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Understanding One-Way ANOVA in Psychological Research: Comparing Multiple Groups

This material covers the concept of One-Way Independent ANOVA as a statistical method for comparing differences among three or more groups or conditions. It explains how ANOVA generalizes t-tests, allowing researchers to test hypotheses involving multiple categories, such as different therapies or treatments. The lecture also delves into key aspects such as variance measures, the F-distribution, and conditions for homogeneity of variance. The application of ANOVA in real studies, including its assumptions and reporting results, is systematically presented for better understanding.

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Understanding One-Way ANOVA in Psychological Research: Comparing Multiple Groups

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  1. PSYC 6130 One-Way Independent ANOVA

  2. Generalizing t-Tests • t-Tests allow us to test hypotheses about differences between two groups or conditions (e.g., treatment and control). • What do we do if we wish to compare multiple groups or conditions simultaneously? • Examples: • Effects of 3 different therapies for autism • Effects of 4 different SSRIs on seratonin re-uptake • Effects of 5 different body orientations on judgement of induced self-motion. PSYC 6130, PROF. J. ELDER

  3. Reinterpreting the 2-Sample t-Statistic PSYC 6130, PROF. J. ELDER

  4. Reinterpreting the 2-Sample t-Statistic PSYC 6130, PROF. J. ELDER

  5. Example PSYC 6130, PROF. J. ELDER

  6. F distribution for 2 groups of size n=13 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 The F Distribution PSYC 6130, PROF. J. ELDER

  7. Within and Between Variances • Recall that the variance is, by definition, the mean squared deviation of scores from their mean. • Since the numerator of the t2 statistic estimates the variance from the deviations of group means, it is called the mean-square-between MSbet. • Since the denominator of the t2 statistic estimates the variance from the deviations within groups, it is called the mean-square-within MSW. • These definitions allow us to generalize to an arbitrary number of groups. PSYC 6130, PROF. J. ELDER

  8. Generalizing to > 2 Groups PSYC 6130, PROF. J. ELDER

  9. Degrees of Freedom • Recall that the sample variance follows a scaled chi-square distribution, parameterized by the degrees of freedom. • Thus the F distribution is a ratio of two chi-square distributions, each with different degrees of freedom. PSYC 6130, PROF. J. ELDER

  10. F distribution for 3 groups of size n=13 1 0.8 p(F) 0.6 0.4 0.2 0 0 2 4 6 8 10 Testing Hypotheses PSYC 6130, PROF. J. ELDER

  11. PSYC 6130, PROF. J. ELDER

  12. When k=2 • ANOVA will give exactly the same result as two-tailed t-test. • One-tailed tests must be done using t-tests. PSYC 6130, PROF. J. ELDER

  13. Example From the Canadian Generalized Social Survey, Cycle 6 (1992) PSYC 6130, PROF. J. ELDER

  14. Example PSYC 6130, PROF. J. ELDER

  15. Reporting Results • A one-way ANOVA demonstrates that frequency of contact with clinical psychologists depends on marital status. Widowed individuals had the least contact (M=0.082). Married individuals (M=0.185) had somewhat more contact. Single (M=0.620) and separated or divorced (M=0.900) had substantially more contact. F(3,11807)=33.3, MSE = 7.8, p<.001. PSYC 6130, PROF. J. ELDER

  16. Summary Table (SPSS) PSYC 6130, PROF. J. ELDER

  17. Interpreting the F Ratio PSYC 6130, PROF. J. ELDER

  18. Effect Size and Proportion of Variance Accounted For PSYC 6130, PROF. J. ELDER

  19. (Approxiately) Unbiased Effect Size PSYC 6130, PROF. J. ELDER

  20. Reporting Results • A one-way ANOVA demonstrates that frequency of contact with clinical psychologists depends on marital status. Widowed individuals had the least contact (M=0.082). Married individuals (M=0.185) had somewhat more contact. Single (M=0.620) and separated or divorced (M=0.900) had substantially more contact. F(3,11807)=33.3, p<.001. However, the size of the effect was relatively small: PSYC 6130, PROF. J. ELDER

  21. Planning a Study: ANOVA and Power PSYC 6130, PROF. J. ELDER

  22. Example • You are interested in whether there is a link between PSYC 6130 final grades and the professor teaching the section. • Grades typically have a standard deviation of about 15% • There are typically 3 sections, each with around 12 students. • What is the probability you would pick up an effect if the standard deviation of the mean grade is around 5%? PSYC 6130, PROF. J. ELDER

  23. Advantages of ANOVA • Avoid inflation in error rate due to multiple comparisons • Can detect an effect of the treatment even when no 2 groups are significantly different. PSYC 6130, PROF. J. ELDER

  24. 6-Step Process for ANOVA • State the hypotheses • Select the statistical test and significance level • Select the samples and collect the data • Find the region of rejection • Calculate the test statistic • Make the statistical decision PSYC 6130, PROF. J. ELDER

  25. Sums of Squares Approach PSYC 6130, PROF. J. ELDER

  26. ANOVA Assumptions • Independent random sampling • Normal distributions • Homogeneity of variance PSYC 6130, PROF. J. ELDER

  27. More on Homogeneity of Variance PSYC 6130, PROF. J. ELDER

  28. Levene’s Test: Basic Idea • SPSS reports an F-statistic for Levene’s test • Allows the homogeneity of variance for two or more variables to be tested. PSYC 6130, PROF. J. ELDER

  29. What to do if Homogeneity of Variance Assumption is Rejected • Some adjustment procedures are available in SPSS (e.g., Welch 1951). • We will not cover the theory behind these adjustments. PSYC 6130, PROF. J. ELDER

  30. Fixed vs Random Effects • Fixed Effects: interested only in the specified levels of the independent variable (e.g., single/married/divorced/widowed) • Random Effects: interested in a large number of possible levels of the independent variable – randomly sampling only a few of these. e.g., • Does the order of questions on a questionnaire effect the results? • Does the order of stimuli in a psychophysical experiment effect the results? PSYC 6130, PROF. J. ELDER

  31. Fixed vs Random Effects • One-Way Independent ANOVA calculation is the same for fixed and random effect designs. • Power and effect size calculations differ. • More complex ANOVA designs differ. • We restrict our attention in this course to fixed effect designs. PSYC 6130, PROF. J. ELDER

  32. Qualitative vs Quantitative Independent Variables • In principle, ANOVA can be applied to either qualitative or quantitative variables. • If IV is quantitative and effect is roughly linear, usually have more power using regression (only using up 2 degrees of freedom, instead of k). • If effect is complex (e.g., non-monotonic): • Use a higher-order regression model (e.g., quadratic) • Use ANOVA (makes no smoothness assumptions) PSYC 6130, PROF. J. ELDER

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