Hello @Grishma,

o.c. there are ways how to handle this.

But first, please let us know which test you applied to check on homoscedasticity?

If the latter mentioned is severely violated, you may consider gls {nlme}, where variance of groups or individuals can be adjusted.

E.g. via gls(..., weights = varIdent(...))

All the best, Marian

@Marian

Thank you so much for taking time to help me out.

I used the Levene's test to check for the homoscedasticity.

I'm very new to the data analysis softwares so I was wondering if the commands mentioned by you are for SPSS or SAS or some other software.

Thanks again.

https://bolt.mph.ufl.edu/6050-6052/unit-4b/module-13/details-for-non-parametric-alternatives/

Hello Grishma Prabhukhot

sry, I somehow assumed you would use R (but I think, this software was mentioned in another question...)

for SPSS this solution could help you out:

https://www.youtube.com/watch?v=enPK_SXILnA

By the way, our statistical supervisor teached us a "rule of thumb": When variance is calculated for each group, then the ratio between minimum Variance of some group to maximum variance of some other group must not exceed 3.5

If this ratio is lower, it might be ok to proceed with standard tests, without consideration of heteroscedasticity.

hth, Marian

I have a few pieces of advice. Hopefully they will be helpful.

1) I would advise against using a statistical test to assess the assumptions of a model. One issue with these tests is that they are sensitive to sample size. That is, with large sample sizes, these tests will report e.g. small deviations from a normal distribution as a significant difference.

It’s better to trust your eyes and judgement. For homoscedasticity, I would plot the model residuals vs. the predicted values from the model. But there are other ways to plot this that some prefer. Or as Marian Schönauer suggests, you can look at the actual variance values of the groups.

2) If heteroscedasticity is the problem, there are options for handling this. a) Some software will allow you to use an approach that doesn’t assume homoscedasticity. Welch’s t-test or Welch’s anova can be used for two-sample or one-way designs. For more complex designs, a White-adjustment (or, White-Huber adjustment) can be applied. b) As Marian Schönauer suggested, you could also use a weighted regression [anova], which requires a little more manual work.

I don’t know if any of these are easy to do in SPSS.

Marian Schönauer

Thank you for your reply.

I plan on doing the method mentioned as a rule of thumb. I have done the calculation for 'ratio between minimum Variance of some group to maximum variance'. I have attached an excel sheet to this reply for your reference. Please let me know if I'm understanding your explanation. Please let me know if I have done an incorrect calculation.

Thank you so much once again!!

Thank you Salvatore S. Mangiafico for your reply. I really appreciate you taking sometime to go through my question. I'm very very new to this field and I'm so grateful that you are willing to help. If possible could you please take a look at the excel file I have attached so that I'm applying the 'rule of thumb(mentioned by Marian Schönauer )' correctly.

Oh, I guess I could try to answer the question in the title....

The traditional non-parametric two-way anova is the Scheirer–Ray–Hare test. I have mixed feelings about this test. As far as I can tell, it isn't well documented, and isn't used too much. I've also noticed that it often won't detect a significant interaction when other tests will. (Citations, Sokal, R.R. and F.J. Rohlf. 1995. Biometry, 3rd ed., or see https://rcompanion.org/handbook/F_14.html).

Other approaches that may work are aligned-ranks anova or anova after normal scores transformation. There are also approaches for permutation anova and general factorial designs.

Grishma Prabhukhot , well, I think that the rule of thumb would compare all nine groups to all other groups (assuming you have an interaction term in your model).

In your Excel example, there really is quite some difference in the variance among groups. I've attached a plot of the residuals vs. the predicted values from a two-way anova with interaction.

There are obvious difference among groups, but I might also note that there's nothing systematic in the differences in variances. In this case, I would probably not be comfortable claiming that the groups are homoscedastic, but since there is nothing systematic in the heteroscedasticity, I don't think there's a problem with the model being mis-specified.

In R, it's easy to apply the White adjustment to the anova table for this model. In this case, the results are not much different with or without the White adjustment.

Hey Grishma Prabhukhot

I checked your excel-data.

the approach is alrigth, but the "rule of thumb" is applied to different estimates.

For example: in columng G (4,10,16) you calculated the variances of P for each group, A, B and C.

Thats perfect.

Then, using these numbers, one could calculate

Var(P_A)/ Var(P_B) , i.e. 32.967/5.367 = 6.143 (example for type P)

Obviously, 6.14 > 3.5

meaning, that variance is really quite different..

Consequently, I suppose to use an adapted model, as the difference of variance between groups is considerable.

Why not use non-parametric tests? There are almost no disadvanteges and proofed mean comparisons are more reliable, and usually it is easier to statistically confirm differences by the right model ;). You should check tests as recommended by Salvatore S. Mangiafico

(Mann-Whitney U-test for example, as a classical rank-correlation)

in F23:F25, you calculated variances for each group, consisting all three tire types.

Please check this ones, since the referred "area" is wrong. I think you just dropped the function to the cells below, but the "overall variance of B" should be "=VAR.S(B10:D15)", for C "=VAR.S(B16:D21)"

I can't assess, whether it is legit to calculate variance over all three tire types (probably yes). This question is up to you ;)

If so, "rule of thumb"-assumption would be met, with 114/63 = 1.81

1.81 < 3.50, allowing for parametric models

phu, I hope this wasn't formulated too confusing, as it is very simple in case.

don't hesitate to ask back, Marian

Marian Schönauer Thank you so very much for taking the time to go through my question. I will be working on my actual data set, now that I have all this information with me. I might get back again on the same thread of questions and answers if something new comes up. Thank you once again!

Hello Salvatore S. Mangiafico

Thank you for your reply. I have a follow-up question. You mentioned "but since there is nothing systematic in the heteroscedasticity, I don't think there's a problem with the model being misspecified. "

Could you please help me understand what does 'nothing systematic in the heteroscedasticity' means?

I'm so sorry once again, my background is really not in statistics and I'm trying to solve a research problem. Thanks again

Grishma Prabhukhot , I mean, if you had a pattern like the following, residuals vs. predicted values, https://rcompanion.org/handbook/images/image104.png , that would suggest that there is something missing from your model or that the model is in some way inappropriate. But if it just happens that some groups have a smaller variance than others, it _could_ be something meaningful, or it could just be that some groups happened to have lower variance than others.

Salvatore S. Mangiafico

Thank you once again for your prompt reply. I will start working on my dataset and generate the residuals vs predicted values plots.

I really appreciate all your help. I might get back on the same thread of questions if something comes up that I'm not sure about how to proceed.

Thank you and have a good one!!

may be kruskal wallis or mann whitney u test

there is no way in the frequentist framework to support "normality".

The idea is that the data that you are comparing is unimodal pretty much and symmetric

avoid nonparametric tests such as KW and MW as suggested by Muhammad Rizwan since they are very often misinterpreted. They are in fact are tests of stochastic dominance and the statistic used is far from being straightforward.

You really need to set your research question straight Grishma Prabhukhot : what are you trying to estimate? Do you want to model an outcome or what?

Test-driven analysis makes little sense

normality and homoskedasticity are not requirements, just used methods that are robust to those deviations, e.g. using resampling methods

Also, remember: removing outliers is a sensible matter, do not remove what you think is an outlier and perform your analysis as if that value was never measured

@ Grishma Prabhukhot

I couldn't look at your data, but the answer to your question "Which non-parametric tests are equivalent to 'two-way anova with replication'?" is Repeated Measure Anova on Ranks (RM Anova on Ranks), in other mean Friedman Test you can choose. good luck