Correlation

1
Correlation Causal Comparative Research

• Class 6

2
This Weeks Schedule

• Today Review and continue w/ statistical
  analysis
• Tuesday
• 3 individual meetings 9-10am
• Full class (Stats Method) 10-11am
• 3 individual meeting 11am-12noon
• Wednesday
• 9am-10am Music ed history (Skype w/ Eastman
  Class) Show and tell!!
• 1000am-1100am Qualitative Research-Full class
• 1100am-12noon-3 individual meetings

3
This Weeks Schedule

• Thursday
• 4 Project Presentations (20 minutes)
• 2 qualitative/historical 5ish minute
  presentations (in pairs a trio)
• Disseminating research
• Friday
• 5 Project Presentations
• 2 qualitative/historical 5ish minute
  presentations (in pairs a trio)

4
Assignments

• Tuesday Work on presentations, projects, etc.
• Wednesday
• Read Queen Bees and Wanna Bees chapt. 1 OR 6
• Read one historical article from the Journal of
  Historical Research in Music Education. Be
  prepared to write or discuss
• Chapter 3 Method
• Thursday Friday
• Project Presentations (20 minute w/ 5 minutes for
  questions discussion)
• Informal presentation in pairs of a qualitative
  or an historical article
• Monday, July 22 by 5pm Final Project Proposal

5
Who When

• Tuesday Meetings
• 9-10 (3)
• 11-noon (3)
• Weds Meetings
• 11-noon (3)

• Thursday
• Project Presentations (4)
• Qual./Hist. presentations (2 pairs)
• Friday
• 5 presentations
• Qual./Hist. trio presentation

6
APA Format

• Headers
• Chapter title Level 1
• Others Level 2 (Flush left)
• Remember title page, page s
• Running Head lt 50 characters total. Goes in the
  header flush left
• Research Question after purpose statement
  before need for study (Header?)
• Commas apples, oranges, and grapes.

7
Types of Data Revisedsimple to complex lowest
to highest

• Nominal/Categorical numbers as labels
• Male/female (1 or 2)
• Sop/Alto/tenor/bass (1, 2, 3, 4)
• Ordinal ranks
• Contest ratings
• Interval Scale (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.

8
Terms

• Inferential statistics
• Parametric vs. non-parametric
• Assumptions/Parameters
• Variances?
• Randomization
• Mean vs. variance
• Used to compare 2 groups and no more?
• Independent vs. dependent (paired or correlated)
• One tail vs. Two tail tests?

9
Terms

• What is I have more than 2 groups? I need a?
• If there is a significant difference in the test
  above, then what do I need to do?
• Why do we test the significance of the difference
  in variances?
• What if the variances are sig. different?

10
Statistical Significance

• Probability that result happened by chance and
  not due to treatment
• Expressed as p
• p lt .1 less than 10 (1/10) probability
• p lt .05 less than 5 (1/20) probability
• p lt .01 less than 1 (1/100) probability
• p lt .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

11
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

12
Statistical Tests

• http//pspp.awardspace.com/ (Windows)
• http//bmi.cchmc.org/resources/software/pspp
  (Mac)
• http//vassarstats.net/

13
See Handout from Friday

• Awareness of non-parametric tests
• 3 groups, ordinal data?
• 2 groups, interval data?
• 2 groups, nominal/categorical data?
• Relationship b/w two groups, ordinal data?

14
Independent Samples t-test

• Used to determine whether differences between two
  independent group means are statistically
  significant
• n lt 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 science tests of inst non-inst. students

15
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 lt 30 for each group
• Application Compare Reading Math test scores
  of Instrumental Students

16
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?

17
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)

18
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)

19
ANCOVA Analysis of Covariance

• Statistical control for unequal groups
• Adjusts posttest means based on pretest means.
• example http//faculty.vassar.edu/lowry/VassarSt
  ats.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

20
Effect Size (Cohens d) http//www.uccs.edu/facul
ty/lbecker/es.htm 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.
• 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 M24.6 SD10.7
• Control Group M10.8 SD7.77

21
Cohen's Standard Effect Size Percentile Standing Percent of Nonoverlap
  2.0 97.7 81.1
  1.9 97.1 79.4
  1.8 96.4 77.4
  1.7 95.5 75.4
  1.6 94.5 73.1
  1.5 93.3 70.7
  1.4 91.9 68.1
  1.3 90 65.3
  1.2 88 62.2
  1.1 86 58.9
  1.0 84 55.4
  0.9 82 51.6
LARGE 0.8 79 47.4
  0.7 76 43.0
  0.6 73 38.2
MEDIUM 0.5 69 33.0
  0.4 66 27.4
  0.3 62 21.3
SMALL 0.2 58 14.7
  0.1 54 7.7
  0.0 50 0
22
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

23
Correlation

• Pearson
• Spearman
• Cronbachs alpha (a)

24
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

25
Uses of Correlational Research

• Clarifying and understanding important phenomena
  (relationship b/w variablese.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)

26
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)

27
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)
• lt .30 small
• lt .65 moderate
• gt .65 strong
• gt .85 very strong
• r2 ( of shared variance)
• of 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

28
Application

• Rate your principal school quality on a scale
  of 1-7
• Principal (1highly ineffective 2ineffective
  3somewhat ineffective 4neither effective nor
  ineffective 5somewhat effective 6effective
  7highly effective
• School cleanliness (1very dirty 2dirty
  3somewhat dirty 4neither dirty or clean
  5somewhat clean 6clean 7very clean)
• Type of data? Calculation (Pearson or Spearman?)
• Reliability (Cronbachs alpha) www.gifted.uconn.ed
  u/siegle/research/.../reliabilitycalculator2.xls

29
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 (e.g.,
  pretest/posttest). Correlation also works w/
  different units of measurement.

50 75 9 40 62 14 35 53
20 24 35 45 15 21 58
30
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)
• Non linear relationship of of performances
  anxiety scores 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. (
• Presence of outliers (all)
• Ho/mo/sce/da/sci/ty relationship consistent
  throughout
• Performance anxiety levels off after several
  performances and remains static (relationship
  lacks Homoscedascity)
• Subjects have only one score for each variable

31
Correlational Approaches for Assessing
Measurement Reliability

• Consistency over time
• test-retest (Pearson, Spearman)
• Consistency within the measure
• internal consistency (split-half, KR-20,
  Cronbachs alpha)
• Spearman Brown Prophecy formula
• 2r/(1 r)
• Among judges
• Interjudge (Cronbachs Alpha)
• Consistency b/w one measure and another
• (Pearson, Spearman)

32
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 Cronbachs alpha
• Measure of internal consistency
• Extent to which responses on individual items
  correspond to each other

33
Spearman Brown Prophesy Formula

• Reliability ___n x r___
• 1(n-1)r
• nnumber of times we multiply items to get new
  test length (10 item to 20 item n2)
• For a test of 10 items w/ reliability (a) of .60
• (15 items) 1.5 x .60/1(1.5 - 1).60 reliability
  for test 1.5x size
• (20 items) 2 x .60/1(2-1).60 reliability for a
  test 2x size
• (25 items) 2.5 x .60/1(2.5 1).60 reliability
  for test 2.5x size
• (5 items) .5 x .60/1(.5 1).60 reliability
  for test .5 size