Regression Analyses

Regression is a statistical tool used to investigate relationships between variables, and in

particular assess the predictive value of one variable, or set of variables over and above the other specified variables which allows us to make rational decisions about the effect of adding

additional information on the accuracy of prediction. The ratio of cases to independent variables should ideally be 20:1, and should certainly be no lower than 5:1, in these data the case to independent variable ratio was 14:1.

In all of the subsequent tables R² indicates the total variance predicted by the regression model in question and the R² Change statistic (R²∆) shows the variance uniquely contributed by the predictor variables entered at the last step of the regression equation.

To facilitate this more detailed analysis of these relationships a succession of fixed-order hierarchical regression analyses were performed (Table 9). Hierarchical regression models were used as this allowed the input of predictor variables into the equation in a specific order based upon past research in this field. Therefore, these models assessed the amount of unique variance in mathematics scores predicted by each of the individual working memory measures after

79 statistically controlling for age related variance and any variance pertaining to the Performance Measures.

5.8.1.1 Reception

In this first series of hierarchical regression analyses Model 1: Reception demonstrated that the WM model accounted for 28% of the unique variance in mathematics scores at Reception age (R²

∆=.28, p=<.0001, ANOVA [f (7, 62) =11.21, p=<.0001]) after eliminating any age related variance from the model (Table 9), with a significant beta value identified for Odd One Out (NV-WM). The adjusted R² figure indicates how well the findings can be generalised, and the minimal shrinkage from R² =.56, to adj R² =.51 is .05, and as such if this model were derived from the wider

population it could be expected to account for 5% less variance in the outcome measure (Maths 5). Durbin-Watson³ test was checked and within acceptable parameters (2.27).

The beta values tell us to what degree each predictor variable affects the outcome measure if all other predictors were held constant. There are two significant beta values indicating that both Age and Odd One Out (NV-WM) are significant independent predictors of Mathematics 5. The standardised beta coefficient (β) is measured in standard units, meaning that they are directly comparable with one another. The β for age and Odd One Out are .29 and .39 respectively. This indicates that in this model Odd One Out is slightly more important than Age in predicting scores on Mathematics 5. In real terms one would expect to see an increase of 1.40 in scores on

Mathematics 5 with every standard deviation increment on Odd One Out (and an increase of 1.23 in Mathematics 5 scores for Age), the caveat of course being that these interpretations only hold true if all other predictor variables remain the same. Problems of multicollinearity were also checked for using the variance inflation factor (VIF) which quantifies the severity of

multicollinearity, providing an index that measures how much the variance (the square of the estimate's standard deviation) of an estimated regression coefficient is increased because of

3 Durbin-Watson test is a test to the assumption of independence of the residuals. The test statistic is between 0 and 4 and a value of 2 means that the residuals are uncorrelated. As a general rule of thumb a value of between 1 and 3 will not give rise to cause for concern.

80 collinearity. At 1.57, the largest VIF was well below 5, and the average VIF was under 1.49,

similarly the tolerance data are all well within acceptable boundaries (all greater than 0.1).

One case was identified as an outlier, however scrutiny of the casewise diagnostics (Cooks’

Distance: none greater than1; average leverage = 0.1) indicates that this outlier is not having an undue effect upon the model and that our sample appears to conform to what would be expected for a fairly accurate model.

5.8.1.2 Year One

In Year One (Model 2) working memory significantly accounted for 36% of the variance in

mathematics scores (R² ∆=.36, p=<.001; ANOVA [F (9, 60) =17.66, p=<.001]) after eliminating both age related and non-verbal performance related variance from the model (Table 9). The adjusted R² figure shows that if this model were resultant from the wider population it could be expected to account for 4% less variance in Mathematics 6 (difference between R² =.73 and adj R² =.69 is .04) again Durbin-Watson test was checked and inside satisfactory bounds (1.80).

The β values inform us that both Listening Recall (V-WM) and Nonword Recall (V-STM) are both significant independent predictors (β=.38, and .26 respectively), with Odd One Out (NV-WM) approaching significance levels (β=.18, p=<.06). There were no issues of multicollinearity detected.

5.8.1.3 Year Two

In the final year (Year Two) working memory contributed 17% (p=<.005) of the unique variance in Mathematics 7 scoring (Model 3). The V-WM measure Listening Recall contributed significant independent variance to maths performance (β=.30, p=<.01) even after all the other variables had been partialled out. As shown none of the other WM variables contributed significant variance.

From the Performance Measures Block Design was also a significant unique predictor with β=.30, p=<.01. No concerns about multicollinearity were found.

The regression analyses find that at no point is either of the measures of nonverbal short-term memory significantly predicting mathematics test performance as a whole.

81 Table 9. Hierarchical regression models predicting mathematics performance with WM measures, controlling for age.

Predictor Variables :

ANOVA ANOVA [f(7,62)=11.21, p=<.0001] ANOVA [f(9,60)=17.66, p=<.001] ANOVA [f(9,60)=9.03, p=<.001]

Mathematics 5 * p=<.0001; Performance measures not assessed Mathematics 6 * p=<.01, **p=.005, ***p=<.001, a p=.06 Mathematics 7 * p=.05, **p=.005, ***p=<.001, a p=.06

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