Mediation Modelling Methodology

The methodology for the testing period for the mediation modelling is the same as that described in Chapter 4 and previously in this chapter. With respect of the data analyses a decision was taken to only analyse the data from the final year of testing. There are several reasons for this decision.

Firstly, in Reception year Performance Measures were not assessed, and for completeness in the mediation model it is necessary to account statistically for as much of the variance attributable to Problem Solving as possible. Secondly, in Reception year of testing there were only five questions asked as part of the Problem Solving strand, and on only one occasion was a calculation required.

Thirdly, in Year One from the regression analyses a potential issue with the Problem Solving data was identified. There were 13 Problem Solving questions in total and the scoring on this strand ranged from 1-3, with a mean of 1.84 (sd =.97), therefore this represents something of a floor

140 effect in the raw scores. As such the following mediation model is presented as a snapshot of what is occurring between the predictor, the mediator and the outcome variables at Year Two.

8.6.1 Statistical Background

The primary goal of mediation analysis is to explain the mechanism that underlies an observed relationship between an independent variable and a dependent variable by including a third explanatory variable, known as a mediator variable. Modern approaches to statistical mediation analysis focus on estimation and inference about the indirect and direct effects of generally accepted cause X on presumed effect Y through proposed intervening variable M. The early causal steps approach that was described by Baron and Kenny (Baron & Kenny, 1986) is now reported to

“leave a lot to be desired” (Hayes & Preacher, 2011). They argue this for two main reasons; firstly that it is “one of the lowest power methods available …” (p.4 Hayes & Preacher, 2011), and secondly that the approach does not emphasise the explicit quantification and inferential testing of the indirect effect. Hayes and Preacher (2011) discuss in detail the reasons why the Baron and Kenny method is now out of favour, and as such this chapter operates with in the more current statistical thinking about mediation analysis, utilising one of the Hayes and Preacher methods. To analyse this data PROCESS method was used (Hayes, 2013; Hayes & Preacher, 2011) , and for simplicity the PROCESS.spd add-in for SPSS 20 was deployed (Hayes, n.d) as this provides a clear graphical user interface and allowed the input of covariates and choose appropriate

bootstrapping methods. A composite working memory score was derived by summing the scores across the working memory tests and obtaining the mean. This was necessary as one of the limitations of using PROCESS is that only one predictor variable is allowed to be input. A composite score of Performance Measures was also used as this model is not overly concerned with the relationship between Performance Measures and Problem Solving.

Also of benefit, this modelling technique allows for theoretically driven models, whereas in older mediation model methods predictor variables must conform to significance at each step before a mediator can be identified. That said causal inference can be strengthened if the researcher can

141 argue or demonstrate that the variables have been modelled in the appropriate causal sequence.

There are two effects of X that are of primary interest in mediation analysis. Firstly if interest is the direct effect between the X and Y variables as depicted in Fig.5, but most central to mediation models is the indirect effect of X, on Y when M is statistically accounted for.

Figure 5. Simple diagram representing a direct effect (c) between an independent variable (X) and a dependent variable (Y)

This is quantified as the product of coefficients a and b. This product, ab, is interpreted as the amount by which two cases that differ by one unit on X are estimated to differ on Y as a result of the effect of X on M which in turn affects Y. This can be visually represented as in Fig.6.

Figure 6. Diagram representing an indirect effect (ab) between an independent variable (X) and a dependent variable (Y), where c-c’ represents the magnitude of the indirect effect.

M

X Y

a b

c direct effect

c’ (magnitude of indirect effect) indirect effect

X Y

c direct effect

142 The indirect effect of X serves as a quantitative tangible example of the mechanism through which X influences Y. But it is not the only path of influence from X to Y. X can also influence Y directly, independent of its indirect effect via M. C prime (c') quantifies how much two cases who differ by one unit on X but who are equal on M are estimated to differ on Y and is represented as the magnitude of the indirect effect.

The terms indirect effect and mediating effect are often used interchangeably in the mediation literature, the preferred term for the remainder of the chapter is indirect effect. The full mediation model can be visually represented as in Fig. 6.

When performing the mediation analyses there were some differences in how the data was used compared to the regression analysis. Problem Solving and Calculation raw scores were used as opposed to standardized (z) scores, as for this section there is no attempt to meaningfully compare the resulting data with that of other chapters in this thesis. This section is much more focused upon the detailed analysis of the variables and their relationships with one another.

8.6.2 Bootstrapping

A brief note on bootstrapping samples indicates that:

“Bootstrapping is a technique from which the sampling distribution for a statistic is estimated by taking repeated samples from the dataset. In effect this treats the data as a population from which smaller samples are taken. The statistic of note is the beta

coefficient (β), and this is calculated for each sample, from which the sampling distribution of the statistic is estimated. The standard error of the statistic is estimated as the standard deviation of the sampling distribution created from the bootstrap samples. From this confidence intervals and significance tests can be calculated”, (adapted from Field, 2009 p.782).

The resulting confidence intervals are identified as significant if the upper and lower levels confidence intervals do not pass through zero (denoted as CL LL and CL UL).

143