#Adding a power term (quadratic) to regression via Aiken & West mean centering or residual centering

#Aiken and West test of quadratic effect of IV1

Wk10nm$c_IV1 <- Wk10nm$IV1 - mean(Wk10nm$IV1)
Wk10nm$c_IV1sq <- Wk10nm$c_IV1^2


fit2 <- lm(DV1 ~ IV1 + c_IV1sq, data=Wk10nm)
summary(fit2) # show results

## 
## Call:
## lm(formula = DV1 ~ IV1 + c_IV1sq, data = Wk10nm)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -30.130  -9.650   0.110   9.177  32.168 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 31.034791   2.210865  14.037   <2e-16 ***
## IV1          0.333432   0.033575   9.931   <2e-16 ***
## c_IV1sq      0.002338   0.001349   1.733   0.0846 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 13.88 on 197 degrees of freedom
## Multiple R-squared:  0.3437, Adjusted R-squared:  0.3371 
## F-statistic: 51.59 on 2 and 197 DF,  p-value: < 2.2e-16

confint(fit2, level=0.95) # CIs for model parameters 

##                     2.5 %       97.5 %
## (Intercept) 26.6747903964 35.394791286
## IV1          0.2672186455  0.399645615
## c_IV1sq     -0.0003224178  0.004998864

library(QuantPsyc)
lm.beta(fit2)

##       IV1   c_IV1sq 
## 0.5736473 0.1001114

#Residual centering test of quadratic effect of IV1

Wk10nm$IV1sq<-Wk10nm$IV1^2

fit3 <- lm(IV1sq ~ IV1, data=Wk10nm)

#The two steps below save the residual back to your data frame

Wk10nm$IV1sq_resid <- NA

Wk10nm$IV1sq_resid <- fit3$resid

#residual centering approach

fit4 <- lm(DV1 ~ IV1 + IV1sq_resid, data=Wk10nm)
summary(fit4) # show results

## 
## Call:
## lm(formula = DV1 ~ IV1 + IV1sq_resid, data = Wk10nm)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -30.130  -9.650   0.110   9.177  32.168 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 32.920507   1.924591  17.105   <2e-16 ***
## IV1          0.335789   0.033548  10.009   <2e-16 ***
## IV1sq_resid  0.002338   0.001349   1.733   0.0846 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 13.88 on 197 degrees of freedom
## Multiple R-squared:  0.3437, Adjusted R-squared:  0.3371 
## F-statistic: 51.59 on 2 and 197 DF,  p-value: < 2.2e-16

confint(fit4, level=0.95) # CIs for model parameters 

##                     2.5 %       97.5 %
## (Intercept) 29.1250606729 36.715953439
## IV1          0.2696295262  0.401947855
## IV1sq_resid -0.0003224178  0.004998864

lm.beta(fit4)

##         IV1 IV1sq_resid 
##   0.5777016   0.1000293