Multiple Regression and Model Diagnostics · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Many predictors, honest reading

Mathematics · Data & Statistics · ages 21-22
Name ______________________   Date ____________

  1. A fitted model is y equals 2 plus 1.5 x1 minus 0.5 x2. Predict y at x1 equals 4, x2 equals 2.

    Answer: ______________

  2. Two predictors are strongly collinear. What does that do to the fitted coefficients?

    • Coefficients turn unstable with wide standard errors
    • R squared drops to zero
    • All residuals become zero

  3. In y equals 2 plus 1.5 x1 minus 0.5 x2, how is the 1.5 read?

    • x1 causes y to rise by exactly 1.5
    • The correlation of x1 and y is 1.5
    • Predicted y rises by 1.5 per unit of x1 with x2 held fixed

  4. A fitted model is y = 2 + 1.5 x1 - 0.5 x2. Predict y at x1 = 4, x2 = 2.

    Answer: ______________

  5. Two predictors are strongly collinear. What does that do to the fitted coefficients?

    • Coefficients turn unstable with wide standard errors
    • R squared drops to zero
    • All residuals become zero
    • Predictions freeze at the mean

  6. In y = 2 + 1.5 x1 - 0.5 x2, how is the coefficient 1.5 read?

    • Predicted y rises by 1.5 per unit of x1 with x2 held fixed
    • Predicted y rises by 1.5 no matter what happens to x2
    • x1 causes y to rise by exactly 1.5
    • The correlation between x1 and y is 1.5

  7. One observation drags the whole fitted surface toward itself. What is it called?

    • A held fixed coefficient
    • A collinear predictor
    • An influential point

  8. Nadia picks the model with the highest R squared and stops there. Is that sound model choice?

    Circle one:   True   False

  9. A slope on observational data is positive. The headline claims proof that raising x1 raises y. What is wrong?

    • Fitted slopes show association with the rest held fixed, not proven cause
    • Cause follows from any p value
    • Slopes never describe associations

  10. A team keeps adding predictors until raw R squared looks great, then reports victory. What is the flaw?

    • Adjusted measures are decorative
    • They overfit: in sample polish without better predictions
    • Collinearity improves stability
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Many predictors, honest reading W1-mt_TUx1tDF0jv-s1

  1. 7 · Insert the values: 2 plus 6 minus 1 is 7.
  2. Coefficients turn unstable with wide standard errors · Collinear predictors share information, so credit splits arbitrarily and slopes swing.
  3. Predicted y rises by 1.5 per unit of x1 with x2 held fixed · A multiple slope is a conditional effect: it needs the other predictors held fixed.
  4. 7 · Insert x1 = 4 and x2 = 2: 2 + 1.5 x 4 - 0.5 x 2 = 2 + 6 - 1 = 7.
  5. Coefficients turn unstable with wide standard errors · Collinear predictors carry nearly the same information, so the fit cannot split credit: slopes swing wildly with wide errors while fitted values stay sane.
  6. Predicted y rises by 1.5 per unit of x1 with x2 held fixed · A multiple regression slope is a ceteris paribus effect: the change in fitted y per unit of x1 while the other predictors stay put.
  7. An influential point · Influential points pull the surface, and diagnostics exist to flag them.
  8. False · False. R squared never falls when predictors are added, so chasing it alone invites overfit. Diagnostics, residuals, and purpose must judge with it.
  9. Fitted slopes show association with the rest held fixed, not proven cause · The coefficient describes the fitted surface conditionally; cause needs a stronger design.
  10. They overfit: in sample polish without better predictions · Chasing in sample polish builds a model tuned to noise that degrades on new data.
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