effects of missing values on the analysis of the ab/ ba crossover trial

Effects of Missing Effects of Missing Values on the Values on the

Analysis of the AB/ Analysis of the AB/ BA Crossover TrialBA Crossover Trial

Lauren RodgersLauren RodgersSupervisor: Prof JNS MatthewsSupervisor: Prof JNS Matthews

University of Newcastle upon University of Newcastle upon TyneTyne

OutlineOutline

Crossover ModelCrossover Model Missing DataMissing Data SimulationsSimulations ConclusionsConclusions Future WorkFuture Work

PERIOD 1:

PERIOD

2:

Randomise Trial Subjects

SEQUENCE 1

TREATMENT A

SEQUENCE 2

TREATMENT B

TREATMENT B

TREATMENT A

AB/ BA Crossover ModelAB/ BA Crossover Model

What can we estimate?What can we estimate? treatment effect, treatment effect, period effect, period effect, subject effectsubject effect

ProblemsProblems carryover effect carryover effect

Within subject estimate of treatment Within subject estimate of treatment effecteffect between subject variability is eliminatedbetween subject variability is eliminated

AB/ BA Crossover ModelAB/ BA Crossover Model

ijkikjkjdijkY ],[

Subject i in period j of sequence k Two treatment sequences indexed by k= 1,

2

i= 1,…, mk – patients in sequence k

j=1, 2 – treatment period

d[j, k]{A, B} – treatment allocated in period j of sequence k

General Mean

Treatment effect

Period effect Subject effects

of subject i in sequence k

Random error term ~ 20 wN ,

Subject EffectsSubject Effects

Fixed EffectsFixed Effects general level of each subject has a general level of each subject has a

fixed valuefixed value find MLE for find MLE for ikik

produce profile log-likelihood model to produce profile log-likelihood model to remove parameterremove parameter

n

ikikiikkmkmkk YYYYY

1212,11211 ,,,,...,,,,

kikiikik

YYf 21 ,0


ikik is a function of subject is a function of subject ii’s period 1 and ’s period 1 and

period 2 responseperiod 2 response when subject when subject ii has no response in any period has no response in any period

this MLE cancels out the remaining termsthis MLE cancels out the remaining terms

Model which includes only those with Model which includes only those with complete datacomplete data effectively exclude all data from a subject if any effectively exclude all data from a subject if any

missing datamissing data closed form for treatment estimate even in closed form for treatment estimate even in

presence missing data presence missing data


Random EffectsRandom Effects subject effect has some distribution – subject effect has some distribution – include all available datainclude all available data can be fitted using a Linear Mixed can be fitted using a Linear Mixed

Effects modelEffects model

No Missing Data – both models No Missing Data – both models produce same resultsproduce same results

),(~ 20 bik N

PERIOD ONEPERIOD ONE PERIOD TWOPERIOD TWO

NA 1.850

0.778 -0.529

0.345 0.327

0.651 NA

0.741 -2.505

-0.065 0.357

NA NA

0.877 NA

-0.829 -2.478

-0.804 -0.496

1.923 0.401

1.222 1.643

-2.749 -3.176

-2.006 -1.947

-0.696 -1.125

NA -1.429

-0.310 0.276

2.872 1.440

2.359 NA

NA -0.222


1.642 1.850

0.778 -0.529

0.345 0.327

0.651 1.501

0.741 -2.505

-0.065 0.357

-0.817 -3.671

0.877 1.136

-0.829 -2.478

-0.804 -0.496

1.923 0.401

1.222 1.643

-2.749 -3.176

-2.006 -1.947

-0.696 -1.125

-0.795 -1.429

-0.310 0.276

2.872 1.440

2.359 -0.114

2.786 -0.222

Missing DataMissing Data

Generate dataGenerate data shown for sequence shown for sequence

AB onlyAB only Introduce MCAR Introduce MCAR

missing datamissing data


- 1.850

0.778 -0.529

0.345 0.327

0.651 -

0.741 -2.505

-0.065 0.357

- -

0.877 -

-0.829 -2.478

-0.804 -0.496

1.923 0.401

1.222 1.643

-2.749 -3.176

-2.006 -1.947

-0.696 -1.125

- -1.429

-0.310 0.276

2.872 1.440

2.359 -

- -0.222

Missing DataMissing Data Fixed subject effectFixed subject effect

remove all data if remove all data if subject has any subject has any missingmissing

Random subject Random subject effecteffect keep all available keep all available

datadata


- -

0.778 -0.529

0.345 0.327

- -

0.741 -2.505

-0.065 0.357

- -

- -

-0.829 -2.478

-0.804 -0.496

1.923 0.401

1.222 1.643

-2.749 -3.176

-2.006 -1.947

-0.696 -1.125

- -

-0.310 0.276

2.872 1.440

- -

- -

Missing DataMissing Data

20%, 40% and 60% of data missing20%, 40% and 60% of data missing Pattern in sequences and periodsPattern in sequences and periods

equal amounts missing in each equal amounts missing in each sequence and periodsequence and period

data missing from period two onlydata missing from period two only equal amounts missing in each sequenceequal amounts missing in each sequence more missing from second sequencemore missing from second sequence

more data missing in second periodmore data missing in second period more data missing in second sequencemore data missing in second sequence

SimulationsSimulations

ParametersParameters number of subjects in trial: number of subjects in trial: mm= 20, 40, 120= 20, 40, 120 between and within subject variance between and within subject variance

amount and pattern of missing dataamount and pattern of missing data OutputOutput

root mean square error (RMSE)root mean square error (RMSE) estimate of estimate of and 95% CIand 95% CI

222wbb

Effect of on RMSE( )Effect of on RMSE( )2b

1 2 3 4 5 6

0.06

50.

070

0.07

50.

080

0.08

50.

090

0.09

5

b2 0.25 0.2

Index

RM

SE

Fixed EffectsRandom Effects

1 2 3 4 5 6

0.06

50.

070

0.07

50.

080

0.08

50.

090

0.09

5

b2 1 0.5

Index

RM

SE

123456

0:0:0:01:1:1:10:1:0:10:1:0:31:3:1:31:1:3:3

1 2 3 4 5 6

0.06

50.

070

0.07

50.

080

0.08

50.

090

0.09

5

b2 4 0.8

Index

RM

SE

Effect of on RMSE( )Effect of on RMSE( )2w

1 2 3 4 5 6

0.05

0.10

0.15

w2 0.25 0.8

Index

RM

SE


1 2 3 4 5 6

0.05

0.10

0.15

w2 1 0.5

Index

RM

SE

123456

0:0:0:01:1:1:10:1:0:10:1:0:31:3:1:31:1:3:3

1 2 3 4 5 6

0.05

0.10

0.15

w2 4 0.2

Index

RM

SE

Pattern of Missing DataPattern of Missing Data

1 2 3 4 5 6

0.20

0.25

0.30

0.35

0.40

0.45

0.50

20% Data Missing

Index

RM

SE


1 2 3 4 5 6

0.20

0.25

0.30

0.35

0.40

0.45

0.50

40% Data Missing

Index

RM

SE

123456

0:0:0:01:1:1:10:1:0:10:1:0:31:3:1:31:1:3:3

1 2 3 4 5 6

0.20

0.25

0.30

0.35

0.40

0.45

0.50

60% Data Missing

Index

RM

SE

95% CI for Treatment 95% CI for Treatment Effect Effect

No missing data: length of CI sameNo missing data: length of CI same Ratio length fixed: length random – Ratio length fixed: length random –

which is smaller?which is smaller? 20% missing20% missingIndexIndex mm=20=20 mm=40=40 mm=120=120

=0.06=0.06 224466

1.0131.011-

1.0341.0441.039

1.0481.0581.051

=0.5=0.5 224466

0.9500.9560.945

0.9910.9960.994

1.0161.0211.018

=0.8=0.8 224466

0.9330.9340.937

0.9750.9770.976

1.0001.0021.001

40% Missing40% Missing

IndexIndex mm=20=20 mm=40=40 mm=120=120

=0.06=0.06 224466

1.0351.087-

1.0841.1511.100

1.0101.1781.124

=0.5=0.5 224466

0.9580.9860.970

1.0151.0521.025

1.0451.0871.058

=0.8=0.8 224466

0.9320.9300.931

0.9960.9920.986

1.0431.0271.016

60% Missing60% Missing

IndexIndex mm=20=20 mm=40=40 mm=120=120

=0.06=0.06 224466

1.0621.476-

1.1441.6471.200

1.1751.7561.230

=0.5=0.5 224466

0.9521.2220.995

1.0411.3711.082

1.0841.4441.121

=0.8=0.8 224466

0.8981.0110.939

0.9801.1251.007

1.0241.1881.042

ConclusionsConclusions

Between subject variance has no effect on Between subject variance has no effect on fixed effects model but increases RMSE for fixed effects model but increases RMSE for random effects modelrandom effects model

Missing data – some differences for patternMissing data – some differences for pattern 95% CI for treatment effect95% CI for treatment effect

smaller for fixed effects model with small smaller for fixed effects model with small sample sizesample size

as sample size increases random effects model as sample size increases random effects model performs betterperforms better

as amount of missing data increases random as amount of missing data increases random effects model performs bettereffects model performs better

Future WorkFuture Work

MCAR missing data – not particularly MCAR missing data – not particularly usefuluseful

Data missing in period 2 if a correlate Data missing in period 2 if a correlate of period 1 response exceeds some of period 1 response exceeds some threshold threshold

Misspecified modelMisspecified model fit normal model to non-normal datafit normal model to non-normal data

Look at current methods to account Look at current methods to account for missing datafor missing data

ENDEND

effects of missing values on the analysis of the ab/ ba crossover trial

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