1.6 Missing data mechanisms
1.6.2 Is MCAR likely in practice?
We have seen that when data are MCAR, we can set the details of the missingness mechanism to one side and analyse the observed data. That is to say, a sensible analysis simply includes only those patients who have complete data on the variables needed for that analysis. All we have lost is some information.5 We therefore need to consider whether MCAR is likely in practice,
and how we might detect it. Recall that the definition of MCAR data is that the missingness mechanism is unrelated to anything we wish to infer from the data. Assuming that reasonably 5As discussed in later chapters, depending on whether covariates or outcomes are MCAR, it may be possible
1.6 Missing data mechanisms 15 careful follow up arrangements are in place, it follows that the proportion of patients with data MCAR is likely to be small. Further, this proportion will not vary with any of the observed covariates (e.g. intervention group, sex, age, illness severity).
Unfortunately, these points are only consistent with MCAR, they are not sufficient to show that data are definitely MCAR. For example, there could be a variable related to intervention, associated with the chance of a patient withdrawing, which was not measured. In the asthma study, withdrawal of hayfever sufferers might depend on the local pollen count. Withdrawal may be unassociated with any of the data recorded. Thus, if local pollen count is not recorded, data may appear MCAR. But they are not MCAR. Rather, as high local pollen count exacerbates a patient’s asthma, we are left with data from patients who either do not have hayfever, or whose hayfever is better controlled. This is a non-random sample of those enrolled in the study. An extreme example of this would be if withdrawal depended on a sudden, unpredicted, change in the response, e.g. a sudden deterioration in FEV1. Again, looking at the observed data,
patients may appear to be MCAR, but in fact patients who withdraw are systematically different from those who do not — just in an unobserved way.
The last two paragraphs underline how the extra assumptions required for an analysis when data are missing cannot be verified from the data. In spite of what our observed data may suggest, we can never be sure that data are MCAR. Nevertheless, the observed data can rule out MCAR. We can investigate whether there is any relationship between observed data and the occurrence of missing data. If there is, data are not MCAR. We can investigate this more formally. For an example in a longitudinal context, seeDiggle(1989).
EXAMPLE1.2 Asthma trial (ctd)
Are patients MCAR in the asthma trial?6 From Table 1.2 the chance of a patient staying in the trial to the end clearly depends on treatment; those in the placebo or lowest dose arm are much less likely to complete. A chi-square test confirms this, p < 0.001. Patients are clearly not MCAR. Of course, patient withdrawal may also depend on other factors besides treatment. ¤ 1.6.3 Missing at random
In practice trial data are rarely MCAR. Usually there is an association between the chance of patient withdrawal and observations — typically intervention, baseline and (in longitudinal follow-up) measurements prior to withdrawal. In this case, it is not sensible to include in the analysis only those with complete data.
For example, suppose that worse health at baseline is associated both with increased risk of withdrawal and poor response to intervention. Analysing data from the patients who remain to the end of the trial will thus give an over optimistic view of the intervention effect. However, if we can identify those variables which are associated with an increased risk of withdrawal, we can carry out a sensible analysis. We illustrate this key idea with the asthma trial.
EXAMPLE1.2 Asthma trial (ctd)
In the placebo arm, only 35 out of 92 patients completed the trial. The average FEV1of the 92
patients at the start of the trial was 2.050 litres. Suppose we are interested in the FEV1 at 12
6This is a minor abuse of our definitions: more strictly, is the reason for patients’ withdrawal (and hence their
weeks, and whether there is any evidence of a ‘placebo effect’ whereby patients taking a drug improve, even though their drug contains no active ingredients.
If we believe the patients are MCAR at 12 weeks, then a valid estimate of the average FEV1at
12 weeks is obtained from the 35 who complete. Their average is 2.072 litres, suggesting no placebo effect.
However, we need to check if MCAR is plausible. So we need to look to see if any variables in the data are associated with withdrawal. As one suspects that patients with worse asthma initially are more likely to withdraw, an obvious place to look is baseline FEV1. Suppose we
classify baseline FEV1as ‘low’ if it is below 2.015 litres and ‘high’ otherwise. Table1.6shows
how patients with low FEV1 at baseline are much more likely to have withdrawn by 12 weeks.
An analysis that assumes MCAR is not sensible.
Baseline FEV1
low high
present 15 20
At 12 weeks
absent 31 26
Table 1.6: Proportion of placebo patients who have withdrawn by 12 weeks, by baseline FEV1
However, suppose the 15 patients with low baseline FEV1who we see at 12 weeks are drawn
randomly from the 46 patients with low baseline. In other words, within the group of patients with low baseline FEV1, patients are MCAR. Then, for those patients with low baseline, a
sensible estimate of the average 12 week FEV1is given by averaging the 12 week values for the
15 patients who we see. This is 1.861 litres.
Likewise, suppose the 20 patients with high baseline FEV1who we see at 12 weeks are drawn
randomly from the 46 patients with high baseline. Arguing in the same way, for those patients with high baseline, a sensible estimate of the average 12 week FEV1is given by averaging the
12 week FEV1for these 20 patients who we see. This is 2.230 litres.
Figure1.2shows this graphically. At 12 weeks, we assume that we observe a random selection of the patients in the ‘low’ and ‘high’ group. A sensible estimate of FEV1 at 12 weeks within
these groups is therefore the average of the observed values in these groups.
The overall average 12 week FEV1 can thus be sensibly estimated by averaging the estimates
from the ‘low’ and ‘high’ groups, allowing for there being 46 patients in the low group and 46 in the high group. This is
46 × 1.861 + 46 × 2.230
92 = 2.046 litres.
Comparing this with the average baseline FEV1 of 2.050 litres, confirms that there is no evi-
dence of a ‘placebo effect’. ¤
1.6 Missing data mechanisms 17
Baseline FEV1 (litres)
F E V1 (litres) at 12 weeks low high 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 15/46 observed at 12 weeks
Average of 15 observed: 1.861 (l) Average of 21 observed: 2.230 (l)20/46 observed at 12 weeks
Figure 1.2: Graphical illustration: within the two groups defined by ‘low’ and ‘high’ baseline FEV1, we assume that we observe a random selection of patients at 12 weeks
1. identify a fully observed variable whose values predict the occurrence of missing data; 2. within groups defined by this variable, assume data are MCAR;
3. within these groups, sensible estimates can thus be obtained from the observed data, and 4. to obtain a sensible estimates overall, average the estimates from the groups in (3), allow-
ing for there being different numbers of patients in the different groups.
When we can find fully observed variables which define groups within which the data are MCAR, we say data are missing at random (MAR). In the asthma trial, within the two base- line FEV1groups, response at 12 weeks is MCAR. The expression ‘MAR’ is thus a convenient
shorthand. Instead of saying ‘assume there exists a variable, say baseline FEV1, and that among
those with the same baseline FEV1, the FEV1at 12 weeks is MCAR’ we can just write ‘assume
FEV1at 12 weeks is MAR’.
The asthma example also enables us to introduce the term conditional, common in missing data literature. Instead of saying ‘among those with the same baseline FEV1, the FEV1at 12 weeks
is MCAR’ we say conditional on baseline FEV1, the FEV1at 12 weeks is MCAR.
Before going on, we underline an aspect of MAR data implicit in what has gone before. In the asthma study, we wanted an overall average 12 week FEV1, not an average in the ‘low’
and ‘high’ baseline group. We call this overall average marginal. If no patients withdrew, the marginal average is simply the average of the values for the 92 patients.
With MAR data, averaging the values of the 35 patients who do not withdraw is wrong. We can no longer go directly to the marginal average. Instead we have to calculate averages conditional on baseline, and then take a further step to estimate the marginal average.
We repeat the previous paragraph more abstractly. If values of a variable Y are MAR, then statistics calculated using only observed values of Y, (e.g. mean, standard deviation, confidence intervals, p-values) are wrong. As values are MAR we cannot go directly to marginal statis- tics. Suppose that conditional on the fully observed variable X, Y is MCAR. Then we have to calculate statistics conditional on X, and then take a further step to estimate the marginal statistics.
Generally, variables that we condition on do not have to take on discrete values (as baseline FEV1 was forced to in the example above). When we are interested in averages, and have
quantitative variables, we can condition using regression. The regression of Y on X estimates the average of Y conditional on X. To do this it estimates the numbers α and β so that
average value of Y = α + β × X. (1.1)
For example, if α = 1 and β = 2, then the average value of Y conditional on X = 5 is 1+2×5 = 11.
We now use the same idea from the asthma example again. Recall that Y is partially observed, X fully observed, and conditional on X, Y is MCAR. We estimate α and β by fitting the regression to the subset of individuals on whom both Y and X are observed. Once we have α and β we can use (1.1) to get the conditional average of Y for each value of X for which Y is missing. Then we average these conditional values with the observed Y values to give the marginal average of Y.
EXAMPLE1.2 Asthma trial (ctd)
Above, we split baseline FEV1 into two groups to introduce the ideas. However this is not
necessary. Suppose 12 week FEV1is MAR; specifically that conditional on baseline FEV1, 12
week FEV1is MCAR. Then we can estimate the average 12 week FEV1conditional on a value
of baseline FEV1by fitting the regression model
Average 12 week FEV1= α + β × baseline FEV1
to the 35 patients on whom we observe both. Doing this, we find that
Average 12 week FEV1= 0.923 + 0.535 × baseline FEV1 (1.2)
Thus, conditional on a baseline FEV1of 2.0 litres the average 12 week FEV1is 0.923 + 0.535 ×
2 = 1.993 litres
The average 12 week FEV1 is obtained by (i) calculating the conditional 12 week FEV1 for
each of the 57 patients with missing 12 week FEV1, and (ii) averaging these 57 values and the
35 observed values. Numerically, this gives: 1
92{(0.923 + 0.535 × baseline FEV1of 1st patient with missing 12 week FEV1) + (0.923 + 0.535 × baseline FEV1of 2nd patient with missing 12 week FEV1) + · · ·
+ (0.923 + 0.535 × baseline FEV1of 57th patient with missing 12 week FEV1)
+ 12 week FEV1of 1st patient with observed 12 week FEV1+ · · ·
+12 week FEV1of 35th patient with observed 12 week FEV1}
1.6 Missing data mechanisms 19 So, assuming 12 week FEV1is MCAR given baseline FEV1, a sensible estimate of average 12
week FEV1 is 2.019 litres, down from the baseline mean of 2.050 litres. This is more accu-
rate than the estimate obtained before, when we lost information by splitting the quantitative
variable, baseline FEV1into two groups, ‘high’ and ‘low’. ¤
Notice that if we say a partially observed variable is MAR, then that means that we have fully observed variables, conditional on which the partially observed variable is MCAR. In other words, conditional on these fully observed variables, the reason for the missing values does not depend on the unobserved values themselves. This is the aspect of MAR that is usually men- tioned in quick descriptions in the literature. We emphasise that this is a conditional statement. If data are MAR, the reason for the missing values will often depend on the unseen values. However, conditional on other fully observed values this association will be broken.
EXAMPLE1.2 Asthma study (ctd)
Consider again the placebo arm of the asthma study. It is plausible that the probability of the 12 week FEV1being missing is higher the lower the value of that (unseen) observation. Assuming
MAR though, conditional on a patient’s baseline FEV1, the probability of them being missing
at 12 weeks no longer depends on their FEV1at 12 weeks. ¤
In summary, if we have a fully observed variable whose values affect the chance of seeing missing data, those missing data are not MCAR. But, if conditional on this fully observed variable, we assume the chance of seeing the partially observed variable does not depend on its values, the data are MAR. The important word is assume. Usually we do not know whether MAR is actually true or not.
The final implication of MAR is that the statistical distribution of potentially missing data is the same (conditionally) for all patients who share the same observed data, whether or not they withdraw. This is implicit in the example above, where in the placebo group we:
1. estimated the conditional distribution of week 12 FEV1given baseline FEV1from the 35
patients on whom both was observed;
2. assumed this distribution was identical in the 57 patients whose week 12 FEV1 was not
observed, and
3. then used this distribution to estimate the mean week 12 FEV1for these 57 patients.
Or, more generally in a longitudinal study design, under MAR subjects who withdraw share the same conditional statistical behaviour in their (unobserved) future, given their observed past, as those who do not withdraw. It is this property that allows principled methods of analysis, like those based on likelihood, to make the appropriate adjustments for withdrawal under MAR. EXAMPLE1.2 Asthma study (ctd)
In the previous example, we fitted the regression relating average 12 week FEV1 to baseline
baseline observations, the conditional distribution of a patient’s 12 week FEV1given their base-
line FEV1 is x, say, is normal, with estimated mean 0.923 + 0.535x, and estimated variance
0.213.
The MAR assumption means that the distribution of 12 week FEV1given baseline FEV1for the
57 patients with missing 12 week FEV1 is the same, and that its parameters are consistently7
estimated by values given above, which are calculated using the data from the 35 patients on
whom both are observed. ¤
This facet of MAR means that (assuming data are MAR) joint modelling of complete and par- tially observed response data, conditional on fully observed data, is a natural way to approach the analysis. This motivates the approach we develop in Chapter3.