Muath -

This is not a subject matter field for me so I'm not familiar with your variables, and I am not entirely clear that scores you are averaging are equivalent, and thus can be meaningfully 'averaged.' Thus, the scoring and averaging being done sounds a bit arbitrary, though the major categories are clear.

At any rate, with regard to linear regression assumptions, I think your primary concern is the degree of heteroscedasticity. Normality is with regard to the estimated residuals, or rather the random factors of the estimated residuals in weighted least squares regression. Normality there is desirable, but not essential. All structure of interest can, I think, be examined by looking at a graphical residual analysis. Plotting predicted y values on the x-axis, and estimated residuals on the y-axis helps examine model fit, and is the first step in examining the degree of heteroscedasticity, which is a natural phenomenon where larger predicted values are associated with larger sigma for the estimated residuals. Note also that to avoid overfitting to your specific sample, a cross-validation of some kind might help.

So, you could start with a "graphical residual analysis." You could research that term for information on making assessments from such scatterplots.

For heteroscedasticity, there are several examples demonstrating how to determine a reasonable coefficient of heteroscedasticity, which are discussed in the following:

https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity

The following tool can be used to do this with your data, and find the regression weights. Hopefully your software will let you enter regression weights. The idea is that the regression coefficients will have been found by minimizing the sum of weighted squared estimated residuals, with respect to those regression coefficients, with equations solved simultaneously.

https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx

Cheers - Jim

If you are going to do analysis with tools which holds the assumptions of normality, it is better to check so. The first two questions can be answered using correlation and regression.

Hello Muath,

The assumptions for hypothesis tests with categorical explanatory (independent) variables (ANOVA) are that the response (dependent) variable is distributed normally with homogeneous and independent residuals. So there is no need to check the explanatory (independent) variables.

For the response (dependent) variable the assumptions for the general linear model (including regression, ANOVA, and ANCOVA) are that the errors (residuals) are normal and homogeneous (Eisenhart 1947, Seber 1966, Neter et al 1983 pp 31& 49, Quinn and Keogh 2002 pp 110 & 280). Evaluation of assumptions in advanced texts, where it occurs, often entails a residual versus fit plot (for homogeneity) and a normal score plot or Quantile-Quantile plot (for normality of the residuals).

The statistical literature warns against statistical tests to evaluate assumptions and advocates graphical tools (Montgomery & Peck 1992; Draper & Smith 1998, Quinn & Keough 2002). Läärä (2009) gives several reasons for not applying preliminary tests for normality, including: most statistical techniques based on normal errors are robust against violation; for larger data sets the central limit theory implies approximate normality; for small samples the power of the tests is low; and for larger data sets the tests are sensitive to small deviations (contradicting the central limit theory).

That preliminary evaluation is insufficient can come as a surprise, as can advice against statistical tests to evaluate assumptions. But I think best practice, as in the literature, needs to be said.

To check assumptions for residuals see:

https://www.datanovia.com/en/lessons/anova-in-r/#check-assumptions-1

Good luck with your research,

~David S

Chatfield, C. 1995. Problem Solving: A statistician's guide, Second edition (Chapman & Hall/CRC Texts in Statistical Science) 2nd Edition,

Draper, NR and H. Smith 1998 Applied Regression Analysis, 3rd Edition. Wiley

Eisenhart, C. 1947. Biometrics 3:1-21

Gelman, A C Pasarica & R. Dodhia (2002) Let's Practice What We Preach, The American Statistician, 56:2, 121-130, DOI: 10.1198/000313002317572790

Läärä, E. 2009. Statistics: reasoning on uncertainty, and the insignificance of testing null. — Ann. Zool. Fennici 46: 138–157.

Montgomery, DC EA Peck, GG Vining 2012 Introduction to Linear Regression Analysis, 5th Edition. Wiley