Inferential Statistics

1. Inferential Statistics Class 5

2. For Monday • Submit first draft of your literature review (Chapter 2). • It should read like one document rather than a series of abstracts. • Use transition sentences. • Group studies into a logical order. • You should include at least ten research-based articles/dissertations. • Include a Reference list in perfect APA format.

3. Review: Effect of Intensive Instruction on Elementary Students’ Memory for Culturally Unfamiliar Music (2013) • Previous researchers have found that both adults and children demonstrate better memory for novel music from their own music culture than from an unfamiliar music culture. It was the purpose of this study to determine whether this “enculturation effect” could be mediated through an extended intensive instructional unit in another culture’s music. Fifth-grade students in four intact general music classrooms (two each at two elementary schools in a large U.S. city) took part in an 8-week curriculum exclusively concentrated on Turkish music. Two additional fifth-grade classes at the same schools served as controls and did not receive the Turkish curriculum. Prior to and following the 8-week unit, all classes completed a music memory test that included Western and Turkish music examples. Comparison of pretest and posttest scores revealed that all participants (N = 110) were significantly more successful overall on the second test administration. Consistent with previous findings, participants were significantly less successful remembering items from the unfamiliar music culture, a result that was consistent across test administrations and between instruction and control groups. It appears that the effect of enculturation on music memory is well established early in life and resistant to modification even through extended instructional approaches.

4. Identify or State: • Independent Variable • Dependent Variable • Treatment Group • Control Group • Diagram experimental design (O & X) • Write a hypothesis & null hypothesis • Paraphrase findings • Implications for the classroom? Did the authors reject or not reject the null hypothesis?

5. Internal Validity (Usefulness/Meaningfulness) -Control of Extraneous Variables: Time Bound Factors • What happens within the experiment • History – What happens b/w pretest and posttest (private lessons, change in practice routine) • Maturation – is change result of treatment natural result of repetition and improvement over time?) • Mortality – Loss of participants may cause imbalance b/w groups

6. Internal Validity – Sampling & Measurement Factors • Testing – pretest affect posttest. Ceiling and floor effects (eliminate outliers?) • Instrumentation – changes in measurement or observers (judges at contest from one site to the next) • Statistical regression – students who score extremely high (ceiling) or low (floor) on pretest may regress to the mean on posttest • Selection – participants do not represent normal population (also affects external validity) • Interactions – influence of a combination of the above factors

7. Internal Validity • John Henry Effect • Control group performs beyond usual level because they perceive they are in competition with the experimental group

8. External Validity – Generalizability Population Validity Extent sample is representative of the population to which the researcher wishes to generalize the results. Ecological Study conditions and setting are representative of the setting in which the researcher would like to apply the findings Replication Results can be reproduced (problem w/ Mozart effect) Detailed description of the sample needed in study Important regardless of sampling method ‘Next best thing’ if not a large, random sample – often the case in music ed. research Consider demographic questions in descriptive research

9. Other Threats to External Validity • Effect or interaction of testing (testing will not occur in natural setting) • Sample does not reflect population • Discuss in research report • Reactive effects of sample • Hawthorne Effect • Effects due simply to subjects’ knowledge of being in a study • Teacher or Researcher interactions different than in population • Subconsciously encouraging or discouraging a group • Research setting does not reflect typical settings (ecological validity) • A university lab school

10. Types of Data • Nominal/Categorical = numbers as labels • Male/female • Sop/Alto/tenor/bass • Ordinal = ranks • Contest ratings • Interval = equal distance b/w each number • Contest scores (1-100) • Lack of meaningful zero (0 on test = no knowledge?, 0 temperature = arbitrary) or meaningful ratios (2x as smart?) • Ratio = • Equal interval data • True zero possible (0 decibels, 0 money) • Ratios can be calculated in a meaningful way [2x as loud, ½ money, height, weight, depth (a lake can dry up) (?), etc.]

11. Inferential Statistics • Statistic = number describing a variable • Descriptive statistics = describe population • Inferential statistics = used when making inferences about a population based on the sample • Stat. used based on type of data and other assumptions • Stats used to compare and find differences

12. Two Types of Inferential Stats • Parametric • Interval & ratio data • Normal or near normal curve (distribution) • Equal variances (Levin’s test) • Sample reflects pop. (randomized) • Most powerful • Non-Parametric • Nominal & Ordinal data • Not normal distribution (skewness or kurtosis) • Unequal variances • Less powerful • More conservative

13. Statistical Significance Probability that result happened by chance and not due to treatment Expressed as p p < .1 = less than 10% (1/10) probability… p < .05 = less than 5% (1/20) probability… p < .01 – less than 1% (1/100) probability… p < .001 – less than .1% (1/1000) probability… Computer software reports actual p alpha level = probability level to be accepted as significant set b/f study begins Statistical significance does not equal practical significance

14. Statistical Power Likelihood that a particular test of statistical significance will lead to the rejection of null hypothesis Parametric tests more powerful than nonparametric. (Par. more likely to discover differences b/w groups. Choice depend on type of data) The larger the sample size, the more likely you will be to find statistically significant effects. The less stringent your criteria (e.g., .05 vs. 01 vs. 001), the easier it is to find statistical significance

15. Review-Type I and Type II Error Type I Error is erroneously claiming statistical significance or rejecting the null hypothesis when in fact, it’s true (claiming success when experiment failed to produce results) Possible w. incorrect statistical test Or when conducting multiple tests on same data (i.e. comparing 2 groups on multiple variables (achievement test parts). [solution, lower alpha level] Type II Error is when a researcher fails to reject the null hypothesis when it is in fact false The smaller the sample size, the more difficult it is to detect statistical significance In this case, a researcher could be missing an important finding because of study design

16. Statistical Tests http://pspp.awardspace.com/ (Windows) http://bmi.cchmc.org/resources/software/pspp (Mac) http://vassarstats.net/

17. Parametric Assumptions • Interval Data • Normality - Scores are normally distributed in each group • Homogeneity of Variance - The amount of variability in scores is similar between each group (Levin’s test) • If assumption are not met – Use non-parametric statistics • Ordinal or nominal data • Likert scales (esp. short scales) • Small sample size

18. One- vs. Two-Tailed Tests If a hypothesis is directional in nature it is one-tailed The chunking method will be more effective than the whole song method If a hypothesis is not directional in nature it is two-tailed There will be a no difference in effectiveness between the chunking method and the whole song method Two-tailed tests are most commonly used since specific hypotheses are rare in music education research. If study is designed knowing that results can only go one direction (e.g., beginning violin), a one tail test is OK. If treatment can only lead to positive results (improvement) use a one tail test. If treatment could result in positive or negative results, use a two tail test. One Tailed test more powerful. If your experiment led to improvement but a two tail test only comes close to significance, try a one tail test. (specify which you used in your study)

19. Independent Samples t-test Used to determine whether differences between two independent group means are statistically significant n = < 30 for each group. Though many researchers have used the t test with larger groups. Groups do not have to be even. Only concerned with overall group differences w/o considering pairs [A robust statistical technique is one that performs well even if its assumptions are somewhat violated by the true model from which the data were generated. Unequal variances = alternative t test or better Mann-Whitney U] Application: Explore Data Compare reading tests of inst & non-inst. students

20. Correlated (paired, dependent) Samples t-test Used to determine differences between two means taken from the same group, or from two groups with matched pairs are statistically significant e.g., pre-test achievement scores for the whole song group vs. post-test achievement scores for the whole song group Group size must be even (paired) N = < 30 for each group Application: Compare Reading & Math test scores of Instrumental Students

21. Compare 2 means • Need sample of at least 10 • Work like Independent and dependent t tests • Independent • Mann Whitney U • Application: Data set #3. Is there a sig. diff. b/w Final ratings at Site 1 vs. site 2? • Pairs or dependent samples • Wilcoxon signed ranks • Application: Data set #2. Is there a sig. difference b/w rating of judges 1 & 2?

22. ANOVA • Analyze means of 2+ groups • Homogeneity of variance • Independent or correlated (paired) groups • More rigorous than t-test (b/w group & w/i group variance). Often used today instead of T test. • F statistic • One-Way = 1 independent variable • Two-Way/Three-Way = 2-3 independent variables (one active & one or two an attribute)

23. One-Way ANOVA • Calculate a One-Way ANOVA for data-set 1 – All non-instrumental tests • Post Hoc tests • Used to find differences b/w groups using one test. You could compare all pairs w/ individual t tests or ANOVA, but leads to problems w/ multiple comparisons on same data • Tukey – Equal Sample Sizes (though can be used for unequal sample sizes as well) • Sheffe – Unequal Sample Sizes (though can be used for equal sample sizes as well)

24. Non-Parametric ANOVAs • Friedman – Related (correlated) Samples • Application: Data Set #2 – Sig. dif. b/w judges? • Kruskal-Wallis – Independent Samples • No post hoc equivalent to Tukey or Sheffe. Music do series of Mann-Whitney U or Wilcoxon for each pair of groups • Bonferroni Correction • Used to adjust α (p) for multiple comparison • .05/N comparisons

25. 2 Way Factorial Designs (2 independent variables [often one manipulated, one attribute)

26. Interpreting Results of 2x2 ANOVA • (columns) Kodaly was more effective than Traditional methods for both bilingual and non-bilingual students • (rows) Bilingual students scored significantly higher than non-bilingual students, regardless of teaching method • Could be a significant interaction between language and teaching method • If there was significant interaction, we would need to do post hoc Tukey or Sheffe do determine where the differences lie.

27. Three Way (2x2x2) ANOVA

28. ANCOVA – Analysis of Covariance • Statistical control for unequal groups • Adjusts posttest means based on pretest means. • [example] http://faculty.vassar.edu/lowry/VassarStats.html • [The homogeneity of regression assumption is met if within each of the groups there is an linear correlation between the dependent variable and the covariate and the correlations are similar b/w groups]

29. Effect Size (Cohen’s d) http://www.uccs.edu/~faculty/lbecker/es.htm • [Mean of Experimental group – Mean of Control group/average SD] • The average percentile standing of the average treated (or experimental) participant relative to the average untreated (or control) participant. • Use table to find where someone ranked in the 50th percentile in the experimental group would be in the control group • Good for showing practical significance • When test in non-significant • When both groups got significantly better (really effective vs. really really effective! • Calculate effect size: • Treatment group: M=24.6; SD=10.7 • Control Group: M=10.8; SD=7.77

30. Chi-Squared • Measure statistical significance b/w frequency counts (nominal/categorical data) • http://www.quantpsy.org/chisq/chisq.htm • Test for independence: Compare 2 or more proportions • Goodness of Fit: compare w/ you have with what is expected • Proportions of contest ratings (I, II, III or I & non Is) • Agree vs. Disagree • Weak statistical test