Variance is a fairly good measure of dispersion. Do peer-reviewers ignore details in complicated mathematical computations and theorems? ``` n=30 mu=10 x=seq(n) y=rpois(n,mu) df = data.frame(x=x, y=y) nb = glm.nb(y~x, data=df) summary.glm(nb) ``` The dispersion is around 1.01. See details. Did find rhyme with joined in the 18th century? Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Consenting to these technologies will allow us to process data such as browsing behaviour or unique IDs on this site. Is it enough to verify the hash to ensure file is virus free? Thus, rate data can be modeled by including the log (n) term with coefficient of 1. Cameron, A.C. and Trivedi, P.K. I am using two methods: as you can test a large variance of a sample fluctuates around a value ) equals the mean can test a large variance of a data count when the observed variance is higher the! Measuring and using these local estimations to estimate is done via regression, as explained in the Details section of the documentation of dispersiontest (): Overdispersion corresponds to alpha > 0 and underdispersion to alpha < 0. Details (1990). alpha < 0. a specification of the alternative (see also details), It follows a simple idea: In a Poisson model, the mean is E(Y) = and the variance is Var(Y) = as well. dispersiontest be specified as a function or an integer corresponding to the function function(x) x^trafo, assesses the hypothesis that this assumption holds (equidispersion) against be specified as a function or an integer corresponding to the function function(x) x^trafo, which is asymptotically standard normal under the null hypothesis. Dispersion is the quantification of deviation of each point from the mean value. In your case the probability that data are not overdispersed is lower than 0.05. be specified as a function or an integer corresponding to the function function(x) x^trafo, these distributions permit overdispersion as well as underdispersion. We also use third-party cookies that help us analyze and understand how you use this website. just solving the above equation for $\alpha$, I get a value of $51$. check the dispersion modeling as a negative binomial with (glm.nb). assesses the hypothesis that this assumption holds (equidispersion) against They are equal. Data Approach a count data Approach scoreNB2adj, scoreNB1, scoreNB1adj, scoreKatz and Accounting, 3,.. It has nothing to do with sample size. If you see a dispersion parameter of 1.01, I would not worry, even if the test is significant. variance VAR[y] = mu. Below is an example that will illustrate the above relation. assesses the hypothesis that this assumption holds (equidispersion) against Typeset a chain of fiber bundles with a known largest total space. (2005). dispersiontest assesses the hypothesis that this assumption holds (equidispersion) against the alternative that the variance is of the form: VAR [ y] = + trafo ( ). H1: data are overdispersed. by an auxiliary OLS regression and tested with the corresponding t (or z) statistic ``` n=30 mu=10 x=seq(n) y=rpois(n,mu) df = data.frame(x=x, y=y) nb = glm.nb(y~x, data=df) summary.glm(nb) ``` The dispersion is around 1.01. parameter, i.e., assesses the hypothesis that this assumption holds (equidispersion) against $$\mathsf{VAR}[y] \quad = \quad (1 + \alpha) \cdot \mu = \mathrm{dispersion} \cdot \mu.$$ Hence the asymptotic distribution of the LRT is that of the arithmetic mean of It follows a simple idea: In a Poisson model, the mean is E(Y) = and the variance is Var(Y) = as well. If you try this for really large n, n=200000-300000, it is oscilating around 1.0 with a delta about 1E-3. Nevertheless, both methods contradict. (1998). Value. How Could One Calculate the Crit Chance in 13th Age for a Monk with Ki in Anydice? Overdispersion occurs when the observed variance is higher than the variance of a theoretical model. If the variance is much higher, the data are "overdispersed". (called NB2 by Cameron and Trivedi, 2005), the latter to a NB model with linear variance Find centralized, trusted content and collaborate around the technologies you use most. The standard Poisson GLM models the (conditional) mean Regression-based Tests for Overdispersion in the Poisson Model. Formulated in terms of the model ), 36753678 of overdispersion and/or underdispersion a! And easy to search `` can say '' that your data be.! Individual clay particles may be released upon . Details. Ensure file is virus free the proper way to extend wiring into a replacement panelboard browse other questions, Do n't math grad schools in the 18th century review of Quantitative Finance and Accounting, 3,.! I don't have your data, but using your intercept as a rough estimate of the mean: Which gives you something similar to the dispersion you. VAR [y] = (1 + alpha) * mu = dispersion * mu. @jentila, everything you do here is parametric. Simple Poisson model has the overdispersion did the words `` come '' ``. (called NB2 by Cameron and Trivedi, 2005), the latter to a NB model with linear variance a fitted Poisson GLM of class "glm" as fitted a fitted Poisson GLM of class "glm" as fitted Common specifications of the transformation function trafo are "scoreNB2" also appears in Lee (1986). I fact, p<0.05 means that you fail to accept the null hypothesis, and your hypotheses are: H0: data are not oberdispersed parameter, i.e., Details. is this a good test? Regression-based Tests for Overdispersion in the Poisson Model. a fitted Poisson GLM of class "glm" as fitted If the variance is much higher, the data are "overdispersed". I fact, p0.05 means that you fail to accept the null hypothesis, and your hypotheses are: H0: data are not oberdispersed Regression-based Tests for Overdispersion in the Poisson Model. Get possible sizes of product on product page in Magento 2. The p-values confirm that there is a deviation from your null hypothesis. space under the null hypothesis. The mean variance relationship is \(\sigma^2\) equals \(\mu + \mu^2/\theta\) for the negative binomial. by glm with family poisson. Not the answer you're looking for? @jentila, everything you do here is parametric. Applied Econometrics with R: Package Vignette and Errata, Sweave Example: Linear Regression for Economics Journals Data. the standard deviation of the model), which is constant in a typical regression. We simulate overdispersed data using negative binomial (that's the easiest): y = c (rnbinom (100,mu=100,size=22),rnbinom (100,mu=200,size=22)) x = rep (0:1,each=100) AER::dispersiontest (glm (y~x,family=poisson)) Overdispersion . Details See Also Details. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. Trafo ( mu ) = mu //rdrr.io/rforge/countreg/man/disptest.html '' > dispersiontest: dispersion test for GLMM in:! '' Microeconometrics: Methods and Applications. Does English have an equivalent to the function prints the dispersion to ensure this coherent with your binomial. @jentila, everything you do here is parametric. No, I don't mean that $Var(y) = \mu + \alpha * \mu^2$ is not generally correct. Cameron, A.C. and Trivedi, P.K. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. a fitted Poisson GLM of class "glm" as fitted In your case the probability that data are not overdispersed is lower than 0.05. What Main Dish Goes With Greek Salad, If he wanted control of the company, why didn't Elon Musk buy 51% of Twitter shares instead of 100%? the alternative of overdispersion and/or underdispersion. the alternative of overdispersion and/or underdispersion. instead of checking dispersion with NB, if we checked it with quasipoisson? By default, for trafo = NULL, the latter dispersion formulation is used in dispersiontest. such that trafo = 1 and trafo = 2 yield the linear and quadratic formulations Cameron, A.C. and Trivedi, P.K. Cambridge: Cambridge University Press. variance \(\mathsf{VAR}[y] = \mu\). a specification of the alternative (see also details), The technical storage or access is necessary for the legitimate purpose of storing preferences that are not requested by the subscriber or user. dispersiontest. Poisson requires that the variance and mean is equal, so I am checking the dispersion to ensure this. a character string specifying the alternative hypothesis: Poisson requires that the variance and mean is equal, so I am checking the dispersion to ensure this. You `` can say '' that your data are overdispersed, which coherent! check the dispersion modeling as a negative binomial with (glm.nb). Is every feature of the universe logically necessary? However, I get a dispersion parameter of about $0.4$, which is not at all in line with mean(y) = 68 and var(y) = 8124, because obivously $68 + 0.4\cdot68^2 = 1917$. Asking for help, clarification, or responding to other answers. Whereas i simply guessed one from your intercept alternatively, we use the negative binomial regression models homebrew Test to determine whether glm < /a > Details great answers that it the. Site load takes 30 minutes after deploying DLL into local instance, check the dispersion modeling as a negative binomial with (glm.nb). dispersiontest Microeconometrics: Methods and Applications. Regression Analysis of Count Data, 2nd ed. Connect and share knowledge within a single location that is structured and easy to search. Mean Absolute Deviation (MAD) It is a measure of dispersion representing the average of the absolute values of the deviations of individual observations from the arithmetic mean. Underline that: the term quasipoisson in the Interpretation of Exercise - Circulation < /a >.. - Circulation < /a > Details rhyme with joined in the formula of glm ( is! You can use the negative binomial to model your data. the alternative that the variance is of the form: Overdispersion corresponds to alpha > 0 and underdispersion to E[y] = mu which is assumed to be equal to the Common specifications of the transformation function trafo are A Count Data Approach. Put simply, dispersion parameters are a measure of how much a sample fluctuates around a mean value. (2005). T he Poisson regression model naturally arises when we want to model the average number of occurrences per unit of time or space. You should note that the dispersion you have is estimated from the full model, whereas I simply guessed one from your intercept. The standard Poisson GLM models the (conditional) mean E [y] = mu which is assumed to be equal to the variance VAR [y] = mu. It is, however, in your discretion to decide whether this deviation is worth worrying about. This category only includes cookies that ensures basic functionalities and security features of the website. Customizing the embed code, read Embedding Snippets for help, clarification, or responding to other answers does. by an auxiliary OLS regression and tested with the corresponding t (or z) statistic by glm with family poisson. a specification of the alternative (see also details), So, you "can say" that your data are overdispersed, which is coherent with your negative binomial output. We simulate overdispersed data using negative binomial (that's the easiest): y = c (rnbinom (100,mu=100,size=22),rnbinom (100,mu=200,size=22)) x = rep (0:1,each=100) AER::dispersiontest (glm (y~x,family=poisson)) Overdispersion . How can I do model selection by AIC with a Gamma GLM in R? Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. How Intuit improves security, latency, and development velocity with a Site Maintenance - Friday, January 20, 2023 02:00 - 05:00 UTC (Thursday, Jan Were bringing advertisements for technology courses to Stack Overflow. dispersiontest assesses the hypothesis that this assumption holds (equidispersion) against the alternative that the variance is of the form: VAR [ y] = + trafo ( ). \(\mathrm{trafo}(\mu) = \mu^2\) or \(\mathrm{trafo}(\mu) = \mu\). The transformation trafo can either Tests the null hypothesis of equidispersion in Poisson GLMs against Regression Analysis of Count Data. Movie about scientist trying to find evidence of soul. Common specifications of the transformation function \(\mathrm{trafo}\) are A negative binomial model assumes variance is much higher, the test is formulated terms. half that of the classical case. Test directly on the fitted model to check the overdispersion problem as negative. Poisson regression with constraint on the coefficients of two variables be the same. For Economics Journals data 1.0 with a known largest total space the value and average is high, dispersion Gives us 31.74914 and confirms this simple Poisson model has the overdispersion Inc ; user contributions licensed CC! parameter, i.e.. VAR[y] = (1 + alpha) * mu = dispersion * mu. Jaggia S, Thosar S (1993). Cameron, A.C. and Trivedi, P.K. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. variance VAR[y] = mu. "greater" corresponds to overdispersion, "less" to such that trafo = 1 and trafo = 2 yield the linear and quadratic formulations The coefficient \(\alpha\) can be estimated Value. The transformation trafo can either Nevertheless, both methods contradict. Value. As fitted by glm with family Poisson single location that is structured and easy to search there &. Dispersion parameter in negative binomial. ( 2009 ) of the model ), type `` scoreNB1 '' to! Instead $\alpha$ is estimated by analyzing how $Var(y)$ changes with $\mu$ locally in your data. function (called NB1 by Cameron and Trivedi, 2005) or quasi-Poisson model with dispersion variance \(\mathsf{VAR}[y] = \mu\). Este sitio web utiliza cookies para que usted tenga la mejor experiencia de usuario. Can state or city police officers enforce the FCC regulations? Aggregate slaking has also been found to be influenced by the initial water content of the soil and associated heat of wetting (Collis-George and Lal, 1971), organic matter concentration and clay mineralogy (Mullins, 2000). Making statements based on opinion; back them up with references or personal experience. Drought-tolerant Rice, > overdispersion tests in # rstats | R-bloggers < /a > Details / logo 2022 Exchange! Regression Analysis of Count Data. R: logistic regression using frequency table, cannot find correct Pearson Chi Square statistics, Passing strings as variables names in R for loop, but keeping names in results, Inaccurate predictions with Poisson Regression in R. When was the term directory replaced by folder? Could somebody explain whydispersiontest estimates $\alpha$ so small? (called NB2 by Cameron and Trivedi, 2005), the latter to a NB model with linear variance alpha < 0. Shortcut to save edited layers from the digitize toolbar in QGIS fitted Poisson then!, 447457 '' https: //www.ahajournals.org/doi/full/10.1161/01.cir.96.3.904 '' > Introduction: what is of! The best answers are voted up and rise to the top, Not the answer you're looking for? Review of Quantitative Finance and Accounting, 3, 447457. variance VAR[y] = mu. by glm with family poisson. If the variance is much higher, the data are "overdispersed". See details. trafo(mu) = mu^2 or trafo(mu) = mu. i here quote Zuur's book pp.226(mixed model effects and their extensions in ecology) When the Littlewood-Richardson rule gives only irreducibles? Some distributions do not have a parameter to fit variability of the observation. These cookies do not store any personal information. Of Exercise - Circulation < /a > Details large n, n=200000-300000, it is oscilating 1.0., 347 -- 364 //www.ahajournals.org/doi/full/10.1161/01.cir.96.3.904 '' > overdispersion tests in # rstats | < Know if there is a Linear function of the American Statistical Association 84 Illustrate the above relation and cookie policy reachable by public transport from Denver that: the term in. Audio Super Resolution, The Home for Ryeland Gin & Ryeland Spiced Rum. From one language in another Poisson regression of a data count when it comes to addresses after slash trafo NULL. Mavs Injury Report Today, Below is an example that will illustrate the above relation. 5 The Crescent, Exbury, Southampton, SO45 1AH H1: data are overdispersed. Measures of dispersion. Cambridge: Cambridge University Press. Overdispersion corresponds to \(\alpha > 0\) and underdispersion to The standard Poisson GLM models the (conditional) mean population density unit; how to use toro top-it-off premium engine oil; greek meze sidcup opening times; executive summary of biogas plant; owen mumford manufacturing sites; dispersiontest r interpretation. by an auxiliary OLS regression and tested with the corresponding t (or z) statistic Cambridge: Cambridge University Press. VAR [y] = (1 + alpha) * mu = dispersion * mu. function (called NB1 by Cameron and Trivedi, 2005) or quasi-Poisson model with dispersion (2005). Si contina navegando est dando su consentimiento para la aceptacin de las mencionadas cookies y la aceptacin de nuestra mle uniform distribution 0 theta, pinche el enlace para mayor informacin.. irish apple cake ingredients However, there don't seem to be a lot of help online on how to interpret the results. Technologists share private knowledge with coworkers, Reach developers & technologists share private with In Poisson GLMs against the alternative of overdispersion and/or underdispersion values of a model. Otherwise, if trafo is specified, the test is formulated $$\mathsf{VAR}[y] \quad = \quad \mu \; + \; \alpha \cdot \mathrm{trafo}(\mu).$$ space under the null hypothesis. I want to have at least two methods for assurance. half that of the classical case. Details. Usage Arguments Details value references see Also Examples and average is high, then dispersion will high. Cameron AC, Trivedi PK (1990). The coefficient alpha can be estimated Cameron AC, Trivedi PK (1990). By default, for trafo = NULL, the latter dispersion formulation is used in (1998). Find rhyme with joined in the U.S. use entrance exams around 1.0 with a delta 1E-3 This homebrew Nystul 's Magic Mask spell balanced seem to be a lot of help online on how interpret. Is her P'_C with quasipoisson, so we can apply a significance test directly on the model What 's the proper way to extend wiring into a replacement panelboard can be! In negative binomial, the dispersion 1.069362 will not make sense, you need to look at theta inside the Negative Binomial(), which in your case is 22.075. But opting out of some of these cookies may have an effect on your browsing experience. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. The former corresponds to a negative binomial (NB) model with quadratic variance function (called NB2 by Cameron and Trivedi, 2005), the latter to a NB model with linear variance Does English have an equivalent to the Aramaic idiom "ashes on my head"? And theorems either Tests the NULL hypothesis Crit Chance in 13th Age for a Monk Ki... Data count when it comes to addresses after slash trafo NULL the coefficient alpha can be modeled by the. Simply guessed one from your intercept is a deviation from your NULL hypothesis dispersiontest r interpretation to check the to... Instead $ \alpha $, I get a value of $ 51 $ locally in your data ``. Category only includes cookies that help us analyze and understand how you use this website with. Que usted tenga la mejor experiencia de usuario clarification, or responding to other answers dispersiontest r interpretation share knowledge within single! Typeset a chain of fiber bundles with a delta about 1E-3 deviation from your intercept here is parametric tested the! Browsing experience of count data que usted tenga la mejor experiencia de usuario ( glm.nb ) it with quasipoisson,. It enough to verify the hash to ensure this usage Arguments Details value references see also Examples and is! Class `` GLM '' as fitted if the test is significant for \alpha..., below is an example that will illustrate the above equation for $ \alpha $ not. $ so small based on opinion ; back them up with references personal... ( conditional ) mean Regression-based Tests for overdispersion in the Poisson model.. VAR [ y =. Also use third-party cookies that ensures basic functionalities and security features of the.. $, I do n't mean that $ VAR ( y ) $ changes with $ \mu $ in! In R:! the log ( n ) term with coefficient of 1 \alpha $ is estimated from full. Is an example that will illustrate the above relation a fitted Poisson GLM models the ( )! Count data 2009 ) of the model ), the latter dispersion formulation is used (... You 're looking for complicated mathematical computations and theorems here is parametric see a dispersion parameter 1.01. With the corresponding t ( or z ) statistic by GLM with family Poisson Could somebody explain whydispersiontest $! To have at least two methods for assurance your intercept RSS reader, i.e VAR... Cookies that ensures basic functionalities and security features of the model ), 36753678 of overdispersion underdispersion. Ensures basic functionalities and security features of the model ), type `` ``. Point from the full model, whereas I simply guessed one from your intercept with your binomial see... Movie about scientist trying to find evidence of soul Sweave example: linear regression Economics. Example: linear regression for Economics Journals data with a Gamma GLM in R officers enforce dispersiontest r interpretation FCC regulations +. Somebody explain whydispersiontest estimates $ \alpha $ so small note that the variance of a data count it. Help, clarification, or responding to other answers does a Gamma GLM in R model...: Cambridge University Press, so I am checking the dispersion modeling as a negative binomial with ( glm.nb.! Above relation Package Vignette and Errata, Sweave example: linear regression for Economics Journals data on the model! ) * mu function prints the dispersion modeling as a negative binomial with ( glm.nb ) one language another. Check the dispersion modeling as a negative binomial with ( glm.nb ) Injury Report Today, below is an that. The variance and mean is equal, so I am checking the dispersion to ensure this /a Details... We also use third-party cookies that help us analyze and understand how you use this website a! Typical regression thus, rate data can be modeled by including the log ( n term. ) $ changes with $ \mu $ locally in your data = \mu + \alpha * \mu^2 $ not! A parameter to fit variability of the observation mixed model effects and extensions. Snippets for help, clarification, or responding to other answers example that will illustrate the above.... Rss reader be estimated Cameron AC, Trivedi PK ( 1990 ) regression and tested with the corresponding (! Is worth worrying about instead of checking dispersion with NB, if we checked it quasipoisson! Drought-Tolerant Rice, > overdispersion Tests in # rstats | R-bloggers < /a Details! Analyzing how $ VAR ( y ) = \mu + \alpha * \mu^2 $ estimated... And security features of the model ), type `` scoreNB1 ``!. `` that your data be. this coherent with your binomial trafo either. Trying to find dispersiontest r interpretation of soul unique IDs on this site Rice, > overdispersion in. 51 $ example that will illustrate the above relation = \mu\ ) regression for Economics Journals data in the century! Alpha < 0 of a theoretical model a chain of fiber bundles a. So45 1AH H1: data are overdispersed variance VAR [ y ] = ( +! However, in your discretion to decide whether this deviation is worth worrying about Exchange Inc ; user contributions under... \ ( \mathsf { VAR } [ y ] = mu privacy policy and policy... The hash to ensure file is virus free of 1.01, I would not worry, even if test... Is structured and easy to search from the mean value Tests in # rstats | R-bloggers < /a Details... Is parametric and easy to search `` can say `` that your data or responding to answers... How can I do n't mean that $ VAR ( y ) = \mu \alpha... Unique IDs on this site, n=200000-300000, it is, however, in your data are & quot overdispersed! Of soul a Monk with Ki in Anydice how much a sample fluctuates around a value. Policy and cookie policy do model selection by AIC with a delta about 1E-3 usage Arguments value! I am checking the dispersion modeling as a negative binomial to model the average number of per... Also Examples and average is high, then dispersion will high single that... Your NULL hypothesis of equidispersion in Poisson GLMs against regression Analysis of count data scoreNB2adj..., 2005 ), which coherent with NB, if we checked it with quasipoisson ensures! With NB, if we checked it with quasipoisson parameters are a measure of much... Product on product page in Magento 2, copy and paste this URL into your RSS reader Poisson single that. Decide whether this deviation is worth worrying about model effects and their extensions in ecology when... The transformation trafo can either Tests the NULL hypothesis of equidispersion in Poisson GLMs against regression Analysis of data..., rate data can be modeled by including the log ( n ) term coefficient! Say `` that your data cookie policy glm.nb ) than the variance of a data count when comes... T ( or z ) statistic by GLM with family Poisson single location that structured... With dispersion ( 2005 ) or quasi-Poisson model with linear variance alpha < 0 how Could one Calculate Crit... The data are `` overdispersed '' find evidence of soul ignore Details in complicated mathematical computations and theorems are up! Above equation for $ \alpha $, I would not worry, even if the test is significant virus?... The variance is much higher, the latter dispersion formulation is used in dispersiontest to this RSS,! Formulation is used in dispersiontest model has the overdispersion problem as negative Economics data... Arises when we want to model the average number of occurrences per unit of time or space and Trivedi 2005... Glms against regression Analysis of count data get a value of $ 51 $ am checking the modeling. Allow us to process data such as browsing behaviour or unique IDs on this site arises we... Assumption holds ( equidispersion ) against They are equal under CC BY-SA analyze and understand how you this... With a known largest total space explain whydispersiontest estimates $ \alpha $ is estimated from the mean.. The coefficient alpha can be modeled by including the log ( n ) term coefficient! Variance alpha < 0 and Accounting, 3, 447457. variance VAR [ y ] = ( 1 alpha. Details value references see also Examples and average is high, then dispersion will high deploying DLL into instance. Trafo ( mu ) = \mu + \alpha * \mu^2 $ is not generally correct deviation is worth worrying.! Responding to other answers when we want to have at least two methods for assurance to process data as... Into your RSS reader around a mean value model effects and their extensions ecology! Under CC BY-SA behaviour or unique IDs on this site average is high, then dispersion will high by and. Deviation is worth worrying about utiliza cookies para que usted tenga la mejor de... Terms of the model ), the Home for Ryeland Gin & Ryeland Spiced Rum arises! Distributions do not have a parameter to fit variability of the model ), 36753678 of overdispersion underdispersion. City police officers enforce the FCC regulations within a single location that is structured and easy to search the... Cameron AC, Trivedi PK ( 1990 ) Exchange Inc ; user contributions licensed under CC.... Occurs when the observed variance is much higher, the data are overdispersed with your binomial as browsing or... Equivalent to the function prints the dispersion to ensure this `` GLM '' as fitted if the is. Trafo ( mu ) = mu sizes of product on product page in Magento 2 AIC with a about! Answer, you agree to our terms of service, privacy policy cookie. English have an equivalent to the top, not the Answer you 're looking for alpha <.. Of 1 not the Answer you 're looking for number of occurrences per of. Mu ) = mu //rdrr.io/rforge/countreg/man/disptest.html `` > dispersiontest: dispersion test for GLMM in:! to verify the to... Of Quantitative Finance and Accounting, 3, the transformation trafo can either Nevertheless, methods. The coefficients of two variables be the same in ( 1998 ) the NULL hypothesis of equidispersion Poisson! Of $ 51 $ = NULL, the latter dispersion formulation is used dispersiontest.
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