do something sensible about that.)

There is certainly not a one-size-fits all solution, especially when
the problem arises from design that did not foresee it.


It is highly unlikely that the hit-and-run questioner from 2014
is still reading this group. So don't expect to hear what he did.

If anyone is interested, the original two replies (from Bruce and
from me) are at

https://groups.google.com/forum/#!topic/comp.soft-sys.stat.spss/ZCppmHoKMNE

They read pretty well.

Googling showed me a similar question in another forum, but I
haven't looked at it.
--
Rich Ulrich
Bruce Weaver
2020-05-13 20:15:53 UTC
Permalink
On Wednesday, May 13, 2020 at 3:28:43 PM UTC-4, Rich Ulrich wrote:
--- snip ---
Post by Rich Ulrich
It is highly unlikely that the hit-and-run questioner from 2014
is still reading this group. So don't expect to hear what he did.
If anyone is interested, the original two replies (from Bruce and
from me) are at
https://groups.google.com/forum/#!topic/comp.soft-sys.stat.spss/ZCppmHoKMNE
Good idea to post that link. It did not occur to me, because I am reading this via Google Groups, and so I could see all of the posts right back to the original. People reading via some other news server may not have access to all of the old posts.
Post by Rich Ulrich
They read pretty well.
I thought so too. I have nothing to add to what we posted in 2014.

Bruce
y***@hotmail.com
2020-05-14 00:17:53 UTC
Permalink
Post by Rich Ulrich
Post by y***@hotmail.com
Post by l***@gmail.com
HI,
I hope someone could help.
within subject (time) with 3 level
between subject (condition) with 2 levels
In each condition 10 subjects (total 20)
I have different DP variables but I want to analyse these one by one.
Unfortunately my data are not normally distributed (as expected)and even with the correction I cannot achieve this assumption. The idea to use a mixed anova is not possible. For sure I could use MANN-WHITNEY test.
I was wondering if there is a different way to analyse the data or a sort of nonparametric GLM.
Thank you very much
L
Hey I was wondering how did you solve your problem at the end?
In fact, I have a very similar problem right now where I have a design in which it has both between subject and within subject component, and the distribution are not at all normal. I was struggling in deciding which non-parametric test to use.
How did you tackle the stat at the end in your research? because if we tease apart the groups and within subject level, we can't detect interaction effect.
best
J
To the Questioner: Why is your distribution "not at all normal"?
More importantly, do equal point-differences describe "equal
intervals" of whatever is important in outcome? (If not, why
not, and can't you do something sensible about that.)
There is certainly not a one-size-fits all solution, especially when
the problem arises from design that did not foresee it.
It is highly unlikely that the hit-and-run questioner from 2014
is still reading this group. So don't expect to hear what he did.
If anyone is interested, the original two replies (from Bruce and
from me) are at
https://groups.google.com/forum/#!topic/comp.soft-sys.stat.spss/ZCppmHoKMNE
They read pretty well.
Googling showed me a similar question in another forum, but I
haven't looked at it.
--
Rich Ulrich
Hello Rich,

it was very nice of you to reply my question, as I actually didn't expect any response since it was a 6 year-old post.

The OP's question basically hit most of the concerns I have with my current data analysis. unfortunately, I don't have a strong background in statistic and am in a process in self-learning most of the statistic knowledge. So i was having hard time understanding both your and Bruce's comments.

I have used the Kolmogorov-Smirnov normality test.

the reason the data was not normal largely because there are a lot of zeros in the data (continuous numeric data with absolute zero), which make it skewed positively. My supervisor advice me to try clean up the outlier, if possible, and then try data transformation. however, I have done both and yet the most of the dependent variables still violate assumption of normality.

so I turned to non-parametric solution. my research has between subject component (2 groups), and time (3 time points) as within subject component.
Since there is no non-parametric equivalence of mixed design ANOVA, I have to find a solution that is similar to what parametric ANOVA does and self-learn how to do that on SPSS.

I have examined Friedman's test, Mann-Whitney U test, Kruskal-Wallis H test, and Wilcoxon signed rank test. as i saw it, most of them are based on the rank of the dataset and is only partial solution to my analyzing goal. So I was trying to find if there is any way to work around that. i wonder if i still can analyze the interaction effect (group x time) under this context?

If run the between group and within group tests separately on my data, what problems/issue would follow by doing so?

that's why I post the question and try to see how other researchers usually deal with these kind of situation.

John
Bruce Weaver
2020-05-14 18:07:31 UTC
Permalink
On Wednesday, May 13, 2020 at 8:17:55 PM UTC-4, ***@hotmail.com wrote:

--- snip ---
Post by y***@hotmail.com
I have used the Kolmogorov-Smirnov normality test.
As Graeme Ruxton (2006) put it, "it is generally unwise to decide whether to perform one statistical test on the basis of the outcome of another (Zimmerman 2004 and references therein)." He was talking specifically about a preliminary test of homogeneity of variance prior to a t-test, but the basic principal holds generally, and is certainly true about testing for normality as a precursor to another test. I have a short conference presentation on that topic that may interest you.

Ruxton (2006): https://academic.oup.com/beheco/article/17/4/688/215960

My presentation on testing for normality:
https://www.researchgate.net/publication/299497976_Silly_or_Pointless_Things_People_Do_When_Analyzing_Data_1_Testing_for_Normality_as_a_Precursor_to_a_t-test
Post by y***@hotmail.com
the reason the data was not normal largely because there are a lot of zeros in the data (continuous numeric data with absolute zero), which make it skewed positively. My supervisor advice me to try clean up the outlier, if possible, and then try data transformation. however, I have done both and yet the most of the dependent variables still violate assumption of normality.
Your outcome variable is not a count by any chance, is it? If so, you should very likely be using some kind of count regression model (e.g., Poisson or negative binomial regression). Thanks for clarifying.

Bruce
Rich Ulrich
2020-05-14 18:26:51 UTC
Permalink
Post by y***@hotmail.com
Hello Rich,
it was very nice of you to reply my question, as I actually didn't expect any response since it was a 6 year-old post.
The OP's question basically hit most of the concerns I have with my current data analysis. unfortunately, I don't have a strong background in statistic and am in a process in self-learning most of the statistic knowledge. So i was having hard time understanding both your and Bruce's comments.
I have used the Kolmogorov-Smirnov normality test.
the reason the data was not normal largely because there are a lot of zeros in the data (continuous numeric data with absolute zero), which make it skewed positively. My supervisor advice me to try clean up the outlier, if possible, and then try data transformation. however, I have done both and yet the most of the dependent
variables still violate assumption of normality.
Post by y***@hotmail.com
so I turned to non-parametric solution. my research has between subject component (2 groups), and time (3 time points) as within subject component.
Since there is no non-parametric equivalence of mixed design ANOVA, I have to find a solution that is similar to what parametric ANOVA does and self-learn how to do that on SPSS.
I have examined Friedman's test, Mann-Whitney U test, Kruskal-Wallis H test, and Wilcoxon signed rank test. as i saw it, most of them are based on the rank of the dataset and is only partial solution to my analyzing goal. So I was trying to find if there is any way to work around that. i wonder if i still can analyze the
interaction effect (group x time) under this context?
Post by y***@hotmail.com
If run the between group and within group tests separately on my data, what problems/issue would follow by doing so?
that's why I post the question and try to see how other researchers usually deal with these kind of situation.
The most common thing that researchers and statisticians
do about non-normality is IGNORE IT. And for pretty
good reasons.

Taking a rank-transformation of the scores is the starting
point for (all?) those tests you mention. When you replace
your scores with their rank-transformed versions ... DO you
get a set of numbers that improve the "interval" distances
between what you think those original scores should
represent? If the original scores look better, more "equal
interval", then you don't want the loss of detail from converting
to ranks.

Non-parametric tests.
- By the way, their complicated formulas (in their simple form)
include the assumption that there are no ties. So, your
data, with many zeroes, also fail to meet the assumptions
for the "exact" nonparamentric tests. However, there are
"approximations" available. About them -

WJ Conover showed in the 1980s that most of the rank-
order tests with complicated formulas can be replaced by
performing ANOVA on the rank-transformed numbers.
Conover showed that the ANOVA on ranks can be better
(more accurate tests) than the approximations in use by
stat packages, especially when there are many ties.

Outliers and zeroes.
You mention "outlier" as if (maybe) you have just one.
Should you (maybe) drop that case, and mention it only
as a case report, because the score is so very atypical?
Or should one (or more) extreme be drawn in, scored as
the next-highest value? What makes sense?

"Many zeroes" is sometimes the justification to rescore
everything as 0/1. Does that lose much sense in your
data? Do those other values matter? I don't know of
anybody giving advice on this topic, but if half your
scores are zero, I think you have a good case to (at
least) try out this alternate scoring.

Non-normality.
The problem with non-normality in the residuals of the
model-fit is that the resulting F-test might not be accurate;
it might reject too often, or it might reject too seldom.
But ANOVA is pretty robust. Analyzing a 0/1 variable is
not a problem between 20%-80%. And it is not a
problem for rank-transformed data, if the transformation
does not screw up the intervals more than it helps them.

Now, if you have one or more cases where all their scores
are zero, that could distort the picture. "Too many zeroes"
in repeated measures is where the Greenhouse–Geisser
correction is used.


Hope this helps.
--
Rich Ulrich
sasa ZHAO
2020-09-26 13:17:22 UTC
Permalink
Post by y***@hotmail.com
Post by Rich Ulrich
Post by y***@hotmail.com
Post by l***@gmail.com
HI,
I hope someone could help.
within subject (time) with 3 level
between subject (condition) with 2 levels
In each condition 10 subjects (total 20)
I have different DP variables but I want to analyse these one by one.
Unfortunately my data are not normally distributed (as expected)and even with the correction I cannot achieve this assumption. The idea to use a mixed anova is not possible. For sure I could use MANN-WHITNEY test.
I was wondering if there is a different way to analyse the data or a sort of nonparametric GLM.
Thank you very much
L
Hey I was wondering how did you solve your problem at the end?
In fact, I have a very similar problem right now where I have a design in which it has both between subject and within subject component, and the distribution are not at all normal. I was struggling in deciding which non-parametric test to use.
How did you tackle the stat at the end in your research? because if we tease apart the groups and within subject level, we can't detect interaction effect.
best
J
To the Questioner: Why is your distribution "not at all normal"?
More importantly, do equal point-differences describe "equal
intervals" of whatever is important in outcome? (If not, why
not, and can't you do something sensible about that.)
There is certainly not a one-size-fits all solution, especially when
the problem arises from design that did not foresee it.
It is highly unlikely that the hit-and-run questioner from 2014
is still reading this group. So don't expect to hear what he did.
If anyone is interested, the original two replies (from Bruce and
from me) are at
https://groups.google.com/forum/#!topic/comp.soft-sys.stat.spss/ZCppmHoKMNE
They read pretty well.
Googling showed me a similar question in another forum, but I
haven't looked at it.
--
Rich Ulrich
Hello Rich,
it was very nice of you to reply my question, as I actually didn't expect any response since it was a 6 year-old post.
The OP's question basically hit most of the concerns I have with my current data analysis. unfortunately, I don't have a strong background in statistic and am in a process in self-learning most of the statistic knowledge. So i was having hard time understanding both your and Bruce's comments.
I have used the Kolmogorov-Smirnov normality test.
the reason the data was not normal largely because there are a lot of zeros in the data (continuous numeric data with absolute zero), which make it skewed positively. My supervisor advice me to try clean up the outlier, if possible, and then try data transformation. however, I have done both and yet the most of the dependent variables still violate assumption of normality.
so I turned to non-parametric solution. my research has between subject component (2 groups), and time (3 time points) as within subject component.
Since there is no non-parametric equivalence of mixed design ANOVA, I have to find a solution that is similar to what parametric ANOVA does and self-learn how to do that on SPSS.
I have examined Friedman's test, Mann-Whitney U test, Kruskal-Wallis H test, and Wilcoxon signed rank test. as i saw it, most of them are based on the rank of the dataset and is only partial solution to my analyzing goal. So I was trying to find if there is any way to work around that. i wonder if i still can analyze the interaction effect (group x time) under this context?
If run the between group and within group tests separately on my data, what problems/issue would follow by doing so?
that's why I post the question and try to see how other researchers usually deal with these kind of situation.
John
Hi John,

I have met the similar problem as you(non-normal distribution, 2 way or 3 way mixed Design, trying to analyze the interaction effect). And I found some information which may be helpful:

1) Wilcox’s robust ANOVA (R, the WRS2 package)
https://www.researchgate.net/publication/333525893_Robust_statistical_methods_in_R_using_the_WRS2_package
https://dornsife.usc.edu/labs/rwilcox/software/

2) GLMM(Generalized linear mixed models) (R's LME4 package)

Sasa

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