A psychologist investigating the relationship between SAT score and academic success found that students with high SAT score were more likely to experience academic success. You pointed out to her that students who have high IQ’s tend to have high SAT scores and also tend to experience academic success. You explain that the relationship between SAT score and academic success might be due to the fact that both academic success and SAT are related to IQ. If the psychologist has IQ scores for her sample, how could she estimate the relationship between SAT scores and grades for students who are equivalent on IQ?
Correlation is used to assess the relationship between two variables, as the researcher did when she looked at the relationship between academic success and GPA. When we introduce a third variable, simple correlation is no longer adequate.
One-way ANOVA can be used to test hypotheses regarding the equality of means for three or more groups. In this example, we do not have distinct groups to compare, but rather we are interested in the relationship between two variables while controlling for the effects of a third variable.
ANCOVA (Analysis of Covariance) is used to remove/control for the effects of confounding variables (such as IQ in relation to SAT score) when we have a categorical independent variable with three or more levels. All variables in this example are continuous.
We are interested in the relationship between two variables with the effects of a third variable removed.
We have continuous measures of all three variables. Partial correlation is a statistical technique that is used to control for the effects of one or more confounding variables. For our example, the procedure would statistically equate everyone at the same IQ level and then produce an estimate of the correlation between academic success and SAT score for people who are equivalent on IQ. The effects of IQ are said to be “removed,” leaving an estimate of correlation that is not affected by IQ levels. If the correlation remains high, then we could conclude that academic success is related to SAT scores even for people of equivalent IQ. If the correlation between SAT and success is large but the partial correlation is near zero, we would conclude that the correlation between SAT and success could be largely explained by differences in IQ levels.
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The Expert says…
We would like to assess the relationship between two variables with the effects of a confounding variable statistically removed. The general procedure is to ‘co-vary’ out the nuisance variable. If we had an analysis of variance model, we could use analysis of covariance.
When examining the relationship between two continuous variables, we can remove the effects of a third continuous variable with partial correlation.
Multiple regression could also be used, whereby the confounding variable would be entered into the model first. Then contribution of a second predictor variable is an indication of the relationship between this second predictor variable and the dependent Y variable with the effects of the confounding variable removed.
We should examine all of the two-way plots to assess whether the assumptions of correlation have been met. In particular, we should be alert for possible outliers, nonlinear relationships, or for uneven error variances.
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