1. Introduction

The Danish mathematician and statistician Georg Rasch (Andersen & Wohlk Olsen, 2001; Andrich, 2005) is well known for his measurement theory based on the principle of invariant comparisons (Rasch, 1960, 1961). In contrast, he is hardly known for his studies in physiological growth based on the same principle, both of which are consistent with formulations of relationships amongst variables in the natural sciences. His approach identifies a transformation of time, called the meta-metre, which characterises the growth for all members of a population. Then within the meta-metre, individual rates of growth can vary, with each individual’s rate of growth linear and therefore invariant across the meta-metre (Rasch, 1977). Andrich, Marais and Sappl (2023) adapted Rasch’s approach to the measurement of growth on variables of the social sciences, and in particular intelligence and attainment tests in reading and mathematics. Although they provided estimates of the meta-metre and rate of growth of individuals within the meta-metre, they did not provide an estimate of the error variance of an individual’s estimate of the rate of growth. This paper is concerned with this topic.

Before proceeding we note that the approach is novel, and had only been applied by Rasch (Wohlk Olsen, 2003) and Rao (1958) to the growth in weight of young animals and human  babies respectively. We recognise the presence of current methods of studying growth in the social sciences, for example structural equation modelling summarised by Wang & Nydick (2020), but there is no space to compare and contrast these methods in this paper. An approach to growth on social science variables that is analogous to that of the natural sciences has also been advocated by, among others, Williamson (2018). However, those methods involve modelling in the form of polynomial regression and are primarily descriptive and group based, rather than explanatory and individual focussed.

1.1 Some distinctive features of the meta-metre approach to growth

Because the meta-metre approach is novel, we summarise some of its key features before proceeding to the main purpose of the paper. First, both the function of the estimated meta-metre and the rates of growth of individuals and subpopulations become of interest. For example, Andrich et al. (2023) showed that the estimated meta-metres of two intelligence, two reading and two mathematics tests, all had decelerating growth, indicating that the rate of growth was inversely proportional to a function of time. This implies that it becomes more and more difficult to grow on the quantitative scale, and sets up a quest for understanding this phenomenon, broached in Andrich  et al. (2023), but which is also beyond the scope of this paper. Their analyses of other longitudinal achievement data showed the same characteristic.

Second, two insights from the decelerating meta-metre were highlighted. One was that the rate of growth in the early years is relatively rapid compared to the later years and that this confirms that this is the time in which to intervene for students at risk; the other, that a small difference on the quantitative scale in the early years, which can imply a small difference on the common grade scale used in schools, invariably implies a greater difference on the grade scale in later years. Often it is implicitly assumed that growth on a quantitative scale is more or less linear, and that it should also be linear on the grade scale (Andrich et al., 2023, Ch. 6).

Third, having, within a meta-metre, the rate of growth characterized by a single value has both conceptual and statistical elegance. For example, it permits standard analysis of variance (ANOVA) procedures to be used in the comparison of mean rates of growth among defined subgroups of a population. Andrich et al. (2023) conducted such analyses comparing performances based on gender and demographic groups. In ANOVA, the error variance of the estimate of each individual is absorbed within the variance among individuals permitting it to proceed without having an estimate of the individual error variance. Establishing an estimate of the error variance of an individual’s rate of growth in a meta-metre is the subject of this paper.

1.2 Adaptations of the meta-metre approach to variables of the social sciences

There are two issues that needed addressing in adapting Rasch’s concept of a meta-metre of growth to social science variables. First, Rasch and Rao could assume a natural origin for weight. However, in social science tests, the origin is arbitrary, and Andrich et al. (2023) successfully adapted Rasch’s approach to account for this feature. Second, in Rasch’s formulation of a stochastic process of growth, it was assumed that the successive differences of measurements, which are used in the estimation and which are summarised below, were independent. This independence is tenable in part under two general conditions: first, if, relative to the range of measurements, the measurement error is negligible; second, if the rate of change of individuals between time points shows very little variation. Both might be tenable for the growth of babies or animals in their first year. In social science tests however, as illustrated below, neither the magnitude of the measurement error nor the potential variation in the amount of growth between time points of an individual can be ignored across a relatively large time scale. Both properties show features of general regression to a true value that cannot be ignored in establishing an error variance for the rate of growth of an individual.

Although the methods of estimation of the individual rate of growth are analogous to those of standard regression analyses, there are two main differences. First, because of repeated measures, successive differences rather than observed measurements are used in the estimation; second, in addition to the study of the rate of growth in means of a population, the focus is on estimating the rate of growth of each individual. It is noted that the studies of this paper, though derived directly from the equation for estimating an individual’s rate of growth, are in part heuristic, taking advantage of modern computing facilities for conducting simulations. They are not seen as the final word on the estimates of the errors of the rates of growth, or indeed on the estimates of the rates of growth themselves for test data typically found in social science instruments. They are intended to generate further studies on both topics.

The rest of this paper is structured as follows. Section 2 summarises the adaptation of Rasch’s formalisation of a meta-metre of growth for social science variables, Section 3 provides variances of the estimates of an individual’s rate of growth, Section 4 shows the results of two simulation studies and Section 5 suggests further research. In addition, two Appendices are provided, one shows the derivation of the estimation equations; the second presents a general discussion on the generic concept of regression to the true value in the presence of random variation.

Although it is elaborated at the end of Appendix 2 because of context, it is anticipated here that the general theme of the analyses presented in this paper is that of probabilistic regression to the true value, a subtle and ubiquitous property of replications in the presence of random variation. Although related to, it is not the same concept to the description of fitting a model typically referred to as a regression analysis.

2. The meta-metre of growth and its estimate

Formally, let Ynt  be a measurement of an individual on some variable, such as on an intelligence, reading or mathematics test, at time t,t=1,2,...T  with an arbitrary origin. Although the origin may be arbitrary, it is assumed that the measurements are of the form commonly known as interval level which follow, for example, if measurements are estimated by a Rasch model of modern test theory (Andrich & Marais, 2019; Rasch, 1960). Then

Ynt=βnt+εnt=an+bnτt+εnt,ε.t∼N(0,σt2)                          (1)

is the linear growth of individual n  as a function of time t  in the meta-metre τ(t) , βnt  is the true value of individual n  at time t , εnt  is measurement error assumed to be homogeneous among individuals and times of measurement, and normally distributed. We return to this point later in the paper.

An important feature of Eq. (1) is that it characterises dynamic consistency in which the function of growth of an individual is the same as the function of growth of the population (Keats, 1982). Thus, taking the mean of a population, and replacing the subscript over which the mean is taken by “.”, gives

Y.t=a.+b.τ(t) ,                                                                                   (2)

where b.,a. respectively characterise the mean rate of growth and initial status of the population. Eq. (2) is used to estimate the meta-metre for the population.

2.1 The meta-metre for growth in weight and intelligence and attainment tests

In the examples of growth on some intelligence and attainment tests, Andrich et al. (2023) were able to characterise growth in the meta-metre by

τ(t)=ln(θt+λ) ,                                                                               (3)

where θt  is the value of the time variable at each occasion of measurement, t,t=1,2,...T.

This function is identical to the one used by Rasch to characterise growth in weight. The estimates of λ  were in the range -4<λ<10  for the intelligence tests and -0.35<λ<1.2  for the reading and mathematics tests. The kind of growth in these tests showed consistent deceleration, thus making the logarithmic function an obvious one to consider for the meta-metre. From graphical representations and the values of the regression correlations of the order of R2≥0.95 , together with other evidence, it was concluded that the meta-metre for each population was estimated fully successfully.

The parameter λ  has two roles. First, it governs the rate of deceleration, with the greater the value of λ  the smaller the deceleration; second, it sets a form of origin, permitting the first time point to take some arbitrary, convenient value. For example, in growth in educational attainment tests, the first time can be set as either a grade (e.g. 1 for Grade 1) or an age (e.g. 6 for children being on average six years old), whichever is deemed convenient.

2.2 An adaptation of Rasch’s method of estimation

Rasch’s method of estimation of the parameters in Eq. (2) is that of least squares, which we follow. There are alternative approaches within this general method, but the one used for this paper is shown in Appendix 1. It is somewhat more efficient than that applied in Andrich et al. (2023). In summary, to estimate λ  for the population for the meta-metre of Eq. (3), advantage is taken of dynamic consistency, and the means Y.t,t=1,2,3,...,T  are used as the relevant data. Then the estimation equations are

λ:t=2T(y.t-b.[τt])(θt-1-θt)(θt+λ)(θt-1+λ)=0,                                                                        (4)

b.=t=2Ty.tτtt=2Tτt2,                                                                                           (5)

where y.t=Y.t-Y.t-1,t=2,3,...,T  and τt=τ(t)-τ(t-1)  are successive differences of the means and of the meta-metre respectively. The derivation of the equations, which must be solved iteratively, are shown in Appendix 1.

Having confirmed the adequacy of the meta-metre with the estimate of λ  for the population, the least squares estimate of the rate of growth bn  of individual n  is given by Eq. (6) (Andrich et al., 2023):

bn=t=2Tyntτtt=2Tτt2                                                                                         (6)

The estimate of the intercept an  for individual n , given λ  and bn , can be obtained from the standard regression equation

an=Yn.-bnτ(.) .                                                                                 (7)

Because this paper is concerned with estimates of bn  and their standard errors, the rest of the paper does not refer to an .

3. The general form of the error variance of the rate of growth

From Eq. (6),

V[bn]=Vt=2Tyntτtt=2Tτt2 ,                                                                           (8)

and because t=2Tτt2  is a constant given τt ,

V[bn]=V[t=2Tyntτt](t=2Tτt2)2.                                                                              (9)

3.1 Covariances of successive differences

Given the denominator of Eq. (9) is a constant, we first expand its numerator giving

Vt=2Tyntτt  

=t=2TV[yntτt]+2t=2T-1t'=3TCOVyntτt,ynt'τt'  

=t=2Tτt2V[ynt]+2t=2T-1t'=3Tτtτt'COVynt,ynt'  

=t=2Tτt2σynt2+2t=2T-1t'=3Tτtτt'ξynt2  ,                                              (10)

where σynt2=V[ynt];ξynt2=COV[ynt,ynt'] .

In summary

V[b.]=t=2Tτt2σynt2+2t=2T-1t'=3Tτtτt'ξynt2(t=2Tτt2)2 .                                                   (11)

The values σynt,2ξynt2  are within person variances and covariances of successive differences. Clearly if the differences (ynt,ynt')  were statistically independent, then ξynt2=0.

However, because of regression effects of measurement error and variable growth between time points, both of which are assumed to be normally distributed in the population, the covariance, ξynt2  between two successive differences ynt,t=2,3,...,T,  is most unlikely to be zero. To avoid a break in the continuity of the derivations, a discussion on the principle of using differences ynt  for the estimation, and the reasons they lead to regression, are discussed in more detail in Appendix 2.

In summary, regression appears because if a measurement on the first occasion is say noticeably greater than its true value, then on a replication, the second measurement is more likely to have a value less extreme than the first measurement and generally closer to the true value than the first measurement. This is because in a normal distribution in particular, the probability of a large discrepancy from the true value is smaller than the probability of a small discrepancy (Andrich & Pedler, 2019).

3.2 The presence and evidence of a measurement error

For further exposition, we elaborate our terminology. From Eq. (1) βnt  is the unknown, presumed true value of an individual on the variable being measured at time θt . We refer to this value as the measure of individual n.  Then the observed value of the variable, Ynt  of Eq. (1) we refer to as a measurement. This measurement has an error, denoted εnt  with variance V[Ynt]=σnt2 , which formalises Eq. (1) and for convenience is reproduced as Eq. (12):

Ynt=βnt+εnt=an+bnτ(t)+εnt.                                                  (12)

To be consistent with the measurement context considered below, for the remainder of the exposition we replace Ynt  with the estimate βnt .

In the typical social science context, with an estimate βnt , its error variance σnt2  is provided. For example, in the context of educational, psychological, or health outcomes assessment, individuals respond to a test or questionnaire composed of a set of items assessing different aspects of the same variable. These tests may be attainment assessments such as mathematics, reading or intelligence tests, referred to above in Andrich et al. (2023), or tests or questionnaires that might be used in health outcomes, for example in Christensen, Kreiner and Mesbah (2013). If the responses are analysed according to models of modern test theory (Van der Linden, 2016), and in particular Rasch measurement theory (Rasch, 1960, 1961; Andrich & Marais, 2019), an estimate βnt  with a standard error of this estimate, σnt , are provided. These are provided by the theory of maximum likelihood (ML) estimation typically employed. Because of the large literature on estimation in the Rasch model, we do not elaborate on this topic here. However, for completeness, Appendix 1 has a summary of the ML estimation equations for the estimate of individual parameters and their standard errors in the dichotomous Rasch model for measurement, which is used in the simulation studies in Section 4. It also confirms, from the simulation, the accuracy of these equations.

Thus, we take that each individual has a measurement βnt  and an estimate of its error variance σnt2 , where this variance varies among individuals. Therefore, when the only error is measurement error, we can write

Vynt=Vβnt-βnt-1=Vβnt+Vβnt-1

=σnt2+σnt-12

=σeynt2 .                                                                                                 (13)

We distinguish the notation for V[ynt]  when obtained directly within an individual over times of measurement and when derived from measurement error alone: the former, introduced in Eq. (11), is σynt2;  the latter introduced above in Eq. (13) is further subscripted by e , σeynt2.

Eq. (13) shows that, in the context with which we are concerned, it provides an estimate of a component of the first term in the numerator of Eq. (11), that of the within individual error variances, V[ynt]=σynt2 . However, there is no obvious estimate for an element of the second term, ξynt2 , that which involves the within individual covariances. We study further analysis of this term, augmented with simulated data, and suggest a possible estimate of ξynt2 . Incidentally, we note that for the variance V[y.t]  across individuals, we again need to concern ourselves with the covariance COV[y.t,y.t'] .

4. Simulation studies

This section provides the results of two simulation studies which illustrate the validity of Eq. (11) in the presence of measurement error and some variable rates of growth across time for each individual. Unlike in real data, in simulated data the measures βnt  are known, and we use this knowledge in interpreting the results of the simulations.

4.1 Simulation 1: only source of variance is measurement error

The variance associated with the successive differences ynt  can in principle come from two sources: first, as indicated above, from the measurement error; second from different rates of growth between times of measurement within an individual. To provide a frame of reference for the study of Vbn  in the realistic context where both forms of variance are present, the first simulation has the measurement error as the only source of variance. Simulation 2 then has, in addition to measurement error, variation within each individual’s rate of growth across times of measurement.

4.1.1 Input parameters for Simulation 1

First, 2500 individuals with measures βnt  were drawn from a normal distribution with a specified mean and standard deviation, (β.t,σβt) , at time t=1.  Second, each individual had the same linear rate of growth in the same meta-metre between the successive time points, giving measures at each successive time point. The parameters of the meta-metre and measures are based on examples of real data (Andrich et al., 2023), with measurements taken on seven occasions, beginning in Kindergarten and occurring at varying times with the final assessment taken in Grade 12. In anticipation of the notation for Simulation 2, we note that the correlation between successive measures is 1.0, and denote Simulation 1 by ρt,t-1=1.0 .

Table 1 shows the specified value of the meta-metre, λ  and distribution of individuals at each time of measurement.

Table 1.

The meta-metre and specified distribution for the grades θt  at each time of measurement t .

Figure 1 shows the interpolated, decelerating growth in means of the measurements as a function of the grade. In anticipation of the estimate of the meta-metre, Panel 2 of Figure 1 shows the linear rate of growth in the estimated meta-metre.

4.1.2 Observing estimates of measures and their standard errors

To study errors of measurement from typical social science contexts, each individual with measure βnt  was simulated to respond to a test composed of 60 items, scored correct or incorrect, according to the dichotomous Rasch model (Rasch, 1960). In the design, it was ensured that the range of item difficulties was wide enough that the measurements were not contaminated by floor or ceiling effects. These responses were then analysed with the same Rasch model giving individual estimates and their standard errors, (βnt,σnt) . These analyses were conducted using the software RUMM2030Plus (Andrich, Sheridan & Luo, 2023). Again, to avoid breaking continuity, the results of the summary statistics, which confirm the consistency of the analyses with the specified parameters, and the accuracy of the estimates of parameters, are also shown in Appendix 1.

The key feature of Simulation 1 is that, although the individuals have different initial measures, because the rate of growth of all individuals is identical, from the perspective of the rate of growth the individuals are replications of each other. We take advantage of this feature in the study. To further confirm the validity of the results, a number of simulations with the same principles as just described, but different parameters, were carried out. Because the results were identical across these simulations, for illustrative purposes only that of one pair of simulations is reported in this paper.

We stress that we are not attempting to obtain an empirical distribution of the error estimates in the individual rate of growth by carrying out multiple simulation studies in this paper, but to give analytic and illustrative direction to studying a specific feature in the estimation of this error, namely the effect of regression between successive measurements. A sample of 2500 was considered sufficiently large to provide stability in the estimates to illustrate these effects.

4.1.3 Results of analyses of error variances and covariances of the differences ynt

The variances and covariances, σynt,2ξynt2  respectively, are the key statistics of Eq. (11) for estimating the variance of an individual’s estimated rate of growth, V[bn] . Table 2 shows these from both the measures βnt and their estimates βnt . For the measures, as expected, the variances and covariances are zero. However, in addition to positive variances, the covariances of the successive differences in the estimates are noticeably negative. This is a direct result of regression to the mean, mentioned above and discussed in some detail in Appendix 2. Interestingly, the covariances of the estimates that are not successive are virtually zero and in practice could be ignored. It is also evident from Table 2 that the mean of the ML estimates of the individual error variances σeynt2  give a very accurate estimate of the actual variances, σey.t2≈  σy.t2 .

Table 2.

Variances V[y.t]=σy.t2  and covariances COV[ynt,ynt']=ξynt2  of ynt  among individuals in measures below, and measurements above, the diagonal, together with the mean of error variances σey.t,2  derived from Eq. (13).

4.1.4 Estimate of λ

Table 3 shows the population estimates of λ  and b.  from both the simulated measures β  of the individuals and the estimated measurements β . The estimates are shown correct to three decimal places with the solution to Eqs. (4) and (5) being correct to four decimal places. The estimate λ  is close to the specified value, and the mean rate of growth b.  of the individual estimates is identical to the population estimate b.  with predictably, those from the measures somewhat more accurate than from their estimates, which include random errors of measurement. Figure 1, Panel 2, showed the linear growth in the means as a function of the estimated population value of λ  in the meta-metre with R2=1.000 . Therefore, together with the details shown in Appendix 1, the simulation was considered valid and sufficiently stable to illustrate the study of the estimates of error variance of individuals’ rates of growth.

Table 3.

Estimates of λ  from the measures and estimates together with summary statistics for three estimates of V[bn] , where ρt,t-1=1.0 .

4.1.5 Analyses of error variances of estimates bn

Table 3 also shows three estimates of individual variance V[bn] .

(i) V[b.] , where the subscript (.) again replaces the index across which the variance is taken, is simply the observed variance of the estimates bn,n=1,2,...,2500.  It will be recalled that because the rate of growth of individuals is identical making the 2500 cases replications of each other, this variance among individuals, V[b.] , is also an estimate of the variance within each individual, V[bn] . Therefore, from the measures which have no measurement error, V[b.]=0.  However, because of measurement error, from the estimates, V[b.]=0.055 . This estimate is used as a criterion for the accuracy of the two other estimates.

(ii) V[b.]|σynt,2ξynt2  is obtained by inserting into Eq. (11) the observed variances and covariances, σy.t2,ξy.t2  of ynt  among individuals. As a result, it is the same value for all individuals, justified because the individuals are replications of each other. This estimate depends only on the observed differences ynt . Although calculated differently, V[b.]|σynt,2ξynt2=V[b.]=0.055 , indicating that V[b.]|σynt,2ξynt2  is a correct estimate of V[b.]=0.055

(iii) V[bn]|σeynt,2ξynt2 , is the mean of the estimates of V[bn]  derived using the error variances within individual estimates. It was calculated as follows. The first term of the numerator of Eq. (11), t=2Tτt2σynt2 , was calculated from the error variances available for each individual from Eq. (13), giving t=2Tτt2σeynt2 . This value was not identical across individuals. The second term of the numerator of Eq. (11), 2t=2T-1t'=3Tτtτt'ξynt2 , was obtained from the covariance among individuals. Again, because individuals are replications of each other, the covariance among individuals is an estimate of the covariance within individuals. This value, therefore, was identical across individuals.

The observed variance, V[bn]|σeynt,2ξynt2=0.056 , differs by only 0.001 from the other two estimates. The closeness of these two values further confirms, first the accuracy of the ML within individual error variances V[βn] , and second, the need to use covariances ξynt2  in estimating the error variance of the latter, using either V[b.]|σynt,2ξynt2  or V[bn]|σeynt,2ξynt2 .