Hi Dawn

I really do not know how and why people may disagree on the original approach of using % (or proportion) cover as measurement unit and then performing ANOVAs (or t-test or any other parametric or non-parametric method). There are literally tons of papers using this approach for many many decades. This is just a random paper I googled:

http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0000711

Of course the only problem is to ensure that the data meets the parametric assumptions. Cover data often is not normal, and the regular recommended transformation is the so-called 'angular transformation' (arcsine of the square root of the proportion). Check out this paper:

http://onlinelibrary.wiley.com/store/10.4319/lo.2006.51.1_part_2.0569/asset/lno2006511part20569.pdf?v=1&t=ipuchlkw&s=62cb5d8cb30864bb8bb276379dc4c8210d247bc3

Even if the angular transformation does not work, ANOVA is very robust to violations to normality (but not to heterocedasticity).

So, asking each of your questions

1) Yes, I agree with you colleague, using areas instead of original point-data (converted to % cover) was not really an improvement, is redundant. After all is exactly the same point-data treated in a different way. Moreover the whole idea of using point-data is that simplifies the sampling, focusing on specific points that can be easily inspected by any observer. These are points, and cannot be converted to area. A very different situation would be if you have photographic records, so you could estimate the area covered by each algae in your plots. That would provide you a direct estimation of the area (cm2) covered by each algae.

2) I partially disagree with your colleague. Point-data from quadrants is a very common and classic  method used by both marine and terrestrial ecologists for decades. Bias introduced by observers is always a possibility (when it does not?), but you can make the case that all observers were well trained students, with good experience in the identification of algae. In addition, you could run a few trials to quantify the % of error introduced by the use of different observers and just reported it. 

3) I really disagree. Of course point-data is not truly continuous, but using a 100 point quadrant ensures a good approximation. Besides, using 100 point quadrant is a standard method for the bulk of ecological studies in coastal rocky shores. Again, do a simple google search and you will find 1000's of papers using exactly the same approach. Your colleague's critique is really unfair, because it is more at the core the historic approach used by the discipline rather than to your specific research. And even if we decide to obey to the "talibans of the statistics", there are methods to deal with non-continous data (e.g. generalized linear models with Poisson errors).

In summary, I recommend you to go back to the % cover (proportion data), use the angular transformation, test the parametric assumptions, and if the data is still homocedastic go the classic ANOVA. Be sure of citing a bunch of classic marine ecology papers to justify your methods. 

Good luck with your research

Marcelo

Hi Dawn

I really do not know how and why people may disagree on the original approach of using % (or proportion) cover as measurement unit and then performing ANOVAs (or t-test or any other parametric or non-parametric method). There are literally tons of papers using this approach for many many decades. This is just a random paper I googled:

http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0000711

Of course the only problem is to ensure that the data meets the parametric assumptions. Cover data often is not normal, and the regular recommended transformation is the so-called 'angular transformation' (arcsine of the square root of the proportion). Check out this paper:

http://onlinelibrary.wiley.com/store/10.4319/lo.2006.51.1_part_2.0569/asset/lno2006511part20569.pdf?v=1&t=ipuchlkw&s=62cb5d8cb30864bb8bb276379dc4c8210d247bc3

Even if the angular transformation does not work, ANOVA is very robust to violations to normality (but not to heterocedasticity).

So, asking each of your questions

1) Yes, I agree with you colleague, using areas instead of original point-data (converted to % cover) was not really an improvement, is redundant. After all is exactly the same point-data treated in a different way. Moreover the whole idea of using point-data is that simplifies the sampling, focusing on specific points that can be easily inspected by any observer. These are points, and cannot be converted to area. A very different situation would be if you have photographic records, so you could estimate the area covered by each algae in your plots. That would provide you a direct estimation of the area (cm2) covered by each algae.

2) I partially disagree with your colleague. Point-data from quadrants is a very common and classic  method used by both marine and terrestrial ecologists for decades. Bias introduced by observers is always a possibility (when it does not?), but you can make the case that all observers were well trained students, with good experience in the identification of algae. In addition, you could run a few trials to quantify the % of error introduced by the use of different observers and just reported it. 

3) I really disagree. Of course point-data is not truly continuous, but using a 100 point quadrant ensures a good approximation. Besides, using 100 point quadrant is a standard method for the bulk of ecological studies in coastal rocky shores. Again, do a simple google search and you will find 1000's of papers using exactly the same approach. Your colleague's critique is really unfair, because it is more at the core the historic approach used by the discipline rather than to your specific research. And even if we decide to obey to the "talibans of the statistics", there are methods to deal with non-continous data (e.g. generalized linear models with Poisson errors).

In summary, I recommend you to go back to the % cover (proportion data), use the angular transformation, test the parametric assumptions, and if the data is still homocedastic go the classic ANOVA. Be sure of citing a bunch of classic marine ecology papers to justify your methods. 

Good luck with your research

Marcelo

Dawn,

You should be using the % cover values in your analysis. However, you should analyse all your % cover data for violations of the basic assumptions of normality and homogeneity of variance implied in all linear model stats like the ANOVA. Any basic stats book or stat software will tell you what tests to use. If the assumptions have not been violated then there is no need to transform your data sets and just use it as is. If you got violations of either of the assumptions than the ArcSine transformation is the one that is appropriate for percentages (0 to 100%) or proportions (0 to 1). For some of the % cover that I analysed in the past the arcsine square root transformation worked the best: p' = arcsin √p

Make sure you re-test the assumptions once you transform the raw % cover data. If the transformation did not work try using non-parametric stats.

Hope this helps a bit.

Tom

Hi Marcelo, Tom

Thank you for your responses. They were very helpful, but I do have one question still. My understanding is that the ArcSine transformation was required to convert percentage data, as it is proportional data, which is not considered continuous data since there is a fixed upper limit (1 or 100, depending). The students, as Marcelo pointed out, essentially had area data, which is not limited (except by the size of the quadrat). This why the idea to use area emerged. The area data could then be tested for the usual assumptions of normality, etc. From what Tom wrote, it seems as though this is a plausible approach, except that he said the percentage cover data is what should be used. I was wondering why is this so.

Dawn

ANOVA can be applied f you want to differentiate between two sets of data

Hi Dawn,

Your question: "....percentage cover data is what should be used. I was wondering why is this so."

If this is correct: " Using a 1m by 1m quadrat subdivided into 100 squares they estimated percentage cover of different species of algae (i.e., percentage of each quadrat occupied by each species). Their resolution was 0.25% (quarter of a square)."

The data that was collected was already an 'estimate" % data. I am guessing the students did not actually measure the areas of the 4 substrate types in each 100cm2 in actual area units, but 'estimated' the % cover through some reference system they used.

Converting the original % cover data into actual area is jus another 'estimation' it is not data transformation to address the assumptions. For example converting an estimated 50% cover by substrate type 1 into X mm2 is not the same as doing data transformation such as: p' = arcsin √p

Coral literature is full of examples that you should find useful.

Tom

Looking at percentages (or arcsine-transformed percentages) may be ok if your data fall near the middle, but if your data fall near the extremes (e.g. close to 100% or close to 0%) then you're not going to get meaningful results, as you will have e.g. confidence intervals going above 100% or below 0%. The arcsine transform doesn't always correct this issue; see e.g. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2613284/.

I'm not familiar with your particular research area so I'm not sure what problems there might be with this proposal, but one common option would be to use a generalized linear mixed model (i.e., the mixed-effects equivalent of logistic regression). Instead of using the 1m by 1m quadrants as your unit of analysis and percent coverage as your outcome variable, you could use the 10cm by 10cm squares as your unit of analysis (with Quadrant as a random effect) and a binary outcome variable (i.e. {has vs. does not have} that kind of algae).

Dear Stephen

That's a very interesting alternative that I will check out.

Hi Dawn,

First let me try to see if I understand:

There are 4 types of substrate, an unknown number of different algae ( I guess in the order of 5-10) and 5 replicates to each substrate. The quadrants subdivision is merely an aid to help quantification underwater, so each species has a potential ‘score’ of 0 to 100 in integer steps, which represents 0 to 10.000 cm2 in steps of 10 cm2. You want to know if substratum (four categories) are a significant predictor for individual algae cover.

Would it not be more interesting to look at this as a ‘many to many’ problem? I would attempt multivariate analyses to tease out the difference in algal community structure amongst the different substrata first. This could be done with PERMANOVA in PRIMER 6 or in the VEGAN package in R or something similar.

Regarding the univariate analysis it really depends on the data.  Make them plot their data first. If they get loads of zeros or hundreds or if there is large variation between the 5 different scores per stratum then think of transforming or use a non -parametric test (Kruskal-Wallis etc.) first. Arc-sine transformation is done to get rid of the ‘edge’ effects in percentage data: There is not much relative difference between 50% and 54% but there is an enormous difference between 1% and 5%.

If you have no scores at 100 you can use a straightforward Generalized Linear Model with a poisson, or a quasi-poisson, or negative binomial error distribution, depending on the distribution of the data, I guess. I like Stephen’s idea of using a mixed effect to account for the unknown variation introduced by the different quadrants. It is an elegant solution and avoids many of the problems with transformation etc. In your case you do not have any continuous predictor variables such as depth. The only problem is that the implementation can be quite tricky for the novice.

Hope this helps!

Sven

Hi Sven

Thank you for the suggestions – fascinating, to the extent that I understand them (I'm not good at stats). I will try to make space of a couple of days to read on the suggestions some more, absorb and understand them better, especially since I have never tried some of those analyses, so there is no experience to fall back on. Hopefully the students might be interested in working some more on the data.

Thanks for taking the time and sharing everyone

Dawn

No data is "really continous", nothing is "really normal distributed". These are abstract mathematical concepts underlying the mathematical tools we use that are useful to cope with some real phenomena. We always have to check that the real phenomena have properties that are not clearly violate such assumptions of the tools we use (if so, then we need to look for tools that would be more appropriate).

Assuming continuity and normal distribution is usually not a vary bad assumption for proportions (percentages) when these proportions are somewehere between 0.3 to 0.7 (more or less). If the get close to 0 or close to 1, the data usually does not fit well to the assumption of a normal distribution. However, for statistical tools that deal with expected values ("means") it is also relevant how many data are used to calculate the means (i.e. the sample size). As another coarse rule of thumb one can say that there is usually no need to think about the distributional assumption when the sample size (per group) exceeds 30 or so.

What in my eyes is most relevant is the interpretability of the results. In your case the main results (as I see it) should be the estimates of how much the occupation of the species are changing depending on the substrate type. You must think in what unit such estimates would be meaningful, sensible, interpretable. Only then can you judge if the changes you observed could be "significant" (you can calculate a p-value anyway, but correctly interpreting a p-value requires that you also have an idea about the size of the effect. A p-value "in outer space" is worthless, although it is commonly seen in publications!).

Effect estimates in the form like "there was 5% higher occupation of species X on substrate B compared to A" is a bit difficult to interpret in terms of relevance. The meaning of this value depends on the proportion of species. To make an extreme point: if the pecies was absent on A but had a 5% occupancy on B, it shows that A and B show some extremely relevant (significant) difference w.r.t. the needs of species X. But of the proportion changed from 40% to 45% there is nothing much relevantly different. A good way to express this is log odds ratios are aused instead of differences between proportions. A standard linear model will estimate such log odds ratios if the used response values are log odds (therefore I would prefer a log odds transformation over the arcsine ("angular") transfromation).

As a last point I would mention that you do actually have binomial data, and there is no problem to fit a binomial model. There is no need to look for safety-belt solutions to use anovas and t-tests - just use the appropriate method for such kind of data. These methods are available and you have access to computers, so the calculations cannot be the problem.

Even if you don't want to consider the numbers of squares samples and you had only the information about the proportion (values between 0 and 1) you could use the beta-regression, what would be far better suited for proportions data than then anova/t-test methods. 

So you can do what Marcello suggested, but that's ignoring that there was quite a bit of development in the last 100 years. It's a bit like using a raft to cross a river if today we have modern ferries and even bridges right next to you.