Andrea, I disagree with:
" As I said, I really like this idea but would like to see a variety of examples in publications and I am not sure how easy it is to do this objectively (not purely based on judgement) and exhaustively. I must be able to detect the effect of a potential bias on all the questions I am investigating (it may many) as well as all the assumptions of the models in the analyses. I think Philipp mentioned path analysis as an option, which is something I'll have to learn more about." Philipp's point is mathematically correct, and does not require one or more arbitrary empirical examples comparing ME directions and signal to validate that the point stands. Both statisticians and morphometricians have known of this for some time. The approach he advocated is also completely objective, and defined based solely on the biological question being investigated. I do not see the process either as subjective or under-developed theoretically. However, I do think that empiricists have under-utilized the approach when they study ME, meaning that our empirical understanding of ME remains in its infancy. I read Philipp's email is a 'call to arms' to correct this, and hope that empiricists interested in ME correct this deficiency in their studies going forward. The issue is simple: if ME is random, all it does is obscure a biological signal. Thus statistically we may 'miss the signal', meaning ME is a cause for type II error. Apart from some potential disappointment in missing a biological signal, that is not a big deal. But what is a big deal is if the ME is systematically-biased. If the direction of that bias aligns with the direction of the signal I am investigating, then the ME has now accentuated the pattern. Conversely, if it runs counter to the pattern (the opposite direction), it obscures it incorrectly. Philipp was absolutely clear and correct on this point. So the question is, how does one check this? Well, one simply identifies the direction of signal that one is investigating in the dataset and compare this with the direction of ME. For instance, say one is examining sexual dimorphism. The direction of SD signal is the phenotypic vector connecting male and female means. Then, one can obtain the ME component of shape variation, identify its primary direction (PC1 of ME), and evaluate the difference in angular direction. For decades we in the GMM world have been doing similar angular comparisons for all kinds of things: comparing the direction of phenotypic trajectories, ecological differences, genotype-phenotype comparisons, estimated selection gradients vs. the direction of phenotypic variation, etc. Examples of the technique abound for all kinds of empirical questions. Also, for certain (non-ME) implementations, these vector direction comparison methods are found in existing software (geomorph, MorphJ, RRPP, and others). Note that while a 'canned' function with the name 'check.ME.direction' may not be present, the mathematical machinery underlying geomorph could be extracted for this purpose (e.g., in trajectory.analysis and other functions). In fact, if one has two row-vectors: one for the biological signal, and the other for the ME, obtaining the difference in angular direction is as simple as: Myangle <- acos(RRPP:::vec.cor.matrix(rbind(MEvector[[j]],signal.vector)))*180/pi So I agree with Philipp that this is an important consideration, and his suggestion makes good use of a well-known method that GMM folks have used for other purposes for decades. Going forward, I assert that for any good study where ME is a quantified and estimated as a component (e.g., via replicated digitizing etc.), and if downstream analyses are subsequently used on other factors in that study (species, sex differences, etc.), then one really should investigate the extent to which the direction of ME aligns with directions in the signal (population differences, sex differences, allometry, etc.). To not do this is to relegate ME to just a magnitude, which could lead to incorrect inferences. Dean Dr. Dean C. Adams (he/him) Distinguished Professor of Evolutionary Biology Department of Ecology, Evolution, and Organismal Biology Iowa State University https://faculty.sites.iastate.edu/dcadams/ phone: 515-294-3834 -----Original Message----- From: [email protected] <[email protected]> On Behalf Of andrea cardini Sent: Monday, November 7, 2022 3:09 PM To: [email protected] Subject: Re: [MORPHMET2] Measurement error in geometric morphometrics Hi Mike, you wrote something I am sure I misunderstood: "small measurement error can look larger when individuals are not that different in shape or large measurement error can look small if they are" ME is relative and scaling it to the variation in the sample is precisely what we need to do: if 'true' variation is large (say, in a macroevolutionary analysis with many different species and genera), a certain amount of ME might be negligible; if 'true' variation is small (with the same configuration, I may be studying small geographic variation among closely related populations), that same amount of ME could be too much. That will be the same for any type of measurement (CS, shape, traditional morphometrics). Your simulation seems to aim at relating the relative estimate of ME I get from the Procrustes ANOVA (the two R2s) to absolute variation due to digitization error on the original specimens. Am I correct? Philipp's point, in my interpretation, was about something else, which is to focus on whether ME bias results in relation to the very specific study question(s). If random, he argued, I could tolerate a very large ME and potentially even an ME that is larger than variation among individuals in a sample. For instance, he says that, if ME is random, it will end up in the last PCs and, if I use only the first ones, results will be unaffected. Even if I see the point from a statistical perspective, from a biological one, I worry about analyzing data that are mostly noise. If that's the only option, I might do it with due acknowledgement of the problem and big caveats: can I really be sure that such a huge error is not affecting results in subtle ways? Is the error in shape really random after the superimposition? For the within a configuration analyses of modularity/integration certainly it is not, as I showed with my simulations of isotropic variance: the bias will always be there, sometimes negligible and sometimes not (who knows?). Even with a simple PCA when p/N is huge, especially with slid semilandmarks, I get a pattern (dominant PC1) from isotropic noise. Again assuming I am correct, Philipp suggests to look for structure in the error component (the differences between replicates) to understand if it may bias results for the specific question I am asking. As I said, I really like this idea but would like to see a variety of examples in publications and I am not sure how easy it is to do this objectively (not purely based on judgement) and exhaustively. I must be able to detect the effect of a potential bias on all the questions I am investigating (it may many) as well as all the assumptions of the models in the analyses. I think Philipp mentioned path analysis as an option, which is something I'll have to learn more about. I feel more confident, for now, with Philipp's other point, which is the basic rationale of my approach: "one can argue that if measurement error is very small, then randomness and homogeneity across groups are less of an issue. But in this case the error really needs to be negligibly small, not just smaller than the individual variation" This is why I tend to use a fairly conservative approach and I am not happy with relying on a P value. The R2 of the effect I am interested must be much larger than that of ME and replicates should cluster tightly around each individual. If the design used for collecting the replicates is accurate, that should take care of the bias. If not, it is tricky because I could have an apparently small ME with an important bias. The example I mentioned of strong group structure in data collected in two chunks shows what happens with a flawed design: the colleague who measured the skull was confident that ME was negligible because he tested it in the first chunk of data collection; however, the way to assess ME correctly would have been having at least a subsample of the same individuals measured in both chunks of data collection. That would have included the effect of the long time between the data collections as well as the different Microscribe used for landmarking. I bet results would have been different (much larger ME relative to sample variance) from the original assessment of ME in the first data collection. With a well design set of replicates, in fact, we could have estimated the different sources of error, found where the problem was and maybe tried a correction. Unfortunately, he had replicates only for the first dataset. The way I found the problem is the same philosophy as in Philipp's message and, with limitations, that may sometimes work even without replicates: there was a pattern in the data (clear group separation on PC1 in relation to the time of data collection) and the most parsimonious explanation was ME. In the end, the bias was obvious and data could not be used. Nice discussion. Cheers Andrea On 04/11/2022 18:59, Mike Collyer wrote: > I agree with Philipp’s main point that it can be dangerous to quantify > measurement error as a value based on (likely a ratio including) the > variation among individuals on which the variation between repeated > digitizations is also made, if it is not clear how variable those individuals > are. I was seeking some examples to demonstrate that small measurement error > can look larger when individuals are not that different in shape or large > measurement error can look small if they are. I was not very successful > before Philipp responded. However, I did play with the “mosquito” data set > in geomorph, which led me in a different direction. I chose this data set > because it contains two replicate configurations for each individual. > > For context, here is the analysis I considered: > >> library(geomorph) >> data("mosquito") >> >> # use just one side for demonstration # resdual SS can be considered >> basis for measurement error >> >> lmks <- mosquito$wingshape[,, which(mosquito$side == 1)] ind <- >> mosquito$ind[ which(mosquito$side == 1)] GPA <- gpagen(lmks, >> print.progress = FALSE) summary(procD.lm(coords ~ ind, data = GPA)) > > Analysis of Variance, using Residual Randomization Permutation > procedure: Randomization of null model residuals Number of > permutations: 1000 Estimation method: Ordinary Least Squares Sums of > Squares and Cross-products: Type I Effect sizes (Z) based on F > distributions > > Df SS MS Rsq F Z Pr(>F) > ind 9 0.069286 0.0076984 0.62764 1.8729 2.6261 0.006 ** > Residuals 10 0.041105 0.0041105 0.37236 > Total 19 0.110390 > --- > Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > Call: procD.lm(f1 = coords ~ ind, data = GPA) > > It might be alarming that the residual Rsq is 0.37236, which is the portion > of variation attributed to multiple measurements on the same individuals. > That might seem high. I grew quickly tired of searching for a similar data > set with contrasting results and decided that maybe I could just simulate > measurement error and ask if the residual SS here was large compared to what > I simulated. I thought about this as a process and came to the conclusion > that one could simulate landmark wobble (like a shaky hand) by making the > standard deviation of wobble sampled from a normal distribution proportional > to a fraction of the centroid size. For example, a 5% error could mean that > the standard deviation for the distribution from which a random value is > sampled (x, y, or z coordinate) is 0.05 * CS for that configuration. (The > shakiness scales with the size of the object. > > I ended up making a function that could simulate a measurement error outcome. > Here is the function, in case anyone might find it useful (I have not tested > this, so please expect clunkiness…). One adds a set of coordinates (assumed > to be a 3d array), the number of replicates to simulate (the observed counts > as 1), and the percentage of centroid size to use to vary the sd of a random > sample from a normal distribution. It performs ANOVA for the simulated data > (following GPA). > > > makeME <- function(coords, reps = 2, per.error = 0.05){ # per.error means sd > = per.error * Csize > if(reps < 2) > stop("Must have more than 1 replicate to run this.\n") > dims <- dim(coords) > n <- dims[3] > p <- dims[1] > k <- dims[2] > > nms <- dimnames(coords)[[3]] > if(is.null(nms)) nms <- paste("spec", 1:n, sep = "") > Coords <- lapply(1:n, function(x) as.matrix(coords[,, x])) > nnms <- paste(rep(nms, each = reps), 1:reps, sep = ".rep") > > newCoords <- rep(Coords, each = reps) > names(newCoords) <- nnms > initGPA <- gpagen(coords, print.progress = FALSE, max.iter = 1) > Csize <- rep(initGPA$Csize, each = reps) > err <- rep(c(0, rep(per.error, reps - 1)), n) > for(i in 1:length(err)) newCoords[[i]] <- newCoords[[i]] + rnorm(p * k, sd > = err[i] * Csize [i]) > newCoords <- simplify2array(newCoords) > > GPA <- gpagen(newCoords, print.progress = FALSE) > > ind <- factor(rep(1:n, each = reps)) > return(summary(procD.lm(coords ~ ind, data = GPA))) } > > And as an example application, using the same data as above: > >> makeME(mosquito$wingshape[,, which(mosquito$side == 1)]) > > Analysis of Variance, using Residual Randomization Permutation > procedure: Randomization of null model residuals Number of > permutations: 1000 Estimation method: Ordinary Least Squares Sums of > Squares and Cross-products: Type I Effect sizes (Z) based on F > distributions > > Df SS MS Rsq F Z Pr(>F) > ind 19 0.91707 0.048267 0.56455 1.3647 3.2186 0.002 ** > Residuals 20 0.70736 0.035368 0.43545 > Total 39 1.62442 > --- > Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > Call: procD.lm(f1 = coords ~ ind, data = GPA) > > So I might conclude from this that if I allowed my digitizing to vary > by 5% of centroid size, it appears my observed digitization has a > measurement error less than that, which might help me to feel > confident. In case I worry that this one random outcome is not fully > representative, the following function allows me to run many > simulations (100 as an example) > > > simulate.makeME <- function(coords, reps = 2, per.error = 0.05, nsims = 100) { > result <- sapply(1:nsims, function(j) { > cat("sim:", j, "... ") > res <- makeME(coords, reps, per.error) > res$table$Rsq[2]} > ) > cat("\n\n") > names(result) <- paste("sim", 1:nsims, sep = ".") > result > } > >> ME.sims <- simulate.makeME (mosquito$wingshape[,, which(mosquito$side >> == 1)], reps = 2, per.error = 0.05, nsims = 100) >> summary(ME.sims) # just Rsq > Min. 1st Qu. Median Mean 3rd Qu. Max. > 0.4264 0.4423 0.4474 0.4476 0.4533 0.4729 > > So now I feel really confident that measurement error is probably not a > worry, based on results from a process that imposes a certain level of > measurement error. > > I might also start to wonder when imposing the randomness starts to approach > what I see in my empirical example. > >> makeME(mosquito$wingshape[,, which(mosquito$side == 1)], per.error = >> 0.03) > > Analysis of Variance, using Residual Randomization Permutation > procedure: Randomization of null model residuals Number of > permutations: 1000 Estimation method: Ordinary Least Squares Sums of > Squares and Cross-products: Type I Effect sizes (Z) based on F > distributions > > Df SS MS Rsq F Z Pr(>F) > ind 19 0.49153 0.025870 0.62935 1.7873 5.7972 0.001 ** > Residuals 20 0.28948 0.014474 0.37065 > Total 39 0.78101 > --- > Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > Call: procD.lm(f1 = coords ~ ind, data = GPA) > > These results mimic my observed empirical results pretty well. Maybe I can > infer from this that my digitizing could off by as much as 3% and produce > results like I observed? > > This is a different way of approaching the problem than calculating and > trying to make sense of statistic that might resemble an effect size, but it > feels more informative to me. I am not sure that it is smart to scale the > amount of variation with centroid size — one might have large and small > individuals but can zoom in or out to better capture landmark locations — so > the function could be rewritten to not include centroid size as variable. > This was done so that the simulated error was made for digitized specimens, > but could be done on configurations already constrained to be unit size > (after GPA). I am also not sure that it is smart to sample from a normal > distribution. Maybe sampling from a uniform distribution would better > resemble digitizing shakiness. I only wandered so far into the weeds with > this. > > I think this might qualify as an additional exploratory approach and agree > with Philipp that making sense of the magnitude and directions between > repeated measures, even if only viewed in a PC plot, is rather important. > I’m sure this could be improved if someone wants to play more with other data > sets. > > Cheers! > Mike > >> On Nov 4, 2022, at 10:38 AM, [email protected] <[email protected]> >> wrote: >> >> Dear all, >> >> I like to challenge this view on measurement error, as summarized by Andrea, >> a bit more generally. >> >> Clearly, measurement error should be "small," but I disagree that "the idea >> is that differences among individuals (averaged replicates) in a >> representative sample should be larger than differences between replicates >> of the same individual". First, the between-individual variance (or mean sum >> of squares, MSS) depends on the choice of individuals. For instance, if the >> sample comprises different species, the MSS between individuals is much >> larger than for a sample of a single species, and the error MSS in relation >> to the individual MSS is much smaller in the multi-species sample. Hence, >> whether or not the error MSS is larger than the between-individual MSS is >> somewhat arbitrary and of secondary importance anyway. "Controlling for main >> effects," as suggested by Andrea, is possible but it removes the actual >> signal against wich I may want to compare the error. In either case, the >> p-value of the MANOVA is uninformative because the underlying H0 is >> irrelevant. >> >> In my opinion, it is more important that the error is unrelated to the >> signal of interest ("random"), rather than that it is small in terms of some >> summary statistic. For instance, if in a growth study the measurement error >> is uncorrelated with the age effects, the error "averages out" (if sample >> size is large enough) and does not bias the average growth trajectory, even >> if the error is large. The same applies to group differences. MANOVA does >> not inform about this independence. Moreover, it pools over all shape >> coordinates. For instance, it does not inform us if the error is large for >> shape features of interest (those that differ between groups or correlate >> with age, etc.) or for shape features of less interest. >> >> Note also that most morphometric analyses are based on a few principal >> components (or similar statistics) of the shape coordinates. PCs are linear >> combinations, i.e., weighted averages, of the shape coordinates. Hence, >> group means in a PC plot are averages over all cases AND all variables, so >> that random error can be expected to be small. Anther issue to consider: If >> measurement error is indeed approximately isotropic, it has a similar >> magnitude for all shape features (all directions of shape space). The >> individual variance, however, typically is much greater for large-scale >> shape features than for small-scale features, and the relative magnitude of >> measurement error decreases with increasing spatial scale. PCs typically >> capture large-scale shape variation, where the relative error is expected to >> be smaller. The same applies to the symmetric vs. asymmetric components, the >> latter of which has much smaller individual variance and hence greater >> relative measurement error. >> >> The situation is slightly different in studies that compare shape variances, >> not means, between groups, between symmetric and asymmetric components, or >> among spatial scales. In contrast to mean estimates, measurement error does >> not average out for these variance estimates. It is thus important that >> magnitude and pattern of measurement error are constant (not necessarily >> small) across groups or components so that observed differences in variance >> are attributable to biological factors rather than systematic differences in >> measurement error. Measurement error is most challenging when comparing >> entire variance-covariance matrices. But again, MANOVA is not the way to >> assess homogeneity of measurement error across groups. >> >> If the sample is properly randomized before measurement, it is >> reasonable to assume that measurement error is approximately >> uncorrelated with the signal of interest. But there can be >> exceptions. For instance, younger and smaller individuals can be >> harder to measure than older and larger individuals. Measurement >> error can thus correlate with age. I discussed this in Mitteroecker >> P, Stansfield E (2021) A model of developmental canalization, applied >> to human cranial form. PLOS Computational Biology 17 (2): e1008381 >> >> Clearly, one can argue that if measurement error is very small, then >> randomness and homogeneity across groups are less of an issue. But in this >> case the error really needs to be negligibly small, not just smaller than >> the individual variation. >> >> Instead of somewhat meaningless scalar summary statistics (like the F-ratio >> or some multivariate version of it), I thus prefer an exploratory approach. >> In the simplest case, a PCA of the data, including the replicated specimens, >> can show the magnitude and directionality of measurement error in relation >> to the signal of interest (e.g., group differences, growth trajectories). >> Measurement error can also be correlated with external variables (e.g., age) >> or compared among groups, but to my knowledge little work has been done in >> this direction in geometric morphometrics. An alternative are >> errors-in-variables models and structural equation models that implement >> estimates of measurement error in the first place. >> >> Best, >> >> Philipp M. >> >> >> >> >> >> [email protected] <http://gmail.com/> schrieb am Donnerstag, 3. November >> 2022 um 16:36:21 UTC+1: >>> Dear All, >>> beside the excellent review by Carmelo, I suggest a few other papers >>> on ME in geometric morphometrics: >>> Arnqvist, G., Martensson, T. Measurement error in geometric >>> morphometrics: empirical strategies to assess and reduce its impact >>> on measures of shape. Acta Zoologica Academiae Scientiarum >>> Hungaricae, 1998, 44: 73–96. (A bit outdated but still wonderfully >>> accurate in how they explain different sources of ME). >>> Klingenberg, C.P., Barluenga, M., Meyer, A. Shape Analysis of >>> Symmetric Structures: Quantifying Variation Among Individuals and >>> Asymmetry. Evolution, 2002, 56: 1909–1920. (From where most of us >>> have borrowed the protocol for assessing ME). >>> Viscosi, V., Cardini, A. Leaf Morphology, Taxonomy and Geometric >>> Morphometrics: A Simplified Protocol for Beginners. PLoS ONE, 2011, 6: >>> e25630. >>> Galimberti, F., Sanvito, S., Vinesi, M.C., Cardini, A. “Nose-metrics” >>> of wild southern elephant seal (Mirounga leonina) males using image >>> analysis and geometric morphometrics. Journal of Zoological >>> Systematics and Evolutionary Research, 2019, 57: 710–720. >>> >>> There's also another one I like, by the Viennese morphometricians >>> (in a paper on human mandibles, or teeth, symmetric and asymmetric >>> variation, if I remember well), but I can't find it now. >>> >>> >>> In general, the idea is that differences among individuals (averaged >>> replicates) in a representative sample should be larger than >>> differences between replicates of the same individual (the estimate >>> of ME). This is what is tested by 'individual' in the Procrustes >>> ANOVA in MorphoJ. It might be important to control for main effects >>> in the analysis. For instance, by including species and sex before >>> individual in the hierarchical analysis, I 'statistically remove' >>> (with some >>> assumptions) the average effect of these factors before comparing >>> individual variation to ME, which makes the test more conservative >>> (NB whether this is OK or not it depends on the question one is >>> asking in her/his study). >>> For shape data, even if the P value of individual vs residual is >>> significant, I would not conclude that ME is negligible for sure. >>> I'd check that the individual Rsq is much larger than the ME >>> (residual) Rsq and also that shape distances between replicates of >>> the same individual are smaller than distances among different >>> individuals (if this is true, replicates should cluster 'within >>> individual' in a UPGMA phenogram). Then, I feel a bit more confident >>> that ME might be negligible. >>> >>> If ME is large, it may happen that its Rsq is larger than the >>> individual Rsq (or, which is the same ME SSQ > individual SSQ). For >>> the F ratio, however, one should look at the mean SSQ, which take df >>> into account. From the MSSQ, one computes F. >>> The F ratio in MorphoJ employs an isotropic model but, with large >>> samples (relative to the number of variables), the software also >>> provides P values using Pillai, that does not depend (if I recall >>> well!) on an isotropic model. That N is large and the sample >>> representative is crucial if one is using a subsample in the >>> assessment of ME to avoid replicate measurements of all individuals, >>> which would be better but might take too long if one has hundreds or >>> thousands individuals. >>> In R, I generally use adonis that employs an F test (same as in >>> MorphoJ, for a simple design) but uses permutations instead of >>> parametric tests. The use of permutations was also suggested as >>> desirable in Klingenberg et al., 2002. Other packages I suspect >>> might do something similar, although maybe using different >>> permutational approaches. I am sure it is explained in their help files. >>> >>> Cheers >>> >>> Andrea >>> >>> On 03/11/2022, ying yi <[email protected] <>> wrote: >>>> Dear all, >>>> I used the “procD.lm” function in the geomorph package to test the >>>> measurement error. I was surprised to find that the within-groups >>>> ANOVA sum >>>> >>>> of squares I got was greater than the among-groups ANOVA sum of >>>> squares. I >>>> >>>> wonder if something went wrong. What does it mean for “procD.lm” >>>> function to get an F value <1? >>>> I would be very happy if someone could help me. >>>> Yours, >>>> Sam >>>> >>>> References are as follows: >>>> >>>> -- >>>> You received this message because you are subscribed to the Google >>>> Groups "Morphmet" group. >>>> To unsubscribe from this group and stop receiving emails from it, >>>> send an email to [email protected] <>. >>>> To view this discussion on the web visit >>>> https://groups.google.com/d/msgid/morphmet2/06065841-c42e-4a58-a5d3-a96eb3c5787dn%40googlegroups.com. >>>> >>> >>> >>> -- >>> E-mail address: [email protected] <>, [email protected] <> >>> WEBPAGE: https://sites.google.com/view/alcardini2/ >>> or https://tinyurl.com/andreacardini >> >> >> -- >> You received this message because you are subscribed to the Google Groups >> "Morphmet" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected] >> <mailto:[email protected]>. >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/morphmet2/9f7a7818-f6c2-446c-aec8-f66f5f2c730cn%40googlegroups.com >> >> <https://groups.google.com/d/msgid/morphmet2/9f7a7818-f6c2-446c-aec8-f66f5f2c730cn%40googlegroups.com?utm_medium=email&utm_source=footer>. > -- Dr. Andrea Cardini Researcher, Dipartimento di Scienze Chimiche e Geologiche, Università di Modena e Reggio Emilia, Via Campi, 103 - 41125 Modena - Italy tel. 0039 059 4223140 Adjunct Associate Professor, Centre for Forensic Anthropology, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia E-mail address: [email protected], [email protected] WEBPAGE: https://sites.google.com/view/alcardini2/ or https://tinyurl.com/andreacardini -- You received this message because you are subscribed to the Google Groups "Morphmet" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/morphmet2/0b88bb60-18a1-325b-9234-7bca5013c09f%40gmail.com. -- You received this message because you are subscribed to the Google Groups "Morphmet" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/morphmet2/CO6PR04MB8427F5070653BBF200A29215A23C9%40CO6PR04MB8427.namprd04.prod.outlook.com.
