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LME (Assumptions (Homoscedasticity within each random effect group,…

- - - - Informative variables
    - - Occur for 2 reasons:
        
        Uninformative variables
        
        designed experiments with different spatial or temporal scales.
        
        observational studies with hierarchical (nested) structure.
      - Random effects occur within a group and make the observations of that group be correlated. This is violating our assumption of independence of errors.
      - Definition: a source of non-independence that affects the variance in a group and that needs to be accounted for.
- - - - model1 <- glmer(y~x + (1|A/B/C), family = binomial)
    - - Random Intercept Model
        
        How to code in R
        
        RIKZ<-read.table("D:\yourWD\RIKZ.txt",header=T)
        library(nlme)
        RIKZ$fBeach <- factor(RIKZ$Beach)
        Mlme1 <- lme(Richness ~ NAP, random = ~1 | fBeach, data=RIKZ)
        
        Summary (Mlme1)
        
        Intepretation
        
        The output is split between random and fixed effects.
        
        The residual variance is R^2.
        
        Variance of random intercept is (StdDev Intercept) ^2
        
        The interpretation of the fixed part is what we are used to.
        
        The predictions are composed of a fixed component (as usual the intercept plus the slope times the value of NAP)
        
        The random component “shifts” this average fixed prediction for each beach according to the variance estimated (2.9442).
      - Random Intercept and Slope
        
        The previous model allowed to shift in parallel up and down the fixed predictions.
        •On top of changing the intercept we might want to change the slope.
        
        ignoring it will lead to systematic variation being absorbed by the residuals, potentially leading to biased inference.
        
        To be more efficient with degrees of freedom we can use a LME with a random intercept and a random slope.
        
        How to code in R
        
        Mlme2 <- lme(Richness ~ NAP, +
        random = ~1 + NAP | fBeach, data = RIKZ)
        summary(Mlme2)
        
        The correlation between random intercept and slope is high (-0.99), indicating that beaches with large positive intercept have large negative slope.
      - To compare models, we can observe that AIC
      - Random Effects Model
        
        There are no fixed effects and only random effects
        
        Richness is now modeled as an intercept and a random effect bi that can differ per beach. The AIC for the model is higher, 267. The code is:
        
        Mlme3 <- lme(Richness ~ 1, random = ~1 | fBeach, data = RIKZ)
        summary(Mlme3)
- - - - plot(Mlme2)
        plot(Mlme2, resid(., scaled=TRUE) ~ fitted(.), abline = 0)
      - Each of the groups of the random effects
        
        plot(Mlme2, resid(., scaled=TRUE) ~ fitted(.) | fBeach, abline = 0)
      - as a function of the explanatory variables:
        
        plot(Mlme2, fBeach ~ resid(., scaled=TRUE))
        plot(Mlme2, resid(., scaled=TRUE) ~ NAP | fBeach, abline = 0)
    - - qqnorm(Mlme2, ~ resid(., type = "p") | fBeach, abline = c(0, 1))
    - - qqnorm(Mlme2, ~ranef(.))
  - - - refit the model first
        
        library(lme4)
        Mlme2 <- lmer(Richness ~ NAP + (NAP | fBeach), data = RIKZ)
        
        library(lattice)
        reAndSE <- ranef(Mlme2, condVar = TRUE)
        dotplot(reAndSE)
- - - - REML is preferred because it gets a better estimator of the variance than ML, that tends to underestimate it.
        
        ML does not correct for the degrees of freedom that need to be used to estimate the parameters of the model.
        
        However, REML cannot be used to compare models where the fixed structure has changed.
      - REML when simplifying the random effects part of the model
      - ML when simplifying the fixed part of the model (we are comparing models with different fixed structures).
  - - - Fix the Fixed effects; use REML to compare the nested random structures
        
        try several random effects structures and compare them with each other.
        
        AIC or BIC
        
        We will choose a random effects part that gets the lowest AIC.
        
        use likelihood ratio tests using the anova() command to simplify the fixed part of the model
        
        The ML function is invariant to one-to-one re-parameterizations of the fixed effects. For instance changing the contrasts representing a categorical variable. The REML function, on the other hand, varies.
        
        1 more item...
- - - - r.squaredGLMM()
        
        reports the marginal R2 (the variance explained by the fixed effects of the model) and the conditional R2 (variance explained by both fixed and random effects).
    - - library(MuMIn)
        r.squaredGLMM(Mlme1)
        
        R2m = variance explained by Fixed Effects; R2c = varianced explained by Random intercept model.