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Repeated measures data arise when observations are taken on each experimental unit on a number of occasions, and time is a factor of interest.
<P>
Repeated measures data arise when observations are taken on each experimental
unit on a number of occasions, and time is a factor of interest. <p>
Many techniques can be used to analyse such data. <p>
Antedependence modelling is a recently developed method which models the
correlations between observations at different times. <p>
<em> Repeated measures </em> refers to observations of a variable which
are repeated, usually over time. However, it can refer to any situation where a factor
of interest has a special intrinsic structure. Soil depth is an example.
<p>
<img align=middle SRC="../images/nonrand1.gif" WIDTH=431 HEIGHT=72> Block 1 <p>
Usually, treatment factors are randomised between blocks.
This is assumed in ANOVA.
<p>
<img align=middle SRC="../images/nonrand2.gif" WIDTH=430 HEIGHT=72> Day 1 <p>
This cannot be done with time, and so time cannot be treated as just another
factor in ANOVA.
<p>
The times, if they are regarded as factor levels, cannot be
<a href="rand.htm">randomised</a>,
an essential requirement of ANOVA. They must follow their natural sequence.
<p>
Randomisation allocates experimental material randomly to different
treatments.
<p>
This means that results for different treatments are independent, and allows
ANOVA.
<p>
The example below shows the random allocation of 6 treatments to 6
experimental units. If you have Java enabled you can see a different
randomisation by clicking on the image.
<p>
<applet codebase="../../../java" code="Ran6.class" width=200 height=50>
<pre> <b> <big> 6 4 1 5 3 2 </big> </b> </pre>
</applet>
<p>
There are a number of ways of analysing repeated measures data:
<ul>
<li> Analysing a summary of the time profile.
<li> Split-plot ANOVA - sometimes this <em> is </em> valid, if certain
assumptions are satisfied.
<li> Multivariate ANOVA.
</ul>
This module describes a recently developed method, <em> Antedependence modelling </em>,
which explicitly models the correlation over time, and then adjusts for this
in assessing treatment effects. <p>
It should not be seen as superior to the other methods mentioned above, but as
an additional technique which may help data understanding. <p>
It is most likely to be useful when there are many (say 8 or more) time points,
and we are interested in general development over time, rather than behaviour
during particular stages.
<p>
Antedependence modelling estimates the correlation between the observations
at different time points, and then builds this correlation structure into
tests for the significance of treatment effects. We could estimate the
correlations between all pairs of time points, but it is likely that a
simpler model would describe what is going on. We will assume that the effect
of the 'past' on an observation is contained in just a few preceding observations.
Using antedependence modelling requires two stages:
<ol>
<li> We decide the <em> order </em> of the model
<li> We assess treatment effects.
</ol>
<p>
For further statistical details about the method see
<a href="ref.htm"> Kenward (1987) </a>
<p>
The order is the number of previous times having an effect beyond that carried
by intermediate times.<p>
<img align=middle SRC="../images/order2.gif" WIDTH=541 HEIGHT=99><br clear=all>
<p>
Here, observations depend only on the previous two observations (although these
will depend on earlier observations, producing an indirect effect).
<p>
This is a model of order 2.
<p>
An experiment was conducted to examine the effects of three pasture types on
growth rates of lambs. <p>
Lambs were available from ewes which had taken part in a diet supplement
trial (5 diets) and it was thought that there may be a carry-over effect
of the ewe's diet. <p>
In addition, there was interest in whether dosing the
lambs for parasites affected their growth. <p>
<table>
<tr>
<td> <br> Design : </td> <td> <a href="slide06a.htm"> <img alt="plan"
src="../images/expt2.gif" WIDTH=500 HEIGHT=200> </a> </td>
</tr>
</table>
<p>
<a href="lambs.htm"> <img alt="lambs" src="../images/lambs2.gif" WIDTH=100 HEIGHT=70>
</a> Eighteen pairs of twin lambs born to ewes on each diet were
randomly allocated to the three pasture types. <p><br>
Within each pair, one lamb was dosed for parasites. <p>
<a href="slide06b.htm"> <img alt="plot" src="../images/lambpl2.gif">
</a> To assess the effect of pasture and dosing on lamb growth, fortnightly
weights were recorded. <p><br>
NB : To view in more detail please click on image
<p>
The 180 lambs were made up of 90 pairs of twins.
We would expect the variation between pairs to be different from that
between the lambs in a pair. The pasture and ewe diet treatments are
assessed by comparing pairs, while the dosing compares lambs within
a pair.
<p>
This experimental structure is termed the blocking, and must be specified
in the analysis. In this case there is a single block factor :
<ul>
<li> Pair
</ul>
<p>
The effect of pasture and dosing was the main interest in this experiment. <p>
There may be a carryover effect of the ewe diet treatment, but we do not
expect it to interact with the present treatments. <p>
Therefore, the treatment factors in the analysis of variance are : <p>
<ul>
<li> Pasture
<li> Dosed
<li> Pasture x Dosed
<li> Diets
</ul>
<p>
<pre>
ANTORDER [block=pair; treat=pasture*dosed+ewediet] wt[]
**** Sequential comparison of ante-dependence structures ****
Unadjusted Adjusted
Chi-square Adjustment Chi-square
Statistic factor Statistic d.f. Prob
Order 0 v. order 1 4998.05 0.441 2203.34 9 0.001
Order 1 v. order 2 189.21 0.434 82.09 8 0.001
Order 2 v. order 3 67.54 0.426 28.80 7 0.001
Order 3 v. order 4 11.29 0.418 4.72 6 0.580
Order 4 v. order 5 37.14 0.412 15.28 5 0.009
Order 5 v. order 6 8.87 0.404 3.58 4 0.465
Order 6 v. order 7 11.33 0.397 4.50 3 0.212
Order 7 v. order 8 6.30 0.392 2.47 2 0.291
Order 8 v. order 9 1.49 0.386 0.58 1 0.448
**** Comparison of ante-dependence structures with max order ****
Unadjusted Adjusted
Chi-square Adjustment Chi-square
Statistic factor Statistic d.f. Prob
Order 0 v. order 9 5331.23 0.421 2244.27 45 0.001
Order 1 v. order 9 333.17 0.416 138.69 36 0.001
Order 2 v. order 9 143.96 0.411 59.24 28 0.001
Order 3 v. order 9 76.42 0.407 31.08 21 0.072
Order 4 v. order 9 65.13 0.402 26.20 15 0.036
Order 5 v. order 9 28.00 0.398 11.13 10 0.347
Order 6 v. order 9 19.13 0.393 7.53 6 0.275
Order 7 v. order 9 7.79 0.390 3.04 3 0.386
Order 8 v. order 9 1.49 0.386 0.58 1 0.448
</pre>
</CENTER>
<p>
The first part compares each model order with the next order, to see
whether this next order is a significant improvement in describing
the data.
<p>
We see that a model of order 3 is needed, but that order 4 is not an
improvement on order 3. However, order 5 might be an improvement.
<pre>
Chi-square Adjustment Chi-square
Statistic factor Statistic d.f. Prob
Order 0 v. order 1 4998.05 0.441 2203.34 9 0.001
Order 1 v. order 2 189.21 0.434 82.09 8 0.001
Order 2 v. order 3 67.54 0.426 28.80 7 0.001
Order 3 v. order 4 11.29 0.418 4.72 6 0.580
Order 4 v. order 5 37.14 0.412 15.28 5 0.009
Order 5 v. order 6 8.87 0.404 3.58 4 0.465
Order 6 v. order 7 11.33 0.397 4.50 3 0.212
Order 7 v. order 8 6.30 0.392 2.47 2 0.291
Order 8 v. order 9 1.49 0.386 0.58 1 0.448
</pre>
The second part compares each order with order 9 (the maximum). Again,
order 3 is called for, or possibly order 5.<p>
<pre>
Chi-square Adjustment Chi-square
Statistic factor Statistic d.f. Prob
Order 0 v. order 9 5331.23 0.421 2244.27 45 0.001
Order 1 v. order 9 333.17 0.416 138.69 36 0.001
Order 2 v. order 9 143.96 0.411 59.24 28 0.001
Order 3 v. order 9 76.42 0.407 31.08 21 0.072
Order 4 v. order 9 65.13 0.402 26.20 15 0.036
Order 5 v. order 9 28.00 0.398 11.13 10 0.347
Order 6 v. order 9 19.13 0.393 7.53 6 0.275
Order 7 v. order 9 7.79 0.390 3.04 3 0.386
Order 8 v. order 9 1.49 0.386 0.58 1 0.448
</pre>
For simplicity, we will try a model of order 3.
<p>
ASSESSING TREATMENT EFFECTS
<P>
<pre>
ANTTEST [ORDER=3; BLOCK=pair;TREAT=pasture*dosed+ewediet] wt[]
**** tests of pasture
assuming ante-dependence structure of order 3 ****
Adjusted Adjusted
Chi-square Chi-Square
Time Statistic d.f. Prob Statistic d.f. Prob
per Timepoint Overall
1 0.199 2 0.905 0.199 2 0.905
2 3.488 2 0.175 3.708 4 0.447
3 0.025 2 0.988 3.710 6 0.716
4 11.631 2 0.003 15.533 8 0.050
5 6.068 2 0.048 21.679 10 0.017
6 35.745 2 <0.001 58.345 12 <0.001
7 8.341 2 0.015 66.636 14 <0.001
8 4.154 2 0.125 70.557 16 <0.001
9 4.480 2 0.106 74.870 18 <0.001
10 9.836 2 0.007 84.758 20 <0.001
**** tests of dosed
assuming ante-dependence structure of order 3 ****
Adjusted Adjusted
Chi-square Chi-Square
Time Statistic d.f. Prob Statistic d.f. Prob
per Timepoint Overall
1 0.301 1 0.584 0.301 1 0.584
2 1.878 1 0.171 2.188 2 0.335
3 0.269 1 0.604 2.447 3 0.485
4 0.705 1 0.401 3.150 4 0.533
5 0.146 1 0.703 3.272 5 0.658
6 0.596 1 0.440 3.866 6 0.695
7 0.232 1 0.630 4.084 7 0.770
8 0.021 1 0.884 4.080 8 0.850
9 0.051 1 0.821 4.114 9 0.904
10 1.398 1 0.237 5.544 10 0.852
**** tests of pasture . dosed
assuming ante-dependence structure of order 3 ****
Adjusted Adjusted
Chi-square Chi-Square
Time Statistic d.f. Prob Statistic d.f. Prob
per Timepoint Overall
1 0.492 2 0.782 0.492 2 0.782
2 0.570 2 0.752 1.063 4 0.900
3 2.008 2 0.367 3.089 6 0.798
4 2.332 2 0.312 5.445 8 0.709
5 1.205 2 0.547 6.645 10 0.759
6 0.007 2 0.996 6.613 12 0.882
7 0.504 2 0.777 7.095 14 0.931
8 0.549 2 0.760 7.624 16 0.959
9 1.307 2 0.520 8.945 18 0.961
10 0.639 2 0.726 9.572 20 0.975
**** tests of ewediet
assuming ante-dependence structure of order 3 ****
Adjusted Adjusted
Chi-square Chi-Square
Time Statistic d.f. Prob Statistic d.f. Prob
per Timepoint Overall
1 8.525 4 0.074 8.525 4 0.074
2 8.248 4 0.083 16.772 8 0.033
3 2.691 4 0.611 19.394 12 0.079
4 4.863 4 0.302 24.228 16 0.085
5 11.664 4 0.020 36.089 20 0.015
6 3.267 4 0.514 39.241 24 0.026
7 6.158 4 0.188 45.385 28 0.020
8 1.273 4 0.866 46.434 32 0.048
9 2.715 4 0.607 49.031 36 0.072
10 2.533 4 0.639 51.464 40 0.106
</pre>
Next, an interpretation....
<p>
INTERPRETING THE TREATMENT TESTS
<P>
There is a table for each factor or interaction assessed.
<p>
For each time point, there is a test for an effect <em> at </em> that
time point, and another for an effect in all the data <em> up to </em>
that time.
We can see
<ul>
<li> For <tt> pasture</tt>, an effect has appeared by the 4th time point,
and can be seen at most subsequent times. No effect can be seen before the
4th time point. <p>
</ul>
<pre>
**** tests of pasture assuming ante-dependence
structure of order 3 ****
Adjusted Adjusted
Chi-square Chi-Square
Time Statistic d.f. Prob Statistic d.f. Prob
per Timepoint Overall
1 0.199 2 0.905 0.199 2 0.905
2 3.488 2 0.175 3.708 4 0.447
3 0.025 2 0.988 3.710 6 0.716
4 11.631 2 0.003 15.533 8 0.050
5 6.068 2 0.048 21.679 10 0.017
6 35.745 2 <0.001 58.345 12 <0.001
7 8.341 2 0.015 66.636 14 <0.001
8 4.154 2 0.125 70.557 16 <0.001
9 4.480 2 0.106 74.870 18 <0.001
10 9.836 2 0.007 84.758 20 <0.001
</pre>
<ul>
<li> For <tt>dosing</tt> and the <tt>pasture.dosing</tt> interaction,
there is no evidence of any effect on weight.<p>
</ul>
<pre>
**** tests of dosed assuming ante-dependence
structure of order 3 ****
Adjusted Adjusted
Chi-square Chi-Square
Time Statistic d.f. Prob Statistic d.f. Prob
per Timepoint Overall
1 0.301 1 0.584 0.301 1 0.584
2 1.878 1 0.171 2.188 2 0.335
3 0.269 1 0.604 2.447 3 0.485
4 0.705 1 0.401 3.150 4 0.533
5 0.146 1 0.703 3.272 5 0.658
6 0.596 1 0.440 3.866 6 0.695
7 0.232 1 0.630 4.084 7 0.770
8 0.021 1 0.884 4.080 8 0.850
9 0.051 1 0.821 4.114 9 0.904
10 1.398 1 0.237 5.544 10 0.852
**** tests of pasture . dosed
assuming ante-dependence structure of order 3 ****
Adjusted Adjusted
Chi-square Chi-Square
Time Statistic d.f. Prob Statistic d.f. Prob
per Timepoint Overall
1 0.492 2 0.782 0.492 2 0.782
2 0.570 2 0.752 1.063 4 0.900
3 2.008 2 0.367 3.089 6 0.798
4 2.332 2 0.312 5.445 8 0.709
5 1.205 2 0.547 6.645 10 0.759
6 0.007 2 0.996 6.613 12 0.882
7 0.504 2 0.777 7.095 14 0.931
8 0.549 2 0.760 7.624 16 0.959
9 1.307 2 0.520 8.945 18 0.961
10 0.639 2 0.726 9.572 20 0.975
</pre>
<ul>
<li> For the <tt> ewediet </tt> treatment, there are only some borderline
p-values, and these point towards effects early on, as might be expected.
</ul>
<pre>
**** tests of ewediet assuming ante-dependence
structure of order 3 ****
Adjusted Adjusted
Chi-square Chi-Square
Time Statistic d.f. Prob Statistic d.f. Prob
per Timepoint Overall
1 8.525 4 0.074 8.525 4 0.074
2 8.248 4 0.083 16.772 8 0.033
3 2.691 4 0.611 19.394 12 0.079
4 4.863 4 0.302 24.228 16 0.085
5 11.664 4 0.020 36.089 20 0.015
6 3.267 4 0.514 39.241 24 0.026
7 6.158 4 0.188 45.385 28 0.020
8 1.273 4 0.866 46.434 32 0.048
9 2.715 4 0.607 49.031 36 0.072
10 2.533 4 0.639 51.464 40 0.106
</pre>
<p>
CONCLUSIONS
<P>
You should now be able to use antedependence modelling for any repeated
measures data
<ul> <li> it is easy to apply;
<li> most suited to situations with many (> 5) time points;
<li> gives insights not readily available with other methods. </ul>
<p>
APPLICATION SOFTWARE
<P>
Antedependence modelling can be done using two <a target="_top" href="http://www.nag.co.uk/">
GENSTAT 5 </a> procedures : <p>
<ul> <li> the <a href="slide14b.htm"> ANTORDER procedure </a> to test the
time order of the model; <p>
<li> the <a href="slide14c.htm"> ANTTEST procedure </a> to test for
treatment effects; <p>
</ul>
<a href="slide14a.htm"> Instructions </a> for reading the lamb
<a href="../example/lamb.htm"> data </a> into GENSTAT 5.
<p>
GENSTAT CODE TO READ LAMB DATA
<P>
<pre>
FACTOR [LEVELS=90] pair
FACTOR [LEVELS=3] pasture
FACTOR [LEVELS=5] ewediet
FACTOR [LEVELS=2; LABELS=!t(Yes,No)] dosed
FACTOR [LEVELS=2; LABELS=!t(M,F)] sex
OPEN 'lambwts.dat'; CHANNEL=2
READ [CHANNEL=2] pair, pasture, ewediet, dosed, sex, wt[1...10]
</pre>
As implemented in Genstat, antedependence modelling can only cope with
missing values which are 'dropouts': i.e. if an observation for an
individual is missing, so too are also subsequent observations on that
individual. To ensure this is so, we need another four lines of
Genstat:
<pre>
FOR i=2...10
CALC iminus1=i-1
CALC wt[i]=wt[i]+wt[iminus1]*0.0
ENDFOR
</pre>
<p>
THE ANTORDER PROCEDURE
<P>
We can use the Genstat procedure ANTORDER to help decide the order of
the model:
<p>
<big> <pre>
ANTORDER [block=pair; \
TREAT=pasture*dosed+ewediet] wt[]
</pre> </big>
<p>
THE ANTTEST PROCEDURE
<P>
We can now use the Genstat procedure ANTTEST to assess treatment effects.
<p>
<big> <pre>
ANTTEST [ORDER=3; BLOCK=pair; \
TREAT=pasture*dosed+ewediet] wt[]
</pre> </big>
<p>
DESIGNING REPEATED MEASURES EXPERIMENTS
<P>
In any experiments with repeated measures, there are three aspects
that need to be considered, and these are summarised here :
<ul>
<li> The <a href="slide15a.htm"> subjects </a>
<li> The <a href="slide15b.htm"> treatments</a>
<li> The <a href="slide15c.htm"> time points </a>
</ul>
<p>
THE SUBJECTS
<P>
The subjects are the basic experimental material. They will vary,
in their average response to a treatment, and in their profile
over time. An important part of the design of the experiment is to
arrange the subjects so that major sources of variability do not,
as far as possible, interfere with the treatment comparisons. This is
most often accomplished by <em> blocking </em>, in which subjects
which are similar to each other are arranged in groups. Treatment comparisons
are then made within groups. <p>
It can also be useful to measure <em> covariates </em>.
These are variables which explain some of the variability
between subjects in the variable(s) of interest in the experiment. This
variability can then be accounted for in the analysis so that it doesn't
interfere with treatment comparisons. Covariates are most often measured
before any treatments are applied to the subjects. <p>
An important choice in the experiment is how many subjects there should
be. <font color="#FF0000"> It should not be thought that lots of measurements on each subject
is equivalent to having lots of subjects. </font> Between subject variability is
what matters when comparing treatments applied to different subjects. It
can be a useful discipline to do some <em> power calculations </em> when
choosing the number of subjects. This will involve calculating the
probability of detecting a treatment effect of a specified magnitude
in an experiment with a given number of subjects.
<p>
THE TREATMENTS
<P>
Subjects will receive different treatments. That is what the experiment
is investigating. Often, the treatment can be considered as affecting
the subject for the duration of the experiment. If the treatment is
varied <em> within </em> the subjects (i.e. a cross-over trial), then
different sequences can be considered as different treatments, and the
treatment effect is partly assessed by examining the time effect.
<p>
If all treatment sequences are to be compared, then these may be randomised
among subjects (e.g. by a latin square) and the design considered as
standard, and a repeated measures approach is not needed (although some
allowance for carryover effects may be required).
<p>
A combined approach
is needed when all sequences are considered, but each subject is
measured a number of times on each treatment.
<p>
THE TIME POINTS
<P>
We need to choose the time points at which we are going to observe
or measure the subjects. The more times the better will be our picture
of each subjects profile, but the greater will be the cost in time, effort
or money. In some experiments, taking measurements might interfere a little
with what is being observed (e.g. a behaviour experiment in which blood
samples are taken), and so there is further incentive for reducing
the number of observations.
<p>
It is not possible to give a set of rules for choosing time points.
The choice will most often be a matter of common sense and judgement.
Here we simply draw attention to a number of points
which need to be considered.
<ul>
<li> It is not necessarily an ideal that the time points be uniformly
spaced. It may be practical to have more frequent observations
when rapid change in the variables being measured is expected.<p>
<li> It is best if all subjects have the same schedule of observation
times, although this can sometimes not be realised. <p>
<li> Observations may be aggregated over an interval of time, rather
than measured in an instant. This should cause no difficulties. The
measurements may be allocated to the centre of the time interval.<p>
<li> If different individuals are observed on different days, weeks etc.,
then between-day differences may arise. It is important that blocking and
randomisation be used to guard against such differences being confused
with treatment effects.<p>
<li> When observations cannot be made at exactly the time scheduled
because several have to collected in sequence, then randomisation
between treatments should be used to guard against biases.<p>
<li> Where possible, between time effects caused by, for example, different
observers, should be avoided.<p>
<li> The data collected can be analysed statistically before the experiment
is completed, but caution is required if the results of such analysis are
allowed to influence the conduct of the remainder of the experiment.
</ul>
<p>
]]>