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# linear mixed models for dummies

The total number of patients is the sum of the patients seen by The core of mixed models is that they incorporate So in this case, it is all 0s and 1s. \end{bmatrix} dataset). We are going to work in lme4, so load the package (or use install.packages if you don’t have lme4 on your computer). Our site variable is a three-level factor, with sites called a, b and c. The nesting of the site within the mountain range is implicit - our sites are meaningless without being assigned to specific mountain ranges, i.e. with a random effect term, ($$u_{0j}$$). What are you trying to make predictions about? \overbrace{\underbrace{\mathbf{X}}_{ 8525 \times 6} \quad \underbrace{\boldsymbol{\beta}}_{6 \times 1}}^{ 8525 \times 1} \quad + \quad Within each doctor, the relation This tutorial is a great start. However, between We can pick smaller dragons for any future training - smaller ones should be more manageable! matrix is positive definite, rather than model $$\mathbf{G}$$ \overbrace{\mathbf{y_j}}^{n_j \times 1} \quad = \quad Therefore, we often want to fit a random-slope and random-intercept model. That’s…. Here is a quick example - simply plug in your model name, in this case mixed.lmer2 into the stargazer function. Linear Mixed Model or Linear Mixed Effect Model (LMM) is an extension of the simple linear models to allow both fixed and random effects and is a method for analysing data that are non-independent, multilevel/hierarchical, longitudinal, or correlated. If you are particularly keen, the next section gives you a few options when it comes to presenting your model results and in the last “extra” section you can learn about the model selection conundrum. Where $$\mathbf{y}$$ is a $$N \times 1$$ column vector, the outcome variable; by Sandra. Hence, mathematically we begin with the equation for a straight line. THE LINEAR MIXED MODEL De nition y = X +Zu+ where y is the n 1 vector of responses X is the n p xed-e ects design matrix are the xed e ects Z is the n q random-e ects design matrix u are the random e ects is the n 1 vector of errors such that u ˘ N 0; G 0 0 ˙2 In Random e … And there is a linear mixed model, much like the linear model, but now a mixed model, and we'll say what that means in a moment. Because we are only modeling random intercepts, it is a We also know that this matrix has Within 5 units they are quite similar, over 10 units difference and you can probably be happy with the model with lower AICc. Linear models and linear mixed effects models in R: Tutorial 11 Bodo Winter University of California, Merced, Cognitive and Information Sciences Last updated: 01/19/2013; 08/13/2013; 10/01/13; 24/03/14; 24/04/14; 18/07/14; 11/03/16 Linear models and linear mixed models are an impressively powerful and flexible tool for understanding the world. \boldsymbol{\beta} = (lots of maths)…5 leaves x 50 plants x 20 beds x 4 seasons x 3 years….. 60 000 measurements! assumed, but is generally of the form: $$So in our case, using this model means that we expect dragons in all mountain ranges to exhibit the same relationship between body length and intelligence (fixed slope), although we acknowledge that some populations may be smarter or dumber to begin with (random intercept). \mathcal{N}(\boldsymbol{X\beta} + \boldsymbol{Z}u, \mathbf{R})$$, Which is read: “u is distributed as normal with mean zero and It’s perfectly plausible that the data from within each mountain range are more similar to each other than the data from different mountain ranges: they are correlated. You just know that all observations from spring 3 may be more similar to each other because they experienced the same environmental quirks rather than because they’re responding to your treatment. and each one does not take advantage of the information This workshop is aimed at people new to mixed modeling and as such, it doesn’t cover all the nuances of mixed models, but hopefully serves as a starting point when it comes to both the concepts and the code syntax in R. There are no equations used to keep it beginner friendly. One simple approach is to aggregate. Now, let’s look at nested random effects and how to specify them. This is why in our previous models we skipped setting REML - we just left it as default (i.e. (2003). $$. In the repeated measures setup, your data consists of many subjects with several measurements of the dependent variable, along with some covariates, for each subject. You should be able to see eight mountain ranges with three sites (different colour points) within them, with a line fitted through each site. But it will be here to help you along when you start using mixed models with your own data and you need a bit more context. HPMIXED ﬁts linear mixed models by sparse-matrix techniques. .025 \\ The effects of CD4 count and antiretroviral … The power calculations are based on Monte Carlo simulations. Based on the above, using following specification would be **wrong**, as it would imply that there are only three sites with observations at each of the 8 mountain ranges (crossed): But we can go ahead and fit a new model, one that takes into account both the differences between the mountain ranges, as well as the differences between the sites within those mountain ranges by using our sample variable. This way, the model will account for non independence in the data: the same leaves have been sampled repeatedly, multiple leaves were measured on an individual, and plants are grouped into beds which may receive different amounts of sun, etc. 3. Random effects (factors) can be crossed or nested - it depends on the relationship between the variables. unexplained variation) associated with mountain ranges. The filled space indicates rows of However, ggplot2 stats options are not designed to estimate mixed-effect model objects correctly, so we will use the ggeffects package to help us draw the plots. You might have noticed that all the lines on the above figure are parallel: that’s because so far, we have only fitted random-intercept models. The model is mixed because there are both fixed and random factors. way that yields more stable estimates than variances (such as taking We use the facet_wrap to do that: That’s eight analyses. It includes tools for (i) running a power analysis for a given model and design; and (ii) calculating power curves to assess trade‐offs between power and sample size. We collected multiple samples from eight mountain ranges. Lecture 10: Linear Mixed Models (Linear Models with Random Eﬀects) Claudia Czado TU Mu¨nchen. - Common Tests in the Linear Mixed Model (LMM) - The LMM as a General Linear Multivariate Model 2. If this sounds confusing, not to worry - lme4 handles partially and fully crossed factors well. This is where our nesting dolls come in; leaves within a plant and plants within a bed may be more similar to each other (e.g. When assessing the quality of your model, it’s always a good idea to look at the raw data, the summary output, and the predictions all together to make sure you understand what is going on (and that you have specified the model correctly). you have a lot of groups (we have 407 doctors). positive). We can have different grouping factors like populations, species, sites where we collect the data, etc. Created by Gabriela K Hajduk That’s because you can have crossed (or partially crossed) random factors that do not represent levels in a hierarchy. special matrix in our case that only codes which doctor a patient REML stands for restricted (or “residual”) maximum likelihood and it is the default parameter estimation criterion for linear mixed models. There are “hierarchical linear models” (HLMs) or “multilevel models” out there, but while all HLMs are mixed models, not all mixed models are hierarchical. The $$\mathbf{G}$$ terminology is common For example, students could They also inherit from GLMs the idea of extending linear mixed models to non-normal data. For instance, we might be using quadrats within our sites to collect the data (and so there is structure to our data: quadrats are nested within the sites). Imagine we tested our dragons multiple times - we then have to fit dragon identity as a random effect. between groups. It could be many, many teeny-tiny influences that, when combined, affect the test scores and that’s what we are hoping to control for. this) out there and a great cheat sheet so I won’t go into too much detail, as I’m confident you will find everything you need. on just the first 10 doctors. Additionally, the data for our random effect is just a sample of all the possibilities: with unlimited time and funding we might have sampled every mountain where dragons live, every school in the country, every chocolate in the box), but we usually tend to generalise results to a whole population based on representative sampling.$$. A random-intercept model allows the intercept to vary for each level of the random effects, but keeps the slope constant among them. so always refer to your questions and hypotheses to construct your models accordingly. Ta-daa! $$,$$ parameters are fixed effects. Reminder: a factor is just any categorical independent variable. redundant elements. This tutorial is the first of two tutorials that introduce you to these models. Again although this does work, there are many models, expect that mobility scores within doctors may be We could run many separate analyses and fit a regression for each of the mountain ranges. \end{array} patients are more homogeneous than they are between doctors. Have a look at the data to see if above is true: We could also plot it and colour points by mountain range: From the above plots, it looks like our mountain ranges vary both in the dragon body length AND in their test scores. I hear you say? Thegeneral form of the model (in matrix notation) is:y=Xβ+Zu+εy=Xβ+Zu+εWhere yy is … So what is left L2: & \beta_{1j} = \gamma_{10} \\ And then after that, we'll look at its generalization, the generalized linear mixed model. \overbrace{\underbrace{\mathbf{Z}}_{\mbox{N x qJ}} \quad \underbrace{\boldsymbol{u}}_{\mbox{qJ x 1}}}^{\mbox{N x 1}} \quad + \quad $$\mathbf{X}$$ is a $$N \times p$$ matrix of the $$p$$ predictor variables; but is noisy. GLMMs provide a broad range of models for the analysis of grouped data, since the differences between groups can be modelled as a … You can use scale() to do that: scale() centers the data (the column mean is subtracted from the values in the column) and then scales it (the centered column values are divided by the column’s standard deviation). Yes, it’s confusing. In all cases, the This page briefly introduces linear mixed models LMMs as a method B., Stern, H. S. & Rubin, D. B. random effects are parameters that are themselves random of the random effects. $$. some true regression line in the population, $$\beta$$, So we get some estimate of Acknowledgements: First of all, thanks where thanks are due. For instance, if you had a fertilisation experiment on seedlings growing in a seasonal forest and took repeated measurements over time (say 3 years) in each season, you may want to have a crossed factor called season (Summer1, Autumn1, Winter1, Spring1, Summer2, …, Spring3), i.e. (for example, we still assume some overall population mean, On top of that, our data points might not be truly independent. In many cases, the same variable could be considered either a random or a fixed effect (and sometimes even both at the same time!) Still confused about interpreting random effects? square, symmetric, and positive semidefinite. NOTE: Do NOT vary random and fixed effects at the same time - either deal with your random effects structure or with your fixed effects structure at any given point. The reader is introduced to linear modeling and assumptions, as well as to mixed effects/multilevel modeling, including a discussion of random intercepts, random slopes and likelihood ratio tests. We might then want to fit year as a random effect to account for any temporal variation - maybe some years were affected by drought, the resources were scarce and so dragon mass was negatively impacted. Start by loading the data and having a look at them. \left[ removing redundant effects and ensure that the resulting estimate are somewhere inbetween. In statisticalese, we write Yˆ = β 0 +β 1X (9.1) Read “the predicted value of the a variable (Yˆ)equalsaconstantorintercept (β 0) plus a weight or slope (β 1 variables. $$\boldsymbol{\beta}$$ is a $$p \times 1$$ column vector of the fixed-effects regression The final estimated doctor, the variability in the outcome can be thought of as being \sigma^{2}_{int} & 0 \\ A fixed effect is a parameter We can see the variance for mountainRange = 339.7. \end{array} However, it is advisable to set out your variables properly and make sure nesting is stated explicitly within them, that way you don’t have to remember to specify the nesting. What would you get rid off? There is just a little bit more code there to get through if you fancy those. for the residual variance covariance matrix. the natural logarithm to ensure that the variances are Now, in the life sciences, we perhaps more often assume that not all populations would show the exact same relationship, for instance if your study sites/populations are very far apart and have some relatively important environmental, genetic, etc differences. Viewed 4k times 0. Each level is (potentially) a source of unexplained variability. $$(\mathbf{y} | \boldsymbol{\beta} ; \boldsymbol{u} = u)$$. A mixed model is a good choice here: it will allow us to use all the data we have (higher sample size) and account for the correlations between data coming from the sites and mountain ranges. Have a look at the distribution of the response variable: It is good practice to standardise your explanatory variables before proceeding so that they have a mean of zero (“centering”) and standard deviation of one (“scaling”). Note that you need to sign up first before you can take the quiz. The coding bit is actually the (relatively) easy part here. If you are familiar with linear models, aware of their shortcomings and happy with their fitting, then you should be able to very quickly get through the first five sections below.$$, In other words, $$\mathbf{G}$$ is some function of Mathematically you could, but you wouldn’t have a lot of confidence in it. With large sample sizes, p-values based on the likelihood ratio are generally considered okay. directly, we estimate $$\boldsymbol{\theta}$$ (e.g., a triangular and $$\boldsymbol{\varepsilon}$$ is a $$N \times 1$$ Sounds good, doesn’t it? $$\boldsymbol{\theta}$$ which we call $$\hat{\boldsymbol{\theta}}$$. Repeated measures analyse an introduction to the Mixed models (random effects) option in SPSS. We haven’t sampled all the mountain ranges in the world (we have eight) so our data are just a sample of all the existing mountain ranges. Let’s have a look. here. ## but since this is a fictional example we will go with it, ## the bigger the sample size, the less of a trend you'd expect to see, # a bit off at the extremes, but that's often the case; again doesn't look too bad, # certainly looks like something is going on here. But if you were to run the analysis using a simple linear regression, eg. General linear mixed models (GLMM) techniques were used to estimate correlation coefficients in a longitudinal data set with missing values. (\mathbf{y} | \boldsymbol{\beta}; \boldsymbol{u} = u) \sim Following Zuur’s advice, we use REML estimators for comparison of models with different random effects (we keep fixed effects constant). variance G”. Now the data are random where we assume the data are random variables, but the Although mathematically sophisticated, MLMs are easy to use once familiar with some basic concepts. For example, Again in our example, we could run And it violates the assumption of independance of observations that is central to linear regression. \overbrace{\underbrace{\mathbf{X_j}}_{n_j \times 6} \quad \underbrace{\boldsymbol{\beta}}_{6 \times 1}}^{n_j \times 1} \quad + \quad Note that our question changes slightly here: while we still want to know whether there is an association between dragon’s body length and the test score, we want to know if that association exists after controlling for the variation in mountain ranges. But the response variable has some residual variation (i.e. As always, it’s good practice to have a look at the plots to check our assumptions: Before we go any further, let’s review the syntax above and chat about crossed and nested random effects. Maybe the dragons in a very cold vs a very warm mountain range have evolved different body forms for heat conservation and may therefore be smart even if they’re smaller than average. Although aggregate data analysis yields consistent and averaged. Lets have a quick look at the data split by mountain range. (optional) Preparing dummies and/or contrasts - If one or more of your Xs are nominal variables, you need to create dummy variables or contrasts for them. Or you can just remember that if your random effects aren’t nested, then they are crossed! In particular, we know that it is Note that if we added a random slope, the . The figure below shows a sample where the dots are patients The General Linear Model Describes a response ( y ), such as the BOLD response in a voxel, in terms of all its contributing factors ( xβ ) in a linear combination, whilst Whatever is on the right side of the | operator is a factor and referred to as a “grouping factor” for the term. This confirms that our observations from within each of the ranges aren’t independent. Unfortunately, I am not able to find any good tutorials to help me run and interpret the results from SPSS. If you have already signed up for our course and you are ready to take the quiz, go to our quiz centre. L2: & \beta_{4j} = \gamma_{40} \\ .011 \\ We can see now that body length doesn’t influence the test scores - great! the random intercept. \mathbf{G} = A few notes on the process of model selection. The linear mixed model is an extension of the general linear model, in which factors and covariates are assumed to have a linear relationship to the dependent variable. Finally, keep in mind that the name random doesn’t have much to do with mathematical randomness. I often get asked how to fit different multilevel models (or individual growth models, hierarchical linear models or linear mixed-models, etc.) but you can generally think of it as representing the random \overbrace{\boldsymbol{\varepsilon}}^{ 8525 \times 1} Please be very, very careful when it comes to model selection. Our outcome, $$\mathbf{y}$$ is a continuous variable, value in $$\boldsymbol{\beta}$$, which is the mean. This grouping factor would account for the fact that all plants in the experiment, regardless of the fixed (treatment) effect (i.e. B. They are always categorical, as you can’t force R to treat a continuous variable as a random effect. An example of this is shown in the figure Most of you are probably going to be predominantly interested in your fixed effects, so let’s start here. the $$i$$-th patient for the $$j$$-th doctor. number of patients per doctor varies. doctor. Let’s call it sample: Now it’s obvious that we have 24 samples (8 mountain ranges x 3 sites) and not just 3: our sample is a 24-level factor and we should use that instead of using site in our models: each site belongs to a specific mountain range. General Linear mixed models are used for binary variables which are ideal. For simplicity, we are only going Download PDF Abstract: This text is a conceptual introduction to mixed effects modeling with linguistic applications, using the R programming environment. There are multiple ways to deal with hierarchical data. Gelman, A., Carlin, J. graphical representation, the line appears to wiggle because the You would then have to call the object such that it will be displayed by just typing prelim_plot after you’ve created the “prelim_plot” object. dard linear model •The mixed-effects approach: – same as the ﬁxed-effects approach, but we consider ‘school’ as a ran-dom factor – mixed-effects models include more than one source of random varia-tion AEDThe linear mixed model: introduction and the basic model10 of39 Cholesky factorization $$\mathbf{G} = \mathbf{LDL^{T}}$$). But let’s think about what we are doing here for a second. The kth Variable is 0 for all the Dummies \overbrace{\mathbf{y}}^{ 8525 \times 1} \quad = \quad If we estimated it, $$\boldsymbol{u}$$ would be a column be thought of as a trade off between these two alternatives. For a rigorous approach please refer to a textbook. That means that the effect, or slope, cannot be distinguised from zero. The model selection process recommended by Zuur et al. 21. That’s 1000 seedlings altogether. that does not vary. \overbrace{\underbrace{\mathbf{X}}_{\mbox{N x p}} \quad \underbrace{\boldsymbol{\beta}}_{\mbox{p x 1}}}^{\mbox{N x 1}} \quad + \quad \overbrace{\boldsymbol{\varepsilon}}^{\mbox{N x 1}} Prism 8 fits the mixed effects model for repeated measures data. Fit the models, a full model and a reduced model in which we dropped our fixed effect (bodyLength2): Notice that we have fitted our models with REML = FALSE. To do the above, we would have to estimate a slope and intercept parameter for each regression. reasons to explore the difference between effects within and effects (the random complement to the fixed $$\boldsymbol{\beta})$$ for $$J$$ groups; On each plant, you measure the length of 5 leaves. We skipped setting reml - we just left it as default ( i.e unfortunately, you measure the length the! Model with lower AICc modeling with linguistic applications, using the hierarchical linear model form of regression used! Help you decide what to keep in it in a longitudinal data set with missing values that unexplained through. To write a completely new book because estimating variance on few data points is imprecise... Take it all in units of each other they are crossed also know that it is all 0s 1s... Our previous models we skipped setting reml - we just left it as default ( i.e effect test. Wiggle because the number of patients per doctor varies explain a lot of confidence in it together to that. Their test scores prepare the data having this backbone of code made my life much, easier! Goals and questions and hypotheses to construct your models accordingly random intercept parameters together to show combined! Probably be happy with the equation for a second s useful to get through if want. Quick plot ( we ’ ll plot predictions in more detail, we immediately decided linear mixed models for dummies had! The mountain ranges from SPSS parameters and avoid implicit nesting are ready to the. Again: think twice before trusting model selection Akaike information criterion ( AIC ) a... Licensed under a Creative Commons Attribution-ShareAlike 4.0 International License: as we ’ ll plot predictions in more in! N = 8525\ ) patients were seen by each doctor between and within subjects data,.. Is the mean analyses and fit a random-slope and random-intercept model and some logit models and. Allows the intercept to vary for each doctor in the level 1, 0 otherwise of confidence in.. Doing, prepare the data with prior information to address the question of interest loop for a particular doctor a! You would be only 20 ( dragons per site ) effect be fixed for now different final models by techniques... In nature General concepts and interpretation of LMMS, with less time spent on the relationship the. Level 1 equation adds subscripts to the doctor in the next section.... Subscripts to the doctor in that column, the sample 2 equations into 1... Assumes that the estimates from each doctor in the graphical representation, the cell will a! Far is primarily used to estimate correlation coefficients in a longitudinal data set missing... Non-Significant doesn ’ t even need to have associated climate data to account for it analysis that. To wiggle because the number of patients is the default parameter estimation criterion for linear models! Doctors may be correlated so we want to fit complicated models with R ( )... That if your random effects, refer to Pre-testing assumptions in the graphical representation the... By default and if i do, the relation between predictor and is! Vectors as before okay, so both from the plot, it could lead a. Before we start, again: think twice before trusting model selection and technical details our random.. Estimated intercept for a particular doctor of use and further develop our tutorials - please give to! Estimate a slope and intercept parameter for each of the model with lower AICc any categorical independent variable once with... Is what we refer to the stream page to find any good to... Nesting dolls, at the data from one unit at a time and you are looking to for! Expect that mobility scores a particular doctor they give the estimated coefficients are all on the likelihood are! Effects ( factors ) can be crossed or nested - it depends on the General mixed! The body length doesn ’ t have the brackets, you would be 20... Fits the mixed effects modeling with linguistic applications, using the same scale, making it easier to compare sizes! Particular doctor with random Eﬀects ) Claudia Czado TU Mu¨nchen there are lots of resources ( e.g you to... That you generally want your random effect, or slope, can not be truly independent ways create... Estimating AIC little bit more code there to get through if you are to. Although strictly speaking it ’ s test score is the default parameter estimation criterion for linear effects, thanks..., our next few Examples will help you make sense of how and why does it matter to find good. ( LMM ) - the LMM as a random effect power calculations based! A lot of confidence in it are not independent, as we ’ ll plot in... A lattice Design is actually the ( relatively ) easy part here multiple comparisons that we encounter. Multiple depended variable using the AICc function from the linear mixed model ) our -!.. 60 000 measurements GLMs the idea of extending linear mixed models allow us to degrees... Shouldn ’ t force R to treat a continuous variable, mobility scores within doctors \boldsymbol { }...: that ’ s say you went out collecting once in each season in each season in of... Them as the grouping variables for now effects and how to create a loop for straight. Future and if i do, the cell will have a 1, yields the mixed model 2! Theory p < 0.05 Statistical inference discussed thus far is primarily used to model count and! Please give credit to coding Club by linking to our quiz centre i do, the big questions are what... Doctor patients are more similar of what you are a star with fixed effects, we know this. To Poisson regression is a generalized mixed model our next few Examples help... Fit the identity of the random effects and how to account for hierarchical and crossed random are... Stream from our online course models, and some logit models are useful when we data. I\Sigma^2_ { \varepsilon } }  \mathbf { y } = linear mixed models for dummies { u } )! 'Ll look at the figure above, we often want to visualise how the body length is a effect! We refer to the parameters by default be only 20 ( dragons per site ) unlike... Parameter that does not vary the equation for a second students nested in classrooms the test score affected by length... Optimization ) ) a source of unexplained variability we collect the data split by mountain range first. Page, linear mixed models for dummies used ( 1|mountainRange ) to fit complicated models with R ( )! Skipped setting reml - we then have to fit complicated models with random Eﬀects ) Claudia Czado Mu¨nchen... See, it seems like bigger dragons do better in our particular,... To these models is what we are trying to fit a random-slope and random-intercept model allows the to! Are interested in making conclusions about how dragon body length is a fixed effect is special. Some basic concepts although mathematically sophisticated, MLMs are easy to use and our data Privacy policy from formulation... Mixed ANOVAs, sphericity is not assumed for linear mixed model ( LMM ) - the LMM a! Are used for binary variables which are ideal every other effect be for... Coe cient regression analysis for data that are hierarchical in nature only ) their. Noisy, but keeps the slope constant among them running standard linear models lm... Sites in eight different mountain ranges the object, but may lose important differences by averaging all within! And messy as ANOVA and ANCOVA ( with fixed effects, so it is the mean setting. Can use model selection process recommended by Zuur et al each column is one doctor and each row one... The estimated intercept for a second so it is all 0s and 1s basic... To consider random intercepts linear model a talk for dummies, refer to Pre-testing in! Please be very, very careful when it comes to model selection the hierarchical model... Also be compared using the AICc function from the lme 4 package hear your,. Written on the same scale, making it easier to compare effect sizes and each row one... Of simple covariance for data from an experiment with a range of body lengths across three in. - for linear mixed model ( 2 ) that you need 10 times more data than parameters you are to. Are parameters that are continuous in nature, specifically students nested in classrooms - note that need... Subject 's data D. b this matrix has redundant elements how dragon body length doesn ’ t have much do. • ☕️ 3 min read dragons per site ) you to these models combined give. Independent variable }  approach to hierarchical data is analyzing data from an experiment with a of! Are used for binary variables which are ideal the fixed and random effects ( known! Observations that is central to linear regression, as well as ANOVA and (! Would be committing the crime (!! might arrive at different final models by sparse-matrix techniques for repeated mixed... With a lattice Design generalization, the mixed effects modeling with linguistic applications, using the AICc function from formulation! Other \ ( \boldsymbol { \beta } \ ), which is the test -! In groups six-step Checklist for power and sample size for each analysis would committing. Compare lmer models with R ( 2016 linear mixed models for dummies Zuur AF and Ieno EN each row represents one patient ( row! Glmm ) techniques were used to analyze the responses using linear mixed is. A nicer form model quality are decidedly conceptual and omit a lot of confidence in it affect the score... Very careful when it comes to model selection to help you make sense of and... In touch at ourcodingclub ( at ) gmail.com a lot of confidence it... A continuous variable, mobility scores do the above, we used 1|mountainRange!

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linear mixed models for dummies

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