G89.2247 Lecture 5 1

G89.2247Lecture 5

• Example fixed

• Repeated measures as clustered data

• Clusters as random effects

• Intraclass correlation

• ANOVA approach

• PROC MIXED Approach

G89.2247 Lecture 5 2

Whoops! A Mistake Fixed

• Checking data is importantPreparing charts for today alerted us to the fact that

POMS in the Comparison Group was on a 1-5 scale, while the Bar Exam Group was 0-4.

Two Groups: Anxiety over Four Weeks

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G89.2247 Lecture 5 3

Repeated Measures as Clustered Data

• There are many examples when clusters of data are collectedSiblings within a familyChildren within a classroomHouseholds within a Primary Sampling Unit

• Repeated measures are a special case of clustered dataTimes within a personBut ... measurements are ordered by time

G89.2247 Lecture 5 4

Thinking generally about clusters

• Suppose we sampled study groups of size fourEach group has four measurement (persons in this

case)Measurements are not usually orderedObservations within a cluster may be more similar

than observations sampled across clusters

• If we have 135 study groups, is it fair to analyze the 135*4=540 persons as though they were independent observations?

G89.2247 Lecture 5 5

Clusters as Random Effects

• Sampling clusters are often considered to be Random EffectsClusters are informative about overall populationActual choice of a specific cluster is due to chanceCluster effects are best thought in variance terms

• Snijders and Bosker call the clusters Macro level unitsElements within the cluster are called Micro level

units

G89.2247 Lecture 5 6

A One-way Random Effects Model• According to S&B, the observation Y for the ith

observation in the jth cluster (macro level) is Yij = + Uj + Rij

where is the overall mean of the population,Uj is the effect of randomly selected macro-unit jRij is the effect of randomly selected micro-unit i in randomly selected macro-unit j.

Define Var(Uj) = Var(Rij) = Var(Yij) = (assuming Corr(U,R)=0)

• Bryk and Raudenbush notation (sort of)• A randomly chosen observation varies as a function of

cluster variation and within cluster variation.

G89.2247 Lecture 5 7

One-way random effects interpreted

• Yij = + Uj + Rij

• Suppose clusters are monozygotic twins and Y is a measure of eye colorAll of the variation in Y would be due to between twin

effects (macro-unit U). R would reflect measurement error only

• Suppose clusters were pairings of persons who report for subject pool studiesThere might be some cluster effects due to subtle

personality differences in when people volunteer, but most variation in Y would be due to micro-unit R

G89.2247 Lecture 5 8

How much of Var(Y) is due to Macro-level variation?

• The Intraclass correlation is used to quantify how much of Var(Y) is due to Var(U).

• Assume we can get estimates of Var(U)= and Var(R)=. These will come from either ANOVA or special software.

• ICC =

G89.2247 Lecture 5 9

ICC interpreted as a correlation

• The correlation between any two observations within a cluster

U*j

Y*1j Y*

2j

Corr(Y1j, Y2j)=

G89.2247 Lecture 5 10

Example of ICC from ANOVA

• Suppose we consider the 135 persons from the examinee and comparison groups to be clusters with four replicationsIgnore the ordering of replicationsLet's think of the replications as random effects

• SPSS Reliability can give us the estimate of Intraclass Correlation

G89.2247 Lecture 5 11

SPSS example

RELIABILITY /VARIABLES=week1 week2 week3 week4

/SCALE(persons)=ALL/MODEL=ALPHA

/STATISTICS=DESCRIPTIVE SCALE ANOVA

/ICC=MODEL(ONEWAY) CIN=95 TESTVAL=0 .

G89.2247 Lecture 5 12

Source of Variation Sum of Sq. DF Mean Square F Prob.

Between People 357.0247 134 2.6644

Within People 80.0973 405 .1978

Between Measures 11.0192 3 3.6731 21.3754 .0000

Residual 69.0781 402 .1718

Total 437.1220 539 .8110

Grand Mean 1.0291

Intraclass Correlation CoefficientOne-way random effect model: People Effect Random

Single Measure Intraclass Correlation = .7572

95.00% C.I.: Lower = .6995 Upper = .8091

F = 13.4720 DF = ( 134, 405.0) Sig. = .0000 (Test Value = .0000 )

Analysis of Variance Table

G89.2247 Lecture 5 13

Where Are the Variance Estimates?

• The ANOVA table shows where to get the estimate of Var(R)=. We use the "Within people" Mean Square, which is MSW=.1978. (S&B Eq. 3.10)

• To get Var(U)= is a bit more work.E(MSB) = 4

stimate((MSB-MSW)/4• (2.6644-.1978)/4 = .6166

• ICC = .6166/(.6166+.1978) = .7572

G89.2247 Lecture 5 14

Interpreting ICC

• In this example, 76% of the variance of the anxiety scores is due to macro-unit differences

• Some of the macro-unit variation may be due to examinee/comparison differencesWithin the comparison group the ICC is still .75

• But the estimates of and are smaller than overall

Within the examinee group the ICC is .59• The within macro-unit variance is relatively large in this

case.

G89.2247 Lecture 5 15

Studying Random Effects using SAS PROC MIXED

• The ANOVA procedure may be familiar, but it is not the easiest way to study the one way random effects model

DATA anxgrps;infile 'bothanxst.dat';input id 1-4 week 5-7 group 8-10 anx 11-15;id = id+100*group; *assign unique IDs to subjects;week=week-2.5; *center week at week 2.5;Proc sort; by id;Proc mixed covtest NOCLPRINT ; Class id; MODEL anx= /s; RANDOM Intercept /Subject=ID g; run;

G89.2247 Lecture 5 16

The Mixed Procedure Dimensions Covariance Parameters 2 Subjects 135 Max Obs Per Subject 4 Observations Used 540 Observations Not Used 0 Total Observations 540 Covariance Parameter Estimates Standard ZCov Parm Subject Estimate Error Value Pr ZIntercept id 0.6163 0.08141 7.57 <.0001Residual 0.1977 0.01389 14.23 <.0001 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 1.0294 0.07023 134 14.66 <.0001

G89.2247 Lecture 5 17

Comparison of Random Effects and Means

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G89.2247 Lecture 5 18

Extension 1: Fixed Cluster Effects

• We can carry out the equivalent of a two sample t test.Group is a Fixed Effect

Proc mixed covtest NOCLPRINT ;

Class id;

MODEL anx=group /s;

RANDOM Intercept /Subject=ID g ;

run;

G89.2247 Lecture 5 19

PROC MIXED Two Group Results

Covariance Parameter Estimates

Standard Z

Cov Parm Subject Estimate Error Value Pr Z

Intercept id 0.3597 0.05028 7.15 <.0001

Residual 0.1977 0.01389 14.23 <.0001

Solution for Fixed Effects

Standard

Effect Estimate Error DF t Value Pr > |t|

Intercept 1.5335 0.07756 133 19.77 <.0001

group -1.0155 0.1101 405 -9.22 <.0001

G89.2247 Lecture 5 20

Extension 2: Time, Group and Time-by-Group Considered

Proc mixed covtest;

Class id;

MODEL anx=week group group*week /s;

RANDOM Intercept week /Subject=ID type=un g gcorr;

run;

G89.2247 Lecture 5 21

PROC MIXED Results Estimated G Correlation Matrix Row Effect id Col1 Col2 1 Intercept 1 1.0000 0.4222 2 week 1 0.4222 1.0000  Covariance Parameter Estimates

Cov Parm Subject Estimate S Error Z Value Pr ZUN(1,1) id 0.3828 0.05022 7.62 <.0001UN(2,1) id 0.0361 0.01154 3.13 0.0018UN(2,2) id 0.0191 0.005236 3.65 0.0001Residual 0.1049 0.009032 11.62 <.0001  Solution for Fixed Effects

Effect Estimate S Error DF t Value Pr > |t|Intercept 1.5335 0.07756 133 19.77 <.0001week 0.2706 0.02428 133 11.14 <.0001group -1.0155 0.1101 270 -9.22 <.0001week*group -0.2942 0.03446 270 -8.54 <.0001

G89.2247 Lecture 5 22

Creating a data File with One Line Per Observation in SPSS

write outfile='bothanxst.dat' records=4

/1 id 1-4 ' 1 ' sample 9-10 week1 (f5.2)

/2 id 1-4 ' 2 ' sample 9-10 week2 (f5.2)

/3 id 1-4 ' 3 ' sample 9-10 week3 (f5.2)

/4 id 1-4 ' 4 ' sample 9-10 week4 (f5.2).

execute.

G89.2247 Lecture 5 23

Reading a data file with four observations per line in SAS

data new;

infile 'G2247_1.dat';

time=1;

input id 1-4 group 5-6 supp 13-14@@; output;

time=2;

input id 1-4 group 5-6 supp 15-19@@; output;

time=3;

input id 1-4 group 5-6 supp 20-24@@; output;

time=4;

input id 1-4 group 5-6 supp 25-29 ; output;

data new2;

set new;

week=time-2.5;

id=id+100*group;