ANOVA ONE WAY REPEATED MEASURES AND CORRELATION, Schemes and Mind Maps of Psychological Data

Download ANOVA ONE WAY REPEATED MEASURES AND CORRELATION and more Schemes and Mind Maps Psychological Data in PDF only on Docsity! Lopez Reyes one-way ANOVA: between-subjects design 1. one independent variable (IV): nominal with two or more categories. These categories give rise to the different groups that are compared. 2. one dependent variable (DV): interval/ratio level of measurement (i.e., “scorable” or quantitative). Groups are compared with respect to their average DV scores. 3. between-subjects design: participants belonging to one group are all different from the participants belonging to the other groups. (i.e., we have a between-subjects design) 4. assumptions: a. random sampling; independent observations b. normal distributed populations c. homogeneity of (population) variance repeated measures ANOVA [one-way ANOVA: within-subjects design] 1. one independent variable (IV): nominal with two or more categories. These categories give rise to the conditions that are compared. 2. one dependent variable (DV): interval/ratio level of measurement (i.e., “scorable” or quantitative). Groups are compared with respect to their average DV scores. 3. repeated measures or within-subjects design: there is only one group of participants. This group undergoes all categories of the IV. 4. A block/matched design is analyzed in the same way was as a repeated measures design but the block/matched design is not covered in this handout. Overall F-Test Null hypothesis: H0: μ1=μ2=…=μk, where k = number of categories of the IV Alternative hypothesis: At least two population means that are not equal to each other. In other words, some of the population means differ. It is wrong to say that: HA: μ1≠μ2≠…≠μk Rule for rejecting the null hypothesis If the p-value corresponding to the F-statistics computed data is less than or equal to α then, reject the null hypothesis. If the p-value is greater than α, then do not reject the null hypothesis. Some post hoc t-tests (upon rejection of the overall F-test) When the null hypothesis for the overall F-test is rejected, we know that there are some population means are not equal, but the F-test does not indicate what these means are. To know which means are different, we can conduct multiple comparison tests. This handout discusses a specific kind: pairwise comparison tests, where means are compared two at a time. Examples of pairwise 1 Lopez Reyes comparison tests are the Tukey HSD (honestly significant difference) test and the LSD (least significant difference) test. Example 1: one-way analysis of variance (between-subjects design) Problem: Levin & Fox (2003), pp. 277, no. 6 On the following random samples of social class, test the null hypothesis that neighborliness does not vary by social class. (Note: Higher scores indicate greater neighborliness.) Data Entry: JAMOVI Output: ANOVA Table: NOTE: Ignore the intercept line 2 Social Class Neighborliness lower 8 lower 4 lower 7 lower 8 working 7 working 3 working 2 working 8 middle 6 middle 5 middle 5 middle 4 upper 5 upper 2 upper 1 upper 3 Lopez Reyes 5 Should the null hypothesis be rejected? No, do not reject the null hypothesis F(3, 12) = 2.72, MSE = 3.96, p = .09. 6 Are there significant differences among the means? No. There are no significant differences among the means. 7 Do you recommend doing pairwise comparison tests? No. Example 2: one-way analysis of variance (between-subjects design) Problem: Levin & Fox (2003), pp. 259 - 268 A researcher is interested in comparing the degree of life satisfaction among adults with different marital statuses. She wants to know if single or married people are more satisfied with life, and whether separated or divorced adults do in fact have a more negative view of life. She selects at random 5 middle- aged adults from each of the following four categories: widowed, divorced, never married, and married. The researcher then administers to each of the 20 respondents a 35-item checklist designed to measure satisfaction with various aspects of life. The scale scores range from 0 for dissatisfaction with all aspects of life to 35 for satisfaction with all aspects of life. Data Entry: Marital Status Life Satisfaction widowed 5 widowed 6 widowed 4 widowed 5 widowed 0 divorced 16 divorced 5 divorced 9 divorced 10 divorced 5 never married 23 never married 30 never married 20 never married 20 never married 27 married 19 married 35 married 15 married 26 married 30 5 Lopez Reyes JAMOVI Output: ANOVA Table (with assumption tests): NOTE: Ignore the intercept line = Table of Means: = Graph of Means: = Tukey Test: (to generate through JAMOVI, click on “post hoc tests”) Significant differences display (The values in the cells of the table are the p-values for the different pairwise comparison tests. There are 6 p-values, because there is a total of 6 pairwise comparisons.) = If we want to test all 6 pairwise comparisons, we arrive at the following conclusions: The mean life satisfaction of the following groups are not significantly different:  Widowed and divorced not significantly different (p=0.459959)  Never married and married not significantly different (p=0.990309) There are significant differences among the following means:  Widowed and divorced on the one hand and never married and married on the other. o On the average people who have never married or are married are more satisfied with their lives than are people who are widowed or divorced. (Based on the obtained sample means) Homogeneous groups display (This display reports the same tests as does the significant differences display.) In this display, so-called homogeneous groups are formed wherein means belonging to the same groups are not significantly different from each other and means belonging to different groups are significantly different. Thus in the following display: o means for widowed and divorced are not significantly different (they both belong to homogeneous group 1; note the asterisks under the column labeled 1) o means for never married and married are not significantly different (they both belong to homogeneous group 2; note the asterisks under the column labeled 2) 6 Lopez Reyes o means under homogenous group 1 are significantly different from means under homogeneous group 2 Questions regarding results of analyses: 1 What is the independent variable? Marital status 2 What is the dependent variable? Life satisfaction 3 What is the null hypothesis for the F- test? H0: μwidowed=μdivorced= μnevermarried=μmarried 4 What is the alternative hypothesis for the F-test? There are at least two population means that are unequal. 5 Should the null hypothesis be rejected? p =.000011 (In the ANOVA table, look at the row for marital status.) F(3, 16) = 20.24, MSE = 27.75, p = .00 6 Are there significant differences among the means? Yes. Therefore conclude the alternative hypothesis. Some marital status groups differ significantly in their mean life satisfaction scores. 7 Do you recommend doing pairwise comparison tests? Yes, so we know which marital statuses are different with respect to life satisfaction 8 Based on the Tukey test, which means are different? If there are differences, state which group has a higher life satisfaction mean. On the average, the never married and the married groups have higher life satisfaction means than the widowed and divorced groups. There is no significant difference between the means of the never married and the married groups. Likewise, there is no significant difference between the means of the widowed and the divorced groups. 9 Based on the Games-Howell test, which means are different? Same pattern of results as in Tukey. 7 Lopez Reyes Example 1: repeated measures ANOVA Problem: undocumented source (modified) Dion, Berscheid, and Walster (1972) examined the stereotypes we hold about attractive people. Subjects looked at four types of photographs of persons all unknown to them: one of a physically attractive person, one of a person of average attractiveness, one of an unattractive person, and one of a highly attractive person. Subjects rated each photographed person according to how socially desirable the person is. Higher scores reflect greater social desirability. Data Entry: Subject Unattractive Average Attractive Highly Attractive 1 35 40 65 68 2 39 39 74 75 3 45 46 47 55 4 50 50 65 67 5 30 40 72 72 6 37 33 74 78 7 50 49 69 75 8 44 45 55 60 9 59 60 49 62 10 49 49 57 60 11 51 53 65 65 12 49 48 62 66 13 47 47 67 68 14 47 48 66 70 15 36 37 59 59 16 39 40 70 75 10 Lopez Reyes JAMOVI Output: ANOVA Table NOTE: Use the p-value for “ATTRACTI”. Again, ignore the intercept line. Table of Means; Graph of Means Tukey Test 11 Lopez Reyes Post Hoc Tests Post Hoc Comparisons - Physical Attractiveness Comparison Physical Attractiveness. Physical Attractiveness Mean Difference SE af t Prukey Unattractive - Average 1.06 0.750 15.0 -142 0,509 - Attractive -19.31 3.404 15.0 567 <.001 - Highly Attractive ~28.00 2.920 15.0 -7:88 —<.001 Average - Attractive -18.25 3.282 15.0 “558 © <.001 Highly Attractive ~21.94 2.834 15.0 “7.74 <.001 Attractive - Highly Attractive 3.69 0.850 15.0 4.34 0,003 LSD test; variable DV_1 (Spreadsheet11) Probabilities for Post Hoc Tests. Error: Within MS = 53.599, df = 45.000 ATTRACTI {i} Py} 3} {4} Cell No. 44.188 | 45.250 | 63.500 | 67.187 Unattractive 0.683398 | 0.000000) 0.000000 Average 0.683398 0.000000) 0.000000 Attractivel 0.000000 0.000000 0.161165 Highly Attractive} 0.000000 0.000000 0.161165 BOM] LSD test; variable DV_1 (Spreadsheet11) Homogenous Groups, alpha = .05000 Error: Within MS = 53.599, df 000 ATTRACTI Dv_1 1,2 Cell No. Mean Unattractive| 44.18750) **** Averagel 45.25000 **** Attractive! 63.50000 ee Highly Attractive] 67.18750 eee Bape Tukey HSD Test Tukey HSD test; variable DV_1 (Spreadsheet11) Homogenous Groups, alpha = .05000 Error: Within MS = 53.599, df = 45.000 ATTRACTI DV_1 1|2 Cell No. Mean Unattractive] 44.18750 "* Average] 45.25000 **"* Attractive] 63.50000: wee Highly Attractive] 67.18750 wee BO) ]e Lopez Reyes Table of Means = Graph of Means = Tukey Test = Questions regarding results of analyses: 1 What is the independent variable? 2 What is the dependent variable? 3 What is the null hypothesis?? H0: 4 What is the alternative hypothesis? 5 Should the null hypothesis be rejected? 6 Are there significant differences in mean scores among the different tests? 7 Do you recommend doing multiple comparison tests? 8 Based on the Tukey HSD test, which means are different? 9 Based on the LSD test, which means are different? 15 Lopez Reyes Exercises Problem 1 In a field experiment, cars either had all-male passengers, all-female passengers, both male and female passengers, male driver with no passenger, female driver with no passenger (15 instances per type of passengers). All cars in the field experiment didn’t go when a traffic light turned green. The experimenters measured how many seconds elapsed before the driver of the following car honked the horn. Can the experiment data be analyzed using one-way ANOVA (between-subjects)? Yes Problem 2 Local officials wanted to study the evaluations of various stakeholders (homeowners, business establishments, and community maintenance workers) of the management of an existing community waste recycling program. A questionnaire was administered to samples of stakeholders, 30 respondents per type of stakeholder for a total of 90 respondents. When scored, the questionnaire yields a score ranging from 10 to 50, with higher scores indicating more positive evaluation. One-way ANOVA Problem 3 Data were obtained from 100 students who took all of the following courses: a creative writing course, a technical writing course, and a grammar course from a review center. Analysis is conducted to determine whether mean grades from the three courses differ significantly. Can these data be analyzed using repeated-measures ANOVA? Why or why not? Yes. One group (students) with three conditions (courses) to be compared to each other. It satisfied the data assumption. Problem 4 As a study of healthy and positive human processes, the relatively new field of positive psychology maintains that human happiness is attainable and that many individuals in fact have achieved and maintained an adequate level of happiness. Researchers in positive psychology give empirical evidence on what makes people happy (cf. Sheldon & King, 2001). One possible predictor of human happiness is whether one is positively, negatively, or neutrally engaged in one’s main occupation, with positive engagement associated with greater happiness and negative engagement associated with lesser happiness. Suppose that in past research you already had created (a) a 100-point scale for measuring human happiness and (b) a scheme for classifying individuals as positively, neutrally, or negatively engaged in one's occupation. 16 Lopez Reyes Think of one study, in which you will use the happiness scale and the classification scheme, that would test whether different types of involvement are associated, on the average, with varying degrees of happiness. For your study: 1. State the variables of your study, indicating whether they are dependent or independent variables and what their levels of measurement are. Independent variable – classification scheme/ nominal data Dependent variable – happiness scale/ interval data 2. What statistical test/s will you use? One-way ANOVA (Between-subjects) 3. Enumerate the statistics you will you generate. Means Variances Degrees of freedom Sums of squares Mean squares F-statistic Critical value/ p-value 4. What statistical result would indicate that individuals with different types of engagement in their occupation, on the average, differ in their extent of experienced happiness? F-statistic Critical value/ p-value 5. If there are differences among these three types of individuals in their extent of experienced happiness, what statistical result would indicate which type is happiest and which type is least happy? Tukey HSD test LSD test Mean of each k 17