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

r2 = the r squared r2 = extent to which variables share

common properties or characteristics r2 = 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 r2 = .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 r2 = .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 r2 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.