1. Effects of Missing Values on the Analysis of the AB/ BA Crossover Trial
Lauren Rodgers
Supervisor: Prof JNS Matthews
University of Newcastle upon Tyne

2. Outline Crossover Model Missing Data Simulations Conclusions
Future Work

3. Randomise Trial Subjects
SEQUENCE 1
SEQUENCE 2
PERIOD 1:
TREATMENT A
TREATMENT B
PERIOD 2:
TREATMENT B
TREATMENT A

4. AB/ BA Crossover Model What can we estimate? Problems
treatment effect, t
period effect, p
subject effect, x
Problems
carryover effect
Within subject estimate of treatment effect
between subject variability is eliminated

5. AB/ BA Crossover Model Random error term ~ General Mean
Treatment effect
Period effect
Subject effects of subject i in sequence k
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

6. Subject Effects Fixed Effects
general level of each subject has a fixed value
find MLE for xik
produce profile log-likelihood model to remove parameter

7. Subject Effects
xik is a function of subject i’s period 1 and period 2 response
when subject i has no response in any period this MLE cancels out the remaining terms
Model which includes only those with complete data
effectively exclude all data from a subject if any missing data
closed form for treatment estimate even in presence missing data

8. Subject Effects Random Effects subject effect has some distribution –
include all available data
can be fitted using a Linear Mixed Effects model
No Missing Data – both models produce same results

9. Missing Data Generate data Introduce MCAR missing data
PERIOD ONE
PERIOD 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
PERIOD ONE
PERIOD TWO NA
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
Generate data
shown for sequence AB only
Introduce MCAR missing data

10. Missing Data Fixed subject effect Random subject effect
PERIOD ONE
PERIOD 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
PERIOD ONE
PERIOD 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
Fixed subject effect
remove all data if subject has any missing
Random subject effect
keep all available data

11. Missing Data 20%, 40% and 60% of data missing
Pattern in sequences and periods
equal amounts missing in each sequence and period
data missing from period two only
equal amounts missing in each sequence
more missing from second sequence
more data missing in second period
more data missing in second sequence

12. Simulations Parameters Output
number of subjects in trial: m= 20, 40, 120
between and within subject variance
t = tA - tB
amount and pattern of missing data
Output
root mean square error (RMSE)
estimate of t and 95% CI

13. Effect of on RMSE( )

14. Effect of on RMSE( )

15. Pattern of Missing Data

16. 95% CI for Treatment Effect
No missing data: length of CI same
Ratio length fixed: length random – which is smaller?
20% missing
Index
m=20
m=40
m=120
r=0.06 2 4 6
1.013
1.011 -
1.034
1.044
1.039
1.048
1.058
1.051
r=0.5
0.950
0.956
0.945
0.991
0.996
0.994
1.016
1.021
1.018
r=0.8
0.933
0.934
0.937
0.975
0.977
0.976
1.000
1.002
1.001

17. 40% Missing Index m=20 m=40 m=120 r=0.06 2 4 6 1.035 1.087 - 1.084
1.151
1.100
1.010
1.178
1.124
r=0.5
0.958
0.986
0.970
1.015
1.052
1.025
1.045
1.058
r=0.8
0.932
0.930
0.931
0.996
0.992
1.043
1.027
1.016

18. 60% Missing Index m=20 m=40 m=120 r=0.06 2 4 6 1.062 1.476 - 1.144
1.647
1.200
1.175
1.756
1.230
r=0.5
0.952
1.222
0.995
1.041
1.371
1.082
1.084
1.444
1.121
r=0.8
0.898
1.011
0.939
0.980
1.125
1.007
1.024
1.188
1.042

19. Conclusions
Between subject variance has no effect on fixed effects model but increases RMSE for random effects model
Missing data – some differences for pattern
95% CI for treatment effect
smaller for fixed effects model with small sample size
as sample size increases random effects model performs better
as amount of missing data increases random effects model performs better

20. Future Work MCAR missing data – not particularly useful
Data missing in period 2 if a correlate of period 1 response exceeds some threshold l
Misspecified model
fit normal model to non-normal data
Look at current methods to account for missing data

21. END