In the presence of measurement errors, we cannot directly fit the regression model (3) or (4). We can, however, replace the true scores with the measured scores, thus fitting model

$$\begin \varDelta T = \gamma _0+\gamma _P P+} \end$$

or

$$\begin \varDelta T = \gamma _0^*+\gamma _P^* P+\gamma _^*T_0+}^*. \end$$

To assess the bias in the coefficients \(\gamma _P\) and \(\gamma _P^*\), with respect to the possible target parameters \(\beta _P\) and \(\beta _P^*\), we use standard results for normal distributions (see “Appendix”), and obtain

$$\begin \gamma _P=b_1d_1-b_0d_0 \end$$

and

$$\begin \gamma _P^*=b_1d_1-b_0d_0\frac. \end$$

We note that, if \(b_0=b_1=b\), \(r=1\), \(d_0=d_1=1\) and \(\rho =0\), as assumed by Eriksson and Häggström [6] and Farmus et al. [2], then the coefficient \(\gamma _P^*\) simplifies to \(\frac\). This is identical to the expression in equation (6) by Eriksson and Häggström [6].

If we consider \(\beta _P\) as the target parameter, then, from the expressions above, we have the biases

$$\begin \gamma _P-\beta _P=b_1(d_1-1)-b_0(d_0-1) \end$$

(10)

and

$$\begin \gamma _P^*-\beta _P=b_1(d_1-1)-b_0\frac \end$$

(11)

for \(\gamma _P\) and \(\gamma _P^*\), respectively. If we instead consider \(\beta _P^*\) as the target parameter, then we have the biases

$$\begin \gamma _P-\beta _P^*=b_1(d_1-1)-b_0(d_0-r) \end$$

(12)

and

$$\begin \gamma _P^*-\beta _P^*=b_1(d_1-1)-b_0\frac \end$$

(13)

for \(\gamma _P\) and \(\gamma _P^*\), respectively.

These bias expressions are complex functions of the parameters in models (2) and (9), and there is no general hierarchy between the biases. As an example, Fig. 4 shows the biases of \(\gamma _P\) (solid lines) and \(\gamma _P^*\) (dashed lines) with respect to \(\beta _P\) (left panel) and \(\beta _P^*\) (right panel) as functions of \(d_0=d_1=d\), for parameter values \(b_0=0.4\), \(b_1=0.8\), \(s^2=\sigma ^2=1\), \(r=0.7\) and \(\rho =0.2\). We observe that all biases are monotonically increasing in d, negative for d close to 0 and positive for d close to 2. However, the switch from negative to positive bias occurs at different values of d for the four combinations of \((\gamma _P,\gamma _P^*)\) and \((\beta _P,\beta _P^*)\). Thus, for some values of d, the biases of \(\gamma _P\) and \(\gamma _P^*\) have opposite signs, so that one of them underestimates the target parameter whereas the other overestimates it. Furthermore, for some values of d, the absolute bias of \(\gamma _P\) is larger than the absolute bias of \(\gamma _P^*\), whereas for other values of d it is the other way around.

This example shows that, regardless of whether \(\beta _P\) or \(\beta _P^*\) is the target parameter, the choice of whether or not one should adjust for the measured baseline score \(T_0\) is generally non-trivial, and requires careful thinking about possible values of the model parameters.

Biases of \(\gamma _P\) (solid lines) and \(\gamma _P^*\) (dashed lines) with respect to \(\beta _P\) (left panel) and \(\beta _P^*\) (right panel) as functions of \(d_0=d_1=d\), for parameter values \(b_0=0.4\), \(b_1=0.8\), \(s^2=\sigma ^2=1\), \(r=0.7\) and \(\rho =0.2\)

We proceed by considering the important special case when the measurement errors are not systematic, i.e., \(d_0=d_1=1\) and \(\rho =0\). As argued above, this may be a fairly reasonable model simplification for the study by Tajik-Parvinchi et al [7]. For this special case, the bias expressions in (10)–(13) simplify to

$$\begin \gamma _P-\beta _P=0,\\ \gamma _P^*-\beta _P=-b_0\frac,\\ \gamma _P-\beta _P^*=-b_0(1-r) \end$$

and

$$\begin \gamma _P^*-\beta _P^*=b_0r\frac. \end$$

Since \(\gamma _P\) has zero bias with respect to \(\beta _P\), whereas \(\gamma _P^*\) generally has non-zero bias, the conclusion is clear: If we consider \(\beta _P\) as the target parameter, and we are willing to assume that the measurement errors are not systematic, then we should not adjust for the measured baseline score.

For the target parameter \(\beta _P^*\), the conclusion is less trivial. From the expressions above, it follows that \(\gamma _P^*\) has smaller absolute bias than \(\gamma _P\), with respect to \(\beta _P^*\), if

$$\begin 1-\frac\cdot \frac>0, \end$$

(14)

but has higher absolute bias otherwise. The left-hand side of this inequality decreases monotonically with the correlation r and with the variance ratio \(\sigma ^2/s^2\). Thus, if r is small, or \(\sigma ^2\) is small relative to \(s^2\), then the parameter \(\gamma _P^*\) is likely to have smaller bias than \(\gamma _P\), with respect to \(\beta _P^*\).

The contour plot in Fig. 5 shows the left-hand side of the inequality in (14) as a function of r and \(\sigma ^2/s^2\). We observe that, unless \(\sigma ^2/s^2\) is close to 0, the contour lines are close to vertical, so that the left-hand side of the inequality depends mainly on r. Specifically, if \(\sigma ^2/s^2\) is larger than \(\sim 0.5\), then \(\gamma _P^*\) has smaller absolute bias than \(\gamma _P\) if r is smaller than \(\sim 0.75\), independently of \(\sigma ^2/s^2\). We thus reach the conclusion: if we consider \(\beta _P^*\) as the target parameter, and we are willing to assume that (a) the measurement errors are not systematic, (b) \(\sigma ^2\) is at least \(\sim 50\%\) of \(s^2\), and (c) r is at most \(\sim 0.75\), then we should adjust for the measured baseline score. We emphasize that a violation of the condition in (b) and/or in (c) does not imply that we should not adjust for the measured baseline score, but it implies that the threshold for r at which adjustment becomes beneficial depends on the value of \(\sigma ^2/s^2\), as seen in the bottom part of Fig. 5.

Whether these assumptions are plausible or not is of course highly context dependent. We don’t have enough subject matter knowledge to firmly judge their plausibility for the study by Tajik-Parvinchi et al. [7]; however, we do suspect that even a standardized assessment tool for emotion regulation may give quite large (non-systematic) measurement errors, and that emotion regulation may vary considerably over 10 week periods within children with autism. If so, then one may tentatively guess that \(\sigma ^2/s^2\) was not close to 0 and r was not close to 1 in the study by Tajik-Parvinchi et al. [7], in which case the authors would possibly have benefited from adjusting for the measured baseline score, had they been interested in the parameter \(\beta _P^*\).

The left-hand side of the inequality in (14) as a function of r and \(\sigma ^2/s^2\)