2016-06-14

A Graph of Burt’s Data

> library(car)
> Burt$class <- with(Burt, factor(class, levels=c("low", "medium", "high")))
> scatterplot(IQbio ~ IQfoster | class,  data=Burt, 
+             boxplots=FALSE, smooth=FALSE)

The regression lines for the three classes are not very different, but they are also remarkably parallel.

Burt’s Regression

> burt.mod <- lm(IQbio ~ IQfoster*class, data=Burt)
> summary(burt.mod)

Call:
lm(formula = IQbio ~ IQfoster * class, data = Burt)

Residuals:
    Min      1Q  Median      3Q     Max 
-14.479  -5.248  -0.155   4.582  13.798 

Coefficients:
                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)           7.20461   16.75126   0.430    0.672    
IQfoster              0.94842    0.18218   5.206 3.69e-05 ***
classmedium          -6.38859   31.02087  -0.206    0.839    
classhigh            -9.07665   24.44870  -0.371    0.714    
IQfoster:classmedium  0.02414    0.33933   0.071    0.944    
IQfoster:classhigh    0.02914    0.24458   0.119    0.906    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.921 on 21 degrees of freedom
Multiple R-squared:  0.8041,    Adjusted R-squared:  0.7574 
F-statistic: 17.24 on 5 and 21 DF,  p-value: 8.31e-07
> Anova(burt.mod)
Anova Table (Type II tests)

Response: IQbio
               Sum Sq Df F value    Pr(>F)    
IQfoster       4674.7  1 74.5132 2.382e-08 ***
class           175.1  2  1.3958    0.2697    
IQfoster:class    0.9  2  0.0074    0.9926    
Residuals      1317.5 21                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

There is no evidence of an interaction, and only weak evidence of a class main effect. There is very strong evidence for an IQfoster “effect.”

Why Should We Strongly Suspect that the Data Are Cooked?

The p-value for the interaction is very close to 1, indicating that the three regression lines are almost perfectly parallel. Data this parallel would occur by chance less than 1 percent of the time, even if there were no interaction in the population.