Regression Assumptions - Correlations Between the Dependent Variable - Statistics Assignment Help

Assignment Task:

The first rule of data is to look at the data. If outliers (subject or unit of analysis that has extreme values on a variable) are identified, it should first be determined if these occurred due to data entry errors or not. If data entry error was not the case, then it should be determined whether or not that data point should be deleted or not. There should not be any deletions in this dataset.

Yes, multiple linear regression is appropriate in the context of the research question and the data. There is only one continuous dependent variable (injury) and multiple independent variables (strength measures of different muscle groups such as quads, glutes, abs, etc.) that have a linear relationship with the dependent variable. 

Regression assumptions are important to test because they can be seen as prerequisites that should be met prior to running the analysis. If the assumptions are not met, p-values and parameter estimates are likely to be incorrect, and thus the significance of your results will be questioned. 

In order to see if relevant variables have been included and irrelevant variables have been excluded, the multivariate correlation must be done to test the correlations between the dependent variable (injuries) and the predictor variables. After this test is done, the Pearson correlations with the injuries variables must be examined. The pearson correlations in Figure 1 show that strength variables all have significant negative correlations with the injuries variable, with glutes having the strongest negative correlation (r = -0.3929) and grip having the weakest negative correlation (r = -0.0988). However, the correlation between quads and injury is not significant, nor is the correlation between grip and injury. Thus, quads and grip will be excluded as variables.

With an N of 100, the requirements for the assumption of sample size appropriateness have been met. With 3 predictors (2 variables being excluded), the rule of 20 cases per predictor requires at least 60 cases, and 100 exceeds this amount. If this were violated, an obvious alternative would be to increase the sample size. In the event of this not being possible, a nonparametric test could also be run. Additionally, reducing the number of predictor variables could increase the power of the analysis, although it would not typically be advisable to remove variables just to produce a more significant effect.

As seen in the scatterplots (Figure 2), there is a clear linear relationship between injury and each of the predictor variables, with the most pronounced linear relationship being between injury and glutes. 

A. Homoscedasticity refers to if variances of the residuals remain constant over the dependent variable. If the variances are not constant, the standard errors, confidence intervals, and significance tests will be incorrect. 

B. After plotting the residuals (Figure 3), the plots for each independent variable showing residuals by predicted plot show the scatterplots distributed around the 0 residual value. The “noise” in the relationship between the independent variables and the dependent variable is the same across all values of the independent variable. Therefore, this assumption is not violated.

A. Multicollinearity means that two or more of the predictor variables are highly correlated. This can be checked by checking bivariate relationships between the independent variables, looking at the VIF, or looking at tolerance.

B. The VIF values for glutes, abs, and arms are 1.425, 1.312, and 1.130 respectively (Figure 4). As these values are near 1, the assumption has not been violated.

A. Normality of residuals is an assumption stating that the residuals of the regression should follow a normal distribution. It can be tested using the histogram overlay method or the normal q-q plot method.

B. According to the normal q-q plot method, the distribution is normal if the plot forms a straight line. As seen in the Figure 5, the vast majority of the points follow the diagonal line, and so this assumption is not violated.

A. The difference is that regression outliers are extreme values found within the regression itself and therefore affect the slope of the regression, whereas general outliers deal only with extreme values within the totality of values for one specific variable (univariate). 

B. The three types of regression outliers that need to be watched out for are leverage, discrepancy, and influence. A leverage outlier can be detected using a bivariate scatterplot. A leverage outlier can be identified if it would change the mean of the independent variable it is an outlier on. Discrepancy can be detected by calculating studentized residuals, with values > 2.6 generally being cause for concern. Influence can be detected by looking at the Cook’s D value (conventional cut-off point is 4/n). As seen in Figure 6, along with row 76 and columns 10-13 of the dataset itself, ID number 76 violates the leverage parameters, the discrepancy parameters (value of -2.66, which is > 2.6), and the influence parameters (value of 0.09, which is > 0.04). 

C. First, I would recommend trying to investigate this further to make sure that data entry was accurate. Assuming that the data entry was correct, I believe that this outlier (ID #76) should be excluded, on the basis that it violates the parameters for every type of regression outlier. Additionally, given that the sample size is not especially large, excluding it could have an effect on the analysis.

This Statistics Assignment has been solved by our Statistics Experts at My Uni Paper. Our Assignment Writing Experts are efficient to provide a fresh solution to this question. We are serving more than 10000+ Students in Australia, UK & US by helping them to score HD in their academics. Our experts are well trained to follow all marking rubrics & referencing style.

Be it a used or new solution, the quality of the work submitted by our assignment experts remains unhampered. You may continue to expect the same or even better quality with the used and new assignment solution files respectively. There’s one thing to be noticed that you could choose one between the two and acquire an HD either way. You could choose a new assignment solution file to get yourself an exclusive, plagiarism (with free Turnitin file), expert quality assignment or order an old solution file that was considered worthy of the highest distinction.