The approach using a dummy variable and looking for interaction terms will work. It only works for comparing models of the same type (transformed or not). The pooling idea is the start of trying randomization tests (or computer intensive methods in general), but it will not work well if used in the way that you suggest.

The randomization test starts by treating the data as if the null hypothesis was true. One then calculates the probability of withdrawing two samples from your data that have a difference that is equal to or greater than the observed difference. So you resample the data, calculate differences, order these differences, and see how many are equal to or larger than the observed difference. Divide that by the total number of randomizations, and you have your answer.

Robert -

First, are you sure you get linear relationships? Perhaps, as in calibration of equipment in analytical chemistry, they are only approximately linear in a given range? I especially think that because you said you get intercepts, and if regression does not go through (actually to) the origin, you have to have snails with size and no mass, or mass and no size.

A similar question was asked on RG about fish. The question was about a linear relationship between length and mass, but that was, as in some of your cases, actually semilog linear. I suggested then that the problem was that there is length, width, and depth.  You can use logs, though I would avoid transformations if possible, to make interpretation of results more straightforward. (A good book that you might find helpful would be Transformation and Weighting in Regression by Carroll and Ruppert.)

So when you note that you have size measures, what are they? Is it a volume measure? That sounds best. Actually it sounds easiest to measure that by water displacement. - Eureka!

:-)

Hypothesis testing needs to be done carefully, as a single p-value is a function of sample size and not very informative. (See first attached link. Also, I have a link on my RG page to another document where there is an example of an hypothesis test - Wilcoxon Rank Sum in that case - where I showed an example of the kind of analysis needed - or some other sensitivity analysis.) So I generally encourage the use of confidence intervals, rather than p-values.

If you did have linear regressions going through the origin, then you could compare the slopes in light of their standard errors. The different sample sizes won't interfere with the analysis, though it just won't likely be the optimal situation. You could look at any pair of slopes (b-value estimates) and get a standard error for the difference between them. Then you could form a confidence interval around the difference between those two b-values. If a reasonable such confidence interval (say 90% or 95% or whatever is reasonable for your study) contains zero, then the slopes may not be substantially different. (I like the word "substantial" because it conveys a practical matter, whereas the word "significant" has already been taken - badly. A statistical test in reality cannot generally say this is true or that is true. We really need to know About how close is our hypothesis to the truth? Confidence intervals can do that better.)

So you can look at slopes, but if there is a (substantial?) nonzero intercept, even if it is out of range of the linear portion of the relationship, then slopes are not all there is to this.

I think one of the very best actions you can take is to plot all four of these relationships on the same scatterplot. The visual impact may tell you both a great deal about what you have, and what you want to know next.

You will likely notice heteroscedasticity in your data in such a scatterplot. That is, variance in y becomes larger as x increases. (If estimated residuals are small, this may not show visually as well.) This is quite natural. You would expect this. You would not expect in some other case to see costs of $10,000 plus or minus $100, and then $1000 plus or minus $100, and then $100 plus or minus $100, and $10 plus or minus $100. So you may see that heteroscedasticity is expected. (See the second link please.) For hypothesis tests, these data would need to be transformed from their natural state, but as I think confidence intervals better, you do not need to try to " remove" this from the natural error structure.

At the least then, you could make the scatterplot of all four x-y data sets together, and for each case estimate slope and intercept and their corresponding standard errors. (If an estimated intercept is not much larger than its estimated standard error, it might be considered expendable. If you can use regression through the origin, then the other two links below may be helpful.)

Cheers - Jim   

Article Practical Interpretation of Hypothesis Tests - letter to the...

Article HETEROSCEDASTICITY AND HOMOSCEDASTICITY

Data CRE Prediction 'Bounds' and Graphs Example for Section 4 of ...

Technical Report A Note on Estimating Impact of Data Collection Mode Change f...

Robert - I have made a great deal of use of linear regression, but I still wonder about it a little in the case you have here. Perhaps there is a very nearly linear relationship between mass and volume, but that would imply that as the sea snails grow, the proportion of fat, etc, in their bodies stays the same. Well, perhaps for snails this works. - Can you tell how old they are? Is age a factor? - Perhaps linearity works here. As noted above, scatterplots might help study this. - Jim

Thank you for these answers, I will have to chew this over. Tim, thank you for the info about randomization.  Sounds promising.

Jim, the short answer is no I'm not sure they are all linear.  There are many kinds of critters, not just snails, and the size estimates are mostly just maximum length.  I have little to no idea of their age. We do have data on basic body composition though (% water, %ash) so that might be worth looking at with size. The idea was to do nondestructive measurements underwater that would still enable biomass to be calculated.  Many of the relationships do go through zero, especially the logged ones, but some don't and I think you are right in that growth is more rapid when the critters are small.  We probably are not measuring the really small ones because they are either hard to find or they don't stay small very long.  I like the idea of the scatterplots you suggest, I'll try that. 

James' answer was good, and based on your response I think we need to know the final goal for this project. Are you trying to compare the length to mass relationship between fish and snails, or is it just subtidal inverts, or ??? What are you expecting to find, or what is your null hypothesis.

I guess what I would like to avoid is telling you a bunch of things that you already know. I note that you have worked with mussels, sea urchins, and fish (and probably a few other things like snails). Your current task is something like the effect of nanoparticle size in heavy metal toxicity. So maybe this is trying to get a fast estimate of biomass by using a length to biomass conversion that would then allow you to estimate the toxicity of nanoparticles on a mg toxicant/kg body weight basis. I can't quite figure out why one would want to compare weight-size relationships between organisms in this context, so I am missing pieces.

So the best way is to do an age specific growth model that lets you model age of first reproduction, and senescence. Run a mark-release-recapture experiment using tetracycline so that you can measure growth in calcified pieces. Get size frequency data to estimate recruitment. Combine these pieces together with toxicology data to get an age specific estimate of toxicity. It will only take a small army of graduate students to accomplish.

That sounds like a great project!  This is not related to the nano work.  It is part of a long-term project investigating change in kelp forest/rocky reef communities.  As part of this we are conducting an experiment that manipulates abundance of giant kelp by doing clearings every year that mimic intense storm disturbance.  The basic prediction is that increased storm intensity/frequency will change the communities over time, lowering diversity and food web complexity and favoring macroalgae over sessile invertebrates (which are hypothesized to benefit from shading by the kelp which relieves them from competition with algae). Because we keep track of a wide variety of organisms that vary in body size and abundance, we think biomass is the best, or at least a very useful, measurement of their abundance.  But since we are monitoring the plots long term, we wanted to estimate biomass using techniques that don't disturb the community.  These techniques vary with the type of organism - urchins and snails for example are measured, whereas percent cover is measured for encrusting organisms.  For the latter, we did percent cover measurements on a sample of plots then scraped up the organisms and measured biomass. 

Robert - I would second the idea of using interactions in a linear model.  Other suggestions to at least try out if you haven't already:

- Try a cube root transformation on the weight outcome.  While I too try to avoid transformations, this has some physical grounding since an object needs three dimensions to have a volume/weight.  This may also reduce hetereoscedasticity and result in more accurate hypothesis tests.

- Consider a linear model where you force the intercepts to be 0. Then you would only have intercepts to compare.  Again, this has good physical interpretation since a length of 0 should imply a weight of 0.

- I would also highly suggest plotting the data with and without the suggestions above (mine and the others) to see which ideas fit best.  This sounds like a long-term project, so you may be able to set up some structure now and check it using future data.

PS. I spent 2011-2013 as visiting faculty in the UCSB statistics department and was always jealous of the marine science folks and their field studies - maybe I studied the wrong subject...

Please be careful with regression through the origin. Yes, an organism of zero size should have zero weight. However, at this scale this is a discontinuous function. At the egg stage (if you call that an organism), it might be a 100 micron diameter ball of cells. There is no egg at 5 microns. The organism consists of many more cells before it gets a shell. It may go through a planktonic form before it arrives at the sea floor and gets measured. There is no basis for assuming that the relationship from zero to the smallest measurable organism is the same as for the smallest individual to oldest individual. Sometimes we get "lucky" and the difference is not measurable given the variability in our data.

The zero intercept is creating data at a location where we have no actual measurements. It will therefore distort the 95% confidence interval about estimates involving the smallest measured individuals because there is now an artificial data point that is outside the measurable range in the data.

I need to think about the original question a bit more, given the new information.

So basically you have a huge database of body length measurements and you want to correct these measurements for differences in body shape because one assumes that biomass is a more appropriate variable to use in a community ecology project. You are interested in using the fewest possible number of equations in the ecology model.

I will suggest that you want to lump the fewest number of species into one model as possible. For one species there is error in converting length to biomass because each individual is somewhat unique. This error will increase as you lump species together. Remember the test for significance is only giving you the probability of observing a value as extreme or greater given that the null-hypothesis is true. Failing to reject the null-hypothesis is not the same as proving that the null-hypothesis is true. There is also some measurement error when measuring length. This error is then multiplied when you convert length to weight. Your combined error could start to significantly degrade your model. It might be something to explore with sensitivity analysis ... but that is a project unto itself.

Excellent points Timothy, thanks for your input.