Advanced ANOVA Procedures

Transcription

1 Advanced ANOVA Procedures Session Lecture Outline:. An example. An example. Two-way ANOVA. An example. Two-way Repeated Measures ANOVA. MANOVA. ANalysis of Co-Variance (): an ANOVA procedure whereby the researcher intentionally partitions out the influence of one or more covariates. Covariate: a variable that has a substantial correlation with the DV, and is included in the experiment to adjust results for differences existing among subjects before the start of the experiment. Two major applications: 1. In a true experiment involving random assignment where there is a nuisance variable that is adding noise to the interpretation.. Where it is not possible to conduct a true experiment, can be used to control for variables that may differ between groups. Hypothetical example: A researcher wants to compare driving proficiency on 3 different sizes of cars to test the experimental hypothesis that small cars are easier to handle. Driving Proficiency Mini 0 Sedan 18 4WD 1 Because drivers recruited for the experiment had very different levels of experience, variability was so great that a general one-way ANOVA would fail to reject the null hypothesis. Most variability in performance is attributable to driving experience, which is not related to the question at hand. procedures enable the researcher to partial out the variation that can be attributed to driving experience (covariate). 1

2 The procedure adjusts the differences between group means resulting from the covariate, and then applies ANOVA to the adjusted data. Assumptions: There are many stringent assumptions that should be met before applying an. These can be found in SPSS documentation and on the web. Output: The output from an is the same as that for an ANOVA. can be used in all ANOVA designs. Example: A researcher was interested in determining whether the fitness levels of individuals changed as a result of three types of interventions (aerobic exercise, anaerobic exercise, and no exercise). The weights of the participants was found to vary between the three conditions at the start of the study so the researcher chose to control for participant weight by making it a covariate. Descriptive statistics: Adjusted means with weight held as a covariate. Dependent Variable: FITNESS Mean Std. Deviation N no exercise anaerobic exercise aerobic exercise Dependent Variable: FITNESS 5% Confidence Interval Mean Std. Error Lower Bound Upper Bound no exercise 5.0 a anaerobic exercise a aerobic exercise. a a. Evaluated at covariates appeared in the model: WEIGHT =.1111.

3 ANOVA Output The results of the indicated that there were significant differences among the three adjusted means, F(,5) = 54., p <.001. Tests of Between-Subjects Effects Dependent Variable: FITNESS Type III Sum Source of Squares df Mean Square F Sig. Corrected Model 84.6 a Intercept WEIGHT Error Corrected Post hoc tests could be used to follow up this finding. a. R Squared =.86 (Adjusted R Squared =.) : where the same group is tested more than two times. The procedure is an extension of the Paired Samples t-test, where the means of more than two related samples are compared simultaneously. Hypotheses: H 0 : at the population level, the means of the repeated measures are equal. H 1 : at the population level, at least two of the means of the repeated measures are not equal. Example: A researcher wished to investigate whether balance training would significantly decrease injuries in professional gymnasts. Gymnasts were measured four times across the course of the balance training intervention. The following results were obtained. Observation of results: The following Box plot was constructed using SPSS to provide a preliminary illustration of the data. 0 Trial1 Trial Trial3 Trial4 Participant Participant 11 6 Participant 3 5 Participant 4 4 Participant 5 8 Participant Participant 5 N = TRIAL1 TRIAL TRIAL3 TRIAL4 3

4 The box plot appears to indicate that there is a significant drop in injuries in trial 4. : TRIAL1 TRIAL TRIAL3 TRIAL4 Mean Std. Deviation N Test for Sphericity Assumption Test of Within Subjects Effects Mauchly's Test of Sphericity b Measure: MEASURE_1 Epsilon a Approx. Greenhous Within Subjects Effect Mauchly's W Chi-Square df Sig. e-geisser Huynh-Feldt Lower-bound Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. a. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table. b. Design: Intercept Within Subjects Design: Tests of Within-Subjects Effects Measure: MEASURE_1 Type III Sum Source of Squares df Mean Square F Sig. Sphericity Assumed Greenhouse-Geisser Huynh-Feldt Lower-bound Error() Sphericity Assumed Greenhouse-Geisser Huynh-Feldt Lower-bound Results indicate that there is a significant difference between at least two of the means: F(3,18) = , p <.05. From here you would go on to conduct paired samples t-tests to determine which groups differed significant using Bonferroni correction (.05 / number of tests) to cover for error inflation. Two-way ANOVA compares the means of populations that are classified in two ways. Useful if there are two important independent variables (IV s), and the researcher wants to know if these two IV s influence the dependent variable. 4

5 Factors In experimental designs that incorporate two or more IV s, the IV s are called factors, and the designs are called factorial designs. Factorial designs are labeled according to how many IV s are being investigated, and by how many different existing levels each factor contains. Hypothetical example: A researcher wants to investigate the effect of gender and primary sport on the number of hours competitor s train each week. There are levels of gender, and 3 different sports, making a X3 ANOVA (6 different groups). Numbers refer to the mean number of training hours per week. Runners Cyclists Swimmers Male Female Hypothetical example (): A researcher wants to investigate the effect of student age and year in course on overall grade average across the year. There are two levels of age, and 4 different years, making a X4 ANOVA (8 different groups). Mature Age 1 st year nd year 3 rd year 13 4 th year So there are a total of 3 tests: a test for the effect of A, a test for the effect of B, and one for their interaction (AB). Interactions: E.g. are the number of hours of training per week jointly influenced by both the competitor s sport and their gender? E.g.. are the grades of students jointly influenced by their age and their year of study? From school 11 Assumptions: The two-way ANOVA assumptions are the same as the one-way ANOVA assumptions, randomness, an interval/ratio scale of measurement, normality, and Levene s test of variance. If the assumptions are broken they are also handled in the same way. Output: Three pieces of information are generated: 1. There is a test for the main effect of the first variable (gender or age).. There is a second test for the main effect of the B factor (sport or year of study). 3. Finally, there is a test that determines if these two variables interact with one another. 5

6 So if there is no significance in the interaction you may look at the main differences (effects) for significance. You can also get a visual representation of the two variables by creating an interaction graph of the two groups. However if the interaction if the two variables is significant then you only need to report that. This type of graph plots the means of the two groups to outline any trends in the data. Example: Are there differences in students' writing scores based on their gender and levels of self-esteem? Note: writing scores is the dependent variable and gender and self-esteem are the independent variables. Dependent Variable: Write score Gender Self Esteem Mean Std. Deviation N Female Low self esteem High self esteem Male Low self esteem High self esteem Low self esteem High self esteem Test of Between Subject Effects Provides information about the two main effects and the interaction. Sample Plot Estimated Marginal Means of Write score 300 Tests of Between-Subjects Effects 0 Dependent Variable: Write score Type III Sum Source of Squares df Mean Square F Sig. Corrected Model a 3 Intercept GENDER ESTEEM GENDER * ESTEEM Error Corrected (Adjusted R Squared.) a. R Squared =.1 = Estimated Marginal Means Low self esteem Self Esteem Gender Female Male High self esteem 6

7 The results indicate that the interaction effect was not significant: F(1, 6) = 1.06, p =.31. In this instance, both main effects did not reach significance either (although gender came close). The two-way repeated measures ANOVA is like the one-way repeated measures ANOVA except that you can have more than one independent variable. This form of ANOVA compares the average change of the groups to see if at least one has significantly changed from the rest. If a significant difference is detected, then post hoc tests would outline which of the groups differed. Assumptions: Random selection Interval ratio data Normality Variance calculated by F-max not greater than 3 (variance of largest group / smallest) Mauchly s test of sphericity The interpretation is very similar to the twoway ANOVA in that you need to examine the interaction effects. You can also examine the main effects. MANOVA Multivariate Analysis Of Variance (MANOVA): an analysis of variance procedure where there is more than one DV.