CORRELATIONAL ANALYSES

EDRS 5305 EDUCATIONAL RESEARCH & STATISTICS

 Focus will be on the Pearson r, most commonly used correlation statistic.

 When reading research studies, likely to encounter studies in which rs are reported without reference to type of correlation.

 Most likely a Pearson r or Pearson product-moment correlation coefficient.

 Even though taught Pearson r should be used only on interval/ratio level data, some social science methodologists have argued, convincingly, that the Pearson r may be used even when data satisfy assumptions of ordinal data.

 Commonly used this way with questionnaire/survey data.

What is null hypothesis for r?

 Null hypothesis, in dealing with a single correlation, will simply be a pinpoint statement as to a possible value of the correlation in the population.

 Typically the pinpoint value is no relationship or a .00 correlation.

Meaning of Correlation

 r indicates degree of relationship between two variables  does NOT indicate the strength of association in the data  strength of association MORE important than degree of relationship

Strength of Association

 r 2 = the r squared  r 2 = extent to which variables share common properties or characteristics  r 2 = measure of the proportion of variability in one variable that can be determined from (explained by) the relationship of the other

 If r = .9, then r 2 = .81 or 81%  81% of the variance in variable A is explained or determined by B and 81% of the variance in variable B is explained or determined by A  In this case, the variables, 81% worth, are essentially measuring the same construct(s).

 If r = .5, then r 2 = .25 or 25%  25% of the variance in variable A is explained or determined by B and 25% of the variance in variable B is explained or determined by A  In this case, the variables, 75% worth, are essentially measuring different construct(s).

 Even if a correlation coefficient of .2 is statistically significant, it only accounts for, explains, or determines 4% of the variance---most likely [not always-- medical research] a trivial amount.

Reporting Correlation Results

 r value  sample size  p value  r 2 value  r (167) = -.63, p < .001, 39.69% of variance accounted for

Example of Reporting r Results

 A statistically significant relationship was found between students’ study skills and their locus of control, r (153) = -.63, p < .001. Squaring the correlation revealed that these two variables had 39.69% of the variance in common. Thus, students who exhibited good study skills tended to report more of an internal locus of control than did students with poor study skills.

Another Example

 Use of a Pearson r yielded a statistically significant relationship, r (235) = +.75, p < .01, between scores on the Wechsler IQ and Wechsler achievement measures. The IQ and achievement measures had 56.25% of variance in common, figures which are supported by previous researchers.

Another Example

 A Pearson r, calculated between scores on the Woodcock Basic Reading Test and the WIAT Basic Reading Test, was statistically significant, r (96) = +.35, p < .05. Even so, only 12.25% of the variance in test scores was shared by these measures in which the same construct is supposedly measured.

Reliability and Validity rs

 Can have statistically significant rs for reliabilities and validities that are NOT important NOR meaningful  Important to examine not only p level but, more importantly, the magnitude of the relationship  The lower the reliability of measuring instrument, lower the validity must be.

 For internal consistency reliability or Cronbach’s coefficient alpha, .9 is desirable.

 .9 means that 90% of the test score is true score variance and 10% is error.

 For test-retest reliability, .8 is desirable.

 .8 means that 80% of the test score is true score variance and 20% is error.  Remember that ERROR is always present.



For research purposes

 Nunnally (1978) stated that coefficient alphas above .75 may be viewed as evidence that a scale has acceptable reliability for use in research.