Not sure whether I understand everything, but when running regressions, the errors should be normally distributed, not primarily the dependent variable. So you could check that.

In any case, you could look at the dependent variable to see why they are not normally distributed. For example, they could be censored or truncated, in which case you could use a suiting model, for example a Tobit model, Hurdle model, or truncated regression.

Additionally, since you have two factors, you need to specify which exact effects you are interested in. Are you interested in main effects, i.e. the effect of depth irrespective of season (i.e. aggregated over season) or the other way around. Are you interested in depth differences for certain levels of seasons (conditional effects, resp. simple effects), or are you interested in their interaction, i.e. the difference between the depth effect for season 1 vs. the depth effect for season 2, for example.

All of this could be tested in a regression model like:

DV_i = b_0 + b_1*FV1_i+b_2*FV2_i+b_3*FV1_i*FV2_i + e

where DV is the dependent variable, FV_k the k factor variables, e the error term, b_0 the intercept. b_3 is the estimated parameter for the interaction, whereas, in this model, FV_1 would be the simple main effect of FV1 vs its base category, when FV2 is zero, so it would be a conditional effect (the effect of FV1 conditional on FV2 being zero).

In R, you can check model assumptions visually by using plot(MODEL). However, it is difficult to test for normally distributed error terms with inferential tests, since they often reject the hypothesis of normally distributed data, even when the distribution is as good as normal.

If you want to use a GLM, you still need to specify a distribution for your dependent variable via the link function and the family.

For a nonparametric approach, I'll copy what I posted here: https://www.researchgate.net/post/Does_anyone_know_a_non_parametric_equivalent_to_repeated_measures_ANOVA_WITH_a_between_subject_factor

"For more complicated nonparametric designs, there is the technique of aligned ranks. ARTool appears to be the premier (and free!) software for this. I have almost no experience with it, but it seems to accommodate most common designs including those with repeated measures. It can be used as an R package, or apparently as a stand-alone Windows program. "

You might also consider parametric approaches that don't assume conditionally normal data. These include generalized linear models, such as beta regression or gamma regression or Poisson regression. Also there are techniques like ordinal regression. It depends on the nature of your data.

I know that this is not an answer to your question, but I cannot restrain from adding that using bar plots is not really the best option, [1]. In the case the data is not normally distributed, this is even worse: how does one interpret the standard deviation in those plots, if there is no symmetry?

[1] Article Beyond Bar Graphs: Time for a New Data Presentation Paradigm

Thank you for taking the time to help me out. I have solved the problem using PRIMER multivariate statistics and running an ANOSIM which gives a low R value which ecologically shows no significance and following it up with PERMANOVA showing strong support for significant different with depth and no significance with season which is what I would expect looking at the figure.

Below is the results

PERMANOVA

Permutational MANOVA

Resemblance worksheet

Name: Resem2

Data type: Distance

Selection: All

Resemblance: D1 Euclidean distance

Sums of squares type: Type I (sequential)

Fixed effects sum to zero for mixed terms

Permutation method: Permutation of residuals under a reduced model

Number of permutations: 999

Factors

Name Abbrev. Type Levels

Depth De Fixed 3

Season Se Fixed 2

PERMANOVA table of results

Unique

Source df SS MS Pseudo-F P(perm) perms

De 2 1923 961.49 215.42 0.001 997

Se 1 69.776 69.776 15.634 0.001 998

DexSe 2 102.38 51.192 11.47 0.001 999

Res 443 1977.2 4.4632

Total 448 4072.3

Details of the expected mean squares (EMS) for the model

Source EMS

De 1*V(Res) + 149.67*S(De)

Se 1*V(Res) + 224.5*S(Se)

DexSe 1*V(Res) + 74.832*S(DexSe)

Res 1*V(Res)

Construction of Pseudo-F ratio(s) from mean squares

Source Numerator Denominator Num.df Den.df

De 1*De 1*Res 2 443

Se 1*Se 1*Res 1 443

DexSe 1*DexSe 1*Res 2 443

Estimates of components of variation

Source Estimate Sq.root

S(De) 6.3944 2.5287

S(Se) 0.29093 0.53938

S(DexSe) 0.62445 0.79022

V(Res) 4.4632 2.1126

Clear effect of depth. Weak effect of season. Significant interaction.

Richness higher at depth in winter. Because of variability and awesome power (loads of replicates) we cannot conclude that richness at different depths in different seasons are exactly the same (so ‘significant’ as that’s what the test is testing), but compared to the effect of depth seasonal difference is small.

Anosim gives the same interpretation (though without testing the interaction, as it doesn’t):

ANOSIM

Analysis of Similarities

Two-Way Crossed - AxB

Resemblance worksheet

Name: Resem2

Data type: Distance

Selection: All

Factors

Place Name Type Levels

A Season Unordered 2

B Depth Ordered 3

Season levels

Summer

Winter

Depth levels

5

10

25

Tests for differences between unordered Season groups

(across all Depth groups)

Global Test

Sample statistic (Average R): 0.081

Significance level of sample statistic: 0.1%

Number of permutations: 999 (Random sample from a large number)

Number of permuted statistics greater than or equal to Average R: 0

Tests for differences between ordered Depth groups

(across all Season groups)

Global Test

Sample statistic (Average R): 0.454

Significance level of sample statistic: 0.1%

Number of permutations: 999 (Random sample from a large number)

Number of permuted statistics greater than or equal to Average R: 0

Pairwise Tests

R Significance Possible Actual Number >=

Groups Statistic Level % Permutations Permutations Observed

5, 10 0.473 0.1 Very large 999 0

5, 25 0.557 0.1 Very large 999 0

10, 25 0.322 0.1 Very large 999 0

But the statistic gives a clear value for comparing the effect sizes

Even if significant few would consider an effect size (Anosim R) of less than 0.1 as ecologically meaningful. Depth (ordered factor) is clearly important.

Perhaps a linear regression on depth removing the effect of season would help, but for me it’s all very clear without extra analyses.