The following link can be helpful.

Kevin - 

This isn't a subject matter area for me, but I was curious about the heteroscedasticity.  As a size measure gets larger, sigma for estimated residuals should get larger.  So as group size gets larger, not only will total productivity get larger, but I'd also expect to have greater variance.  However, you are looking at per capita production.  The larger the group, the lower the variance of the mean.  I don't think I'd really call that a reverse heteroscedasticity, it's just that standard error is always reduced with larger sample sizes, which here is the group size. 

You say that "Our question is whether the number of individuals in a group increases per-capita productivity of that group." If you put group size on the x-axis as you do, and total productivity on the y-axis, then if there were a straight-line relationship (not just a linear relationship, which refers to equation form), then the slope would be total productivity per group size, i.e., a constant per capita productivity, and I would expect it to go through (actually to) the origin.  Since you expect greater per capita productivity for a larger group, then you should have a line curving upward.  The tangent at any one point being the per capita productivity.  You could try a polynomial regression before trying nonlinear regression.

Just some thoughts.

Cheers - Jim

Hi James R Knaub , thanks so much for your thoughts. I agree that looking at the shape of the relationship of (non-logged) size vs (non-logged) total productivity would be a way of analyzing this...in fact, that's what some studies do. If a polynomial term is significant, they interpret that as evidence for either increasing or decreasing per capita productivity with size. The predicted coefficient for the squared term is zero if there is no effect of size on per-capita productivity.

Our main goal is to come up with a standard effect size to compare across studies in the meta analysis. I don't see an easy way to quantitatively compare the coefficient of that squared term across studies, as it depends on the scale of the x and y axes for each study and the shape of the distribution. If we standardized them in an attempt to get all variables on the same scale, i think we'd run into issues given how skewed the data often are. The thinking behind the log-log regression is similar, but the slope should be comparable across studies (does an n-fold increase in group size result in less than or greater than an n-fold increase in total productivity?), and the transformed data lose the heteroscedasticity.

I sort of get what you're saying about heteroscedasticity. Groups with more individuals have a better estimate of the true average per-capita output of that group, causing the variance in those big groups to be lower. However, does that matter for the correlation/regression of size vs per capita productivity? I think it depends on whether we view per-capita productivity as a property of an individual or a group, right? If it's a group-property, then i think the data are clearly heteroscedastic. If the group per-capita value is merely an average of the intrinsic productivity values of the individual group members (i.e. no interactions/non-linearities among group members, just a straight sum of their productivities), then maybe there wouldn't be an issue since the underlying variance in individual per capita productivity could be the same across group sizes, we just have better estimates in large groups due to sample size. But i think we don't want to assume that, and it makes more sense for our biological problem if we consider per-capita productivity as a property of the group itself, not the individuals within it.

Thanks again for thinking about it with me, much appreciated!

Best,

Kevin

Hi Kevin,

One way of dealing with heteroscedascity in R is applying the White correction with the Anova function in library(car). i.e. Anova(“your_linear_model” , white.adjust=T). Otherwise, as you already suggested, I would think transforming the dependent variable or covariate(s), or perhaps adding a polynomial term is the way to go. If I understood well your question (and provided the metrics for productivity and population size are standardized across the studies) I think it would be interesting to add an interaction term between species and population size and look at the slopes. In that way, you would see whether the effect of population size in productivity is the same across the different species or not.

Good luck with the study!

Cheers,

Ricardo

The following link can be helpful.

Hey Ricardo Caliari Oliveira , thanks for your thoughts! I'll look into the White correction, i haven't heard of that. I also realized i might be able to just bootstrap the 95% CIs for the correlation coefficients with non-normal distributions, which may be the most straightforward way to go. The meta-analysis/meta-regression framework will be cool for looking at variation across species and effects of phylogeny, colony size, etc, to get at the interspecific variation. Hope you're doing well.

Please note that the line in the first scatterplot appears to be of the form

y = a - bx + e.

So, when the group size is zero, the per capita productivity is predicted to be largest. Of course, this would be a problem. One could say that this was 'out of range,' but as I state below, I think it best not to consider the line there as a regression, though we can.

I just realized that the positive intercept was there, so the size measure is not

y* = bx (where x as the relative size measure would be OK because b is constant), but rather we have

y* = a - bx.

(The best size measure is predicted y, y*.)

Thus the size measure gets larger as you go to the left, not right, on the x-axis, in the plot shown. If you did a graphical residual analysis with y* on the x-axis, and estimated residuals, or y, on the y-axis, this would no longer appear to be 'backward.'

Naturally occurring heteroscedasticity in regression is a result of the varying size of predicted y. (Other model and data issues can also impact heteroscedasticity.) If you predict y from one variable, x, here group size, in this case you can use that to predict total productivity. However, (mean) per capita productivity is not predicted by group size so much as group size can be considered the sample size, in each case, when taking a mean to use as per capita productivity. In other words, instead of thinking of the first scatterplot graph as the results of a regression, it may be considered to be the graphical representation of samples (groups) which have been 'averaged,' and their means plotted on the y-axis, with the group sizes on the x-axis, almost like a type of histogram, in a way. So, group size is a size measure for total productivity in a regression, but for (mean) per capita productivity, group size is better thought of as a sample size. These are different phenomena.

In a simple regression with no intercept term, y = bx + e. Here, we could have y as the total productivity, and x is group size. Group size is on the x-axis. However, for a more complex case, y* could be any prediction of y, perhaps from a multiple linear regression, rather than just y* = bx, or even y* = a - bx + e, and you would plot y* on the x-axis. Often instead of y on the y-axis, you plot e there. These plots are called "graphical residual analyses."

In the first plot, it just appears that we have regression with a negative coefficient of heteroscedasticity***. But I don't think that I would think of it as a regression, although you might use it that way. However, there probably is a natural heteroscedasticity, which "fans out" in the opposite way shown, if you use total productivity as y. (A "fan shape" often occurs, but heteroscedasticity is not always so noticeable. Variability may be too small to easily show, or there may be a decrease in density of estimated residuals in the y-direction as you go to larger predicted values, without an increase in range in the y-direction. - Still, fewer values in the same range means greater sigma.)

Here we can show that the heteroscedasticity is in the usual direction, because the line in the scatterplot in the question is for

y = a - bx + e.

When per capita productivity is related to group size, and we look at that first scatterplot, you want to see if the means (of those means or per capita productivity values) differ, and the line indicates that they do, though I don't think it should be a straight line. The standard deviations within each group likely differ too, but standard errors definitely do differ and become smaller with larger group/sample size, as you divide standard deviations by the square root of n (i.e., sample size) to get standard errors.

--------------------

***For the nature of heteroscedasticity of estimated residuals in regressions of form

y = y* + (e_0)(z^gamma),

for finite populations, see:

https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity, as determined by Ken Brewer.

To estimate the coefficient of heteroscedasticity, see

https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity

and

https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx.

For more information, see

https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression, and updates to that project.

Kevin -

Towards the bottom of your February 11 notes, there is this: 

"If the group per-capita value is merely an average of the intrinsic productivity values of the individual group members (i.e. no interactions/non-linearities among group members, just a straight sum of their productivities), then maybe there wouldn't be an issue since the underlying variance in individual per capita productivity could be the same across group sizes, we just have better estimates in large groups due to sample size." 

Well, it wouldn't be "...the same across group sizes..." if the regression using total productivity is not a straight line.  In my first response, I noted that tangents (first derivatives) along the regression equation, perhaps for a log or exponential or quadratic (now i think perhaps better nonlinear than quadratic linear here), could show per capita results that are different for larger total productivity cases versus smaller ones, but within each group the per capita values are the same, just because "per capita" indicates a mean.  But different "group sizes" could have different per capita values, one each, the values you want to use, one per group, right?   

Your question discussion said "Our question is whether the number of individuals in a group affects the per-capita productivity of that group. Does increasing group size reduce per-capita output?"  I think you can answer that question with the total production regression I mentioned in my first response.  You did not ask about different member productivity in the same group.  Did you?  That sounds like a different question.  But, regarding heteroscedasticity, at the bottom of page 3 in https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity, I did look at infinitesimal building-blocks, equal-sized pieces of an overall group, but each member of your group can be broken down further so that they fit the same scheme. 

The "interactions/non-linearities among group members" impacts how you predict y-values from the x-values, but y-values and predicted y-values are still additive, whether they are clumped or not. 

Perhaps I misunderstand your notes. 

One other point of interest:  On your first plot, which i have been noting, you have a case with only one member in a group, where the productivity is 14.  Further, there is one group of 11, which has a per-capita productivity of just over 1, so for the whole group it must be somewhere near 14.  So those two groups, extreme, each in their own way, both have about the same total productivity.  That, along with the other data in that scatterplot graph, indicate a great deal of variance to me.  Just sayin'. 

Cheers - Jim