1. Maximising the Value of Time Series Data:
Repeated Measures Analysis Using JMP
Daniel Tray1 and Simon Bate2, GlaxoSmithKline, Stevenage, United Kingdom
1API Chemistry and 2Statistical Sciences
Abstract
Data Visualisation
Analysis Methods
In this poster we will assess the benefits of using JMP to perform a mixed model repeated measures based analysis on time course data acquired from the optimisation of a pharmaceutical production process
These include being able to model reaction profiles over time whilst taking into account the inherent correlation structure that is a feature of data generated when taking multiple repeat samples
We will also discuss some advantages of this methodology over alternative analyse strategies, in terms of model reliability and accuracy
Trend plots for the product yield and impurity level at each time point were quickly produced using the Graph Builder platform
This allowed for straightforward data visualisation and identification of potentially outlying runs or data points
The following methods were investigated and their relative benefits evaluated:
Separate univariate analyses at each time point
MANOVA based analysis
Random coefficient regression (RCR) using mixed models
Method
Pros
Cons
Univariate Analysis (Least Squares)
Straightforward analysis
No assessment of time based trends or links between analyses. Increased risk of false positives
Analyses do not take into account correlations over time
Predictions from data acquired at one time point may not be valid at other time points
MANOVA
Discrete Time based effects can be modelled.
Correlations over time can now accounted for. 
Technique sensitive to missing data - if a run is missing a value, then all information for that “run” is lost to the analysis
Assumes Sphericity , i.e. a certain correlation structure
Treats time as a categorical variable
Output can be difficult to interpret
Superseded by more modern methodology
Random coefficient regression using Mixed Models
Time treated as a continuous variable
Time based trends can be modelled
All data are used leading to improved accuracy and reliability of predictions – method is inherently less sensitive to underlying variability
Method is less sensitive to the presence of outliers due to ‘smoothing’
Relies on underlying assumption that within-run trends over time are linear, following transformation if required
Complex analysis to perform in JMP
Denominator degrees of freedom are no longer integers due to application of the Kackar-Harville adjustment
Product
Problem and Objectives
A pharmaceutical production process was investigated with the objectives of maximising the yield of the product whilst simultaneously controlling the levels of a problematic impurity
These objectives were achieved through process optimisation by application of response surface DoE methodology. Four factors were investigated in 30 runs using 16 factorial points, 8 axial points and 6 centre points. The experiment was performed using 3 equal blocks.
As part of the study, data was collected over time from each run at six time points
We wished to maximise the value of the data collected by modelling the reaction profiles over time, instead of analysing the data at each time point separately
Impurity

2. Maximising the Value of Time Series Data:
Repeated Measures Analysis Using JMP
Daniel Tray1 and Simon Bate2, GlaxoSmithKline, Stevenage, United Kingdom
1API Chemistry and 2Statistical Sciences
Results: Univariate Analysis
Results: MANOVA Analysis
Results: Random Coefficient Regression using Mixed Models Methodology
Univariate analysis afforded separate models for product and impurity at each time point. In total twelve individual models were generated, with no time dependent information captured
The example below illustrates the model for the product obtained from the 150 min time point, as well as a surface plot combining both responses of interest
MANOVA based analysis, whilst straightforward to perform, treated time as a categorical variable and led to complex output
Mauchly test of Sphericity performed – both Huynh-Feldt and Greenhouse-Geisser adjustments preferred if Sphericity test is significant
Overall effect of Time and factor interactions with Time assessed as part of output
The plots below show the least squares means predictions for both the product and impurity responses:
A mixed model approach, available in JMP Pro, allows the estimation of time based effects in which Time is treated as a continuous variable
In addition, all data are used for the model predictions, leading to improved accuracy and reliability of predictions
Other advantages of this analysis method are less sensitivity to the presence of both outlying data points (due to smoothing) and missing data
The example below illustrates the surface plots obtained from models where time is treated as a random effect, nested within the individuals runs of the response surface DoE. The repeated covariance structure has been selected as ‘residual’
A logarithmic transformation was applied to the impurity response prior to analysis as the trend over time was non-linear on the original scale
Product
Impurity
Key:
Product
Impurity
Key:
Product Impurity

3. Maximising the Value of Time Series Data:
Repeated Measures Analysis Using JMP
Daniel Tray1 and Simon Bate2, GlaxoSmithKline, Stevenage, United Kingdom
1API Chemistry and 2Statistical Sciences
Results: Random Coefficient Regression using Mixed Models
Comparison of Standard Errors of Predictions for the Yield Response
Conclusions
This work has demonstrated the value of performing a mixed model repeated measures based analysis when analysing time based data
The advantages of this method include improved accuracy and reliability of predictions, less sensitivity to the presence of both outlying data points and missing data, and less sensitivity to the underlying variability of observations due to smoothing over time
The analysis can be carried out by scientists more familiar with a ‘point-and-click’ statistical package, comparing favourably against output obtained using SAS Proc Mixed
The table below demonstrates how this approach has captured and modelled the interactions between Time and other process variables for the product response
This output compares favourably to that obtained using SAS Proc Mixed (Type I SS)
The standard errors (SEs) of the predictions from the RCR models were calculated and compared to those generated from the univariate models
For both responses, the standard errors were generally smaller from the RCR, leading to improved accuracy and reliability of predictions
In the histograms below the absolute differences between the SEs from the univariate and mixed models are presented for each response, where a positive value indicates the univariate analysis SE is bigger than the mixed model SE
Effect Tests: Product
References
Differences in SEs – Product (LHS) and Impurity (RHS)
Kenward, Michael G., and James H. Roger. "Small sample inference for fixed effects from restricted maximum likelihood." Biometrics (1997):
The SAS System for Windows (Release 9.3 TS Level 1M2, SAS Institute, Cary, NC)
Littell RC, Milliken GA, Stroup WW, Wolfinger RD. SAS System for Mixed Models. SAS Institute Inc.: Cary, NC, 1996.
Acknowledgements
The authors would like to thank Sonya Godbert for her contribution to this work