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Introduction

To further analyze and understand the customers’ buying habits, Diligent Consulting Group was requested to analyze the factors that influence the Annual Amount Spent on Organic Food. The first three reports provided a better understanding of customers’ buying habits through simple linear regression analysis. This report further reinforces the purpose of Loving Organic Foods by interpreting additional information, such as explanatory variables, to explain customer motives.

A Brief Comparison of Simple Linear Regression and Multiple Linear Regression

Simple linear regression is used to identify the interdependence of two variables. Alexander et al. (2017) note that linear regression for two variables is based on a linear equation with one independent variable. The equation has the form:

Y = a + b * x, where a and b are constant numbers.

Multiple linear regression is the analysis of the relationship between several independent variables and the dependent variable. In multiple linear regression, the prerequisites of regression analysis and its implementation are completely the same as simple linear regression. A feature of multiple regression is the correlation of independent variables. This method corresponds to the simultaneous processing of all independent variables selected for analysis.

According to Alexander et al. (2017), the multiple linear regression model can be stated by the equation:

y_i = β₀+ β₁X_1i+ β₂X_{2 i}+ ⋯ +β_kX_ki+ ε_i,

where β₀ is the intercept, β_i‘s are the slope between Y and the appropriate X_i, and ε is the error term that captures errors in measurement of Y and the effect on Y of any variables missing from the equation that would contribute to explaining variations in Y.

Therefore, multiple linear regression is one of the most common analyzes because, as a rule, several factors simultaneously influence the final result.

The regression output generated in Excel

Regression Statistics
Multiple R		0, 830460276
R Square		0, 68966427
Adjusted R Square		0, 679232817
Standard Error		2111, 587193
Observations		124

ANOVA
	df			SS		MS		F		Significance F
Regression	4			1179155134		294788783, 6		66, 11392127		2, 44119E-29
Residual	119			530597256, 6		4458800, 476
Total	123			1709752391
	Coefficients		Standard Error		t Stat		P-value		Lower 95%		Upper 95%	Lower 95,0%	Upper 95,0%
Intercept	-1932, 108684		978,5413609		-1, 974478302		0,050643489		-3869,718312		5,500943605	-3869, 718312	5,500943605
Age	14,12302239		11,77664046		1,199240346		0,232816957		-9,195901943		37,44194671	-9,195901943	37,44194671
Annual Income	0,016677283		0,002635295		6,328431		4,54E-09		0,011459136		0,021895431	0,011459136	0,021895431
Number of People in Household	2222,506847		153,2477882		14,50270097		4,52E-28		1919,06092		2525,952773	1919,06092	2525,952773
Gender (0 = Male; 1 = Female)	40,50058317		384,711882		0,105275103		0,916334763		-721,2673668		802,2685331	-721,2673668	802,2685331

Interpretation of the coefficient of determination (r-squared) and the global test for statistical significance (the F-test)

Based on Regression Statistics, the following interpretation can be made. The coefficient of determination (r-squared) is equal to 69 % of the total spread in the Annual Amount Spent on Organic Food.

H₀: none of the beta estimates are statistically significant
H_A: at least one of the beta estimates is statistically significant

When considering ANOVA analysis, the global test for statistical significance (the F-test) reflects the p-value. Based on the indicator 2, 44119367422295E-29, we can reject H₀ and accept H_A. Thus, the overall regression equation is statistically significant.

Interpretation of the coefficient estimates for all the independent variables

Analysis of the coefficient estimate shows that when Age increases by one unit and other independent variables remain fixed, the Annual Amount Spent on Organic Food grows by 14, 12 units. In turn, with an increase in Annual Income by one unit, and other independent variables remain unchanged, the Annual Amount Spent on Organic Food rises by 0, 02 units. Moreover, when the Number of People in Household increases by one unit and the other independent variables remain fixed, the Annual Amount Spent on Organic Food enhances by 2222, 51 units. When Male customers and other independent variables remain fixed, the Annual Amount Spent on Organic Food is negative 1932 units. When Female customers and other independent variables remain fixed, the Annual Amount Spent on Organic Food is negative 1892 units (coef Gender – coef Intercept).

Interpretation of the statistical significance of the coefficient estimates for all the independent variables

H₀: B=0 (coefficient estimate is not statistically significant)
H_A: B is not equal to 0 (coefficient estimate is statistically significant)

Since the p-values correspond to Age and Gender are 0, 23 and 0, 92, respectively, this indicates their insignificant presence in this regression model.

In turn, the p-values corresponding to the Number of People in Household and Annual Income are 4.54 and 4.52, respectively, so the Number of People in Household and Annual Income is significantly present in this model.

The regression equation with estimates substituted into the equation

Annual Amount Spent = -1932 + 14 * Age +0.02 * Annual Income + 2223 * Number of People in Household + 41 * Gender.

An estimate of “Annual Amount Spent on Organic Food” for the average consumer

	Mean
Age	48,23387097
Annual Income	161006,621
Number of People in Household	4,314516129
Gender (0 = Male; 1 = Female)	0,572580645

“Annual Amount Spent on Organic Food” for the average consumer = -1932 + 14 * 48,23 + 0.02 * 161006,62 + 2223 * 4,31 + 41 * 0,57 = 11567,85.

A discussion of whether or not the coefficient estimate on the Age variable in this estimation is different than it was in the simple linear regression model from Module 3 Case

		Coefficients
Module 4	Age	14,12
Module 3	Age	26, 29

The coefficient of the independent variable (Age) in the simple linear regression equation is different from the coefficient of the corresponding variable in the multiple regression equation. Olive (2017) asserts that this is since the influence of all other features taken into account in this equation is excluded in the linear regression equation.

References

Alexander, H., Illowsky, B., & Dean, S. (2017). Introductory business statistics. Openstax. Web.

Olive, D.J. (2017). Multiple Linear Regression. In: Linear Regression. Springer, Cham. Web.

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