Lecture 10a Activity

Unit D · Chapter 10

An in-class activity. Nothing to turn in and no answer key – this one needs other people, which is why it happens in class rather than at home.

The idea. “Is this assumption violated?” is the wrong first question. The right one is “violated for what purpose?” – and the fastest way to see that is for four groups to reach four different verdicts on the same output.

Format. About 10 minutes · fours · no laptop needed.

1. The output (2 minutes)

Two predictors, correlated at \(.95\), \(n = 200\):

Parameter   | Coefficient |   SE |        95% CI | t(197) |     p
-----------------------------------------------------------------
(Intercept) |        0.15 | 0.07 | [ 0.00, 0.29] |   1.97 | 0.051
x1          |        0.43 | 0.25 | [-0.05, 0.92] |   1.76 | 0.079
x2          |        0.31 | 0.25 | [-0.18, 0.79] |   1.25 | 0.214

R2 = 0.32,  F(2, 197) = 46.4, p < .001
VIF: x1 = 10.5, x2 = 10.5

Neither predictor is significant. The model is overwhelmingly significant. Write down, on your own, whether you would call this model usable.

2. Your assigned purpose (5 minutes)

Each group is assigned one of these four purposes. Decide whether the collinearity is fatal, survivable, or irrelevant for yours, and prepare a one-sentence verdict.

1. Predicting \(y\) for a new case whose \(x_1\) and \(x_2\) look like these.

2. Predicting \(y\) for a new case with high \(x_1\) and low \(x_2\).

3. Testing whether \(x_1\) and \(x_2\) together account for variance in \(y\).

4. Arguing that \(x_1\) matters over and above \(x_2\).

3. Compare verdicts (3 minutes)

Each group reads its verdict. Then, as a room:

  • The four verdicts are not the same. Which purposes does the VIF actually speak to, and which is it silent about?
  • Someone proposes dropping \(x_2\). Which of the four purposes does that help, and which does it destroy?
  • Someone proposes centring both predictors. What will that do?

If you have more time: name three plausible confounders that were never measured in a study you care about, give the direction each would bias the effect, and sort them into “fixable by measuring” and “fixable only by redesigning.”

If you were not in class

Collinearity is not a violation of anything. The model is unbiased and its predictions are fine. What it does is inflate the standard errors of the individual coefficients, because the data contain very little information about what \(x_1\) does while \(x_2\) is held constant – you barely observed that combination.

So the four verdicts differ, and all four are right. Purpose 1 is unaffected. Purpose 2 is extrapolation into a region with no data, which is a real problem but not the one the VIF is measuring. Purpose 3 is untouched – the omnibus \(F\) and \(R^2\) are fine, which is why the model can be overwhelmingly significant while neither coefficient is. Purpose 4 is exactly what collinearity destroys, and it is the only one of the four the VIF addresses.

Dropping \(x_2\) rescues purpose 4 at the cost of changing the question: the remaining coefficient is now a zero-order effect and means something different. Centring does nothing at all here – it helps only with the artificial correlation between a predictor and its own interaction or polynomial term, which is a different problem with the same symptom.