Covariance specifications

 

After much reflection, the Doctor specifies variances and covariances over all the quantities of interest as follows:

To facilitate our analysis, suppose that we calculate the corresponding correlations:

Some of the principal features of the Doctor's variance and covariance specifications are as follows:

• Assessing the expectation and variance specified for , we observe that these specifications reflect the Doctor's belief that the ogt-test tends to misdiagnose many elderly patients in that 6.25 is quite close (less than half a standard deviation distant) to the threshhold of 7.0 for a diagnosis of impaired glucose tolerance.
• She is less sure about the two-hour measurements than she is about the fasting measurements; and she has assigned a correlation of about 0.436 between and and between and .
• Her covariances specified between and imply that and are moderately related (a correlation of about 0.554), whereas and are only weakly related, with the correlation between them being about 0.177. She has also specified about the same degree of relationship (a correlation of about 0.182) between and . Observe that is more strongly related to than is

Informally, the strength of correlation between two quantities determines the degree to which one quantity can be used to help learn about another. For example, if the Doctor had specified zero correlations between and , then observing would tell us nothing (in a linear analysis) about . Here she has specified a larger correlation between and than between and , suggesting perhaps that the fasting measurement will be more informative than the two-hour measurement in learning about both unknowns and .