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Chapter 13: Introduction to Analysis of Variance

Chapter 13: Introduction to Analysis of Variance. The One-Factor Independent Measures Design: Part 2. The F ratio: The test statistic for the ANOVA. F = variance between treatments ………………….. F = D ifferences due to treatment effect + differences due to chance

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Chapter 13: Introduction to Analysis of Variance

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  1. Chapter 13: Introduction to Analysis of Variance The One-Factor Independent Measures Design: Part 2

  2. The F ratio: The test statistic for the ANOVA • F = variance between treatments ………………….. • F = Differences due to treatment effect + differences due to chance differences due to chance • F = treatment effect + individual differences + experimental error individual differences + experimental error • F = TE + ID + EE ID + EE Bigger the treatment effect, the ……………… the F value Bigger the differences due to chance, the ……………… the F value.

  3. Table 13-2 (p. 406) Hypothetical data from an experiment examining learning performance under three temperature conditions.

  4. ANOVA formulas: Overview • F = variance between treatments variance within treatments • Recall that s2 = ………….. • See Fig. 13.5 on p. 399 which summarizes what has to be calculated. • ………………

  5. Analysis of Sum of Squares • Calculational method is presented in text (and in Fig. 13.6 p. 400). • I will focus on deviation method. • SS total = total sum of squares. • = ∑(X – GM)2 • …………… • ………………. • SS between treatments. Use all treatment means & GM • = ∑(M – GM)2 x n per sample …………………………. ……………………….. • SS within treatments • = ∑(X – M)2 • ……………… • ……………….. • …………………

  6. Analysis of df • Follows same pattern as SS analysis • For room temperature example df total = • For room temperature df within = ……… • For room temperature df between = …… • ………………………..

  7. Figure 13-7 (p. 411) Partitioning degrees of freedom (df) for the independent-measures analysis of variance.

  8. Calculation of Variances (Mean Squares --MS)& F ratio • In Anova use term Mean Square (MS) instead of variance • Mean square – mean of the squared deviations • SS = sum of squared deviations • MS = SS/df • F = MS between • MS within

  9. Calculation of Variances (Mean Squares --MS)& F ratio • MS Between = SS between /df • = …………… • MS within = SS within /df • = ………………… • Note this is the same value as ………… • F = MS between = ………….. • MS within • Organize in ANOVA SUMMARY TABLE

  10. Evaluating the F value using the F Distribution • Compare obtained F to required F value which depends upon the F distribution. • The F distribution, in turn, depends upon the degrees of freedom for ....................... • Next slide shows part of the F table • How to read • 2 slides down shows F distribution for 2 and 12 degrees of freedom.

  11. Table 13-3 (p. 414) A portion of the F distribution table. Entries in roman type are critical values for the .05 level of significance, and bold type values are for the .01 level of significance. The critical values for df = 2.12 have been highlighted (see text).

  12. Figure 13-8 (p. 413) The distribution of F-ratios with df = 2.12. Of all the values in the distribution, only 5% are larger than F = 3.88, and only 1% are larger than F = 6.93.

  13. Final Steps in ANOVA • After organizing into ANOVA summary table we can make our statistical conclusion and our behavioural interpretation • Statistical conclusion • F (2, 12) required = • F (2, 12) obtained = • So, we …………………….. Ho • Behavioural interpretation (in APA style) • Descriptives (usually in text, table, or a figure) • Inferential (include a report of F value in APA style)

  14. Characteristics of the F Distribution • F values are always positive • Because ………………… • Distribution of F ratios pile up around 1 • Most frequently occurring F value is ……… • > # of scores, > df, closer the expected value of F is to … • Smaller the # of scores, ……………………………. • Larger the dfs, less the value of F required …………

  15. Another Example (different values than in text): The effect of drug treatments on the amount of time (in seconds) a painful stimulus is endured. • Placebo Drug A Drug B Drug C • 0 0 3 8 • 0 1 4 5 • 3 2 5 5 • __________________________________

  16. STEP 1: STATE Hs AND CHOOSE ALPHA LEVEL • Ho: u1=u2=u3=u4. • Samples are all drawn from same population • No treatment effect …………………. • No effect of the drugs on …………………………… • H1: At least 1 of the population means is different from ………….. • …………………….. one of the treatments conditions is more effective for pain tolerance • Choose alpha = .05

  17. STEP 2: DETERMINE CRITICAL VALUE OF F(region for rejecting Ho). • Depends on df Between treatments, df Within treatments, and alpha level • df Between treatments. = • df Within treatments = • Calculate also by calculating df for each group & adding together ………………………… • alpha = .05 • F required (3,8) = …………… • Obtained F must be …………………….

  18. STEP 3: CALCULATE VALUE OF F STATISTIC. • Start with blank ANOVA summary table • Calculate SS values (any order) • SS total • SS between • SS within • Calculate df values • df total • df between • df within • Calculate MS values • MS between treatments • MS within treatments • Calculate F • MS between treatments • F = ---------------------------------- • MS within treatments • F = …………………………

  19. ANOVA SUMMARY TABLE • Source SS df MS F • Between treatments (drug) …………………. • Within treatments (error) …………………….. • Total ……….. __________________________________________ • *p < .05. F required ……………………

  20. STEP 4: MAKE STATISTICAL DECISION • F required …………………….. • F obtained ………………………….. • Reject H0 • of no difference between treatments conditions in the population. • Conclude that there is a difference between treatments in the population, • at least 1 treatment mean……………………… • Need additional comparisons to decide ………………

  21. STEP 5: BEHAVIOURAL INTERPRETATION • At this point, we can only say that there is a significant difference among the drug treatments in pain tolerance. • Additional analyses are necessary to determine …………………… • Pairwise comparisons (later).

  22. Measuring effect size with ANOVA • Remember difference between statistical significance and practical significance. • With a statistically significant effect, we usually measure …………… • Calculate r2 but now called η2 ……………….. • η2 is the percentage (or proportion) of the variation between (all) scores that is accounted for by …………………………………….. • η2 = SS between • ……………. • For previous example = …………………….. • 77 percent of the variation in scores (# seconds withstand pain) is accounted for by …………………………………….

  23. APA reporting of ANOVA • Descriptive statistics (followed by inferential) • E.g. Means and standard deviation in …………….. • Don’t duplicate information ………………. • Note that you can easily calculate variance for each treatment condition: s2 = ……………….. • and standard deviations: s = ……..

  24. Reporting F (APA style) • For pain killer study: Would present means and SD in Table or Figure. • A one-factor, independent measures ANOVA was calculated on the …………………………………………………………….. • The ANOVA indicated a significant difference in pain tolerance in the four conditions ………………………. • Subsequent comparisons would expand upon this. • Would indicate exactly …………………. • Next week’s lab.

  25. Conceptual View of ANOVA • Example 13.2 (p. 412) has no difference btw. treatment means (hence no TE), hence F = ………………………………… • Remember the bigger the differences btw. means, the bigger ……………………………………. • Example 13.3(a) has small Within treatments variance — F = ………………………. • Example 13.3(b) has same difference (12 – 8 = 4) btw. treatment means as 13.3(a), but larger ………………………….. • F = 1.39 is ………………………(since denominator of F ratio is relatively large.) • Remember, the smaller the within treatment variances, the smaller the numerator, the ……………………………. • Illustrated in graphs on p. 414 also ………………………………………………..

  26. MS Within and Pooled Variance • Major point is that the denominators of both t-ratio and F-ratio are based on an average of ………………………………….. • MS Within Treatments in F-test is like the denominator in the ………………………………………………. • Pooled (or average) ………………………………………. • Variance measures the amount of noise or confusion in the data ( as in Figs. on p. 414). • Greater the variance, greater the …………………………… • MS within is known as the ……………………….. • Differences within a treatment must be due to ……………………………

  27. Unequal n example • No problem • Just adjust values of n and df as appropriate. • You work through example in text (p. 416 – 417)

  28. Pairwise Comparisons • General ANOVA simply reveals whether or not …………………….. • If we get a significant F value, we must then determine EXACTLY which …………………………………………………….. • See next slide for my pain killing data • Could use multiple t-tests, but problem with ……………………… • Test-wise vs. …………………………………………….. • Post-hoc tests have been developed that attempt to control the ………………………………… alpha level. and, therefore, the probability of making a ………………………… • Most Post hoc tests do this by ………………………………….for declaring two means significantly different. • 2 examples given in your text: Tukey's HSD and the Scheffe test. • We will concentrate on the Tukey HSD test.

  29. Another Example (different values than in text): The effect of drug treatments on the amount of time (in seconds) a painful stimulus is endured. • Placebo Drug A Drug B Drug C • 0 0 3 8 • 0 1 4 5 • 3 2 5 5 • __________________________________

  30. Tukey’s HSD Method of Pairwise comparisons • Essence is to compute a single value which represents the ………………………………………. that is required for the difference between any pair of means to be considered significant. • This ………………………… is called the honestly significant difference (HSD). • If difference between means exceeds HSD value then we assume that the …………………………………. • Otherwise assume …………………………………. • How to do pairwise comparisons using Tukey’s HSD values. • Work through example for my pain killing data • Anova table re-presented on next slide.

  31. ANOVA SUMMARY TABLE • Source SS df MS F • Between treatments (drug) 54 3 18 9.00* • Within treatments (error) 16 8 2 • Total 70 11 • __________________________________________ • *p < .05. F required (3,8) = 4.07

  32. Reporting Results in APA style • Report means in the text or a Table or a Fig. • Then report ANOVA in APA style (example provided for my pain killer data)

  33. The Relationship between Independent Measures ANOVA & independent measures t-test • With 2 independent groups only F = ….. • Comparing the two tests. • See next slide for an example • With 2 independent groups, the t-test & F test are …………………………….

  34. Figure 13-11 (p. 431) The distribution of t statistics with df = 18 and the corresponding distribution of F-ratios with df = 1,18. Notice that the critical values for  = .05 are t = ±2.101 and that F = 2.1012 = 4.41

  35. ASSUMPTIONS FOR INDEPENDENT MEASURES ANOVA • Same as for independent measures t-test • 1) Observations within each sample ………………….. • That is, one score does not affect …………………………. • 2) Populations from which samples are selected must be ………. • More important with very small …………………… • 3) Populations from which samples are selected should have roughly equal ………………………….. • ……………………. assumption. Test with ……………….. • Do F max test for room temperature and learning data. • 4) Measurement of the dependent variable should be on an …………………………. scale of measurement.

  36. Interpreting SPSS output • An example will be worked in class (if time) • You will have done in your labs also.

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