Aug 8, 2018 · 3. MIXED EFFECTS MODELS 3.1 Overview of mixed effects models When a regression contains both random and fixed effects, it is said to be a mixed effects model, or simply, a mixed model. Fixed effects are those with which most researchers are familiar. Any covariate that is assumed to have the same effect for all responses throughout the Nov 1, 2019 · Therefore, even greater sampling rates will be required when autocorrelation is present to meet the levels prescribed by analyses of the power and precision when estimating individual variation using mixed effect models (e.g., Wolak et al. 2012; Dingemanse and Dochtermann 2013) A random effects model that contains only random intercepts, which is the most common use of mixed effect modeling in randomized trials, assumes that the responses within subject are exchangeable. This can be seen from the statement of the linear mixed effects model with random intercepts.Feb 3, 2021 · I have temporal blocks in my data frame, so I took the effect of time dependency through a random intercept in a glmer model. Now I want to test the spatial autocorrelation in the residuals but I’m not sure if the test procedure based on the residual is the same as for the fixed-effect models since now I have time dependency. The advantage of mixed effects models is that you can also account for non-independence among "slopes". As you said, you may assume more similarity from fish within tanks, but - e.g. - over time ... Mixed-effects models allow multiple levels of variability; AKA hierarchical models, multilevel models, multistratum models; Good references on mixed-effects models: Bolker [1–3] Gelman & Hill [4] Pinheiro & Bates [5]. Growth curve models (possibly Latent GCM) Mixed effects models. 이 모두는 mixed model 의 다른 종류를 말한다. 어떤 용어들은 역사가 깊고, 어떤 것들은 특수 분야에서 자주 사용되고, 어떤 것들은 특정 데이터 구조를 뜻하고, 어떤 것들은 특수한 케이스들이다. Mixed effects 혹은 mixed ...It is a linear mixed model, with log-transformed OM regressed on marsh site (categorical), marsh type (categorical), soil category (categorical), depth (numerical, based on ordinal depth ranges), and the interaction between depth and marsh type; marsh site effects are modeled as random, on which the ICAR spatial autocorrelation structure is ...The advantage of mixed effects models is that you can also account for non-independence among "slopes". As you said, you may assume more similarity from fish within tanks, but - e.g. - over time ... Feb 23, 2022 · It is evident that the classical bootstrap methods developed for simple linear models should be modified to take into account the characteristics of mixed-effects models (Das and Krishen 1999). In ... Oct 11, 2022 · The code below shows how the random effects (intercepts) of mixed models without autocorrelation terms can be extracted and plotted. However, this approach does not work when modelling autocorrelation in glmmTMB. Use reproducible example data from this question: glmmTMB with autocorrelation of irregular times The nlme package allows you to fit mixed effects models. So does lme4 - which is in some ways faster and more modern, but does NOT model heteroskedasticity or (!spoiler alert!) autocorrelation. Let’s try a model that looks just like our best model above, but rather than have a unique Time slope The nlme package allows you to fit mixed effects models. So does lme4 - which is in some ways faster and more modern, but does NOT model heteroskedasticity or (!spoiler alert!) autocorrelation. Let’s try a model that looks just like our best model above, but rather than have a unique Time slope c (Claudia Czado, TU Munich) – 11 – Likelihood Inference for LMM: 1) Estimation of β and γ for known G and R Estimation of β: Using (5), we have as MLE or weighted LSE of βSep 16, 2018 · Recently I have made good use of Matlab's built-in functions for making linear mixed effects. Currently I am trying to model time-series data (neuronal activity) from cognitive experiments with the fitlme() function using two continuous fixed effects (linear speed and acceleration) and several, hierarchically nested categorical random factors (subject identity, experimental session and binned ... The “random effects model” (also known as the mixed effects model) is used when the analysis must account for both fixed and random effects in the model. This occurs when data for a subject are independent observations following a linear model or GLM, but the regression coefficients vary from person to person. Infant growth is a In order to assess the effect of autocorrelation on biasing our estimates of R when not accounted for, the simulated data was fit with random intercept models, ignoring the effect of autocorrelation. We aimed to study the effect of two factors of sampling on the estimated repeatability: 1) the period of time between successive observations, and ...Mixed Models, i.e. models with both fixed and random effects arise in a variety of research situations. Split plots, strip plots, repeated measures, multi-site clinical trials, hierar chical linear models, random coefficients, analysis of covariance are all special cases of the mixed model. Models all contained the same fixed effects, were compared using AIC, and were fitted by REML (to allow comparison of different correlation structures by AIC). I'm using the R package nlme and the gls function. Question 1. The GLS models' residuals still display almost identical cyclical patterns when plotted against time.Recently I have made good use of Matlab's built-in functions for making linear mixed effects. Currently I am trying to model time-series data (neuronal activity) from cognitive experiments with the fitlme() function using two continuous fixed effects (linear speed and acceleration) and several, hierarchically nested categorical random factors (subject identity, experimental session and binned ...GLMMs. In principle, we simply define some kind of correlation structure on the random-effects variance-covariance matrix of the latent variables; there is not a particularly strong distinction between a correlation structure on the observation-level random effects and one on some other grouping structure (e.g., if there were a random effect of year (with multiple measurements within each year ...Apr 11, 2023 · Inspecting and modeling residual autocorrelation with gaps in linear mixed effects models. Here I generate a dataset where measurements of response variable y and covariates x1 and x2 are collected on 30 individuals through time. Each individual is denoted by a unique ID . Because I have 4 observations for each Site but I am not interested in this effect, I wanted to go for a Linear Mixed Model with Site as random effect. However, climatic variables are often highly spatially autocorrelated so I also wanted to add a spatial autocorrelation structure using the coordinates of the sites.Sep 16, 2018 · Recently I have made good use of Matlab's built-in functions for making linear mixed effects. Currently I am trying to model time-series data (neuronal activity) from cognitive experiments with the fitlme() function using two continuous fixed effects (linear speed and acceleration) and several, hierarchically nested categorical random factors (subject identity, experimental session and binned ... the mixed-effect model with a first-order autocorrelation structure. The model was estimated using the R package nlme and the lme function (Pinheiro et al., 2020 ).The advantage of mixed effects models is that you can also account for non-independence among "slopes". As you said, you may assume more similarity from fish within tanks, but - e.g. - over time ... I used this data to run 240 basic linear models of mean Length vs mean Temperature, the models were ran per location box, per month, per sex. I am now looking to extend my analysis by using a mixed effects model, which attempts to account for the temporal (months) and spatial (location boxes) autocorrelation in the dataset.PROC MIXED in the SAS System provides a very flexible modeling environment for handling a variety of repeated measures problems. Random effects can be used to build hierarchical models correlating measurements made on the same level of a random factor, including subject-specific regression models, while a variety of covariance and Apr 15, 2021 · Yes. How can glmmTMB tell how far apart moments in time are if the time sequence must be provided as a factor? The assumption is that successive levels of the factor are one time step apart (the ar1 () covariance structure does not allow for unevenly spaced time steps: for that you need the ou () covariance structure, for which you need to use ... a combination of both models (ARMA). random effects that model independence among observations from the same site using GAMMs. That is, in addition to changing the basis as with the nottem example, we can also add complexity to the model by incorporating an autocorrelation structure or mixed effects using the gamm() function in the mgcv package It is evident that the classical bootstrap methods developed for simple linear models should be modified to take into account the characteristics of mixed-effects models (Das and Krishen 1999). In ...spaMM fits mixed-effect models and allow the inclusion of spatial effect in different forms (Matern, Interpolated Markov Random Fields, CAR / AR1) but also provide interesting other features such as non-gaussian random effects or autocorrelated random coefficient (ie group-specific spatial dependency). spaMM uses a syntax close to the one used ...Linear mixed models allow for modeling fixed, random and repeated effects in analysis of variance models. “Factor effects are either fixed or random depending on how levels of factors that appear in the study are selected. An effect is called fixed if the levels in the study represent all possible levels of theMixed Models (GLMM), and as our random effects logistic regression model is a special case of that model it fits our needs. An overview about the macro and the theory behind is given in Chapter 11 of Littell et al., 1996. Briefly, the estimating algorithm uses the principle of quasi-likelihood and an approximation to the likelihood function of ... Nov 1, 2019 · Therefore, even greater sampling rates will be required when autocorrelation is present to meet the levels prescribed by analyses of the power and precision when estimating individual variation using mixed effect models (e.g., Wolak et al. 2012; Dingemanse and Dochtermann 2013) At this point, it is important to highlight how spatial data is internally stored in a SpatialGridDataFrame and the latent effects described in Table 7.1. For some models, INLA considers data sorted by column, i.e., a vector with the first column of the grid from top to bottom, followed by the second column and so on.GLMMs. In principle, we simply define some kind of correlation structure on the random-effects variance-covariance matrix of the latent variables; there is not a particularly strong distinction between a correlation structure on the observation-level random effects and one on some other grouping structure (e.g., if there were a random effect of year (with multiple measurements within each year ...of freedom obtained by the same method used in the most recently fit mixed model. If option dfmethod() is not specified in the previous mixed command, option small is not allowed. For certain methods, the degrees of freedom for some linear combinations may not be available. See Small-sample inference for fixed effects in[ME] mixed for more ... This is what we refer to as “random factors” and so we arrive at mixed effects models. Ta-daa! 6. Mixed effects models. A mixed model is a good choice here: it will allow us to use all the data we have (higher sample size) and account for the correlations between data coming from the sites and mountain ranges. At this point, it is important to highlight how spatial data is internally stored in a SpatialGridDataFrame and the latent effects described in Table 7.1. For some models, INLA considers data sorted by column, i.e., a vector with the first column of the grid from top to bottom, followed by the second column and so on.Recently I have made good use of Matlab's built-in functions for making linear mixed effects. Currently I am trying to model time-series data (neuronal activity) from cognitive experiments with the fitlme() function using two continuous fixed effects (linear speed and acceleration) and several, hierarchically nested categorical random factors (subject identity, experimental session and binned ...Mixed Models, i.e. models with both fixed and random effects arise in a variety of research situations. Split plots, strip plots, repeated measures, multi-site clinical trials, hierar chical linear models, random coefficients, analysis of covariance are all special cases of the mixed model.Aug 8, 2018 · 3. MIXED EFFECTS MODELS 3.1 Overview of mixed effects models When a regression contains both random and fixed effects, it is said to be a mixed effects model, or simply, a mixed model. Fixed effects are those with which most researchers are familiar. Any covariate that is assumed to have the same effect for all responses throughout the I am seeking advice on how to effectively eliminate autocorrelation from a linear mixed model. My experimental design and explanation of fixed and random factors can be found here from an earlier question I asked: Crossed fixed effects model specification including nesting and repeated measures using glmm in Rlmer (lme4) glmmTMB (glmmTMB) We will start by fitting the linear mixed effects model. data.hier.lme <- lme(y ~ x, random = ~1 | block, data.hier, method = "REML") The hierarchical random effects structure is defined by the random= parameter. In this case, random=~1|block indicates that blocks are random effects and that the intercept should be ...Aug 8, 2018 · 3. MIXED EFFECTS MODELS 3.1 Overview of mixed effects models When a regression contains both random and fixed effects, it is said to be a mixed effects model, or simply, a mixed model. Fixed effects are those with which most researchers are familiar. Any covariate that is assumed to have the same effect for all responses throughout the At this point, it is important to highlight how spatial data is internally stored in a SpatialGridDataFrame and the latent effects described in Table 7.1. For some models, INLA considers data sorted by column, i.e., a vector with the first column of the grid from top to bottom, followed by the second column and so on. The advantage of mixed effects models is that you can also account for non-independence among "slopes". As you said, you may assume more similarity from fish within tanks, but - e.g. - over time ... Is it accurate to say that we used a linear mixed model to account for missing data (i.e. non-response; technology issues) and participant-level effects (i.e. how frequently each participant used ...in nlme, it is possible to specify the variance-covariance matrix for the random effects (e.g. an AR (1)); it is not possible in lme4. Now, lme4 can easily handle very huge number of random effects (hence, number of individuals in a given study) thanks to its C part and the use of sparse matrices. The nlme package has somewhat been superseded ...Mar 15, 2022 · A random effects model that contains only random intercepts, which is the most common use of mixed effect modeling in randomized trials, assumes that the responses within subject are exchangeable. This can be seen from the statement of the linear mixed effects model with random intercepts. Dec 24, 2014 · Is it accurate to say that we used a linear mixed model to account for missing data (i.e. non-response; technology issues) and participant-level effects (i.e. how frequently each participant used ... The “random effects model” (also known as the mixed effects model) is used when the analysis must account for both fixed and random effects in the model. This occurs when data for a subject are independent observations following a linear model or GLM, but the regression coefficients vary from person to person. Infant growth is aAbstract. The ‘DHARMa’ package uses a simulation-based approach to create readily interpretable scaled (quantile) residuals for fitted (generalized) linear mixed models. Currently supported are linear and generalized linear (mixed) models from ‘lme4’ (classes ‘lmerMod’, ‘glmerMod’), ‘glmmTMB’, ‘GLMMadaptive’ and ‘spaMM ...the mixed-effect model with a first-order autocorrelation structure. The model was estimated using the R package nlme and the lme function (Pinheiro et al., 2020 ).1 Answer. Mixed models are often a good choice when you have repeated measures, such as here, within whales. lme from the nlme package can fit mixed models and also handle autocorrelation based on a AR (1) process, where values of X X at t − 1 t − 1 determine the values of X X at t t.Therefore, even greater sampling rates will be required when autocorrelation is present to meet the levels prescribed by analyses of the power and precision when estimating individual variation using mixed effect models (e.g., Wolak et al. 2012; Dingemanse and Dochtermann 2013)Arguments. the value of the lag 1 autocorrelation, which must be between -1 and 1. Defaults to 0 (no autocorrelation). a one sided formula of the form ~ t, or ~ t | g, specifying a time covariate t and, optionally, a grouping factor g. A covariate for this correlation structure must be integer valued. When a grouping factor is present in form ...PROC MIXED in the SAS System provides a very flexible modeling environment for handling a variety of repeated measures problems. Random effects can be used to build hierarchical models correlating measurements made on the same level of a random factor, including subject-specific regression models, while a variety of covariance andTo do this, you would specify: m2 <- lmer (Obs ~ Day + Treatment + Day:Treatment + (Day | Subject), mydata) In this model: The intercept if the predicted score for the treatment reference category at Day=0. The coefficient for Day is the predicted change over time for each 1-unit increase in days for the treatment reference category.in nlme, it is possible to specify the variance-covariance matrix for the random effects (e.g. an AR (1)); it is not possible in lme4. Now, lme4 can easily handle very huge number of random effects (hence, number of individuals in a given study) thanks to its C part and the use of sparse matrices. The nlme package has somewhat been superseded ... What is autocorrelation? Generalized Additive Mixed Effects Models have several components: Smooth terms for covariates; Random Effects: Intercepts, Slopes and Smooths. Categorical Predictors; Interactions of (1)-(3) We can add one more component for autocorrelation: modeling the residuals: Covariance structure for the residuals.Mar 15, 2022 · A random effects model that contains only random intercepts, which is the most common use of mixed effect modeling in randomized trials, assumes that the responses within subject are exchangeable. This can be seen from the statement of the linear mixed effects model with random intercepts. My approach is to incorporate routes and year as random effects in generalized mixed effects models as shown below (using lme4 package). But, I am not sure how well autocorrelation is modeled adequately in this way. glmer (Abundance ~ Area_harvested + (1 | route) + (1 | Year), data = mydata, family = poisson) Although I specified Poisson above ...GLMMs. In principle, we simply define some kind of correlation structure on the random-effects variance-covariance matrix of the latent variables; there is not a particularly strong distinction between a correlation structure on the observation-level random effects and one on some other grouping structure (e.g., if there were a random effect of year (with multiple measurements within each year ...Dec 11, 2017 · Mixed-effect linear models. Whereas the classic linear model with n observational units and p predictors has the vectorized form. where and are design matrices that jointly represent the set of predictors. Random effects models include only an intercept as the fixed effect and a defined set of random effects. Aug 8, 2018 · 3. MIXED EFFECTS MODELS 3.1 Overview of mixed effects models When a regression contains both random and fixed effects, it is said to be a mixed effects model, or simply, a mixed model. Fixed effects are those with which most researchers are familiar. Any covariate that is assumed to have the same effect for all responses throughout the Eight models were estimated in which subjects nervousness values were regressed on all aforementioned predictors. The first model was a standard mixed-effects model with random effects for the intercept and the slope but no autocorrelation (Model 1 in Tables 2 and 3). The second model included such an autocorrelation (Model 2).Recently I have made good use of Matlab's built-in functions for making linear mixed effects. Currently I am trying to model time-series data (neuronal activity) from cognitive experiments with the fitlme() function using two continuous fixed effects (linear speed and acceleration) and several, hierarchically nested categorical random factors (subject identity, experimental session and binned ...What is autocorrelation? Generalized Additive Mixed Effects Models have several components: Smooth terms for covariates; Random Effects: Intercepts, Slopes and Smooths. Categorical Predictors; Interactions of (1)-(3) We can add one more component for autocorrelation: modeling the residuals: Covariance structure for the residuals. To do this, you would specify: m2 <- lmer (Obs ~ Day + Treatment + Day:Treatment + (Day | Subject), mydata) In this model: The intercept if the predicted score for the treatment reference category at Day=0. The coefficient for Day is the predicted change over time for each 1-unit increase in days for the treatment reference category. My approach is to incorporate routes and year as random effects in generalized mixed effects models as shown below (using lme4 package). But, I am not sure how well autocorrelation is modeled adequately in this way. glmer (Abundance ~ Area_harvested + (1 | route) + (1 | Year), data = mydata, family = poisson) Although I specified Poisson above ...Jul 1, 2021 · Mixed Effects Models - Autocorrelation. Jul. 1, 2021 • 0 likes • 171 views. Download Now. Download to read offline. Education. Lecture 19 from my mixed-effects modeling course: Autocorrelation in longitudinal and time-series data. Scott Fraundorf Follow. I am seeking advice on how to effectively eliminate autocorrelation from a linear mixed model. My experimental design and explanation of fixed and random factors can be found here from an earlier question I asked: Crossed fixed effects model specification including nesting and repeated measures using glmm in REight models were estimated in which subjects nervousness values were regressed on all aforementioned predictors. The first model was a standard mixed-effects model with random effects for the intercept and the slope but no autocorrelation (Model 1 in Tables 2 and 3). The second model included such an autocorrelation (Model 2).Apr 12, 2018 · Here's a mixed model without autocorrelation included: cmod_lme <- lme(GS.NEE ~ cYear, data=mc2, method="REML", random = ~ 1 + cYear | Site) and you can explore the autocorrelation by using plot(ACF(cmod_lme)) . I used this data to run 240 basic linear models of mean Length vs mean Temperature, the models were ran per location box, per month, per sex. I am now looking to extend my analysis by using a mixed effects model, which attempts to account for the temporal (months) and spatial (location boxes) autocorrelation in the dataset.Is it accurate to say that we used a linear mixed model to account for missing data (i.e. non-response; technology issues) and participant-level effects (i.e. how frequently each participant used ...PROC MIXED in the SAS System provides a very flexible modeling environment for handling a variety of repeated measures problems. Random effects can be used to build hierarchical models correlating measurements made on the same level of a random factor, including subject-specific regression models, while a variety of covariance and This is what we refer to as “random factors” and so we arrive at mixed effects models. Ta-daa! 6. Mixed effects models. A mixed model is a good choice here: it will allow us to use all the data we have (higher sample size) and account for the correlations between data coming from the sites and mountain ranges.May 22, 2018 · 10.8k 7 39 67. 1. All LMMs correspond to a multivariate normal model (while the converse is not true) with a structured variance covariance matrix, so "all" you have to do is to work out the marginal variance covariance matrix for the nested random-effect model and fit that - whether gls is then able to parameterize that model is then the next ... Eight models were estimated in which subjects nervousness values were regressed on all aforementioned predictors. The first model was a standard mixed-effects model with random effects for the intercept and the slope but no autocorrelation (Model 1 in Tables 2 and 3). The second model included such an autocorrelation (Model 2).The nlme package allows you to fit mixed effects models. So does lme4 - which is in some ways faster and more modern, but does NOT model heteroskedasticity or (!spoiler alert!) autocorrelation. Let’s try a model that looks just like our best model above, but rather than have a unique Time slope
of freedom obtained by the same method used in the most recently fit mixed model. If option dfmethod() is not specified in the previous mixed command, option small is not allowed. For certain methods, the degrees of freedom for some linear combinations may not be available. See Small-sample inference for fixed effects in[ME] mixed for more .... Carfax used cars under dollar2 000
The first model was a longitudinal mixed-effect model with a first-order autocorrelation structure, and the second model was the E-MELS. Both were implemented as described above. The third model was a longitudinal mixed-effect model with a Lasso penalty. Jul 25, 2020 · How is it possible that the model fits perfectly the data while the fixed effect is far from overfitting ? Is it normal that including the temporal autocorrelation process gives such R² and almost a perfect fit ? (largely due to the random part, fixed part often explains a small part of the variance in my data). Is the model still interpretable ? Apr 12, 2018 · Here's a mixed model without autocorrelation included: cmod_lme <- lme(GS.NEE ~ cYear, data=mc2, method="REML", random = ~ 1 + cYear | Site) and you can explore the autocorrelation by using plot(ACF(cmod_lme)) . We conducted a small simulation study to investigate whether an extension of the mixed-effect model that considers between-person differences in the Level 1 variance and the autocorrelation (i.e., the E-MELS) yields more precise forecasts than a standard longitudinal mixed-effect model.Sep 22, 2015 · $\begingroup$ it's more a please check that I have taken care of the random effects, autocorrelation, and a variance that increases with the mean properly. $\endgroup$ – M.T.West Sep 22, 2015 at 12:15 Feb 28, 2020 · There is spatial autocorrelation in the data which has been identified using a variogram and Moran's I. The problem is I tried to run a lme model, with a random effect of the State that district is within: mod.cor<-lme(FLkm ~ Monsoon.Precip + Monsoon.Temp,correlation=corGaus(form=~x+y,nugget=TRUE), data=NE1, random = ~1|State) I have a dataset of 12 days of diary data. I am trying to use lme to model the effect of sleep quality on stress, with random intercept effects of participant and random slope effect of sleep quality. I am not particularly interested in asking whether there was change over time from diaryday 1 to 12, just in accounting for the time variable.Is it accurate to say that we used a linear mixed model to account for missing data (i.e. non-response; technology issues) and participant-level effects (i.e. how frequently each participant used ...A 1 on the right hand side of the formula(s) indicates a single fixed effects for the corresponding parameter(s). By default, the parameters are obtained from the names of start . startMixed Models (GLMM), and as our random effects logistic regression model is a special case of that model it fits our needs. An overview about the macro and the theory behind is given in Chapter 11 of Littell et al., 1996. Briefly, the estimating algorithm uses the principle of quasi-likelihood and an approximation to the likelihood function of ... In R, the lme linear mixed-effects regression command in the nlme R package allows the user to fit a regression model in which the outcome and the expected errors are spatially autocorrelated. There are several different forms that the spatial autocorrelation can take and the most appropriate form for a given dataset can be assessed by looking ...For a linear mixed-effects model (LMM), as fit by lmer, this integral can be evaluated exactly. For a GLMM the integral must be approximated. For a GLMM the integral must be approximated. The most reliable approximation for GLMMs is adaptive Gauss-Hermite quadrature, at present implemented only for models with a single scalar random effect.Models all contained the same fixed effects, were compared using AIC, and were fitted by REML (to allow comparison of different correlation structures by AIC). I'm using the R package nlme and the gls function. Question 1. The GLS models' residuals still display almost identical cyclical patterns when plotted against time.Is it accurate to say that we used a linear mixed model to account for missing data (i.e. non-response; technology issues) and participant-level effects (i.e. how frequently each participant used ...A comparison to mixed models. We noted previously that there were ties between generalized additive and mixed models. Aside from the identical matrix representation noted in the technical section, one of the key ideas is that the penalty parameter for the smooth coefficients reflects the ratio of the residual variance to the variance components for the random effects (see Fahrmeier et al ...1 Answer. Mixed models are often a good choice when you have repeated measures, such as here, within whales. lme from the nlme package can fit mixed models and also handle autocorrelation based on a AR (1) process, where values of X X at t − 1 t − 1 determine the values of X X at t t.Apr 15, 2021 · Yes. How can glmmTMB tell how far apart moments in time are if the time sequence must be provided as a factor? The assumption is that successive levels of the factor are one time step apart (the ar1 () covariance structure does not allow for unevenly spaced time steps: for that you need the ou () covariance structure, for which you need to use ... The nlme package allows you to fit mixed effects models. So does lme4 - which is in some ways faster and more modern, but does NOT model heteroskedasticity or (!spoiler alert!) autocorrelation. Let’s try a model that looks just like our best model above, but rather than have a unique Time slope .
