Q: (Initial status and change) I observe the same variable at two different times. I suspect that the change in the variable may be related to its initial value. For example, I suspect that a student's kindergarten achievement score may help to predict the change in that student's achievement from kindergarten to first grade. Should I regress change on initial status?

A: When measurement error is present, regressing change on initial status is known to produce biased results (Thomson 1924; Blomqvist 1977). Typically, the bias is negative, though it may be toward zero if the true relationship is severely negative. The bias can be corrected, as we discuss a little later.

To understand the bias, suppose

You'd like to estimate the covariance between change and initial status, Cov(Y2-Y1,Y1). But unfortunately Y1 and Y2 are measured with error. Let e1 and e2 be the errors, so that y1 and y2 are the observed values, and y2-y1 is the observed change:

Notice that the error term for y2-y1 is e2-e1, which is positively correlated with e2 and negatively correlated with e1. So if you regress y2-y1 on y1, then the error in the regressor y1 is correlated with the error in the outcome y2-y1. This leads to bias. More formally, if e2 and e1 are independent of each other and independent of Y2 and Y1, then the covariance between y1 and y2-y1 is

So estimating Cov (y2-y1, y1) gives a negatively biased estimate of Cov (Y2-Y1, Y1).

Similarly the variance of y1 is positively biased:

Since the regression of y2-y1 on y1 has a slope of Cov (y2-y1, y1) / Var (y1), there are two sources of bias. The numerator is too small and the denominator is too large.

Since the bias depends on Var(e1), it can be corrected if an estimate of Var(e1) is available from some outside source (Thomson 1924; Blomqvist 1977). For example, if Y1 and Y2 are test scores, then the reliability of the test can be used to deduce the variance of the measurement error (e.g., see Table 1 in Downey, von Hippel, and Broh 2004). This error variance can be incorporated into a structural equation model using packages such as AMOS or LISREL.

Often no information about the error variance is available. In that case, you can try plugging in plausible values for the error variance and see whether there is material change in your results.

References

Blomqvist, Nils. 1977. “On the Relation Between Change and Initial Value.” Journal of the American Statistical Association 72(360):746-749.

Thomson, Godfrey H. 1924. “A Formula To Correct for the Effect of Errors of Measurement on the Correlation of Initial Values with Gains.” Journal of Experimental Psychology 7, 321-324.

Downey, D.B.; von Hippel, P.T.; and Broh, B. 2004. “Are Schools the Great Equalizer? Cognitive Inequality During the Summer Months and the School Year.” American Sociological Review 69(5), 613-635.