Question

Two simple (i.e, one x variable) linear regression models are fit. Model A, using variable x₁ only, has an R² of 0.23. Model B, using variable x₂ only, has an R² of 0.57. What will the R² value be for Model C, which uses both x₁ and x₂?

		Cannot be determined without knowing the correlation between x1 and x2.
		It will be some value less than 0.
		It will equal 0.34
		It will equal 0.80.
		We cannot say without knowing the units of Y.
		It will be 0.57.

Answer 1

Here this cannot be determined without knowing the correlation between x1 and x2.

As using x1 we can describe the 23% of the variation of Y and using x2 we can describe the 57% of the variation of Y, but while combining both x1 and x2 we can't say anything without knowing whether there is any linear relationship between x1 and x2. Because if there is a perfect correlation we shouldn't use both the variables. If there is no correlation we can say that the variability explained by both will be 23+57 = 80%. But if there is a correlation between -1 to 1, then we can't say how many variations can be explained by both of the variables.