4.5 ANCOVA

We’ve found that our experimental conditions do not significantly affect conspicuous consumption:

linearmodel1 <- lm(cc ~ power*audience, data = powercc)
type3anova(linearmodel1)

## # A tibble: 5 x 6
##   term                ss   df1   df2       f pvalue
##   <chr>            <dbl> <dbl> <int>   <dbl>  <dbl>
## 1 (Intercept)    5080.       1   139 4710.    0    
## 2 power             2.64     1   139    2.45  0.12 
## 3 audience          2.48     1   139    2.30  0.132
## 4 power:audience    1.11     1   139    1.03  0.313
## 5 Residuals       150.     139   139   NA    NA

On the one hand, this could mean that there simply are no effects of the experimental conditions on conspicuous consumption. On the other hand, it could mean that the experimental manipulations are not strong enough or that there is too much unexplained variance in our dependent variable (or both). We can reduce the unexplained variance in our dependent variable, however, by including a variable in our model that we suspect to be related to the dependent variable. In our case, we suspect that willingness to spend on inconspicuous consumption (icc) is related to willingness to spend on conspicuous consumption. Even though icc is a continuous variable, we can include it as an independent variable in our ANOVA and this will allow us to reduce the unexplained variance in our dependent variable:

linearmodel2 <- lm(cc ~ power*audience + icc, data = powercc)
type3anova(linearmodel2)

## # A tibble: 6 x 6
##   term               ss   df1   df2      f pvalue
##   <chr>           <dbl> <dbl> <int>  <dbl>  <dbl>
## 1 (Intercept)    209.       1   138 221.    0    
## 2 power            1.88     1   138   1.99  0.16 
## 3 audience         1.14     1   138   1.21  0.274
## 4 icc             19.6      1   138  20.8   0    
## 5 power:audience   1.96     1   138   2.08  0.152
## 6 Residuals      130.     138   138  NA    NA

We see that icc is related to the dependent variable and hence that the sum of squares of the residuals of this model, i.e., the unexplained variance in our dependent variable, is lower (130.32) than that of the model without icc (149.93). The p-values of the experimental factors do not decrease, however. You can report this as follows: “Controlling for willingness to spend on inconspicuous consumption, neither the main effect of power (F(1, 138) = 1.99, p = 0.16), nor the main effect of audience (F(1, 138) = 1.21, p = 0.27), nor the interaction between power and audience (F(1, 138) = 2.08, p = 0.15) was significant.”

We call such an analysis ANCOVA because icc is a covariate (it covaries or is related with our dependent variable).