## Major Concept of Hierarchical Linear Regression

This type of regression forms a basis for comparing models. Thus, several regression models are created by adding variables to the previous model. Further, the addition of variables into the model is not done by the software like the case of stepwise regression. Hierarchical linear regression plays a key role in determining whether adding independent variables lead to an improvement in the coefficient of determination (Verbeek, 2017). In the paper a three-step hierarchical linear regression is carried out and the results are discussed below.

### Results

The results are presented in the attached hierarchical linear regression.spv file.

### Discussion of Results

The dependent variable is awareness of cultural barriers. Social desirability is entered as the independent variable in the first block. Two independent variables were added in the second block. These are experience in mental health and whether or not the practitioners had received training in multicultural counseling. The three variables that were added in the third block are institutional discrimination, ethnic identity exploration, and collectivism. The correlation results show that there is a weak association between awareness of cultural barriers and the explanatory variables. All the correlation coefficients are less than 0.5. Further, the correlation coefficient for social desirability and experience in mental health are not statistically significant. The results for the first block shows that R square is low at 0.002. The value of F-calculated is 0.870, while the significance level is 0.352.

This implies that the overall regression line is not statistically significant. Further, the t-test shows that social desirability is not statistically significant. In the second block, the value of R-square has improved to 0.030. The ANOVA table shows that F-calculated is 3.84, while significance F is 0.010. This indicates that the overall regression line is statistically significant. From the results of the t-test, it can be deduced that attendance of multicultural workshop is the only statistically significant variable. The other two variables are not statistically significant. In the final block, the value of R square increases further to 0.276. This implies that the explanatory power of the independent variables has improved. From the ANOVA table, the value of F-calculated is 22.902, while the significance level is 0.000. This implies that the regression line is statistically significant. The t-test shows that institutional discrimination, ethnic identity exploration, and collectivism are statistically significant. Addition of variables in the regression blocks increases the explanatory power of the independent variables. However, the coefficient of determination shows that all the independent variables explain only 27.6% of the variations in the dependent variable. In addition, collectivism, institutional discrimination, and ethnic identity exploration have a significant effect on awareness of cultural barriers (Meyers, Gamst & Guarino, 2013).

### Strengths and Weaknesses

A major strength of this type of regression is that enables an analyst to evaluate the impact of adding variables in a regression model. Therefore, it is possible to know the contribution of each independent variable. Further, this technique can be used to examine the influence of an independent variable after controlling for other variables. The control is done by computing the change in the value of the adjusted R square and it is done after each regression. A limitation of this technique is that it can only be used to analyze simple relationships. Also, it is suitable for a small sample size (Wooldridge, 2013).

## Major Concept of Multilevel Modelling

Multilevel modelling makes use of data that have a hierarchical structure. This type of data often arises from longitudinal studies. Therefore, this model introduces practicality that is often ignored in single-level models. Therefore, it disregards the assumptions of uniformity of regression slopes (Heck, Thomas & Tabata, 2013).

## Results

The results are shown in the attached multilevel output.spv file.

### Discussion of Results

The first step in multilevel modelling is to determine if the data set provided have a hierarchical structure. This will be achieved by evaluating if there are significant differences among the levels of dealership. Therefore, goodness-of-fit will be measured using the -2 Restricted Log Likelihood technique. The estimated value is 7397.059 and it will be a basis of comparison for other models. From the results, the value of variances in sales that is accounted for by the dealership is 21% (133.86 / (495.74 + 133.86). The value is large enough and it can be interpreted that there is a clustering effect. This implies that multilevel models can be developed using the data. The second step in multilevel modelling entails centering experience which is the only covariate that is provided in the data. This will be accomplished in two levels. The first level entails centering of individual experience scores.

The second level entails centering of the dealership experience scores. Once the two levels of centering are completed, then the boundary between dealership 1 and dealership 2 will be visible. The resulting deviation values for dealership 1 and 2 are -2.23 and -0.19 respectively. The final step entails building multilevel models. This will be achieved by starting with simple and moving to more intricate models. The modelling will be carried out in four stages. In the first model, the value of -2 Restricted Log Likelihood has improved to 7367.343. Therefore, 29.716 units have been lost. This signifies an improvement in goodness of fit. Further, this improvement is statistically significant. The result of the first model further indicates that years of experience in dealership are a significant factor in explaining differences in the dealership. In the second model, -2 Restricted Log Likelihood has improved further to 7338.059. It shows an improvement in goodness of fit because 59 units have been lost.

Further, the results show that the variance of the intercepts is statistically significant and it implies that the mean of the 20 dealerships varies significantly. Secondly, the variance of the slopes is not statistically significant. This implies that the slopes of the 20 dealerships are analogous. Finally, the covariance between the slopes and intercepts is not statistically significant. It implies that the association between slope and intercept is constant across the 20 dealerships. In the third model, the -2 Restricted Log Likelihood has improved further to 7300.958. The results further show that adding sex into the analysis does not change the result for slope, intercept and the interaction of the two from what was observed in the second model. However, it is established that females generated about 14.81 thousand of sales more than the males. In the fourth model, the goodness of fit has improved as indicated by the lower value of the -2 Restricted Log Likelihood (7281.487). Further, the results show that there is no association between sales and dealership experience, controlling for sex and collaboration of sex and group experience (Meyers et al., 2013).

### Strengths and Weaknesses

The first strength of multilevel modelling is that it has the ability of estimating the predictive effect of an explanatory variable and the mean of its group level. This makes it possible to evaluate the direct and contextual effects of the explanatory variable. In addition, the tool gives more accurate predictions than the single-level modelling techniques. This makes it a better statistical tool for analysis than multiple regression analysis. It is worth mentioning that the standard errors are understated while statistical significance is exaggerated in a multiple regression analysis. The first limitation of multilevel modelling is that the use of specific averages of the variables as the group-level mean can be construed as contextual effects. This can lead to distorted conclusions and an analyst can find contextual effects where they do not exist. Another drawback of this tool is that reasonable analysis can be carried out only using a realistically large number of higher-level units (Gujarati, 2014).

## References

Gujarati, D. (2014). *Econometrics by example*. New York, NY: Macmillan Publishers Limited.

Heck, R. H., Thomas, S. L., & Tabata, L. N. (2013). *Multilevel modeling of categorial outcome using IBM SPSS*. New York, NY: Taylor & Francis Group.

Meyers, L. S., Gamst, G. C., & Guarino, A. J. (2013). *Performing data analysis using IBM SPSS*. New Jersey, NJ: John Wiley & Sons, Inc.

Verbeek, M. (2017). *A guide to modern econometrics*. New Jersey, NJ: John Wiley & Sons, Inc.

Wooldridge, J. M. (2013). *Introductory econometrics: A modern approach*. Mason, OH: South-Cengage Learning.