Yvonne L. LaMar
Correlation and Causality
Correlational Research refers to studies in which the purpose is to discover relationships between variables through the use of correlational statistics (r). The square of a correlation coefficient yields the explained variance (r-squared). A correlational relationship between two variables is occasionally the result of an outside source, so we have to be careful and remember that correlation does not necessarily tell us about cause and effect. If a strong relationship is found between two variables, causality can be tested by using an experimental approach.
Advantages of the Correlational Method
The correlational method permits the researcher to analyze the relationships among a large number of variables in a single study. The correlation coefficient provides a measure of degree and direction of relationship. Correlations do not have to be positive to be important, we'll discuss that a little later.
Uses of the Correlational Method
Explore relationships between variables.
Predict scores on one variable from subject’s scores on other variables
Planning A Relationship Study
Basic Research Design
The primary purpose is to identify the causes of effects of important phenomena.
*Defining the problem - identify specific variables that may be important determinants of the characteristics or behavior patterns being studied.
*Review of existing literature is helpful in identifying variables.
*Selection of research participants - only those who can be measured by the variables being investigated.
*Data collection - must be in quantifiable form.
*Data analysis - correlate scores of measured variable (x) that represent the phenomena of interest with scores of a measured variable (y) thought to be related to that phenomena.
Interpretation
One problem with interpretation is the shotgun approach which is when a large number of variables are measured and analyzed without a justifiable rationale for their inclusion. This approach can lead to inconveniencing participants and higher expenses for the time and number of measuring tools or methods. One way to avoid this is to do preliminary research to esrtablish that the variables that you intend to use are the most relevant to your purpose
Limitations of Relationship Studies
1) Correlations do not establish cause and effect relationships between variables.
2) Correlations break down complex relationships into simpler components.
3) Success in many complex activities can probably be achieved in different ways.
Planning a Prediction Study
Prediction studies provide three types of information:
- the extent to which a criterion behavior pattern can be predicted.
- data for developing a theory about the determinants of the criterion behavior pattern.
- evidence about the predictive validity of the test or tests that were correlated with the criterion behavior pattern.
Basic Research Design
1) The problem - this will reflect the type of information that you are trying to predict
2) Selection of research participants - draw from the specific population most pertinent to your study.
3) Data collection - predictor variables must be measured before the criterion behavior pattern occurs.
4) Data Analysis - primary method is to correlate each predictor criterion with the criterion
Useful Definitions
bivariate correlational statistics - expresses the magnitude of relationship between two variables.
multiple regression - uses scores on two or more predictor variables to predict performance on the criterion variables.
Statistical Factors in Prediction Research
Group Prediction
Prediction research is useful for practical selection purposes.
selection ratio - proportion of the available candidates who must be selected.
base rate- percentage of candidates who would be successful if no selection procedures were applied.
Taylor-Russell Tables
combine three factors; predictive validity, selection ratio, and base rate.
Shrinkage
This is the tendency for predictive validity to decrease when a research study is repeated.
Bivariate Correlational Statistics
Product Moment Correlation, r
“r” is computed when the variables that we wish to correlate are expressed as continuous scores.
Correlation Ratio, eta
This computation is used when the relationship between two variables is non-linear.
Adjustments to Correlation Coefficients
Correction for Attenuation
Provides an estimate of what the correlation between the variables would be if measures had perfect reliability.
Correction for Restriction in Range
Applied when researcher knows that the range of scores for a sample is restricted on one or both of the variables being correlated. This application requires the assumption that the two variables are linearly related throughout the entire range.
Part and Partial Correlation
This application is employed to rule out the influence of one or more measured variables upon the criterion in order to clarify the role of other variables.
Multivariate Correlational Statistics
These are used when examining the interrelationship of three or more variables.
Multiple Regression
This method is used to determine the correlation between the criterion variable and a combination of two or more predictor variables. It can be used to analyze data from any quantitative research design.
Multiple correlation coefficient
measure of the magnitude of the relationship between a criterion variable and some combination of predictor variables.
Coefficient of determination - r-squared
expresses the amount of variance that can be explained by a predictor variable or combination of predictor variables.
Discriminant analysis
also involves two or more predictor variables and a single criterion variable, but, is limited to the case where the criterion is a categorical variable (dichotomous?).
Canonical correlation
a combination of several predictor variables is used to predict a combination of several criterion variables.
Path Analysis
Used to test the validity of theories about causal relationships between two or more variables that have been studied in a correlational research design.
Step One: formulate a hypothesis that causally link the variables of interest.
Step Two: select or develop measures of the variables.
Step Three: compute statistics that show the strength of relationship between each pair of variables that are causally linked in the hypothesis.
Step Four: interpret statistics to determine whether they support or refute the theory.
Correlation matrix
an arrangement of rows and columns that make it easy to see how each measured variable in a set of such variables correlates with all the other variables in the set.
Recursive model
considers only unidirectional causal relationships.
Non-recursive
used to test hypotheses that involve reciprocal causation between pairs of variables.
Factor Analysis
Provides an empirical basis for reducing numerous variables that are moderately or highly correlated with each other. A factor represents the variables that are most correlated.
Loading
the individual coefficients of each variable on the factor.
Factor score
a score given for subjects when each factor is treated like a variable
Orthogonal solution
. when factor analysis yields factors that are not correlated with each other.
Oblique solution
when factor analysis yields factors that do correlate with each other.
Structural Equation Modeling, LISREL
Also known as latent variable causal modeling, tests theories of causal relationships between variables and supplies more reliable and valid measures than path analysis.
Latent variables
theoretical constructs of interest in the model
Manifest variables
variables that were actually measured by the researchers.
Interpretation of Correlation Coefficients
Statistical Significance of Correlation Coefficients
Indicates whether the obtained coefficient is different from zero at a given level of confidence. If the coefficient is statistically significant different from zero, the null hypothesis from zero cannot be rejected.
Interpreting the Magnitude of Correlation Coefficient
The closer to one that the correlation coefficient is the stronger the relationship between two variables. The closer to zero, the weaker the relationship. If the correlation coefficient is a negative number, the magnitude is the same only in the opposite direction.
Mistakes Sometimes Made in Doing Correlational Research
The researcher:
-assumes that correlation is proof of cause-effect
-relies on shotgun approach
-selects statistics that are inappropriate
-limit analyses to bivariate when multivariate would be more appropriate
-does not conduct cross-validation study
-uses path analysis or structural equation modeling without checking assumptions
-fails to specify an important causal variable in planning a path analysis
-misinterprets the practical or statistical significance in a study