EDUCACIÓN FISICA

The use of parametric techniques requires the researcher to examine the data set prior to data analysis. Multivariate techniques (stepwise regression and multivariate analysis of variance) have been selected for the data analysis. Hair et al. (1995) noted that, with the delegation of strong analytical power to the analyst, multivariate techniques place a greater burden on the analyst to ensure that the statistical and theoretical underpinning on which they are based are supported. Thus, these statistical and theoretical assumptions were examined to check if they supported the use of multivariate techniques.

The first assumption for multivariate techniques refers to outliers. Outliers are observations with a unique combination of characteristics that are distinctly different from the other observations (Hair et al. 1995). In this study, histograms and boxplots were firstly used to detect univariate outliers (Coakes and Steed, 2001). Hair et al. (1995) and Tabachnick and Fidell (2001) suggested that multivariate outliers could be detected by using ‘Mahalanobis distance’. ‘Mahalanobis distance’ is a measure of the distance in multidimensional space of each observation from the mean centre of the observations (Hair et al., 1995). The process used to identify outliers was the casewise subcommand in the regression procedure. The casewise subcommand also produces a plot of outliers that have a standardised residual greater than 3 (Coakes and Steed, 2001). Hair et al. (1995) noted that the outliers should be retained unless there is demonstrable proof that they are truly aberrant and not representative of any

observations in the population. They further suggested that the alpha level for detecting the outliers using Mahalanobis distance should be very conservative, that is .001.

Firstly, none of the standardised residuals in this study was greater than 3; rather the maximum standardised residual in this study was 2.053. Secondly, examination of the ‘Mahalanobis distance’ values indicated that the values for only four cases were greater than to the chi-square value of 49.726 at an alpha level of .001. That means there were only four outlying cases in this study, which is not unexpected in a sample size of 216. If a large number of outlying cases had been identified, then their inclusion would need to be considered carefully (Coakes and Steed, 2001). Coakes and Steed (2001) further suggested that the removal of outliers from the data set often results in the generation of further outlying cases. Considering this situation, these four outliers were left untouched in the data set. Although four outliers were detected among the cases, considering the large number of sample size, it can be concluded that the assumption relating to outliers was not violated in this study (Coakes and Steed, 2001).

The second underlying assumption of multivariate analysis is that each variable and all linear combinations of the variables are normally distributed. Hair et al. (1995) noted that, if the variation from the normal distribution is sufficiently large then all statistical tests are invalid, as normality is required to use F and t statistics. They further suggested that the researcher should assess the normality for all variables included in the analysis, even though large sample sizes tend to diminish the detrimental effects of normality. Normality can be assessed either statistically or graphically (Hair et al., 1995). The two statistical components for assessing normality are skewness and kurtosis. Skewness and kurtosis refer to the shape of the distribution, and the observed distribution is exactly normal if the values for skewness and kurtosis are zero (Coakes and Steed, 2001). Avkiran (1995) noted that a measure of skewness of 3 (plus or minus) is usually regarded as a strong deviation from normality. Normality can also be determined by examining the residual plots through a graphical method. In order to meet the assumption of normality, the residuals should be normally and independently distributed (Hair et al., 1995).

The assumption of normality was verified in this study by looking at both the residual plots and skewness and kurtosis. The measurement of skewness and kurtosis for the

variables in this study indicated no violation of the assumption of normality. None of the values of skewness and kurtosis for the variables in this study reached the 3 (plus or minus) bar suggested by Avkiran (1995), rather, most of them were close to zero and very few of them were close to 1 (plus or minus). The residual plots in this study also appeared to be normal. Thus, it can be said that the assumption of normality was not violated in this study (Coakes and Steed, 2001; Afifi and Clark, 1996).

The third assumption for multivariate techniques is multicollinearity. In the case of regression analysis, multicollinearity refers to high correlations among the independent variables, and in the case of multivariate analysis, it refers to high correlations among the dependent variables (Coakes and Steed, 2001). The regression procedure assumes that no explanatory variable has a perfect linear relationship with another explanatory variable, and MANOVA assumes that no dependent variable has a perfect linear relationship with another dependent variable (Tabachnick and Fidell, 2001; Coakes and Steed, 2001). Thus, it is necessary to detect the linear relationships among the variables in both cases (regression and multivariate analysis of variance). Berry and Feldman (1985) suggested that intercorrelations among the variables higher than .80 should be considered as evidence of high multicollinearity. In this connection, Hair et al. (1995) noted that, no limit has been set that defines high correlations, values exceeding .90 should always be examined, and many times correlation exceeding .80 can be indicative of problems.

Multicollinearity among the independent variables and dependent variables was tested and compared in this study by analysing a bivariate Pearson product-moment correlation. This bivariate correlation analysis indicated that, in most cases, the correlations among all the independent variables and dependent variables were much lower than .80, and none of them reached the limit of .80 recommended by Berry and Feldman (1985) and Hair et al. (1995). Thus, multicollinearity was not a problem at all in this study.

Normality was checked further in this study by looking at linearity and homoscedasticity as these are also important in multivariate analysis. Linearity is important because one of the underlying assumptions of regression analysis is that the relationships between independent and dependent variables are linear, and the

assumption of MANOVA is that there are linear relationships among all pairs of dependent variables. In addition, if there are substantial non-linear relationships, they will be ignored in the analysis as the correlation (Pearson’s r) only captures the linear association between variables, and this in turn will underestimate the actual strength of the relationship (Hair et a., 1995). The assumption of the homoscedasticity is that the variance of the dependent variable is approximately the same at different levels of the independent variables (Tabachnick and Fidell, 2001).

This study examined the linearity for both the purposes of regression and multivariate analysis of variance by looking at the residual plots and scatterplots. Standardised residuals were plotted against predicted values. Most of the residuals were scattered around the zero point and almost had an oval shape, indicating a null plot. Thus, this examination of residual plots suggested that the assumption of linearity was not a problem in this study (Stevens, 1992; Tabachnick and Fidell, 1989). The examination of scatterplots in this study also indicated linear relationships among all the pairs of dependent variables.

An investigation of the homoscedasticity was examined by looking at a scatterplot of predicted values against residuals. The even distribution of residuals indicated homoscedasticity in this study and provided support for the assumption of the multivariate normality. It should be mentioned that the presence of any heteroscedasticity might weaken the analysis but it will not invalidate the results (Tabachnick and Fidell, 1989).

While the above assumptions and tests are applicable for both regression analysis and multivariate analysis of variance, there are two additional assumptions that are required for multivariate analysis.

One of the key assumptions for multivariate analysis of variance is that the variance- covariance matrices within each cell of the design are sampled from the same population variance-covariance matrix and can reasonably be pooled to create a single estimate of error (Tabachnick and Fidell, 1989). They further suggested that, if the within-cell error matrices are heterogeneous, the pooled matrix is misleading as an estimate of error variance.

Tabchnick and Fidell (1989, p. 379) recommended that, “if sample sizes are equal, robustness of significance test is expected; disregard the outcome of Box’s M test”. They further suggested that the robustness is not guaranteed if the sample sizes are unequal and Box’s M test is significant at p < .001. Bernstein et al. (1988) recommended that if the homogeneity of variance is not significant at an alpha level of .001, the researcher should look into univariate tests for homogeneity of variance for each of the dependent variables. It is important to mention that, Box’s M test is a sensitive test of homogeneity of variance-covariance matrices conducted through SPSS MANOVA. The second key assumption relates to the Levene’s test of equality of error variance. Coakes and Steed (2001) noted that, if the Levene’s test of equality of error variance is significant for any variable, the researcher should look into the univariate F- test for that variable in order to check whether or not it is also significant. If it is significant in the univariate F-test then the researcher must interpret this finding at a more conservative alpha level (.05/3 = .017). This adjustment is called ‘Bonferroni-type adjustment’ (Coakes and Steed, 2001).

In the MANOVA procedure in this study, the Box’s M test was not significant at an alpha level of .001; rather, it was significant at an alpha level of .05. Thus, in this study there was no problem with the homogeneity of variance. The Levene’s test of equality of variances indicated that some of the variables were significant and thus, it was necessary to look into the univariate F-test for these variables. This will be discussed in detail in Chapter 8 along with the MANOVA procedure.

After testing of the above assumptions, it can be concluded that, in this study parametric techniques could be used for data analysis. Thus, the techniques of stepwise regression analysis and multivariate analysis of variance were utilised for testing the hypotheses of this study.