# 5 An interaction model with quantitative predictors

When two or more independent variables are involved in research, there is more to consider than just the ‘main effect’ of each of the independent variables (also called ‘factors’). This means that the effect of one independent variable on the dependent variable of interest may not be the same at all levels of the other independent variable. Another way of saying this is that the effect of one independent variable may depend on the level of the other independent variable.

When the x’s are dependent, we say that $$x_{1}$$ and $$x_{2}$$ interact. Therefore we have to add an interaction term to the simple first-order model and then the model is called an interaction model. The interaction term is just the multiplication of the two x variables. The model is

$E(y) = \beta_{0} + \beta_{1} x_{1} + \beta_{2} x_{2} + \beta_{3} x_{1} x_{2}$

The term $$\beta_{3} x_{1} x_{2}$$ is called the interaction term.

We will have situations when more than two x variables have an influence on y. Below is an example of what the interaction models look like if this is the case.

$E(y) = \beta_{0} + \beta_{1} x_{1} + \beta_{2} x_{2} + \beta_{3} x_{3} + \beta_{4} x_{1} x_{2} + \beta_{5} x_{1} x_{3} + \beta_{6} x_{2} x_{3}$

Example 5.1 We’ll use the marketing data set to predict sales units on the basis of the amount of money spent in the three advertising medias (youtube, facebook and newspaper)

library(datarium)

data(marketing)

head(marketing)
##   youtube facebook newspaper sales
## 1  276.12    45.36     83.04 26.52
## 2   53.40    47.16     54.12 12.48
## 3   20.64    55.08     83.16 11.16
## 4  181.80    49.56     70.20 22.20
## 5  216.96    12.96     70.08 15.48
## 6   10.44    58.68     90.00  8.64

First-order model

m1 <- lm(sales ~ youtube + facebook + newspaper, data=marketing)

anova(m1)
## Analysis of Variance Table
##
## Response: sales
##            Df Sum Sq Mean Sq   F value Pr(>F)
## youtube     1 4773.1  4773.1 1166.7308 <2e-16 ***
## facebook    1 2225.7  2225.7  544.0501 <2e-16 ***
## newspaper   1    0.1     0.1    0.0312 0.8599
## Residuals 196  801.8     4.1
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(m1)
##
## Call:
## lm(formula = sales ~ youtube + facebook + newspaper, data = marketing)
##
## Residuals:
##      Min       1Q   Median       3Q      Max
## -10.5932  -1.0690   0.2902   1.4272   3.3951
##
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept)  3.526667   0.374290   9.422   <2e-16 ***
## youtube      0.045765   0.001395  32.809   <2e-16 ***
## facebook     0.188530   0.008611  21.893   <2e-16 ***
## newspaper   -0.001037   0.005871  -0.177     0.86
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.023 on 196 degrees of freedom
## Multiple R-squared:  0.8972, Adjusted R-squared:  0.8956
## F-statistic: 570.3 on 3 and 196 DF,  p-value: < 2.2e-16

Interaction model

m2 <- lm(sales ~ youtube*facebook + newspaper, data=marketing)

anova(m2)
## Analysis of Variance Table
##
## Response: sales
##                   Df Sum Sq Mean Sq   F value Pr(>F)
## youtube            1 4773.1  4773.1 3708.0160 <2e-16 ***
## facebook           1 2225.7  2225.7 1729.0592 <2e-16 ***
## newspaper          1    0.1     0.1    0.0992 0.7531
## Residuals        195  251.0     1.3
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(m2)
##
## Call:
## lm(formula = sales ~ youtube * facebook + newspaper, data = marketing)
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -7.5515 -0.4780  0.2174  0.7148  1.8011
##
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)
## (Intercept)      8.0740943  0.3039834  26.561  < 2e-16 ***
## youtube          0.0190668  0.0015093  12.633  < 2e-16 ***
## facebook         0.0279917  0.0091412   3.062  0.00251 **
## newspaper        0.0014442  0.0032955   0.438  0.66169
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.135 on 195 degrees of freedom
## Multiple R-squared:  0.9678, Adjusted R-squared:  0.9672
## F-statistic:  1466 on 4 and 195 DF,  p-value: < 2.2e-16

Interaction model 2

m3 <- lm(sales ~ youtube*facebook, data=marketing)

anova(m3)
## Analysis of Variance Table
##
## Response: sales
##                   Df Sum Sq Mean Sq F value    Pr(>F)
## youtube            1 4773.1  4773.1 3723.36 < 2.2e-16 ***
## facebook           1 2225.7  2225.7 1736.22 < 2.2e-16 ***
## Residuals        196  251.3     1.3
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(m3)
##
## Call:
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -7.6039 -0.4833  0.2197  0.7137  1.8295
##
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)
## (Intercept)      8.100e+00  2.974e-01  27.233   <2e-16 ***
## youtube          1.910e-02  1.504e-03  12.699   <2e-16 ***
## facebook         2.886e-02  8.905e-03   3.241   0.0014 **
## F-statistic:  1963 on 3 and 196 DF,  p-value: < 2.2e-16