# Non-parametric Two-way ANOVA

#### brett_a_baxter

##### New Member
Hi Guys,

I wanted to run a TWO-way ANOVA but unfortunately, my variables are non-parametric. I have tried transforming the data but that did not work. I was wondering what the next step was? I was thinking of conducting two separate tests, a Friedman's test as a non-parametric alternative to examine within-group effects and a Kruskal-Wallis test to examine between-group effects. I imagine this increases my chances of error but i cannot think/find another solution. Any help would be extremely appreciated. Thank you in advance,

Brett

#### Karabiner

##### TS Contributor
I wanted to run a TWO-way ANOVA but unfortunately, my variables are non-parametric.
Please notice, there are no nonparametric variables. There are non-parametric statistical tests, though.

Probably you refer to the distribution of your dependent variable. What exactely do you
consider as the problem in this regard? Seemingly, you wanted to perform a mixed ANOVA?

Maybe you should decribe your study in your detail first (study design, variables measured, sample sizes).

With kind regards

Karabiner

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#### brett_a_baxter

##### New Member
Hi Karabiner,

Thank you for your response. Good point about my choice of wording, thank you for picking me up on that, a bad habit I need to get out of! I will explain my study in detail below.

I am measuring numerous physiological variables such as strength, muscle size, and functional ability etc. over 3 periods of time. There are two training groups being compared, a once-weekly group vs. a twice-weekly group. The sample sizes are 10 and 8 in each group, respectively.

So in summary, the DV's are the physiological variables and there are 2 IV's (time and training group).

My data are non-parametric and I would like to know how to best analyse these data.

Any help would be greatly appreciated. Thank you in advance.

#### Karabiner

##### TS Contributor
The concept of non-parametric data doesn't exist, as far as I know.
Non-parametric tests are those which make no assumptions about
underlying distributions of the data.

You maybe want to express that you have reason to assume that your
data in each separete group at each tome point are sampled from
non-normally distributed populations?
I would like to know how to best analyse these data.
This depends to a large part on your precise research question(s).

With kind regards

Karabiner

#### brett_a_baxter

##### New Member
My data are non-normally distributed to better describe the situation.

I would like to know whether there is an effect over time (within analysis), so I was presuming I would use a Friedman test.

I would also like to know whether there were any differences between the training groups at any given time point (between analysis), so I was presuming I would conduct a Krusk-Wallis test.

Would dyou advise on this or are you aware of any other methods that may be more suitable?

#### Karabiner

##### TS Contributor
My data are non-normally distributed to better describe the situation.
In each group, at each time point? If the residuals can roughly
be considered as ampled from a normal distribution, this could
justify a mixed ANOVA. ANOVAs are generally robust against
non-normality.

With kind regards

Karabiner

#### brett_a_baxter

##### New Member
In each group, at each time point? If the residuals can roughly
be considered as ampled from a normal distribution, this could
justify a mixed ANOVA. ANOVAs are generally robust against
non-normality.

Unfortunately yes, that tends to be the case that the variables are more non-normally distributed than normally distributed for both groups and time.

#### brett_a_baxter

##### New Member
Hi Hlsmith,

I did not in my original post, I apologise. The sample sizes are 10 and 8.

#### katxt

##### Active Member
Maybe you are being too hard on your data. What makes them unsuitable?
Try one DV and post a probability plot of the residuals. kat

#### brett_a_baxter

##### New Member
Hi Kat,

thank you for your suggestion, to avoid any subjectivity I have conducted Shapiro-Wilk tests on each DV.

#### Karabiner

##### TS Contributor
What sense does it make to test a cell with n=8 observations?
The test has such a low power in that case that non-significant
results will never lead to meaningful conclusions.

With kind regards

Karabiner

#### katxt

##### Active Member
Shapiro-Wilk tests on each DV
It is the residuals that matter. Did you test them?
I am measuring numerous physiological variables such as strength, muscle size, and functional ability
Perhaps you could consider a manova on all dvs at once, and check the residuals using a normal probability plot.
kat

#### obh

##### Well-Known Member
Hi Hlsmith,

I did not in my original post, I apologise. The sample sizes are 10 and 8.
What do you mean by 10 and 8?
A total of 18 values?
If you want to do two ways ANOVA, how many categories do you have in each IVs?

#### brett_a_baxter

##### New Member
What do you mean by 10 and 8?
A total of 18 values?
If you want to do two ways ANOVA, how many categories do you have in each IVs?
Yes I have a total of 18 values, my IV has two categories, in which there are 8 and 10 in each category.

#### brett_a_baxter

##### New Member
What sense does it make to test a cell with n=8 observations?
The test has such a low power in that case that non-significant
results will never lead to meaningful conclusions.

With kind regards

Karabiner
I am not testing on cells, this sample size is quite normal for this type of study.

#### obh

##### Well-Known Member
Yes I have a total of 18 values, my IV has two categories, in which there are 8 and 10 in each category.
Only one IV? didn't you say two-ways? so two IVs?

#### brett_a_baxter

##### New Member
Only one IV? didn't you say two-ways? so two IVs?
Sorry! My and, yes 2 IV's, once has 2 categories and one has 3

#### brett_a_baxter

##### New Member
It is the residuals that matter. Did you test them?

Perhaps you could consider a manova on all dvs at once, and check the residuals using a normal probability plot.
kat
I tested the residuals and they were still non-parametric, i was thinking of conducting Friedman's tests for within analyses and Kruskal-Wallis tests for between analyses.

#### obh

##### Well-Known Member
Hi Brett,

First, as @Karabiner already wrote, you mean that the residuals don't distribute normally (non-parametric can describe a test not distribution)
It seems that the test power will be very small.

For example, for a large effect size (F=0.59) you may get very low test power, around 0.1.
So unless the effect size is really huge, the test may have low power.

Code:
> pwr.f2.test(u =1, v=2, f2=0.59^2, sig.level =0.05)

Multiple regression power calculation

u = 1
v = 2
f2 = 0.3481
sig.level = 0.05
power = 0.1123456

> pwr.f2.test(u =2, v=2, f2=0.59^2, sig.level =0.05)

Multiple regression power calculation

u = 2
v = 2
f2 = 0.3481
sig.level = 0.05
power = 0.09045044