# Multiple Regression Analysis Research Paper

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Updated: May 15th, 2019

## Introduction

Multiple regression analysis is one of the statistical tools normally being applied in business decision-making processes. Organizations’ managers often use multiple regression analysis in the circumstances where the outcome to be predicted contain more than two variables. In other words, multiple regression analysis is used to predict an outcome whose predicting variables are two or more (Black 312).

In fact, in real business situation, decisions normally involve consideration of several variables. The suppositions in simple linear regression are also applicable in multiple regressions. Even though hypothetical multiple regression analysis might seem simpler, in the real world scenario the procedure is often complex depending on the number of variables used to predict the outcome (Black 315).

However, the concerns of multiple regressions are the assessments of the influence of several variables that in contemporary researches is known as the predictors on criterion, which is the expected outcome (Black 310). The aim of the multiple regressions is to determine the level of percentage of influence the predictor variables have on the dependent variables.

In real business situations, multiple regressions can be applied in the examination of outcomes from varied variables of connecting hypothetical model. Businesses outcomes are normally varied ranging from the production outcome to sales performance. Within the continuum are the profits or returns on investments. All the factors that determine these outcomes are the variables and several factors often determine a single outcome (Groebner 43).

Business managers are often faced with difficulties in deciding the best factor combination that will result in maximum output. For instance, a firm may utilize the multiple regressions in the determination of whether the individual job performances are influenced by aggression and friendly or outgoing behaviors. Therefore, appropriate statistical tool that takes into consideration all the factor variables is required for decision-making processes.

The paper tends to demonstrate how the multiple regressions as a statistical tool are applied by businesses to analyze data to make meaningful decisions. The paper examines the application of regression analysis in three parts. The first part provide decision-making problem that are normally experienced by businesses and offer procedures through which regression analysis is utilized to solve such dilemmas.

The second part examines the literature on how regression is applied in real business situation. The final part tend to examine the effects of violating the ethical procedures in regression analysis such as avoiding the assumptions being made while applying the multiple regression technique in business decision-making process.

## A case involving decisions dilemma in relation to multiple regression

There are various situations where multiple regressions are applied in real business decision-making process. In most cases, the circumstances involve the determination of the correlation existing between the expected outcome and the independent variables. In multiple regressions, the outcomes are determined by several independent variables (Groebner 34).

However, this case involves ABC Corporation wanting to determine whether the employees’ performances in sales are influenced by their intelligence and extraversion. ABC limited is establishing a new retail sales store in one of the various outlets. The company wants to stock the retail outlet with employees that have high chances of succeeding in sales. To attain the objective, the company has to come up with strategies and measures that would increase the chances of getting the required quality in the employees.

In fact, the company decided to examine the intelligence and extraversion among the existing sales staff in one of their stores. The company predicts that friendly and outgoing character is fundamental to succeed in sales. In other words, the high sales performances directly correlate to the intelligence and the extraversion. The problem is how to determine the correlations and the way the organization can come up with the best employment decision.

From the examination of existing sales executives, the company realized that extraversion characteristics, which were described as being friendly and outgoing, determined success in sales.

Moreover, the firm found out that more intelligent worker had increased performances in their sales outcomes. The intelligence and extraversion in employees are directly related to the sales outcomes. In essence, the connection between the two-predictor variables and the expected outcome can be used to forecast the required outcome of the sales performances in employees.

Therefore, the company decided to measure the intelligence and extraversion of new employees to determine their future performances in selling the products. However, studying the intelligence and extraversion is tricky to the company top management. The decision to hire the right people requires a careful study that utilizes the regression analysis. Since the outcome, which are sales is determined by two variables, intelligence and extraversion, the analysis requires multiple regressions.

The company decided to test the employees’ psychology. The psychological study was intended to assess the needed predictor variables in presented sales executives. The final data for the employees will have scores that contain the intelligence score with 50 scores at low level and 150 at the highest scores. The second score contain the extraversion that range between the scales of 15 to 30. The final score, which is the outcome or the sales performance, is quantified in terms of average per dollar that the employees sold per week.

In this case, intelligence and the extraversion are the independent (predictor) variables while sales performance is the dependent (criterion) variable (Weiers 111). The predictor variables are used to forecast the criterion. In other words, the criterion depends on the changes of the predictors.

To come to the final decision, the corporate managers can analyze the data through examining the correlation between each independent variable and the dependent variable. Put differently, the relationship between performance and each of the independent variables can be analyzed separately. The other option is the situation where the relationship between the workers’ performances and the independent variables are analyzed simultaneously (Weiers 112). In essence, multiple regression analysis is applied.

Using the first option where the relationship between the sales performance and the independent variables are analyzed separately, the bivariate correlation between the sales performance and the intelligence R = 0.33 while the bivariate correlation between the sales performance and the extraversion R = 0.55.

In both cases, the correlation is positive indicating strong relationship between these variables and the expected outcome. In essence, there is increase in the sales performances as the workers in the stores become increasingly intellectual and vociferous. The results indicated above can be proved through the application of scatter graphs of the found data for each variable.

Applying the multiple regressions to analyze the correlation of both predictors and the criterion, the equation to represent the above correlation is indicated as Y = α + Ix +Ex where, in this case, Y represent the predicted sales performances, α represent a constant value when both the independent variables are equal to zero. I represent the independent variable intelligence and E represents the extraversion. Using the values as shown in the appendix, the derived equation is (forecasted sales outcome =993.93 + 8.22I + 49.71E.

Once the formula has been derived, it would be easier for the managers to envisage the sales outcomes of the new employees. Through the application of the derived multiple regression formula it would be easier for the management to decide on the right employee with the highest outcome to be employed.

In other words, each applicant employees’ test scores are substituted in the equation to predict the sales performances outcome. As indicated in the appendix, the hypothetical sales job applicants’ test scores are replaced in the equation and the managers obviously decides on the second applicant that have the highest predicted sales outcome.

The decision dilemma involving hiring the right sales executive in the newly created sales outlet of the company has been resolved through the application of the multiple regression analysis. In essence, the multiple regression analysis helps in determining the outcomes of the two or more independent variables.

The decisions involving the correlation between two or more independent variables and the expected outcome are made once the multiple regressions have shown strong and positive relationship. Moreover, in this case, the multiple regressions of the two variables and the expected outcome are derived from the simple linear regressions where the procedures have been followed. The correlations Rs are the major predictors and critical in determining the correlation coefficient of the variables.

## Reviewed literature on multiple regression analysis

Multiple regressions are imperative statistical methods that aid in the prediction of an individual’s gain on single parameter basing the scores on other diverse variables. For instance, in forecasting the amount of fulfillment that the employees achieve in the workplace, a number of variables are considered invaluable.

In this regard, variables including remunerations, academic backgrounds, gender, age as well as the period that individuals take on full time employment are all taken into account. In fact, gathering information on the variables through surveying a number of individuals, the organization is able to forecast correctly the variables that accurately influence the level of employment satisfaction (Weiers 112).

Generally, researchers often utilize multiple regressions in predicting decisive factor variables. Of more importance, multiple regressions emphasize on the assessment of at least two predictors of decisive factors. The justification for such examination is to show the effect of including extra predictor variables on the forecast of the outcome variable. Often, the inclusion of additional predictor variables has the effect of increasing the calculation of the resultant variable.

In addition, multiple regressions are significant in testing the conjectural causal paradigms that have assorted results. Indeed, job performance, heart ailments as well as antagonistic behaviors are just a few examples of theoretical causal prototypes in which the utilization of multiple regressions is useful.

In essence, these outcomes are influenced by the independent variables. Therefore, considering job performance, remunerations as well as the category of profession are significant predictor variables that influence the job output among the organization’s employees (Black 314). Further, multiple regressions encompasses correlation of variables. The possibility of predicting the score of a single variable when an individual knows the value of the other variable depends on the correlation existing between the models.

Based on this principle, the stronger the connection, the forecast becomes more precise since the outcomes plummet on the regression path. The application of this theory is useful in the forecast of human emotions and thoughts due the availability of different predictor variables.

On the analysis of variance, the prediction of the dependent variables such human action is often difficult. Therefore, multiple regressions report the variances based on the observations. For example, in the determination of the factors that greatly influence employee performance at the workplace, it may be probably true that remuneration factor accounts for a higher proportion in predicting an individual’s job performance.

In essence, having the knowledge of a person’s salary is vital in the prediction of occupation output. In other words, multiple regressions measure the logically happening levels of the independent variables and use them to forecast the criterion variable scores as opposed to other models such as ANOVA (Black 328). Since other models such as ANOVA cannot provide clear solution due to the unequivocal and controlled procedures of the hypotheses, the ratio of controlled predictor variables explains the differences.

The statistical method of multiple regressions is utilized in diverse operations. To begin with, in the examination straight-line correlations between the independent and the decisive factor variables the technique is used. Secondly, the technique uses the interval as well as the ratio scales in the prediction of the criterion variable (Groebner 128).

Conversely, the independent variables are calculated and the answers are provided in all statistical measurements. Further, dichotomous nominal independent variables such as sexual category are allowed since there are only two groups that are male and female. In carrying out multiple regressions, the amount of decisive factors often surpasses the quantity of predictor variables extensively. In fact, 10:1 proportion is extra conventional.

In conducting multiple regressions, the standardized regression coefficients known as beta that is measured in standard deviation units quantify the strength of the predictor variable on the criterion variables. For example, a beta quantity of 3.1 translates into a 3.1 standard deviation change in the independent variable results from a unit change in the standard deviation of the predictor variable.

In essence, lower quantities of beta exhibits the low impact of the independent variables on the dependent variables (Groebner 124). Moreover, the presence of a single independent variable in a model equates beta to the correlation coefficient between the forecasting and the criterion variables.

In general, the beta regression coefficient is imperative in coming up with comparisons that aid in the evaluation of strength of correlation between the independent and the dependent variables. Additionally, in the determination of correlation between the variables, the R, R2 and adjusted R2 measures are used. R computes the link that exists between the experiential value and the forecasted value of the criterion variable.

According to the regression model, the deviation proportion of the criterion variables is determined by the R2. Moreover, the adjusted R2 measures the correlation between the variables by taking into account both the number of variables in the regression model as well as the quantity of annotations the model is founded.

## Ethical issues in violation of assumptions in statistics

The application of professionalism in analyzing statistical data has greater significance in influencing major portions of humanity. For instance, the utilization of statistical information in business researches is important in determining and analyzing the performance as well ensuring efficiency among the workers. For this reason, early detection of inefficiencies in the organization depends on the use of sound statistical techniques.

Further, effective management of any organization and economy in general is based on the accessibility of consistent, accurate and correctly construed statistical data (Groebner 57). Therefore, to increase the significance in statistical operations, the protocols used in the analysis must be followed. However, due to the regular changes in the dynamically evolving statistical methodology, researchers have often predetermined statistical results thus violating various assumptions used in analyzing data.

To ensure legitimate interpretations of statistical methods, meeting the set suppositions is critical. However, data do not satisfy most of the stipulated statistical assumptions. A number of reasons have been put forward to explain occurrence.

First, in most occasions, simply the statistics that meet the set requirements are used thereby ignoring the verification of the violations (Black 114). Analysis of statistical data involves several techniques. For instance, the t-procedure, ANOVA as well as regression techniques are some of the procedures of great importance in the operations and evaluations involving statistical data.

Actually, the violation of statistical assumptions have great impact on the type 1 as well as type 2 errors that have the effect of undervaluing and overestimation of the inferential dimensions. In addition, such errors affect the extent of statistical outcomes. In essence, the regular application of conventional techniques without taking into account the related postulations leads to non-replicable outcomes (Black 114).

Further, statisticians prohibit the application of statistical procedures without making inherent hypotheses. In other words, the utilization of implicit assumptions is an ingredient for valid results. Therefore, the procedures used by scholars in the operations must support the analysis of statistical data as well as the sturdiness to the contravention of the techniques.

The contraventions of statistical assumptions have led to the development of overcoming the problems linked to hypothesis violation. For instance, statistical tests such as Levene’s test have been proposed to check for the violations. However, the methods continue to receive oppositions from a number of researchers who argue that the tests are unsuitable to apply before choosing the efficient technique of analysis since such tests composite the likelihood of incurring type 1 error (Black 115).

More importantly, failure to meet the statistical assumptions causes a number of shortcomings. For example, in dealing with the population samples, the population is also often defined though not known. Under such circumstances, the homogeneity inconsistency postulation cannot be determined. The assumption argues that the two population variances should be the same.

Given the strict way under which the postulations are defined, statisticians and scholars have developed a tendency of using samples that slightly contravene the hypotheses. For example, the assumptions of normality and homogeneity of variances are inevitable for the t-tests, ANOVA together with regression.

Nevertheless, researchers continue to contravene such assumptions. Regarding one-way ANOVA, the use of unequal group sizes is a normal violation that influences the procedure negatively. Graphical presentations through normal quartile plots have been developed for checking violation of normality assumptions. Further, preliminary tests are imperative in deciding the best techniques to utilize in analyzing statistical data. However, the preliminary test has the effect of blowing up the likelihood of incurring type one errors.

## Works Cited

Black, Ken. Business Statistics: For Contemporary Decision Making. Hoboken, NJ: Wiley Global Education, 2011. Print.

Groebner, David. Business Statistics: A Decision-making Approach. London, UK: Pearson Education, 2011. Print.

Weiers, Ronald. Introduction to Business Statistics. Farmington Hills, MI: Cengage Learning, 2010. Print.

## Appendices

### Appendix 1: An hypothetical data from the employees test scores

 Sales person Intelligence Extroversion Sales per week 1 89 21 2625 2 93 24 2700 3 91 21 3100 4 122 23 3150 5 115 27 3175 6 100 18 3100 7 98 19 2700 8 105 16 2475 9 112 23 3625 10 109 28 3525 11 130 20 3225 12 104 25 3450 13 104 20 2425 14 111 26 3025 15 97 28 3625 16 115 29 2750 17 113 25 3150 18 88 23 2600 19 108 19 2525 20 101 16 2650

### Appendix 2: Predicted weekly sales for three hypothetical applicants J, N and B

 Name Reg Const Intelligence IQ score Extroversion Ext score Wkly sales J 993.93 + 8.22 85 + 49.71 = 24 2886 N 993.93 + 8.22 127 + 49.71 = 27 3380 B 993.93 + 8.22 103 + 49.71 = 19 2785
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