PSY 210 Exam #3

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  1. Analysis of Variance (ANOVA)
    A procedure used to evaluate the differences between two or moretreatments (groups) – a t-test compares only two groups.
  2. Statistical Hypotheses for ANOVA
    • Assuming there are 3 groups – more μ values if there are more groups
    • H0: μ1 = μ2 = μ3
    • H1: not H0
  3. ANOVA Conceptual Idea
    Image Upload 1
  4. Why variance?
    When there is more than 2 sample means it is impossible to computemean differences (therefore need to use “variance”)
  5. What is variance?
    Variance is simply a method for measuring how big the differences arefor a set of numbers (variance = differences)
  6. The Logic of ANOVA
    • (1) Not all the scores are the same – there is variability (variance)
    • (2) There are 3 types of variability –
    • (a) total
    • (b) between-treatments
    • (c) within-treatments
  7. Total Variability
    • (1) The deviation of each score from the Grand Mean (viz. the average of all the scores)
    • (2) This variability can be partitioned or divided into 2 parts:
    • (a) the amount that score deviates from its group mean
    • (i.e., within-treatment variation)
    • (b) the amount that score’s group mean differs from the grand mean
    • (i.e., between-treatment variation)
  8. Between-Treatments Variability
    • (1) Measures the differences between sample means
    • (2) What causes those differences between means?
    • (a) Treatment Effects: Systematic differences between groups due to the effect of the IV
    • (b) Chance or Experimental Error
    • (3) Sources of chance or experimental error
    • (a) Individual Differences (e.g., sleep, past drug use, IQ, etc.)
    • (b) Measurement Error (e.g., not calibrating a machine, misreading a dial, etc.)
  9. Treatment Effects
    Systematic differences between groups due to the effect of the IV
  10. Within-Treatments Variability Logic
    If all Ss were the same (no experimental error), all the subjects who received thesame treatment (e.g., 10 mg of a drug) would have the same score on the DV
  11. Within-Treatments Variability Conclusion
    The variability of subjects in the same condition gives us an estimate ofchance or experimental error. As such, within-treatments variability is solelycaused by chance or experimental error.
  12. Goal of ANOVA
    To test for treatment effects (did the IV impact the DV). The F-ratio allows youto accomplish this goal.
  13. F-ratio
    • You form a ratio in which Image Upload 2
    • [“conceptual formula”] Image Upload 3

    Image Upload 4
  14. If Ho is true, then there are no treatment effects and the “conceptual formula” reduces to:
    • Image Upload 5
    • This equals 1
  15. If Ho is false, then there are treatment effects and the numerator or the “conceptual formula” gets
    • bigger and bigger (making the overall fraction large).
    • If the fraction (i.e.,value of this test statistic) gets large enough, we will feel that we haveadequate evidence to Reject Ho and conclude that we have treatment effects
  16. ANOVA Notation and Formulas
    • k = the number of treatment conditions (groups)
    • n = the number of scores in each treatment
    • N = the total number of scores in the entire study
    • T = the total for each treatment condition (ΣX) [needed for formulas so memorize]
    • G ("Grand Total") = the sum of all the scores in the study G = ΣT [needed for formulas so memorize]
  17. Total Sums of Squares (SSWithin-treatments) [Total Variability]
    Image Upload 6
  18. Within-Treatments Sums of Squares (SSWithin-treatments) [Within-Treatments Variability]
    • Image Upload 7 (add up all SS values for each group)
    • Image Upload 8
  19. Degrees of freedom for ANOVA
    • (1) dfTotal = N – 1
    • (2) dfWithin-treatments = N – k
    • (3) dfBetween-treatments = k – 1
  20. Why do we Need Mean Squares?
    Need to convert Sums of Squares (SS) into Variance estimates [i.e., Mean Square (MS)values] to use them in Analysis of Variance
  21. What is a Mean Square?
    MS or Variance = SS/df
  22. MS Calculations
    • (1) MSBetween-treatments = SSBetween-treatments / dfBetween-treatments
    • (2) MSWithin-treatments = SSWithin-treatments / dfWithin-treatments
  23. F-ratio Calculation
    F = MSBetween-treatments / MSWithin-treatments
  24. Characteristics of the F-distribution
    • (1) F-values will always be positive (there is never a negative variance)
    • (2) When H0 is true, F is close to 1.00. As such, the distribution piles up around 1.00.
  25. F-distribution Table (Table B.4)
    Purpose: Used to obtain critical value for boundary of your Critical Region
  26. Hypothesis Testing Example
    • * Research Question: Are these drugs equally effective as pain relievers?*
    • DV: # of seconds that a painful stimulus is endured

    • (A) Step #1 – State the hypotheses
    • (1) H0: μ1 = μ2 = μ3 = μ4
    • (2) H1: not H0

    • (B) Step #2 – Set the criteria for a decision
    • (1) α = .05

    • (C)Step #3 Determine and draw boundary of the critical region(from Table B.4)
    • ** Draw graph with the critical regionsPage #5

    • (D) Step #4 – Collect data and compute sample statistics
    • (1) Part 1 = calculate SS values
    • (2) Part 2 = calculate MS values
    • (3) Part 3 = calculate F statistic
    • (4) Part 4 = complete the ANOVA Summary Table
    • (E) Step #5 – Make a decision
    • (1) Draw obtained score (“test statistic”) on the graph
    • (2) Decision Rule:
    • (a) If obtained value is in the critical region, you “Reject H0
    • (b) If obtained value is not in the critical region, you “Fail to Reject H0”
    • (3) Decision = Reject H0 (conclude that these drugs effective are not equally effective as painrelievers -- that is there are differences among the 4 groups)

    Note: Still need to do post-hocs to determine “what differs from what”
  27. Assumptions of the 1-way Between subjects ANOVA
    • (A) Independent scores
    • (B) Normality (the populations from which the samples are selected must be normal)
    • (C) Homogeneity of Variance (the populations from which the
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PSY 210 Exam #3
Statistics Test
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