Inferential Statistics & Correlational Research

1. Inferential Statistics & Correlational Research Days 5 & 6

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. Statistics Review

4. Measurement ScalesLevels of Measurement [NOIR] Nominal Categorical, frequency counts (gender, color, yes/no, etc.) Ordinal Rank-order (Contest Ratings, Likert data??) Interval Continuous scale with consistent distances between points. No meaningful absolute zero (test scores, singing range, Fahrenheit or Celsius temperature, knowledge). Ratio Continuous scale with consistent distances between points and an absolute zero (decibels, money, age) Examples: N-choir robes; O – div. 1 at contest; I – 96/100 score; R – Kelvin scale for temperature (has absolute 0)

5. Normal Curve/Distribution(+/- 1 = 68.26%, 2 = 95%, 3 = 99.7%)

6. Altogether.. describes the shape of a distribution More on distributions.. Normal Curve (bell curve) Most scores clustered at the middle with fewer scores falling at the extreme highs and lows Kurtosis - When the distribution is… …PEAKED = positive kurtosis = leptokurtic = greater than +2 or +3 depending on who you ask… …SMALLER PEAK (flatter) THAN A NORMAL CURVE = negative kurtosis = platykurtic = less than -2 or -3 Skewness - When the scores tend to bunch up… …on the HIGH END = Negative Skew = less than -1 …on the LOW END = Positive Skew = greater than +1 Bi-Modal When there are two humps in the curve, more than one mode

8. Standard Error of the Mean & Confidence Limits • SEM is the standard deviation of infinite number of sample means • Example: If I tested 30 students on music theory • Test 0-100 • Mean 75; SD 10 • Standard Error of the Mean (SEM) would estimate average SD among any number of same size samples taken from the population • SEM = SD/sq root N • Calculate for example on the left. • 95% Confidence Interval • 95% of the area under a normal curve lies within roughly 1.96 SD units above or below the Mean (rounded to +/-2) • 95% CI = M + or – (SEM X 1.96) • 99% CI = M + or – (SEM X 2.76)

9. Confidence Limits/Intervalhttp://graphpad.com/quickcalcs/CImean1.cfm • Attempts to define range of true population mean based on Standard Error estimate. • Confidence level • 95% chance vs. 99% chance • Confidence Limits • 2 numbers that define the range • Confidence Interval • The range b/w confidence limits

10. Confidence • Calculate at a 95% confidence level • ACT Explore scores • Confidence Limits • Interval • How close to your scores represent true population?

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 (Levene’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 .05 standard for significance Only reason to set higher bar is for multiple comparisons Bonferroni Correction Statistical significance does not equal practical significance

14. Review-Hypothesis Testing Research Hypothesis (H) The predicted outcome of the research study The chunking method will be more effective than the whole song method in teaching a song by rote Null Hypothesis (Ho) An outcome in which no difference/relationship exists There will be no difference in effectiveness between the chunking and whole song methods (value free)

15. 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

16. 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

17. Statistical Tests - Parametric http://faculty.vassar.edu/lowry/VassarStats.html

18. 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)

19. 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) GO TO: http://vassarstats.net/

20. Independent Samples t-test[see data set] 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]

21. Correlated (paired, dependent) Samples t-test [see data set] Used to determine whether 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

22. ANOVA(use ACT Explore test data from Day 4) • 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 • Difference b/w subject tests for 2007 Non-band students? (correlated) Implications? • Difference b/w reading scores for non-band students 2007/2008/2009 • 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 • Bonferroni correction – reduce alpha to compensate for multiple comparisons. (.05/N comparisons) • Tukey – Equal Sample Sizes (though can be used for unequal sample sizes as well) [HSD = honest significant difference] • Sheffe – Unequal Sample Sizes (though can be used for equal sample sizes as well)

24. Two Way ANOVA(2X2) [see data set day 5]

25. Interpreting Results of 2x2 ANOVA • (columns) CAI was more effective than Traditional methods for both high and low achieving students • (rows) High Achieving students scored significantly higher than Low achieving students, regardless of teaching method • There was no significant interaction between rows & columns • If there was significant interaction, we would need to do post hoc Tukey or Sheffe do determine where the differences lie.

26. ANCOVA – Analysis of Covariance[data set day 5] • Statistical control for unequal groups • Adjusts posttest means based on pretest means. • [example – see data set] 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]

27. Effect Size (Cohen’s d)http://www.uccs.edu/~lbecker/ • [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. • Effect sizes can also be interpreted in terms of the percent of nonoverlap of the treated group's scores with those of the untreated group. • 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: • Pretest: M=10.8; SD= 5.7 • Posttest: M=24.6; SD=4.3

28. Effect Size (d) Interpretation

29. Correlational Research

30. Correlational Research Basics Relationships among two or more variables are investigated The researcher does not manipulate the variables Direction (positive [+] or negative [-]) and degree (how strong) in which two or more variables are related

31. Uses of Correlational Research Clarifying and understanding important phenomena (relationship b/w variables—e.g., height and voice range in MS boys) Explaining human behaviors (class periods per weeks correlated to practice time) Predicting likely outcomes (one test predicts another)

32. Uses of Correlation Research • Particularly beneficial when experimental studies are difficult or impossible to design • Allows for examinations of relationships among variables measured in different units (decibels, pitch; retention numbers and test scores, etc.) • DOES NOT indicate causation • Reciprocal effect (a change in weight may affect body image, but body image does not cause a change in weight) • Third (other) variable actually responsible for difference (Tendency of smart kids to persist in music is cause of higher SATs among HS music students rather than music study itself)

33. Interpreting Correlations r Correlation coefficient (Pearson, Spearman) Can range from -1.00 to +1.00 Direction Positive As X increases, so does Y and vice versa Negative As X decreases, Y increases and vice versa Degree or Strength (rough indicators) < + .30; weak < + .65; moderate > + .65; strong > + .85; very strong r2 (% of shared variance) Overlap b/w two variables percent of the variation in one variable that is related to the variation in the other. Example: Correlation b/w musical achievement and minutes of instruction is r = .86. What is the % of shared variance (r2)? Easy to obtain significant results w/ correlation. Strength is most important

34. Interpreting Correlations (cont.) Words typically used to describe correlations Direct (Large values w/ large values or small values w/ small values. Moving parallel. 0 to +1 Indirect or inverse (Large values w/small values. Moving in opposite directions. 0 to -1 Perfect (exactly 1 or -1) Strong, weak High, moderate, low Positive, Negative Correlations vs. Mean Differences Groups of scores that are correlated will not necessarily have similar means. Correlation also works w/ different units of measurement. 50 75 9 40 62 14 35 53 20 24 35 45 15 21 58

35. Statistical Assumptions The mathematical equations used to determine various correlation coefficients carry with them certain assumptions about the nature of the data used… Level of data (types of correlation for different levels) Normal curve (Pearson, if not-Spearman) Linearity (relationships move parallel or inverse) Young students initially have a low level of performance anxiety, but it rises with each performance as they realize the pressure and potential rewards that come with performance. However, once they have several performances under their belts, the anxiety subsides. (non linear relationship of # of performances & anxiety scores) Presence of outliers (all) Homoscedascity – relationship consistent throughout Performance anxiety levels off after several performances and remains static (relationship lacks Homoscedascity) Subjects have only one score for each variable Minimum sample size needed for significance

36. Correlational Approaches for Assessing Measurement Reliability Consistency over time test-retest (Pearson, Spearman) Consistency within the measure internal consistency (split-half, KR-20, Cronbach’s alpha) Spearman Brown Prophecy formula 2r/(1 + r) Among judges Interjudge (Cronbach’s Alpha) Consistency b/w one measure and another (Pearson, Spearman)

37. Reliability of Survey • What broad single dimension is being studied? • e.g. = attitudes towards elementary music • Preference for Western art music • “People who answered a on #3 answered c on #5” • Use Cronbach’s alpha • Measure of internal consistency • Extent to which responses on individual items correspond to each other

38. Examples • Calculate the Pearson correlation between each a subject test and combined score on the ACT Explore for 2007-2009. (each take one subject) • Calculate a Spearman Correlation for Contest ratings each judge vs. final rating • Calculate internal consistency (reliability) of all three judges using Cronbach’s alpha. • http://www.wessa.net/rwasp_cronbach.wasp

39. Other Stats

40. Chi-Squaredhttp://vassarstats.net/ • Measure statistical significance b/w frequency counts (nominal/categorical data) • Test for independence/association: Compare 2 or more proportions • Example: Proportion of females to males in choirs at 4 district HSs. • Goodness of Fit: compare w/ you have with what is expected • Example: Proportions of females to males in choir compared to school population

41. Method Discussions