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Multiple Comparisons

Multiple Comparisons. Multiple Comparison Procedures. Suppose that analysis of variance has provided statistical evidence to reject the null hypothesis of equal population means. Fisher’s least significant difference (LSD) procedure can be used to determine where the differences occur.

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Multiple Comparisons

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  1. Multiple Comparisons

  2. Multiple Comparison Procedures • Suppose that analysis of variance has provided statistical evidence to reject the null hypothesis of equal population means. • Fisher’s least significant difference (LSD) procedure can be used to determine where the differences occur.

  3. Fisher’s LSD Procedure • Hypotheses • Test Statistic =

  4. Fisher’s LSD Procedure • Rejection Rule p-value Approach: Reject H0 if p-value <a Critical Value Approach: Reject H0 if t < -ta/2 or t > ta/2 where the value of ta/2 is based on a t distribution with nT - k degrees of freedom.

  5. Reject H0 if > LSD Fisher’s LSD ProcedureBased on the Test Statistic xi - xj _ _ • Hypotheses • Test Statistic = • Rejection Rule where

  6. Fisher’s LSD ProcedureBased on the Test Statistic xi - xj Reed Manufacturing Recall that Janet Reed wants to know if there is any significant difference in the mean number of hours worked per week for the department managers at her three manufacturing plants. Analysis of variance has provided statistical evidence to reject the null hypothesis of equal population means. Fisher’s least significant difference (LSD) procedure can be used to determine where the differences occur.

  7. Fisher’s LSD ProcedureBased on the Test Statistic xi - xj For  = .05 and nT - k = 15 – 3 = 12 degrees of freedom, t.025 = 2.179 MSE value was computed earlier

  8. Reject H0 if > 6.98 = |55 - 68| = 13 Fisher’s LSD ProcedureBased on the Test Statistic xi - xj • LSD for Plants 1 and 2 • Hypotheses (A) = • Rejection Rule • Test Statistic • Conclusion • The mean number of hours worked at Plant 1 is • not equal to the mean number worked at Plant 2.

  9. Reject H0 if > 6.98 = |55 - 57| = 2 Fisher’s LSD ProcedureBased on the Test Statistic xi - xj • LSD for Plants 1 and 3 • Hypotheses (B) = • Rejection Rule • Test Statistic • Conclusion There is no significant difference between the mean number of hours worked at Plant 1 and the mean number of hours worked at Plant 3.

  10. Reject H0 if > 6.98 = |68 - 57| = 11 Fisher’s LSD ProcedureBased on the Test Statistic xi - xj • LSD for Plants 2 and 3 • Hypotheses (C) = • Rejection Rule • Test Statistic • Conclusion The mean number of hours worked at Plant 2 is not equal to the mean number worked at Plant 3.

  11. Multiple Comparisons Practice

  12. Perform Multiple Comparisons =0.05

  13. F Distribution

  14. Type I Errors

  15. Type I Error Rates • The comparison-wise Type I error ratea indicates the level of significance associated with a single pairwise comparison. • The experiment-wise Type I error rate gets larger for problems with more populations (larger k). • The experiment-wise Type I error rateaEW is the probability of making a Type I error on at least one of the (k – 1)! pairwise comparisons. aEW = 1 – (1 – a)(k – 1)!

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