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Hypothesis testing for the GLM

Hypothesis testing for the GLM. The General Linear Hypothesis. Testing the General Linear Hypotheses The General Linear Hypothesis. H 0 : h 11 b 1 + h 12 b 2 + h 13 b 3 +... + h 1 p b p = h 1 h 21 b 1 + h 22 b 2 + h 23 b 3 +... + h 2 p b p = h 2 ...

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Hypothesis testing for the GLM

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  1. Hypothesis testing for the GLM The General Linear Hypothesis

  2. Testing the General Linear Hypotheses The General Linear Hypothesis H0: h11b1 + h12b2 + h13b3 +... + h1pbp = h1 h21b1 + h22b2 + h23b3 +... + h2pbp = h2 ... hq1b1 + hq2b2 + hq3b3 +... + hqpbp = hq where h11, h12, h13, ... , hqp and h1, h2, h3, ... , hq are known coefficients. In matrix notation

  3. Examples • H0: b1 = 0 • H0: b1 = 0, b2 = 0, b3 = 0 • H0: b1 = b2

  4. Examples • H0: b1 = b2 , b3 = b4 • H0: b1 = 1/2(b2 + b3) • H0: b1 = 1/2(b2 + b3), b3 = 1/3(b4 + b5 + b6)

  5. TheLikelihood Ratio Test The joint density of is: The likelihood function The log-likelihood function

  6. Defn (Likelihood Ratio Test of sizea) Rejects H0:q w against the alternative hypothesis H1:q w. when and K is chosen so that

  7. Note The Lagrange multiplier technique will be used for this purpose We will maximize.

  8. We will maximize.

  9. or finally or

  10. Thus the equations for are Now or and

  11. Thus Note Now

  12. Now

  13. Now

  14. Thus

  15. The Likelihood Ratio Test of sizea) Rejects H0:q w against the alternative hypothesis H1:q w. when and K is chosen so that

  16. Thus The LR tests rejects H0:q w when

  17. now

  18. The LR tests rejects H0:q w when

  19. Theorem If is true then has an F distribution with n1 = q d.f. in the numerator and n2 = n – p d.f. in the denominator.

  20. Proof Recall Thus and Hence Also recall

  21. Finally Thus is independent of has an F distribution with n1 = q d.f. in the numerator and n2 = n – p d.f. in the denominator.

  22. An Alternative form of the F statistic

  23. Special Case Testing vs Thus the test would reject when

  24. The ANOVA table:

  25. Testing if a sub vector is zero

  26. Testing if a sub vector is zero (an alternative formulation)

  27. An Alternative form of the F statistic Exercise: Show

  28. The ANOVA table:

  29. Model General linear model with intercept b0

  30. The ANOVA table:

  31. An alternative form of ANOVA table:

  32. Example: General Linear Model with intercept Assume we have collected data on Y, X1, X2, X3 and Y = b0 + b1X1 + b1X2 + b1X3 + e The Data

  33. The Model

  34. The Estimates

  35. The ANOVA Table

  36. The ANOVA Table

  37. SPSS Output

  38. SPSS Output continued

  39. Confidence intervals, Prediction intervals, Confidence Regions General Linear Model

  40. Recall

  41. Thus

  42. i.e. is a (1 – a)100 % confidence interval for Special cases

  43. Confidence intervals for s2

  44. i.e. are (1 – a)100 % confidence interval for s2 and s.

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