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Psychology 10

Psychology 10. Analysis of Psychological Data April 23, 2014. The plan for today. Finishing the Kelly & Nils example. Assumptions of two-way ANOVA. An example of two-way ANOVA calculated from raw data: Summary statistics; Graphics; Computations; Interpretation; Checking assumptions.

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Psychology 10

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  1. Psychology 10 Analysis of Psychological Data April 23, 2014

  2. The plan for today • Finishing the Kelly & Nils example. • Assumptions of two-way ANOVA. • An example of two-way ANOVA calculated from raw data: • Summary statistics; • Graphics; • Computations; • Interpretation; • Checking assumptions.

  3. Example (cont.) • Male TA, male student: T = 583.5. • Male TA, female student: T = 532. • Female TA, male student: T = 582.5. • Female TA, female student: T = 590.5. • n = 7. • S X2 = 193391.8.

  4. TA Sex vs. Student Sex

  5. What are our null hypotheses? • H0 : mNils = mKelly. • H0 : mMales = mFemales. • H0 : There is no interaction between TA sex and student sex.

  6. Sums of Squares

  7. Calculating SSAfor TAs • For Nils, TA= 583.5 + 532 = 1115.5 • For Kelly, TA= 582.5 + 590.5 = 1173.0 • The grand total G is 1115.5 + 1173.0 = 2288.5 • SSA= 1115.52 / 14 + 1173.02 / 14 – 2288.52 / 28 = 118.0804

  8. ANOVA Table Source SS df MS F ----------------------------------- TA Sex 118.080 1 118.080 Student Sex Interaction Within ------------------------ Total

  9. Calculating SSBfor Student Sex • For male students, TB= 583.5 + 582.5 = 1166.0 • For female students, TB= 532 + 590.5 = 1122.5 • The grand total G is still 2288.5 • SSA= 11662 / 14 + 1122.52 / 14 – 2288.52 / 28 = 67.58036

  10. ANOVA Table Source SS df MS F ----------------------------------- TA Sex 118.080 1 118.080 Student Sex 67.580 1 67.580 Interaction Within ------------------------ Total

  11. Calculating the Interaction SS • SScells= 583.52 / 7 + 5322 / 7 + 582.52 / 7 + 590.52 / 7 – 2288.52 / 28 = 312.0982 • SSA×B = Sscells - SSA – SSB • 312.0982 - 118.0804 - 67.58036 = 125.4374

  12. ANOVA Table Source SS df MS F ----------------------------------- TA Sex 118.080 1 118.080 Student Sex 67.580 1 67.580 Interaction 126.437 1 126.437 Within ------------------------ Total

  13. Calculating the Within-Groups SS directly • In the four cells, SX2= 50311.75 (Nils, males), 42289.5 (Nils, females), 49957.75 (Kelly, males), and 50832.75 (Kelly, females) • 50311.75 - 583.52 / 7 = 1672.857 • 42289.5 - 5322 / 7 = 1857.5 • 49957.75 - 582.52 / 7 = 1485.429 • 50832.75 - 590.52 / 7 = 1019.857 • SSW = 1672.857 + 1857.5 + 1485.429 + 1019.857 = 6035.643

  14. ANOVA Table Source SS df MS F ----------------------------------- TA Sex 118.080 1 118.080 Student Sex 67.580 1 67.580 Interaction 126.438 1 126.438 Within 6035.643 24 251.487 ------------------------ Total

  15. SST • SST may be calculated two ways: • 118.080 + 67.580 + 126.438 + 6035.643 = 6347.741 • 50311.75 + 42289.5 + 49957.75 + 50832.75 - 2288.52 / 28 = 6347.741 • The fact that we get the same answer both ways confirms our other calculations.

  16. ANOVA Table Source SS df MS F ----------------------------------- TA Sex 118.080 1 118.080 .47 Student Sex 67.580 1 67.580 .27 Interaction 126.438 1 126.438 .50 Within 6035.643 24 251.485 ------------------------ Total 6347.741 27

  17. Making a decision • For each of those F statistics, the degrees of freedom are 1 in the numerator and 24 in the denominator. • From the table, the critical value is 4.26. • We fail to reject each one of the null hypotheses.

  18. Interpreting the decision • We have not found evidence that the population mean midterm score of students with the male TA differs from the population mean of students with the female TA. • We have not found evidence that the population means of midterm scores for males and females differ. • We have not found evidence that male/female differences depend on which TA the student has.

  19. Main effects and interactions • It may seem strange to talk about Male/Female differences depending on TA when we haven’t actually found evidence of Male/Female differences. • It is possible for the interaction to be present even when no main effects are present.

  20. Interaction, no main effects

  21. Interaction, sex effect, no TA effect

  22. Interaction, no sex effect, TA effect

  23. Assumptions of two-way ANOVA • Independence between groups. • Independence within groups. • Equal variances in all populations. • Normal distributions in all populations.

  24. Two-way ANOVA from scratch • Anemia is a serious third world nutrition problem. • In Ethiopia, traditional iron pots have mostly been replaced by aluminum pots. • A 1999 study randomly assigned 4 lots of three different types of food (meat, legumes, vegetables) to three types of pot (aluminum, clay, iron). • DV = iron content (milligrams iron per 100 grams of food).

  25. Hypotheses • H0 : mMeat = mLegumes = mVeggies . • H0 : mAluminum = mClay = mIron. • H0 : There is no interaction between pot material and type of food.

  26. Cooperative exercise • Calculate summary statistics. • Create an interaction plot. • Test all three null hypotheses. • Interpret the test results. • Evaluate the assumptions.

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