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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
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OutlineOutline
Crossover ModelCrossover Model Missing DataMissing Data SimulationsSimulations ConclusionsConclusions Future WorkFuture Work
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PERIOD 1:
PERIOD
2:
Randomise Trial Subjects
SEQUENCE 1
TREATMENT A
SEQUENCE 2
TREATMENT B
TREATMENT B
TREATMENT A
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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
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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 ,
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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
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Subject EffectsSubject Effects
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
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Subject EffectsSubject Effects
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
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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
PERIOD ONEPERIOD ONE PERIOD TWOPERIOD TWO
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
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PERIOD ONEPERIOD ONE PERIOD TWOPERIOD TWO
- 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
PERIOD ONEPERIOD ONE PERIOD TWOPERIOD TWO
- -
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
- -
- -
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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
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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
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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
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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
Fixed EffectsRandom Effects
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
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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
Fixed EffectsRandom Effects
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
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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
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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
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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
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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
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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
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