16.1.3.2  Imputing standard deviations for changes from baseline

A special case of missing standard deviations is for changes from baseline. Often, only the following information is available:

Note that the mean change in each group can always be obtained by subtracting the final mean from the baseline mean even if it is not presented explicitly. However, the information in this table does not allow us to calculate the standard deviation of the changes. We cannot know whether the changes were very consistent or very variable. Some other information in a paper may help us determine the standard deviation of the changes. If statistical analyses comparing the changes themselves are presented (e.g. confidence intervals, standard errors, t values, P values, F values) then the techniques described in Chapter 7 (Section 7.7.3) may be used.

When there is not enough information available to calculate the standard deviations for the changes, they can be imputed. When change-from-baseline standard deviations for the same outcome measure are available from other studies in the review, it may be reasonable to use these in place of the missing standard deviations. However, the appropriateness of using a standard deviation from another study relies on whether the studies used the same measurement scale, had the same degree of measurement error and had the same time periods (between baseline and final value measurement).

The following alternative technique may be used for imputing missing standard deviations for changes from baseline (Follmann 1992, Abrams 2005). A typically unreported number known as the correlation coefficient describes how similar the baseline and final measurements were across participants. Here we describe (1) how to calculate the correlation coefficient from a study that is reported in considerable detail and (2) how to impute a change-from-baseline standard deviation in another study, making use of an imputed correlation coefficient. Note that the methods in (2) are applicable both to correlation coefficients obtained using (1) and to correlation coefficients obtained in other ways (for example, by reasoned argument). These methods should be used sparingly, because one can never be sure that an imputed correlation is appropriate (correlations between baseline and final values will, for example, decrease with increasing time between baseline and final measurements, as well as depending on the outcomes and characteristics of the participants). An alternative to these methods is simply to use a comparison of final measurements, which in a randomized trial in theory estimates the same quantity as the comparison of changes from baseline.

(1) Calculating a correlation coefficient from a study reported in considerable detail

Suppose a study is available that presents means and standard deviations for change as well as for baseline and final measurements, for example:

An analysis of change from baseline is available from this study, using only the data in the final column. However, we can use the other data from the study to calculate two correlation coefficients, one for each intervention group. Let us use the following notation:

The correlation coefficient in the experimental group, Corr_E, can be calculated as:

;

and similarly for the control intervention, to obtain Corr_C. In the example, these turn out to be

,

.

Where either the baseline or final standard deviation is unavailable, then it may be substituted by the other, providing it is reasonable to assume that the intervention does not alter the variability of the outcome measure. Correlation coefficients lie between –1 and 1. If a value less than 0.5 is obtained, then there is no value in using change from baseline and an analysis of final values will be more precise. Assuming the correlation coefficients from the two intervention groups are similar, a simple average will provide a reasonable measure of the similarity of baseline and final measurements across all individuals in the study (the average of 0.78 and 0.82 for the example is 0.80). If the correlation coefficients differ, then either the sample sizes are too small for reliable estimation, the intervention is affecting the variability in outcome measures, or the intervention effect depends on baseline level, and the use of imputation is best avoided. Before imputation is undertaken it is recommended that correlation coefficients are computed for many (if not all) studies in the meta-analysis and it is noted whether or not they are consistent. Imputation should be done only as a very tentative analysis if correlations are inconsistent.

(2) Imputing a change-from-baseline standard deviation using a correlation coefficient

Now consider a study for which the standard deviation of changes from baseline is missing. When baseline and final standard deviations are known, we can impute the missing standard deviation using an imputed value, Corr, for the correlation coefficient. The value Corr might be imputed from another study in the meta-analysis (using the method in (1) above), it might be imputed from elsewhere, or it might be hypothesized based on reasoned argument. In all of these situations, a sensitivity analysis should be undertaken, trying different values of Corr, to determine whether the overall result of the analysis is robust to the use of imputed correlation coefficients.

To impute a standard deviation of the change from baseline for the experimental intervention, use

,

and similarly for the control intervention. Again, if either of the standard deviations (at baseline and final) are unavailable, then one may be substituted by the other if it is reasonable to assume that the intervention does not alter the variability of the outcome measure.

As an example, given the following data:

and using an imputed correlation coefficient of 0.80, we can impute the change-from-baseline standard deviation in the control group as:

.