Defining and evaluating stability in early years assessment

Figures & data

Table 1. Types of stability and corresponding (linear) multilevel models and assumptions.

Type	Model	Model assumptions	Graphical representation
Strict stability: Intraindividual differences are absent	$Y_{it}={\text{β}}_{0i}+R_{it}$ ${\text{β}}_{0i}={\gamma }_{00}+U_{0i}$	$U_{0i}\sim N\left(0,{\tau }_0\right)$ $R_{it}\sim N\left(0,\sigma \right)$
Parallel stability: Intraindividual differences are the same for each individual	$Y_{it}={\text{β}}_{0i}+{\text{β}}_{1}t+R_{it}$ ${\text{β}}_{0i}={\gamma }_{00}+U_{0i}$	$U_{0i}\sim N\left(0,{\tau }_0\right)$ $R_{it}\sim N\left(0,\sigma \right)$
Linear/monotonic stability: Intraindividual rank orders are constant	$Y_{it}={\text{β}}_{0i}+{\text{β}}_{1i}t+R_{it}$ ${\text{β}}_{0i}={\gamma }_{00}+U_{0i}$ ${\text{β}}_{1i}={\gamma }_{10}+U_{1i}$	$U_{0i}\sim N\left(0,{\tau }_0\right)$ $U_{1i}=a+c{U}_{0i}$ (Linear)^a $U_{1i}=M^{+}\left(U_{0i}\right)$. (Monotonic) $R_{it}\sim N\left(0,\sigma \right)$
Function stability: Intraindividual growth follows the same function for each individual	$Y_{it}={\text{β}}_{0i}+{\text{β}}_{1i}t+R_{it}$ ${\text{β}}_{0i}={\gamma }_{00}+U_{0i}$ ${\text{β}}_{1i}={\gamma }_{10}+U_{1i}$	$U_{0i}\sim N\left(0,{\tau }_0\right)$ $U_{1i}\sim N\left(0,{\tau }_1\right)$ $R_{it}\sim N\left(0,\sigma \right)$

Note: The indices i and t indicate child and measurement occasion, respectively. $M^{+}$ = monotonic increasing function.

^aChoosing $c$ such that $ct=-1$ for any $t$ forms an exception to this rule. The bold line in the figures depicts the mean growth.

Table 2. Mean score and SD for mathematics, split by test.

Grade	Repeated test (prop.)	Mean	(SD)
Mid K	0.025	0.53	1.239
Mid 1	0.035	16	1.158
End 1	0.034	11	1.160
Mid 2	0.026	0.15	1.113
End 2	0.029	0.15	1.118
Mid 3	0.014	0.10	1.092
End 3	0.014	0.10	1.161

Table 3. Linear and function stability estimates for mathematics scores.

	Linear (SD)		Function (SD)
Fixed intercept	0.10	(0.004)	0.11	(0.005)
Old test	0.78	(0.013)	0.74	(0.013)
School variance $\left({\varphi }_0^2\right)$	0.11	(0.003)	0.11	(0.004)
Student variance $\left({\tau }_0^2\right)$	0.74	(0.006)	0.74	(0.015)
Test slope variance $\left({\tau }_1^2\right)$			0.01	(0.000)
Slope int. cor. $\left({\rho }_{U_{0jt},U_{1jt}}\right)$.
Residual variance $\left({\sigma }^2\right)$	0.46	(0.003)	0.40	(0.003)
Deviance $\left(-2LL\right)$	23,750.1	(56.46)	23,431.8	(59.25)
BIC	23,796.0	(56.46)	23,496.1	(59.25)

Note: The reported SDs describe the between-imputation variance.

Table 4. Probability of continuous declines $\le 1{\tau }_1$ and $\le 2{\tau }_1$ under the residual distribution $N\left(0;\sigma \right)$ over a number of test administrations.

	Number of test administrations between measurements
	1	2	3	4	5	6	7
$P\left(x\le \text{–}0.10\ast \mathrm{\#}Test|\sigma =0.63\right)$	0.44	0.37	0.31	0.26	0.21	0.16	0.13
$P\left(x\le \text{–}0.20\ast \mathrm{\#}Test|\sigma =0.63\right)$	0.37	0.26	0.16	0.10	0.05	0.03	0.01

Note: #Test indicates the number of test administrations between measurements.

Figure 1 .#$\mathrm{\Delta }\mathrm{i}\mathrm{n}{\mathrm{d}}_{\mathrm{C}\mathrm{H}{\mathrm{I}}_i}$ Values for language and mathematics. One extreme value (14.12) in mathematics that falls outside the plot range is indicated by a cross and a label.