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Chapter 10

Correlation andRegression:

Explaining Association and Causation

Marketing ResearchText and Cases

byRajendra Nargundkar

Application Areas: Correlation

1. Correlation and Regression are generally performed together. The application of correlation analysis is to measure the degree of association between two sets of quantitative data. The correlation coefficient measures this association. It has a value ranging from 0 (no correlation) to 1 (perfect positive correlation), or -1 (perfect negative correlation).

2. For example, how are sales of product A correlated with sales of product B? Or, how is the advertising expenditure correlated with other promotional expenditure? Or, are daily ice cream sales correlated with daily maximum temperature?

3. Correlation does not necessarily mean there is a causal effect. Given any two strings of numbers, there will be some correlation among them. It does not imply that one variable is causing a change in another, or is dependent upon another.

4. Correlation is usually followed by regression analysis in many applications.

Slide 1

Application Areas: Regression

1. The main objective of regression analysis is to explain the variation in one variable (called the dependent variable), based on the variation in one or more other variables (called the independent variables).

2. The applications areas are in ‘explaining’ variations in sales of a product based on advertising expenses, or number of sales people, or number of sales offices, or on all the above variables.

3. If there is only one dependent variable and one independent variable is used to explain the variation in it, then the model is known as a simple regression.

4. If multiple independent variables are used to explain the variation in a dependent variable, it is called a multiple regression model.

5. Even though the form of the regression equation could be either linear or non-linear, we will limit our discussion to linear (straight line) models.

6. As seen from the preceding discussion, the major application of regression analysis in marketing is in the area of sales forecasting, based on some independent (or explanatory) variables. This does not mean that regression analysis is the only technique used in sales forecasting. There are a variety of quantitative and qualitative methods used in sales forecasting, and regression is only one of the better known (and often used) quantitative techniques.

Methods

There are basically two approaches to regression –A hit and trial approach .

A pre- conceived approach.

Hit and trial Approach

In the hit and trial approach we collect data on a large number of independent variables and then try to fit a regression model with a stepwise regression model, entering one variable into the regression equation at a time.The general regression model (linear) is of the type

Y = a + b1x1 + b2x2 +…….+ bnxn

where y is the dependent variable and x1, x2 , x3….xn are the independent variables expected to be related to y and expected to explain or predict y. b1, b2, b3…bn are the coefficients of the respective independent variables, which will be determined from the input data.

Pre-conceived Approach

The pre-conceived approach assumes the researcher knows reasonably well which variables explain ‘y’ and the model is pre-conceived, say, with 3 independent variables x1, x2, x3. Therefore, not too much experimentation is done. The main objective is to find out if the pre-conceived model is good or not. The equation is of the same form as earlier.

Data1. Input data on y and each of the x variables is required to do a regression analysis. This data is input into a computer package to perform the regression analysis.

2. The output consists of the ‘b’ coefficients for all the independent variables in the model. The output also gives you the results of a ‘t’ test for the significance of each variable in the model, and the results of the ‘F’ test for the model on the whole.

3. Assuming the model is statistically significant at the desired confidence level (usually 90 or 95% for typical applications in the marketing area), the coefficient of determination or R2 of the model is an important part of the output. The R2 value is the percentage (or proportion) of the total variance in ‘y’ explained by all the independent variables in the regression equation.

Recommended usage

1. It is recommended by the author that for exploratory research, the hit-and-trial approach may be used. But for serious decision-making, there has to be a-priori knowledge of the variables which are likely to affect y, and only such variables should be used in the regression analysis.

2. It is also recommended that unless the model is itself significant at the desired confidence level (as evidenced by the F test results printed out for the model), the R² value should not be interpreted.

3. The variables used (both independent and dependent) are assumed to be either interval scaled or ratio scaled. Nominally scaled variables can also be used as independent variables in a regression model, with dummy variable coding. Please refer to either Marketing Research: Methodological Foundations by Churchill or Research for Marketing Decisions by Green, Tull & Albaum for further details on the use of dummy variables in regression analysis. Our worked example confines itself to metric interval scaled variables.

4. If the dependent variable happens to be a nominally scaled one, discriminant analysis should be the technique used instead of regression.

Worked Example: Problem

1. A manufacturer and marketer of electric motors would like to build a regression model consisting of five or six independent variables, to predict sales. Past data has been collected for 15 sales territories, on Sales and six different independent variables. Build a regression model and recommend whether or not it should be used by the company.

2. We will assume that data are for a particular year, in different sales territories in which the company operates, and the variables on which data are collected are as follows:

Dependent Variable

Y =sales in Rs.lakhs in the territory

Independent Variables

X1 = market potential in the territory (in Rs.lakhs).X2 = No. of dealers of the company in the territory.X3 = No. of salespeople in the territory.X4 = Index of competitor activity in the territory on

a 5 point scale (1=low, 5 = high level of activity by competitors).

X5 = No. of service people in the territory.X6 = No. of existing customers in the territory.

Input data:

The data set consisting of 15 observations, is given infig 1. Fig. 1

Data file : REGDATA1.STA (15 cases with 7variables)

1SALES

2POTENTL

3DEALERS

4PEOPLE

5COMPET

6SERVICE

7CUSTOM

1 5 25 1 6 5 2 202 60 150 12 30 4 5 503 20 45 5 15 3 2 254 11 30 2 10 3 2 205 45 75 12 20 2 4 306 6 10 3 8 2 3 167 15 29 5 18 4 5 308 22 43 7 16 3 6 409 29 70 4 15 2 5 3910 3 40 1 6 5 2 511 16 40 4 11 4 2 1712 8 25 2 9 3 3 1013 18 32 7 14 3 4 3114 23 73 10 10 4 3 4315 81 150 15 35 4 7 70

Correlation

First, let us look at the correlations of all the variableswith each other. The correlation table (output fromthe computer for the Pearson Correlation procedure)is shown in Fig. 2. The values in the correlation tableare standardised, and range from 0 to 1 (+ ve and - ve).

Fig.2 : Correlations Table

STAT.MULTIPLEREGRESS.

Correlations (regdata1.sta)

VariablePOTENTL

DEALERS PEOP

LECOMPET

SERVICE CUST

OMSALES

POTENTL 1.00 .84 .88 .14 .61 .83 .94DEALERS .84 1.00 .85 -.08 .68 .86 .91PEOPLE .88 .85 1.00 -.04 .79 .85 .95COMPET .14 -.08 -.04 1.00 -.18 -.01 -.05SERVICE .61 .68 .79 -.18 1.00 .82 .73CUSTOM .83 .86 .85 -.01 .82 1.00 .88SALES .94 .91 .95 -.05 .73 .88 1.00

1. Looking at the last column of the table, we find that except for COMPET (index of competitor activity), all other variables are highly correlated (ranging from .73 to .95) with Sales.

2. This means we may have chosen a fairly good set of independent variables (No. of Dealers, Sales Potential, No. of Customers, No. of Service People, No. of Sales People) to try and correlate with Sales.

3. Only the Index of Competitor Activity does not appear to be strongly correlated (correlation coefficient is -.05) with Sales. But we must remember that these correlations in Fig. 2 are one-to-one correlations of each variable with the other. So we may still want to do a multiple regression with an independent variable showing low correlation with a dependent variable, because in the presence of other variables, this independent variable may become a good predictor of the dependent variable.

4. The other point to be noted in the correlation table is whether independent variables are highly correlated with each other. If they are, like in Fig. 2, this may indicate that they are not independent of each other, and we may be able to use only 1 or 2 of them to predict the dependent variables.

5. As we will see later, our regression ends up eliminating some of the independent variables, because all six of them are not required. Some of them, being correlated with other variables, do not add any value to the regression model.

6. We now move on to the regression analysis of the same data.

Slide 9 contd...

Regression

We will first run the regression model of the following form, by entering all the 6 'x' variables in the model -

Y= a + b1x1 + b2x2 + b3x3 + b4x4 + b5x5 + b6x6 ……………..Equation 1

and determine the values of a, b1, b2, b3, b4, b5, & b6.

Regression Output:

The results (output) of this regression model are in Fig.4 in table form.

Column 4 of the table, titled ‘B’ lists all the coefficients for the model. According to this,

a (intercept) = -3.17298b1 = .22685b2 = .81938b3 = 1.09104b4 = -1.89270b5 = -0.54925b6 = 0.06594

These values of a, b1, b2, ..b6 can be substituted inequation 1 above and we can write the equation(rounding off all coefficients to 2 decimals), as

Sales = -3.17 + .23 (potential) + .82 (dealers) + 1.09(salespeople) - 1.89 (competitor activity) - 0.55(service people) + 0.07 (existing customers)

Before we use this equation, however, we need tolook at the statistical significance of the model, andthe R2 value. These are available from Fig. 3 , theAnalysis of Variance Table, and Fig. 4.

Fig. 3 : The ANOVA Table


Analysis of Variance; Depen.Var: SALES (regdata1.sta)

EffectSums ofSquares df

MeanSquares F

Regress.ResidualTotal

6609.484154.249

6763.733

68

1101.58119.281

57.13269 .000004

From Fig. 3, the analysis of variance table, the lastcolumn indicates the p-level to be 0.000004. Thisindicates that the model is statistically significant at aconfidence level of (1-0.000004)*100 or(0.999996)*100, or 99.9996.

The R2 value is 0.977, from the top of Fig. 4. FromFig. 4, we also note that ‘t’ tests for significance ofindividual independent variables indicate that at thesignificance level of 0.10 (equivalent to a confidencelevel of 90%), only POTENTL and PEOPLE arestatistically significant in the model. The other 4independent variables are individually not significant.

Fig. 4 MULTIPLE REGRESSION RESULTS:

All independent variables were entered in one block

Dependent Variable: SALESMultiple R: .988531605Multiple R-Square: .977194734Adjusted R-Square: .960090784Number of cases: 15F(6, 8) = 57.13269 p< .000004Standard Error of Estimate: 4.391024067Intercept: -3.172982117Std.Error: 5.813394 t(8) = -.5458 p< .600084

contd...


Regression Summary for Dependent Variable: SALESR= .98853160 R2= .97719473 Adjusted R2= .96009078F(6,8)=57.133 p< .00000 Std.Error of Estimate: 4.3910

N=15BETA

St.Err.ofBETA

BSt. Err.

of B t(8) p-level

Intercept -3.1729 5.813394 -.54581 .600084POTENTL .439073 .144411 .22685 .074611 3.04044 .016052DEALERS .164315 .126591 .81938 .631266 1.29800 .230457PEOPLE .413967 .158646 1.09104 .418122 2.60937 .031161COMPET .084871 .060074 -1.89270 1.339712 -1.41276 .195427SERVICE .040806 .116511 -.54925 1.568233 -.35024 .735204CUSTOM .050490 .149302 .06594 .095002 .33817 .743935

However, ignoring the significance of individual variables for now, we shall use the model as it is, and try to apply it for decision making. The real use of the regression model would be to try and ‘predict’ sales in Rs. lakhs, given all the independent variable values.

The equation we have obtained means, in effect, that sales will increase in a territory if the potential increases, or if the number of dealers increases, or if level of competitor’s activity decreases, if number of service people decreases, and if the number of existing customers increases.

The estimated increase in sales for every unit increase or decrease in these variables is given by the coefficients of the respective variables. For instance, if the number of sales people is increased by 1, sales in Rs . lakhs, are estimated to increase by 1.09, if all other variables are unchanged. Similarly, if 1 more dealer is added, sales are expected to increase by 0.82 lakh, if other variables are held constant.

There is one co-efficient, that of the SERVICE variable, which does not make too much intuitive sense. If we increase the number of service people, sales are estimated to decrease according to the –0.55 coefficient of the variable "No. of Service People" (SERVICE).

But if we look at the individual variable ‘t’ tests, we find that the coefficients of the variable SERVICE is statistically not significant (p-level 0.735204 from fig. 4). Therefore, the coefficient for SERVICE is not to be used in interpreting the regression, as it may lead to wrong conclusions.

Strictly speaking, only two variables, potential (POTENTL) and No. of sales people (PEOPLE) are significant statistically at 90 percent confidence level since their p- level is less than 0.10. One should therefore only look at the relationship of sales with one of these variables, or both these variables.

Slide 13 contd...

Making Predictions/Sales Forecasts

Given the levels of X1, X2, X3, X4, X5, and X6 for a particular territory, we can use the regression model for prediction of sales. Before we do that, we have the option of redoing the regression model so that the variables not statistically significant are minimized or eliminated. We can follow either the Forward Stepwise Regression method, or the Backward Stepwise Regression method, to try and eliminate the 'insignificant' variables from the full regression model containing all six independent variables.

Forward Stepwise Regression

For example, we could ask the computer for a Forward stepwise Regression model, in which case the algorithm adds one independent variable, at a time , starting with the one which ‘explains’ most of the variation in sales (y), and adding one more X variable to it , rechecking the model to see that both variables form a good model, then adding a third variable if it still adds to the explanation of Y , and so on. Fig, 5 shows the result of running a forward stepwise Regression, which ends up with only 4 out of 6 independent variables remaining in the regression model.

Fig. 5


Regression Summary for Dependent Variable: SalesR = .98831786 R2 = .97677220 Adjusted R2 = .96748108F = (4,10) = 105.13 p<.00000 Std. Error of estimate: 3.9637

N=15 BETA St. Err.ofBETA

B St. Err. ofB

T (10) p-level

InterceptPeoplePotentlDealersCompet

.390134

.462686

.180700-.081195

.115138

.117988

.102687

.053434

-3.74194 1.02822 .23905 .90109-1.81074

4.847683 .303453 .060959 .512065-1.191624

-.771903.388413.921471.75971-1.51955

.458025

.006904

.002860

.108955

.159589

The 4 variables in the model are PEOPLE (No, of salespeople) POTENTL (sales potential), Dealers (No ofDealers) and COMPET (competitive index). Again wenotice, that the two significant variables (those with p value<.10) at 90 % confidence are only PEOPLE and POTENTL(p- levels of .006904 and .002860).But DEALERS is now at p-level of .108955, very close tosignificance at 90 % confidence level. This could be theequation, instead of the one with 6 independent variables,that we could use. We would be economizing on the twovariables, which are not required if we decide to use themodel from Fig, 5 instead of that from Fig, 4. The F test for the model in Fig, 5 also indicates it is highlysignificant (From top of Fig, 5, F=105.1296, P<.000000)and R² value for the model is 0.9767, which is very close tothe 6-independent variable model of Fig, 4. If we decide touse the model from Fig. 5, it would be written as follows -

Sales = -3.74 + 1.03 (PEOPLE) + .24 (POTETL) + .9 (DEALERS) - 1.81 (COMPET) ……….Equation 2

Backward Stepwise Regression

We could, as another alternative, perform aBackward stepwise Regression, on the same set of 6independent variables. This procedure starts with all6 variables in the model, and gradually eliminatesthose, one after another, which do not explain muchof the variation in ‘Y’, until it ends with an optimalmix of independent variables according to pre-setcriteria for the exit of variables.This results in a model with only 2 independentvariables POTENTL and PEOPLE remaining in theequation. This model is shown in Fig, 6.

Fig. 6

Backward stepwise regression, no, of steps: 4

STATMULTIPLEREGRESS.

Regression Summary for Dependent Variable: SALESR=.97975624 R2=.95992229 Adjusted R2=.95324267F(2, 12)=143.71 P<.00000 Std. Error estimate: 4.7528

N=15 BETA St.Err. ofBETA

B St Err. ofBETA

t(12) P level

InterceptPOTENTLPEOPLE

.470825

.540454.120127.120127

-10.6164.24331.4244

2.659532.062065.316602

-3.991833.919394.49902

.001788

.000728

.000728

The R² for the model has dropped only slightly, to 0.9599, the F-test for the model is highly significant, and both the independent variables POTENTL and PEOPLE are significant at 90 % confidence level (p-levels of .002037 and .000728 from last column, Fig, 6). If we were to decide to use this model for prediction , we only require data to be collected on the number of sales people (PEOPLE) and the sales potential (POTENTL), in a given territory . We could form the equation using the Intercept and coefficients from column “B” in Fig. 6. as follows-

Sales = -10.6164 + .2433 (POTENTL) + 1.4244 (PEOPLE)…………...Equation 3

Thus, if potential in a territory were to be Rs. 50 lakhs, and the territory had 6 salespeople, then expected sales, using the above equation would be = -10.6164 +.2433(50) +1.4244(6) = 10.095 lakhs.Similarly, we could use this model to make predictions regarding sales in any territory for which Potential and No. of Sales People were known.

Slide 17

Additional comments

1. As we can see from the example discussed, regression analysis is a very simple (particularly on a computer), and useful techniques to predict one metric dependent variable based on a set of metric independent variables. Its use, however, gets more complex, for instance, if the independent variables are nominally scaled into two (dichotomous) or more (polytomous) categories.

2. It is also a good idea to define the range of all independent variables used for constructing the regression model. For prediction of Y values, only those X values which fall within or close to this range (used earlier in the model construction stage) must be used, for the predictions to be effective.

3. Finally, we have assumed that a linear model is the only option available to us. That is not the only choice. A regression model could be of any non linear variety, and some of these could be more suitable for particular cases.

4. Generally, a look at the plot of Y and X tells us in case of a simple regression model, whether the linear (straight line) approach is best or not. But in a multiple regression, this visual plot may not indicate the best kind of model, as there are many independent variables, and the plot in 2 dimensions is not possible.

5. In this particular example, we have not used any macroeconomic variables, but in industrial marketing, we may use those types of industry or macroeconomic variables in a regression model. For example, to forecast sales of steel, we may use as independent variables, the growth rate of a country’s GDP, the new construction starts, and the growth rate of the automobile industry.

Slide 18 contd….