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Lecture 5 Linear Mixed Effects Models

Advanced Research Skills. Lecture 5 Linear Mixed Effects Models. Olivier MISSA, om502@york.ac.uk. Outline. Explore options available when assumptions of classical linear models are untenable.

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Lecture 5 Linear Mixed Effects Models

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  1. Advanced Research Skills Lecture 5 Linear Mixed Effects Models Olivier MISSA, om502@york.ac.uk

  2. Outline • Explore options available when assumptions of classical linear models are untenable. • In this lecture: What can we do when observations (and thus residuals) are not strictly independent ?

  3. Classical Linear Models • Defined by three assumptions: • (1) the response variable is continuous. • (2) the residuals (ε) are normally distributed and ... • (3) ... independently and identically distributed. • Today, we will consider a range of options availablewhen we either know or suspect that our data are not strictly independent from each other. • (Departures from the other assumptions will be dealt with later)

  4. A non-linear trend Non-independent Residuals • In previous lectures: • We merely checked the independence of our residuals by inspecting the plot of residualsvs.fitted values. • Example from lecture 2: which suggested that our linear model was probably misspecified

  5. Non-independent Observations • Data collection can often lead to non-independence among your observations. • A few examples: • Repeated (longitudinal) observations on the same "individuals" • (on different days, weeks, months, years) • Collecting data from a few locations (spatial structure) • (surveys conducted in schools/streets, fields/sites/islands) • Collecting data on related individuals • (father & sons, twins, species within the same genus/tribe/family). • If we treat all these observations as fully independent, we are likely to overestimate the number of degrees of freedom. • which may lead us to wrongly reject a null hypothesis (type I error)

  6. Non-independent Observations • Two ways to cope with non-independent observations • When design is balanced ("equal sample size") • We can use factors to partition our observations in different "groups" and analyse them as an ANOVA or ANCOVA. • We already know how to do that (when factors are "crossed") • We just need to figure out how to cope with nested factors. • When design is unbalanced ("uneven sample size") • Mixed effect models are then called for.

  7. control irrigated high medium Block low N P NP Nested Anova • Example: • A designed field experiment on crop yield with three treatments : • irrigation (control, irrigated) • sowing density (low, medium, high) • fertilizer (N, P, NP) Split plot design each block has 18 different subplots

  8. Nested Anova • Example: • A designed field experiment on crop yield with three treatments > yields <- read.table("splityield.txt", header=T) > attach(yields) > names(yields) [1] "yield" "block" "irrigation" "density" "fertilizer" > str(yields) 'data.frame': 72 obs. of 5 variables: $ yield : int 90 95 107 92 89 92 81 92 93 80 ... $ block : Factor w/ 4 levels "A","B","C","D": 1 1 1 1 1... $ irrigation: Factor w/ 2 levels "control","irrigated": 1 1... $ density : Factor w/ 3 levels "high","low","medium": 2 2... $ fertilizer: Factor w/ 3 levels "N","NP","P": 1 3 2 1 3 2...

  9. Nested Anova • Example: • A designed field experiment on crop yield with three treatments > model0 <- aov(yield ~ irrigation*density*fertilizer) ## non-nested version (incorrect !!) > summary(model0) Df Sum Sq Mean Sq F value Pr(>F) irrigation 1 8277.6 8277.6 59.5746 2.813e-10 *** density 2 1758.4 879.2 6.3276 0.0033972 ** fertilizer 2 1977.4 988.7 7.1160 0.0018070 ** irrigation:density 2 2747.0 1373.5 9.8853 0.0002197 *** irrigation:fertilizer 2 953.4 476.7 3.4310 0.0395615 * density:fertilizer 4 304.9 76.2 0.5486 0.7008151 irrigation:density:fertilizer 4 234.7 58.7 0.4223 0.7918283 Residuals 54 7503.0 138.9 Sum 71 23756.44

  10. Nested Anova > model1 <- aov(yield ~ irrigation*density*fertilizer + Error(block/irrigation/density) ) ## Correct nested version, nesting from large to small > summary(model1) Error:block Df Sum Sq Mean Sq F value Pr(>F) Residuals 3 194.44 64.815 Error:block:irrigation Df Sum Sq Mean Sq F value Pr(>F) irrigation 1 8277.6 8277.6 17.59 0.0247 * Residuals 3 1411.8 470.6 Error:block:irrigation:density Df Sum Sq Mean Sq F value Pr(>F) density 2 1758.4 879.18 3.784 0.05318 . irrigation:density 2 2747.0 1373.51 5.912 0.01633 * Residuals 12 2787.9 232.33 Error:Within Df Sum Sq Mean Sq F value Pr(>F) fertilizer 2 1977.4 988.72 11.449 0.000142 *** irrigation:fertilizer 2 953.4 476.72 5.520 0.008108 ** density:fertilizer 4 304.9 76.22 0.883 0.484053 irrigation:density:fertilizer 4 234.7 58.68 0.679 0.610667 Residuals 36 3108.8 86.36 Res Sum 54 7503.00Gd Sum 71 23756.44

  11. control irrigated high medium Block low N P NP Nested Anova • Comparison between nested and non-nested results Non-nested Nested Df F value Pr(>F) F value Pr(>F) irrigation 1 59.5746 2.81e-10 17.5896 0.024725 density 2 6.3276 0.003397 3.7842 0.053181 fertilizer 2 7.1160 0.001807 11.4493 0.000142 irrig:dens 2 9.8853 0.000220 5.9119 0.016331 irrig:ferti 2 3.4310 0.039562 5.5204 0.008108 dens:ferti 4 0.5486 0.700815 0.8826 0.484053 irrig:dens:ferti 4 0.4223 0.791828 0.6795 0.610667

  12. Recognizing Nestedness is key ! • Being able to distinguish crossed factors (independent from each other) from nested factors is essential. • Nestedness occurs most often from spatial structure • Student surveys in different classes from different schools. • Samples from individual branches on sets of trees within a number of forest patches. • But can also occur from temporal structure • Samples taken from the same individuals every fortnight for 2 months on two successive years.

  13. When the design is not balanced • We need a different modelling framework:Mixed Effects Models. • So called because they mix together fixed effectsand random effects. • Until now, we have only used fixed effects in our models, • each effect having an estimated parameter • (intercept, slope, mean, ...). • But in certain circumstances, these parameters may not be very informative and one would be better off trying to "estimate" the underlying distribution they come from. • An example will help clarify the difference between these 2 approaches.

  14. Mixed Effects Modelling • Example: railway rails tested for longitudinal stress. • 6 rails chosen at random and tested three times with ultrasound. > library(nlme) ## package dedicated to mixed effects models > data(Rail) > names(Rail) [1] "Rail" "travel" > stripchart(Rail$travel ~ Rail$Rail, pch=16, ylab="Ultrasonic Travel Time (nanosecs)", xlab="Rail number", vertical =T, col=rainbow(6) ) > abline(h=mean(Rail$travel), col="Gray85", lty=2, lwd=2) Classically in a linear model, we would be able to tell whether the rails differ significantly from each other. But it doesn't help us make predictions about other rails.

  15. Mixed Effects Modelling • Random effects: interested in explaining the variance of the response. • Fixed effects: interested in explaining the response itself. Fixed effects Male & Female Control & Treatment Wet vs. Dry Light vs. Shade Random effects Blocks Individuals withRepeated measures Genotypes Sites

  16. Makes Rail a simple factor (not an ordered one) Mixed Effects Modelling • Back to our example: > Rail2 <- data.frame(travel=Rail$travel, Rail=factor(as.character(Rail$Rail)) ) > Rail.lm <- lm(travel ~ Rail, data=Rail2) ## LINEAR MODEL > summary(Rail.lm) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 54.000 2.321 23.262 2.37e-11 *** Rail2 -22.333 3.283 -6.803 1.90e-05 *** Rail3 30.667 3.283 9.341 7.44e-07 *** Rail4 42.000 3.283 12.793 2.36e-08 *** Rail5 -4.000 3.283 -1.218 0.246 Rail6 28.667 3.283 8.732 1.52e-06 *** --- Residual standard error: 4.021 on 12 degrees of freedom Multiple R-squared: 0.9796, Adjusted R-squared: 0.9711 F-statistic: 115.2 on 5 and 12 DF, p-value: 1.033e-09

  17. Standard Deviation associated with rails Standard Deviation of residuals Grand Average Mixed Effects Modelling > anova(Rail.lm) Analysis of Variance Table Response: travel Df Sum Sq Mean Sq F value Pr(>F) Rail 5 9310.5 1862.1 115.18 1.033e-09 *** Residuals 12 194.0 16.2 ## now as a MIXED EFFECT MODEL > Rail.lme <- lme(travel ~ 1, data=Rail, random= ~1|Rail) > summary(Rail.lme) Linear mixed-effects model fit by REML Data: Rail AIC BIC logLik 128.177 130.6766 -61.0885 Random effects: Formula: ~1 | Rail (Intercept) Residual StdDev: 24.80547 4.020779 Fixed effects: travel ~ 1 Value Std.Error DF t-value p-value (Intercept) 66.5 10.17104 12 6.538173 0

  18. Standard Deviation associated with rails Standard Deviation of residuals Grand Average Mixed Effects Modelling > anova(Rail.lm) Analysis of Variance Table Response: travel Df Sum Sq Mean Sq F value Pr(>F) Rail 5 9310.5 1862.1 115.18 1.033e-09 *** Residuals 12 194.0 16.2 ## now as a MIXED EFFECT MODEL > Rail.lme <- lme(travel ~ 1, data=Rail, random= ~1|Rail) > summary(Rail.lme) Linear mixed-effects model fit by REML Data: Rail AIC BIC logLik 128.177 130.6766 -61.0885 Random effects: Formula: ~1 | Rail (Intercept) Residual StdDev: 24.80547 4.020779 Fixed effects: travel ~ 1 Value Std.Error DF t-value p-value (Intercept) 66.5 10.17104 12 6.538173 0

  19. Mixed Effects Modelling Is the Random effect significant ? > Rail.lme$call ## random effect model lme.formula(fixed = travel ~ 1, data = Rail, random = ~1 | Rail) > AIC(Rail.lme) ## exact AIC version [1] 128.177 > Rail.lm0 <- lm(travel ~ 1, data=Rail2) ## NULL linear model > AIC(Rail.lm0) [1] 167.9265 > Rail.lm$call ## model with Rail as a fixed effect factor lm(formula = travel ~ Rail, data = Rail) > AIC(Rail.lm) [1] 107.8765 Comparing models which only differ in their random effects is easy (with AIC). Comparing models which differ in their fixed effects is a little harder. Can only be done using "maximum likelihood" (not the default method in lme).

  20. Mixed Effects Modelling Applied on the split plot study of crop yield > yields <- read.table("splityield.txt", header=T) > yield.lme <- lme(yield ~ irrigation*density*fertilizer, data=yields, random= ~1|block/irrigation/density) > summary(yield.lme) Linear mixed-effects model fit by REML Data: yields AIC BIC logLik 481.6212 525.3789 -218.8106 Random effects: Formula: ~1 | block (Intercept) StdDev: 0.0006600339 Formula: ~1 | irrigation %in% block (Intercept) StdDev: 1.982461 Formula: ~1 | density %in% irrigation %in% block (Intercept) Residual StdDev: 6.975554 9.292805 ...

  21. Mixed Effects Modelling Fixed effects: yield ~ irrigation * density * fertilizer Value Std.Error DF t-value p-value (Intercept) 80.50 5.893741 36 13.658558 0.0000 irrigirrig 31.75 8.335008 3 3.809234 0.0318 dnslow 5.50 8.216282 12 0.669403 0.5159 dnsmed 14.75 8.216282 12 1.795216 0.0978 fertiNP 5.50 6.571005 36 0.837010 0.4081 fertiP 4.50 6.571005 36 0.684827 0.4978 irrigirrig:dnslow -39.00 11.619577 12 -3.356404 0.0057 irrigirrig:dnsmed -22.25 11.619577 12 -1.914872 0.0796 irrigirrig:fertiNP 13.00 9.292805 36 1.398932 0.1704 irrigirrig:fertiP 5.50 9.292805 36 0.591856 0.5576 dnslow:fertiNP 3.25 9.292805 36 0.349733 0.7286 dnsmed:fertiNP -6.75 9.292805 36 -0.726368 0.4723 dnslow:fertiP -5.25 9.292805 36 -0.564953 0.5756 dnsmed:fertiP -5.50 9.292805 36 -0.591856 0.5576 irrigirrig:dnslow:fertiNP 7.75 13.142011 36 0.589712 0.5591 irrigirrig:dnsmed:fertiNP 3.75 13.142011 36 0.285344 0.7770 irrigirrig:dnslow:fertiP 20.00 13.142011 36 1.521837 0.1368 irrigirrig:dnsmed:fertiP 4.00 13.142011 36 0.304367 0.7626

  22. Mixed Effects Modelling > anova(yield.lme) numDF denDF F-value p-value (Intercept) 1 36 2674.6301 <.0001 irrigation 1 3 30.9207 0.0115 density 2 12 3.7842 0.0532 fertilizer 2 36 11.4493 0.0001 irrigation:density 2 12 5.9119 0.0163 irrigation:fertilizer 2 36 5.5204 0.0081 density:fertilizer 4 36 0.8826 0.4841 irrigation:density:fertilizer 4 36 0.6795 0.6107 ## We should probably remove the three-way interaction ## But if we are fiddling with the fixed effects, we ought ## to fit the model through Maximum Likelihood and base our ## decisions on its AIC values and Likelihood Ratio Tests > yield.lme.ml <- update(yield.lme, ~. ,method="ML") > AIC(yield.lme.ml) [1] 573.5108

  23. Mixed Effects Modelling > yield.lme.ml2 <- update(yield.lme.ml, ~. - irrigation:density:fertilizer) > yield.lme.ml2$method [1] "ML" ## just checking that update() kept using "ML" > AIC(yield.lme.ml2) [1] 569.0046 ## an improvement > anova(yield.lme.ml2) numDF denDF F-value p-value (Intercept) 1 40 2872.7394 <.0001 irrigation 1 3 33.2110 0.0104 density 2 12 4.0645 0.0449 fertilizer 2 40 11.4341 0.0001 irrigation:density 2 12 6.3499 0.0132 irrigation:fertilizer 2 40 5.5131 0.0077 density:fertilizer 4 40 0.8815 0.4837 > yield.lme.ml3 <- update(yield.lme.ml2, ~. - density:fertilizer) > AIC(yield.mle.lm3) [1] 565.1933

  24. Mixed Effects Modelling > anova(yield.lme.ml,yield.lme.ml2) Model df AIC BIC logLik Test L.Ratio p-value yield.lme.ml 1 22 573.5108 623.5974 -264.7554 yield.lme.ml2 2 18 569.0046 609.9845 -266.5023 1vs2 3.49379 0.4788 > anova(yield.lme.ml2, yield.lme.ml3) Model df AIC BIC logLik Test L.Ratio p-value yield.lme.ml2 1 18 569.0046 609.9845 -266.5023 yield.lme.ml3 2 14 565.1933 597.0667 -268.5967 1vs2 4.18877 0.3811 > anova(yield.lme.ml3) numDF denDF F-value p-value (Intercept) 1 44 3070.8771 <.0001 irrigation 1 3 35.5016 0.0095 density 2 12 4.3448 0.0381 fertilizer 2 44 11.2013 0.0001 irrigation:density 2 12 6.7878 0.0107 irrigation:fertilizer 2 44 5.4008 0.0080 > yield.lme.ml4 <- update(yield.lme.ml3, ~. –irrigation:density) > AIC(yield.mle.ml4) [1] 572.9022

  25. ml4 is one simplification too far column matrix Mixed Effects Modelling > anova(yield.lme.m3,yield.lme.ml4) Model df AIC BIC logLik Test L.Ratio p-value yield.lme.ml3 1 14 565.1933 597.0667 -268.5967 yield.lme.ml4 2 12 572.9022 600.2221 -274.4511 1vs2 11.7088 0.0029 > anova(yield.lme.ml4) numDF denDF F-value p-value (Intercept) 1 44 2138.9678 <.0001 irrigation 1 3 24.7281 0.0156 density 2 14 2.6264 0.1075 fertilizer 2 44 11.5626 0.0001 irrigation:fertilizer 2 44 5.5750 0.0069 ## here comes Model Checking > shapiro.test(yield.lme.ml3$residuals[,"fixed"]) # Best Model Shapiro-Wilk normality test data: yield.lme.ml3$residuals[, "fixed"] W = 0.9797, p-value = 0.2958

  26. including all random effects excluding all random effects Mixed Effects Modelling > res <- yield.lme.ml3$resid[,"fixed"] > st.res <- res/sd(res) > qqnorm(st.res, pch=16, main="") > qqline(st.res, col="red", lwd=2) > res <- yield.lme.ml3$resid[,4] > st.res <- res/sd(res) > qqnorm(st.res, pch=16, main="") > qqline(st.res, col="red", lwd=2)

  27. Mixed Effects Modelling > plot(yield.lme.ml3) ## by default Residuals vs Fitted values > plot(yield.lme.ml3, yield ~ fitted(.) ) ## Observed vs Fitted values

  28. Mixed Effects Modelling > qqnorm(yield.lme.ml3, ~resid(.) | block) ## qqplot but broken down by block

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