The Sales Prediction of Abctronics’ based on the Multi-factorial

Linear Model in Terms of Variance Inflation Factor

Zheming Cao

1,† a

and Sheng Luo

2,† b

Art & Science, University of Toronto, Ontario, Canada

Rutgers Business School, Rutgers University, New Brunswick, U.S.A.

sl1558@scarletmail.rutgers.edu

†

These authors contributed equally

Keywords: Multi-Factorial Linear Model, Inflation Factor, Linear Regression.

Abstract: To make the prediction of ABCtronics’ sales, the multi-factorial linear model is used. In terms of variance

inflation factor, a linear regression model can be set up by Minitab for us to analyze and predict the

ABCtronics’ sales. Based on the approach, the estimated multiple regression equation is derived for sales

volume with respect to total market demand, price per chip, and economic situation. According to the analysis,

the total market variable is not statistically significant at the 5% significance level. These results shed light

on processing a more precise prediction by using multi-factorial linear model in terms of variance inflation

factor.

1 INTRODUCTION

ABCtronics is looking for avenues to improve its

manufacturing as well as quality control processes. In

a crucial quarterly review meeting, Jim Morris, the

chief operating officer (COO) of the Chip

Manufacturing Unit, is meeting with the quality

control, manufacturing, and marketing team to decide

the way forward (Irwin, Mcclelland, 2001). The case

provides the relevant data on the plant performance,

detailed description of its operating procedures,

quality control policies and issues currently faced at a

client site. In order to offer a precise prediction of

ABCtronics’ sales, we will use the multifactorial

linear regression and variance inflation factors to set

up and analysis the model. Some scholars have

discussed the multicollinearity illusion in moderated

regression analysis (Disatnik, Sivan, 2016). Numerous

papers in the fields of marketing and consumer

behavior that utilize moderated multiple regression

express concerns regarding the existence of a

multicollinearity problem in their analyses. In most

cases, however, as show in this paper, the perceived

multicollinearity problem is merely an illusion that

arises from misinterpreting high correlations between

https://orcid.org/ 0000-0002-4671-0927

https://orcid.org/ 0000-0002-6721-0200

independent variables and interaction terms (Becker,

Ringle, Sarstedt, Völckner, 2015) and a small sample

performance of the Wald test in the sample selection

model under the multicollinearity problem (Andrews,

Brusco, Currim, Davis, 2010). This paper reviews and

extends the literature on the finite sample behavior of

tests for sample selection bias. According to previous

analysis, the standard regression-based t-test and the

asymptotically efficient Lagrange Multiplier test, are

robust to nonnormality but have very little power

(Kalnins, 2018). There are some miss leadings of

multiple regression models were discussed.

Moderated multiple regression models allow the

simple relationship between the dependent variable

and an independent variable to depend on the level of

another independent variable. The moderated

relationship, often referred to as the interaction, is

modeled by including a product term as an additional

independent variable. Multiple regression models not

including a product term are widely used and well

understood. It is argued that researchers have derived

from this simpler type of multiple regression several

data analysis heuristics that, when inappropriately

generalized to moderated multiple regression, can

result in faulty interpretations of model coefficients

and incorrect statistical analyses.

Cao, Z. and Luo, S.

The Sales Prediction of ABCtronics’ based on the Multi-factorial Linear Model in Terms of Variance Inﬂation Factor.

DOI: 10.5220/0011194700003440

In Proceedings of the International Conference on Big Data Economy and Digital Management (BDEDM 2022), pages 587-591

ISBN: 978-989-758-593-7

587

Based on the theoretical arguments and

constructed data sets, previous literature has discussed

heuristics, discuss how they may easily be misapplied,

and suggest some good practices for estimating,

testing, and interpreting regression models that

include moderated relationships (Yamagata, 2006). In

order to figure out the collinearity impacts on mixture

regression result, we collect some statement form

Becker and Ringle. Mixture regression models are an

important method for uncovering unobserved

heterogeneity. A fundamental challenge in their

application relates to the identification of the

appropriate number of segments to retain from the

data. Prior research has provided several simulation

studies that compare the performance of different

segment retention criteria.

Although collinearity be- tween the predictor

variables is a common phenomenon in regression

models, its effect on the performance of these criteria

has not been analyzed thus far. We address this gap in

research by examining the performance of segment

retention criteria in mixture regression models

characterized by systematically increased collinearity

levels. The results have fundamental implications and

provide guidance for using mixture regression models

in empirical (marketing) studies (Yamagata, Orme,

2005). Based on some of their theories, we will use

multi-factorial linear regression and variance inflation

factors to predict ABCtroncs’ sales. First, we will

calculate date using Minitab and displayed in the

original formula, e.g., multiple linear regression.

Subsequently, a regression model is applied for

ABCtronics’ sales and carry out the analysis based on

the software Minitab. The regression model test is

widely available and has been used in many

investigational studies. Traditionally, simple linear

regression has been assessed by measuring.

Subsequently, the reasons attributed to the prediction

of ABCtronics are analyzed and the approach to

predict it will also be demonstrated. Finally, the future

usage of the opinions in the paper is demonstrated as

well as the application.

2 DATA & METHOD

The research method is a simple linear regression

model for ABCtronics’ sales based on the analysis

software Minitab. The regression model test is widely

available and has been used in many investigational

studies. Traditionally, a simple linear regression has

been assessed by measuring.

𝑌=𝛽



+𝛽



𝑋 + 𝜀,𝜀~ 𝑁𝑜𝑟𝑚𝑎𝑙

(

0, 𝜎



)

(

)

We include the expected term with a normal

distribution to improve the estimate. The simple

linear regression model only explains the slight

change in sales volume, which makes us doubt

whether the model is enough to be a good forecast.

Multiple regression models that do not include

product terms are widely used and understood.

Scholars believe that this simple multiple regression

led to some data analysis heuristics. Moderately

improper expansion of these heuristics in multiple

regressions can lead to misinterpretation of model

coefficients and inaccurate statistical analysis. In this

case, it is necessary to consider multiple regression

models. The main difference between simple linear

regression and multiple linear regression is the

number of independent variables. We decided first to

use all three variables to run the multiple regression

models X1, X2, X3 with the following regression

equation:

𝑌=𝛽



+𝛽



𝑋



+𝛽



𝑋



+𝛽



𝑋



+𝜀,𝜀~ 𝑁𝑜𝑟𝑚𝑎𝑙

(

0, 𝜎



)(

)

Becker et al. claimed that it is usually assumed

that the regression coefficients are constant (Becker,

Ringle, Sarstedt, Völckner 2015). Besides, the same

regression equation is sufficient to describe all

members of the population. Under normal

circumstances, this assumption of similar data

structures does not hold. If not fully explained, non-

uniformity can lead to inaccurate results and

preliminary conclusions. Before drawing conclusions

about the comparison between the two models, the

final step is to test for multicollinearity. If there is

multicollinearity in the latest multiple regression

model, it fails. We test for multicollinearity by

calculating the VIF variance inflation factor, which

indicates the correlation between the two independent

variables.

𝑉𝐼𝐹𝑋



 =

1−𝑅





(

)

Numeric publications in the fields of marketing

and consumer behavior that use mitigated multiple

regression raise concerns about the existence of

multicollinearity issues in analysis. However, in most

cases, as shown in this paper, the perceived

multicollinearity problem is merely an illusion

resulting from a misunderstanding of the high

correlation between the independent variable and the

interaction term. In order to obtain multiple

regression model and simple regression model, VIF,

the Minitab software is utilized to perform

calculations and analysis.

BDEDM 2022 - The International Conference on Big Data Economy and Digital Management

588

3 RESULTS

3.1 Simple Linear Regression

The regression result of simple linear regression is

given as Eq. (4) and Tables. I~III:

𝑆𝑎𝑙𝑒𝑠 𝑉𝑜𝑙𝑢𝑚𝑒 = 0.753 + 0.00614 𝑀𝑎𝑟𝑘𝑒𝑡 𝐷𝑒𝑚𝑎𝑛𝑑

(

)

Table 1: Coefficients- Simple Linear Regression.

Term Coef SE Coef

T-

Value

P-

value

VIF

Constant 0.753 0.779 0.97 0.362

Overall

Market

Demand

0.00614 0.00344 1.79 0.112 1.00

Table 2: Model Summary- Simple Linear Regression.

S R-sq R-sq(adj) R-sq(pred)

0.833884 28.50% 19.56% 0.00%

Table 3: Analysis of Variance- Simple Linear Regression.

Source DF

Adj

F-

Value

P-

Value

Regression 1 2.217 2.2171 3.19 0.112

Overall

Market

Demand

1 2.217 2.2171 3.19 0.112

Error 8 5.563 0.6954

Total 9 7.780

In Table II, R

shows that only 28.5% of the

variation in the sales volume is explained by this

regression model. Then the estimated multiple linear

regression equation of sales volume on overall market

demand, price per chip, and economic condition are

given.

3.2 Multiple Linear Regression

Table 4: Coefficients- Multiple Linear Regression.

Term Coef

Coef

T-

Value

P-

Value

VIF

Constant 8.86 1.35 6.57 0.001

Overall

Market

Demand

-0.00524 0.00258 -2.03 0.089 3.27

Price Per

Chip

-5.505 0.881 -6.25 0.001 2.54

Economic

Condition

1.130 0.342 3.30 0.016 2.44

Table 5: Model Summary- Multiple Linear Regression.

S R-sq R-sq(adj) R-sq(pred)

0.346274 90.75% 86.13% 72.75%

Table 6: Analysis of Variance - Multiple Linear Regression.

Source DF Adj SS

Adj

F-

Value

P-

Value

Regression 3 7.0606 2.3535 19.63 0.002

Overall

Market

Demand

1 0.4930 0.4930 4.11 0.089

Price Per

Chip

1 4.6786 4.6786 39.02 0.001

Economic

Condition

1 1.3089 1.3089 10.92 0.016

Error 6 0.7194 0.1199

Total 9 7.7800

𝑆𝑎𝑙𝑒𝑠 𝑉𝑜𝑙𝑢𝑚𝑒 = 8.86 − 0.00524𝑀𝑎𝑟𝑘𝑒𝑡 𝐷𝑒𝑚𝑎𝑛𝑑

− 5.505 𝑃𝑟𝑖𝑐𝑒

+ 1.130 𝐸𝑐𝑜𝑛𝑜𝑚𝑖𝑐 𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 (5)

The regression result of mutiple linear regression is

given in Eq. (7) and Tables. IV-VI. For Table IV, the

p-value of overall market demand is only 0.089,

which is not statistically significant, i.e., one needs to

perform another regression.

3.3 Regression Analysis on Price and

Condition

Table 7: Coefficients- Analysis on Price and Condition.

Term Coef

Coef

T-

Value

P-

Value

VIF

Constant 6.452 0.766 8.42 0.000

Price Per

Chip

4.136

0.680 -6.08 0.001 1.05

Economic

Condition

0.606 0.269 2.25 0.059 1.05

Table 8: Model Summary- Analysis on Price and

Condition.

S R-sq R-sq(adj) R-sq(pred)

0.416172 84.42% 79.96% 64.42%

The regression analysis results of the price and

condition are given in Eq. (6) and Table VII.

The Sales Prediction of ABCtronics’ based on the Multi-factorial Linear Model in Terms of Variance Inﬂation Factor

589

𝑆𝑎𝑙𝑒𝑠 𝑉𝑜𝑙𝑢𝑚𝑒 = 6.452 − 4.136 𝑃𝑟𝑖𝑐𝑒

+ 0.606 𝐸𝑐𝑜𝑛𝑜𝑚𝑖𝑐 𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 (6)

For more accuracy, we use adjusted 𝑅



in Table

VIII. The adjusted 𝑅



tells you the percentage of

variation explained by only the independent variables

that actually affect the dependent variable.

3.4 Test Multicollinearity

Table 9: Coefficients- Test Multicollinearity.

Term Coef

Coef

T-

Value

P-

Value

VIF

Constant 6.452 0.766 8.42 0.000

Price Per

Chip

-4.136 0.680 -6.08 0.001 1.05

Economic

Condition

0.606 0.269 2.25 0.059 1.05

In Table XI, the VIF of Price Per Chip is 1.05 which

is VIF (X2, X3) <5. Hence, there is no

multicollinearity. Comparing the two results, it can be

seen that Multiple Linear Regression is better than the

simple linear regression model used by ABCtronics.

4 DISCUSSION

ABCtronics uses a simple linear regression model to

forecast sales volume based on aggregate market

demand. For example, according to a regression

model, we take aggregate demand in the market as an

independent variable. Y = 𝞫₀ + 𝞫₁X + 𝛆, 𝛆 ~ normal

(0, 𝛔²). Here, we include the error term of the normal

distribution for estimation accuracy. According to the

evaluation from Minitab, the estimated parameters for

𝞫₀ and 𝞫₁ along with the estimated linear regression

equations. The model summary of the two values

shows that only 28.5% of sales fluctuations are the

cause. This volume regression model explains why it

makes us wonder if this model is sufficient for good

predictions. Yao, & Li argued that linear regression

varies from normal linear regression in that it models

the conditional mean (Yao, Li, 2014). To address this

issue, we need to consider the multiple regression

model proposed by the SMT intern. The main

difference between simple regression and multiple

regression is the number of independent variables.

Here, in Table III, we first run a multiple regression

model using all three variables, and to simplify it,

ABCtronics sales volume Y, total market demand X1,

price per chip X2, and economic situation X3 are

shown, respectively. The corresponding model used

is Y = 𝞫₀ + 𝞫₁X₁ + 𝞫₂X₂ + 𝞫₃X₃ + 𝛆, 𝛆 ~ Normal (0,

𝛔²). Minitab is utilized to regress on X1 and X2,

including error terms. Therefore, the estimated

multiple regression equation for sales volume with

respect to total market demand, price per chip, and

economic situation.

Horssen et al. denotes that prediction ranges

derived from logistic regression model predictions

when uncertainties in explanatory variables and

uncertainty in regression coefficients are considered

(van Horssen, Pebesma, Schot 2002). To make a more

rigorous estimate of the economic situation, we look

further at the statistical significance of each

independent variable, including the p-value, and level

5% of significance. It is verified again to ensure that

the p-value for the market-wide concept is 0.089,

which is greater than 5%. To sum up, we find that the

total market variable is not statistically significant at

the 5% significance level. Before proceeding to

rebuild the multiple regression model, analysis claim

why chose 5% as the standard. One can also find the

p-value for total market demand from the previous

simple linear regression model. This simple model

gives a p-value of 0.112. A market-wide term is

defined as not statistically significant, not to mention

the 5% level, even if the significance level is 10%.

This may work for higher levels of significance, but

overall accuracy makes it less likely to select too high

level of significance for each model. Therefore, if one

omits the demand period, the multiple regression

model to is able to estimate sales volume more

accurately. The modified equation regresses only the

price per chip and the economic situation, and the

equation becomes Eq. (6).

This model uses adjusted 𝑅



, and the summary is

as follows. Approximately 79.96% of sales volume

fluctuations are explained by this regression model.

This is much higher than the data obtained from a

simple regression model. The final step before

concluding the contrast between the two models is to

test multicollinearity. Yamagata and Orme indicate

that when the “multicollinearity problem” is severe,

the t-test based on the Heckman–Greene variance

estimator can be unreliable, but the Likelihood Ratio

test remains powerful, and nonnormality can be

interpreted by Maximum Likelihood methods as

severe sample selection bias, resulting in negative

Wald statistics (Yamagata, 2006). Arturs Kalnins

(Kalnins, 2018) states that even if the variables’ true

effects are minor and of the same sign, calculated beta

coefficients will trend to infinite magnitudes in

opposite directions if they are associated via an

unobservable common component. Diagnostics (e.g.,

VIF) can be used to falsely verify Type 1 mistakes as

genuine results. If the latest multiple regression

BDEDM 2022 - The International Conference on Big Data Economy and Digital Management

590

model has multicollinearity, it will fail. To test

multicollinearity, calculate the VIF variance factor.

This gives a correlation between the two independent

variables, which shows that it is 1.05, which is much

smaller than the limit line of 5. To sum up, there exists

no multicollinearity and we suggest using multiple

linear regression models proposed by STM interns

since it has stronger explanatory power. Takashi

Yamagata shows that the Maximum Likelihood (ML)

estimator is robust to such multicollinearity and can

produce a more reliable estimator in the same

circumstances (Yamagata, 2006).

The limitation of this sales forecasting study is

that the data in the study is not enough. There are only

40 customers in the sample, and there is no in-depth

investigation into the research data of a larger sa[ple

size. Compared with the regional vegetation models

presented in Ref. (van Horssen, Pebesma, Schot,

2002) are based on a database of 306 sample locations

throughout the area. The abundance of 78 wetland

plant species, as well as 21 environmental

characteristics, were recorded at Each location. The

sample of ABCtronics is insufficient. Additionally,

the time span of historical data is that there is not

enough new sales data for the period from 2004 to

2013, i.e., there will be slight differences in feedback

on the recent situation. For the analysis of sales data,

there is no specific analysis to the month, and the

evaluation is also a parameter for one year. This may

make people who need to know more specific to the

month feel that there is not enough detail.

5 CONCLUSIONS

In summary, we construct a model to predict

ABCtronics’ sales based on the Multiple Linear

Regression Model in terms of Variance Inflation

Factor with the analysis software Minitab. According

to the analysis, the model proposed by the SMT

interns predicts the sales figure better than the model

previously used by ABCtronics. ABCtronics use

simple linear regression with low R^2-low accuracy.

SIM interns propose multiple linear regression. Multi-

linear regression gives a more precise prediction --

suggest using this one. The significance of the results

is to help manufacturing companies make better sales

forecasts. Therefore, it turns out that we are unable to

solve for companies with large and complex data

sources. During the research, the multicollinearity

illusion in moderated regression analysis is discussed.

The perceived multicollinearity problem is merely an

illusion that arises from misinterpreting high

correlations between independent variables and

interaction terms. Moreover, based on Multi-linear

regression and VIF, a more accurate sales forecast can

be provided. It is also hoped that there will be more

models of predictions that can help companies make

decisions in the future. These results offer a guideline

for companies to fast respond to market conditions

promptly to adjust their sales strategies.

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The Sales Prediction of ABCtronics’ based on the Multi-factorial Linear Model in Terms of Variance Inﬂation Factor

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