PSYC 6130 One-Way Independent ANOVA

PSYC 6130, PROF. J. ELDER 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 The F Distribution F distribution for 2 groups of size n=

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 t 2 statistic estimates the variance from the deviations of group means, it is called the mean-square-between MS bet. Since the denominator of the t 2 statistic estimates the variance from the deviations within groups, it is called the mean-square-within MS W. 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 Properties of the F Distribution n=2 n=5 n=10 n= n=2 n=5 n=10 n=100

PSYC 6130, PROF. J. ELDER 11 The F Statistic

PSYC 6130, PROF. J. ELDER 12 Testing Hypotheses p(F) F distribution for 3 groups of size n=13

PSYC 6130, PROF. J. ELDER 13

PSYC 6130, PROF. J. ELDER 14 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 15 Example From the Canadian Generalized Social Survey, Cycle 6 (1992)

PSYC 6130, PROF. J. ELDER 16 Example

PSYC 6130, PROF. J. ELDER 17 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 18 Summary Table (SPSS)

PSYC 6130, PROF. J. ELDER 19 Interpreting the F Ratio

PSYC 6130, PROF. J. ELDER 20 Effect Size and Proportion of Variance Accounted For

PSYC 6130, PROF. J. ELDER 21 (Approxiately) Unbiased Effect Size

PSYC 6130, PROF. J. ELDER 22 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 23 Planning a Study: ANOVA and Power

PSYC 6130, PROF. J. ELDER 24 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 25 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 26 6-Step Process for ANOVA 1.State the hypotheses 2.Select the statistical test and significance level 3.Select the samples and collect the data 4.Find the region of rejection 5.Calculate the test statistic 6.Make the statistical decision

PSYC 6130, PROF. J. ELDER 27 Sums of Squares Approach

PSYC 6130, PROF. J. ELDER 28 ANOVA Assumptions Independent random sampling Normal distributions Homogeneity of variance

PSYC 6130, PROF. J. ELDER 29 More on Homogeneity of Variance

PSYC 6130, PROF. J. ELDER 30 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 31 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 32 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 33 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 34 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)