Multilevel Model - A very brief tutorial

Kamarul Imran M

8 December 2015

Kamarul Imran M Multilevel Model - A very brief tutorial 8 December 2015 1 / 12

Motivation

You may collect data from groupsThese groups may be selected from different levels or hierarchyThe elements of members in a particular group share more similarfeatures or characteristics.For example

Level 3 districtsLevel 2 schools (school)Level 1 pupils (case)

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Our data

caseid - pupils in school - Level 1schoolid - schools where pupils belong - Level 2

setwd("/Volumes/DATA/RandSpatialAndArcGISNotes/MixedModels")mydata <- read.table("5.1.txt", sep = ",", header = TRUE)summary(mydata)[, c(1, 2)]

## caseid schoolid## "Min. : 1 " "Min. : 1.0 "## "1st Qu.: 8532 " "1st Qu.:123.0 "## "Median :17318 " "Median :256.0 "## "Mean :18466 " "Mean :254.4 "## "3rd Qu.:29428 " "3rd Qu.:386.0 "## "Max. :38192 " "Max. :511.0 "

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Variables

Dependent variable is scoreCovariate is cohort90 (different cohorts of pupils in schools)

summary(mydata)[,c(3,4)]

## score cohort90## "Min. : 0.00 " "Min. :-6.0000 "## "1st Qu.:19.00 " "1st Qu.:-4.0000 "## "Median :33.00 " "Median :-2.0000 "## "Mean :31.09 " "Mean : 0.2767 "## "3rd Qu.:45.00 " "3rd Qu.: 6.0000 "## "Max. :75.00 " "Max. : 8.0000 "

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DataframeA glimpse of our data

head(mydata,4)[,1:4]

## caseid schoolid score cohort90## 1 18 1 0 -6## 2 17 1 10 -6## 3 19 1 0 -6## 4 20 1 40 -6

tail(mydata,4)[,1:4]

## caseid schoolid score cohort90## 33985 6493 511 20 -6## 33986 6496 511 48 -6## 33987 6494 511 10 -6## 33988 6495 511 0 -6

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Random intercept (RI) model

we will fit an RI model using a linear mixed modelmodel with only 1 covariate (for simplicity) - cohort90

## Loading required package: Matrix

ri1<-lmer(score~cohort90+(1|schoolid),data=mydata,REML=FALSE)fixef(ri1)

## (Intercept) cohort90## 30.559151 1.214955

VarCorr(ri1)

## Groups Name Std.Dev.## schoolid (Intercept) 6.7815## Residual 14.8084

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The RI results

On average, the score at baseline was 30.6On average, with 1 unit increase in cohort, the score increases for 1.22We will see the plot.

The y axis is the score (dependent)The x axis is the cohort (covariate=beta1)

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Each schools have different scores at baseline (SD for intercept=6.8)Each school has the same slope (beta1=1.215)

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Kamarul Imran M Multilevel Model - A very brief tutorial 8 December 2015 8 / 12

Random slope (RS) model

Each schools have different score at baseline (SD for intercept=6.5)Each school has the different slope (SD for beta1=0.40)

rs1 <- lmer(score ~ cohort90 + (1 + cohort90 | schoolid),data = mydata, REML = FALSE)

fixef(rs1)

## (Intercept) cohort90## 30.609633 1.233903

VarCorr(rs1)

## Groups Name Std.Dev. Corr## schoolid (Intercept) 6.54660## cohort90 0.40074 -0.390## Residual 14.68807

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Intercept and slope correlation

The average score for schools at baseline = 30.6 (SD=6.5)The average slope for schools with 1 unit increase in cohort = 1.233(SD=0.39)The intercept-slope correlation = -0.390Negative correlation means

Schools with lower baseline score have higher slopeSchools with higher baseline score have lower slope

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xyplot(predscore ~ cohort90, data = mydata, groups = schoolid, type = c("p","l"), col = "blue",lwd=0.2)

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Summary

Multilevel data needs special treatmentRandom variations between members in different levels

variations at baselinevariations for slope

Linear mixed model provides more useful information than linearregressionLinear mixed model more accurately reflects our data than linearregression

THANKS

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