{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T03:39:24Z","timestamp":1790825964033,"version":"4.1.0"},"reference-count":53,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,8]],"date-time":"2020-01-08T00:00:00Z","timestamp":1578441600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Data"],"abstract":"<jats:p>Germination data are discrete and binomial. Although analysis of variance (ANOVA) has long been used for the statistical analysis of these data, generalized linear mixed models (GzLMMs) provide a more consistent theoretical framework. GzLMMs are suitable for final germination percentages (FGP) as well as longitudinal studies of germination time-courses. Germination indices (i.e., single-value parameters summarizing the results of a germination assay by combining the level and rapidity of germination) and other data with a Gaussian error distribution can be analyzed too. There are, however, different kinds of GzLMMs: Conditional (i.e., random effects are modeled as deviations from the general intercept with a specific covariance structure), marginal (i.e., random effects are modeled solely as a variance\/covariance structure of the error terms), and quasi-marginal (some random effects are modeled as deviations from the intercept and some are modeled as a covariance structure of the error terms) models can be applied to the same data. It is shown that: (a) For germination data, conditional, marginal, and quasi-marginal GzLMMs tend to converge to a similar inference; (b) conditional models are the first choice for FGP; (c) marginal or quasi-marginal models are more suited for longitudinal studies, although conditional models lead to a congruent inference; (d) in general, common random factors are better dealt with as random intercepts, whereas serial correlation is easier to model in terms of the covariance structure of the error terms; (e) germination indices are not binomial and can be easier to analyze with a marginal model; (f) in boundary conditions (when some means approach 0% or 100%), conditional models with an integral approximation of true likelihood are more appropriate; in non-boundary conditions, (g) germination data can be fitted with default pseudo-likelihood estimation techniques, on the basis of the SAS-based code templates provided here; (h) GzLMMs are remarkably good for the analysis of germination data except if some means are 0% or 100%. In this case, alternative statistical approaches may be used, such as survival analysis or linear mixed models (LMMs) with transformed data, unless an ad hoc data adjustment in estimates of limit means is considered, either experimentally or computationally. This review is intended as a basic tutorial for the application of GzLMMs, and is, therefore, of interest primarily to researchers in the agricultural sciences.<\/jats:p>","DOI":"10.3390\/data5010006","type":"journal-article","created":{"date-parts":[[2020,1,9]],"date-time":"2020-01-09T03:07:11Z","timestamp":1578539231000},"page":"6","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["Basic Features of the Analysis of Germination Data with Generalized Linear Mixed Models"],"prefix":"10.3390","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0919-3956","authenticated-orcid":false,"given":"Alberto","family":"Gianinetti","sequence":"first","affiliation":[{"name":"Council for Agricultural Research and Economics\u2014Research Centre for Genomics and Bioinformatics, via S. Protaso 302, 29017 Fiorenzuola d\u2019Arda (PC), Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bewley, J.D., Bradford, K.J., Hilhorst, H.W.M., and Nonogaki, H. (2013). Seeds: Physiology of Development, Germination and Dormancy, Springer. [3rd ed.].","DOI":"10.1007\/978-1-4614-4693-4"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1105\/tpc.9.7.1055","article-title":"Seed germination and dormancy","volume":"9","author":"Bewley","year":"1997","journal-title":"Plant Cell"},{"key":"ref_3","unstructured":"Stroup, W.W. (2013). Generalized Linear Mixed Models: Modern Concepts, Methods and Applications, CRC Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1017\/S0021859600041642","article-title":"The analysis of data from germination tests","volume":"102","author":"Hunter","year":"1984","journal-title":"J. Agric. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Quinn, G.P., and Keough, M.J. (2002). Experimental Design and Data Analysis for Biologists, Cambridge University Press.","DOI":"10.1017\/CBO9780511806384"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"e4794","DOI":"10.7717\/peerj.4794","article-title":"A brief introduction to mixed effects modelling and multi-model inference in ecology","volume":"6","author":"Harrison","year":"2018","journal-title":"PeerJ"},{"key":"ref_7","unstructured":"B\u00e1nyai, J., and Barab\u00e1s, J. (2002). Handbook on Statistics in Seed Testing, International Seed Testing Association."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1017\/S0960258512000025","article-title":"A critique of current trends in the statistical analysis of seed germination and viability data","volume":"22","author":"Sileshi","year":"2012","journal-title":"Seed Sci. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"811","DOI":"10.2134\/agronj2013.0342","article-title":"Rethinking the analysis of non-normal data in plant and soil science","volume":"107","author":"Stroup","year":"2015","journal-title":"Agron. J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1002\/9780470988848.ch5","article-title":"Genetic Aspects of Seed Dormancy","volume":"Volume 27","author":"Bentsink","year":"2007","journal-title":"Seed Development, Dormancy and Germination"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1017\/S0960258508037999","article-title":"Seed dormancy in red rice. XIII. Interaction of dry-afterripening and hydration temperature","volume":"18","author":"Gianinetti","year":"2008","journal-title":"Seed Sci. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1017\/S0960258516000076","article-title":"Anomalous germination of dormant dehulled red rice seeds provides a new perspective to study the transition from dormancy to germination and to unravel the role of the caryopsis coat in seed dormancy","volume":"26","author":"Gianinetti","year":"2016","journal-title":"Seed Sci. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1017\/S0960258511000547","article-title":"How to analyse seed germination data using statistical time-to-event analysis: Non-parametric and semi-parametric methods","volume":"22","author":"McNair","year":"2012","journal-title":"Seed Sci. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1111\/aab.12477","article-title":"Analysing censored data in agricultural research: A review with examples and software tips","volume":"174","author":"Onofri","year":"2019","journal-title":"Ann. Appl. Biol."},{"key":"ref_15","unstructured":"Zar, J.H. (1999). Biostatistical Analysis, Prentice Hall. [4th ed.]."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Gbur, E.E., Stroup, W.W., McCarter, K.S., Durham, S., Young, L.J., Christman, M., West, M., and Kramer, M. (2012). Analysis of Generalized Linear Mixed Models in the Agricultural and Natural Resources Sciences, American Society of Agronomy, Crop Science Society of America, Soil Science Society of America. ACSESS Publications.","DOI":"10.2134\/2012.generalized-linear-mixed-models"},{"key":"ref_17","unstructured":"Sokal, R.R., and Rohlf, F.J. (1969). Biometry: The Principles and Practice of Statistics in Biological Research, Freeman."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1111\/wre.12095","article-title":"Experimental design and parameter estimation for threshold models in seed germination","volume":"54","author":"Onofri","year":"2014","journal-title":"Weed Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1111\/j.1365-3180.2009.00758.x","article-title":"Current statistical issues in Weed Research","volume":"50","author":"Onofri","year":"2010","journal-title":"Weed Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1038\/nmeth.2613","article-title":"Importance of being uncertain","volume":"10","author":"Krzywinski","year":"2013","journal-title":"Nat. Methods"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1017\/S0960258518000156","article-title":"Interpreting biological variation: Seeds, populations and sensitivity thresholds","volume":"28","author":"Bradford","year":"2018","journal-title":"Seed Sci. Res."},{"key":"ref_22","unstructured":"Stroup, W.W., Milliken, G.A., Claassen, E.A., and Wolfinger, R.D. (2018). SAS\u00ae for Mixed Models: Introduction and Basic Applications, SAS Institute, Inc."},{"key":"ref_23","first-page":"191","article-title":"Theoretical bases for establishing latitudes and the optimum number of seeds in germination tests","volume":"IV","author":"Yakushevskaya","year":"1937","journal-title":"Tr. Prikl. Bot. Genet. Sel. Bull. Appl. Bot. Genet. Plant Breed."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"ISTA (2018). Chapter 5: The Germination Test. International Rules for Seed Testing, International Seed Testing Association.","DOI":"10.15258\/istarules.2018.05"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1037\/h0057566","article-title":"Note on a transformation function for proportions and percentages","volume":"19","author":"Zubin","year":"1935","journal-title":"J. Appl. Psychol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1017\/S096025851400021X","article-title":"Modelling seed germination in response to continuous variables: Use and limitations of probit analysis and alternative approaches","volume":"24","author":"Hay","year":"2014","journal-title":"Seed Sci. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1017\/S0960258507841523","article-title":"Seed dormancy in red rice. XII. Population-based analysis of dry-afterripening with a hydrotime model","volume":"17","author":"Gianinetti","year":"2007","journal-title":"Seed Sci. Res."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Baskin, C.C., and Baskin, J.M. (1998). Seeds: Ecology, Biogeography, and Evolution of Dormancy and Germination, Academic Press.","DOI":"10.1017\/CBO9780511525445.004"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.eja.2018.08.011","article-title":"Hydrothermal-time-to-event models for seed germination","volume":"101","author":"Onofri","year":"2018","journal-title":"Eur. J. Agron."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1590\/S0100-84042006000100002","article-title":"de How and why to measure the germination process?","volume":"29","author":"Ranal","year":"2006","journal-title":"Rev. Bras. Bot."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1093\/oxfordjournals.aob.a087534","article-title":"Representing cumulative germination. 1. A critical analysis of single-value germination indices","volume":"61","author":"Brown","year":"1988","journal-title":"Ann. Bot."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1111\/jac.12267","article-title":"A tutorial on the statistical analysis of factorial experiments with qualitative and quantitative treatment factor levels","volume":"204","author":"Piepho","year":"2018","journal-title":"J. Agron. Crop Sci."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Rencher, A.C., and Schaalje, G.B. (2008). Linear Models in Statistics, Wiley-Interscience. [2nd ed.].","DOI":"10.1002\/9780470192610"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1614\/WS-D-13-00159.1","article-title":"Research Methods in Weed Science: Statistics","volume":"63","author":"Ritz","year":"2015","journal-title":"Weed Sci."},{"key":"ref_35","unstructured":"Littell, R.C., Milliken, G.A., Stroup, W.W., Wolfinger, R.D., and Schabenberger, O. (2006). SAS\u00ae for Mixed Models, SAS Institute, Inc.. [2nd ed.]."},{"key":"ref_36","unstructured":"Brown, B. (2019, September 09). What Are the Permitted Uses of SAS University Edition?. Available online: https:\/\/communities.sas.com\/t5\/SAS-Communities-Library\/What-are-the-permitted-uses-of-SAS-University-Edition\/ta-p\/235130."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Gianinetti, A., Finocchiaro, F., Maisenti, F., Kouongni Satsap, D., Morcia, C., Ghizzoni, R., and Terzi, V. (2018). The caryopsis of red-grained rice has enhanced resistance to fungal attack. J. Fungi, 4.","DOI":"10.3390\/jof4020071"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1038\/nmeth.3288","article-title":"The fickle P value generates irreproducible results","volume":"12","author":"Halsey","year":"2015","journal-title":"Nat. Methods"},{"key":"ref_39","unstructured":"High, R. (2014, January 23\u201326). Plotting differences among LSMEANS in Generalized Linear Models. Proceedings of the SAS Global Forum 2014 Conference, Washington, DC, USA."},{"key":"ref_40","first-page":"111","article-title":"Ricerche sui metodi di analisi delle sementi di riso (Oryza sativa L.) nei riguardi della facolt\u00e0 germinativa","volume":"2","author":"Piacco","year":"1954","journal-title":"Ann. Stazione Sper. Risic. Colt. Irrigue Vercelli"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1016\/j.dib.2018.05.040","article-title":"Data on seed germination using LED versus fluorescent light under growth chamber conditions","volume":"19","author":"Sanoubar","year":"2018","journal-title":"Data Brief"},{"key":"ref_42","first-page":"75","article-title":"Applications of hydrotime analysis in seed testing","volume":"26","author":"Bradford","year":"2004","journal-title":"Seed Technol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"921","DOI":"10.1038\/nmeth.2659","article-title":"Error bars","volume":"10","author":"Krzywinski","year":"2013","journal-title":"Nat. Methods"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"281","DOI":"10.15258\/sst.2016.44.2.11","article-title":"Underdispersion of replicate results in germination tests is species and laboratory specific","volume":"44","author":"Deplewski","year":"2016","journal-title":"Seed Sci. Technol."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"656","DOI":"10.1094\/PHYTO-07-17-0234-RVW","article-title":"Twenty-five years of the binary power law for characterizing heterogeneity of disease incidence","volume":"108","author":"Madden","year":"2018","journal-title":"Phytopathology"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1017\/S0960258519000011","article-title":"Exact theoretical distributions around the replicate results of a germination test","volume":"29","author":"Laffont","year":"2019","journal-title":"Seed Sci. Res."},{"key":"ref_47","first-page":"34","article-title":"The incidence of weed seeds in duplicate analyses","volume":"5","author":"Leggatt","year":"1933","journal-title":"Proc. Int. Seed Test. Assoc."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1007\/s00442-016-3576-7","article-title":"Seed heteromorphism in Triticum dicoccoides: Association between seed positions within a dispersal unit and dormancy","volume":"181","author":"Volis","year":"2016","journal-title":"Oecologia"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1079\/SSR2005203","article-title":"Structural, physiological and molecular aspects of heterogeneity in seeds: A review","volume":"15","author":"Matilla","year":"2005","journal-title":"Seed Sci. Res."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1093\/oxfordjournals.aob.a087535","article-title":"Representing cumulative germination. 2. The use of the Weibull function and other empirically derived curves","volume":"61","author":"Brown","year":"1988","journal-title":"Ann. Bot."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"5857","DOI":"10.1080\/03610926.2014.950753","article-title":"On generalized exponential transformations for proportions","volume":"45","author":"Malik","year":"2016","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.eja.2012.10.003","article-title":"Analysis of germination data from agricultural experiments","volume":"45","author":"Ritz","year":"2013","journal-title":"Eur. J. Agron."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Gianinetti, A., Finocchiaro, F., Bagnaresi, P., Zechini, A., Faccioli, P., Cattivelli, L., Val\u00e8, G., and Biselli, C. (2018). Seed dormancy involves a transcriptional program that supports early plastid functionality during imbibition. Plants, 7.","DOI":"10.3390\/plants7020035"}],"container-title":["Data"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2306-5729\/5\/1\/6\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:29:04Z","timestamp":1760362144000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2306-5729\/5\/1\/6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,8]]},"references-count":53,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["data5010006"],"URL":"https:\/\/doi.org\/10.3390\/data5010006","relation":{},"ISSN":["2306-5729"],"issn-type":[{"value":"2306-5729","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,8]]}}}