Chapter 11Correlational Research

Correlational Studies• Correlational Study: An approach to research

that involves measuring different variables to see whether there is a predictable relation among variables

• Finding Relationships– The goal of some correlational studies is simply to

find out if variables are related– Is there a relation between whether students

prefer early or late classes and their grades in those classes?

Are You a Morning Person?Question• Do student grades relate to whether they are

“morning people”?MethodThe researchers correlated two kinds of

variables• Grades in early and late classes• Scores on a scale of “morningness” (13-item

inventory)

Are You a Morning Person?Morningness Scale Sample Items

Assuming normal circumstance, how easy do you find getting up in the morning?At what time in the evening do you feel tired and, as a result, in need of sleep?How long a time does it usually take before you "recover your senses" in the morning after rising from a night's sleep?

Source: Smith, C. S., Reilly, C., & Midkiff, K. (1989). Evaluation of three circadian rhythm questionnaires with suggestions for an improved measure of morningness. Journal of Applied Psychology, 74, 728-738.

Are You a Morning Person?

ResultThere was a modest, significant, correlation

between morningness and grades at 8:00 a.m., r(312) = .21, p < .01

Students who were not morning people did less well at 8:00 and performed better in later classes.

Are You a Morning Person?Conclusion• The pattern did not hold for classes starting at

9:00 or later. So even “evening people” can do well in early classes.

• The study is correlational, so we do not know if being a morning person is the cause of better grades in the morning.

• Source: Guthrie, J. P., Ash, R. A., & Bendapudi, V. (1995). Additional validity evidence for a measure of morningness. Journal of Applied Psychology, 80, 186-190.

Using the Correlational Approach• Correlational Studies involve relationships

among variables. It is not possible to make determinations of causation with a correlational study.

Using the Correlational ApproachProblems in Determining Causes with Correlational

StudiesThird Variable Problem--When two variables are correlated, you cannot say that one causes the other because there may be a third variable you do not know about that affects either or both of the variables you are investigating.Directionality Problem--When two variables are related, you do not know if the first causes the second or the second causes the first.

Correlational Studies• Making Predictions

– Prediction Study: A correlational study in which the goal is to predict the value of one variable (the criterion variable), given the level of another variable (the predictor variable) with the predictor variable occurring first.

• Example: Colleges try to predict the GPA of prospective students (the criterion variable), given their SAT scores (the predictor variable).

– Predictions of behavior are never perfect, but knowing the relation between the predictor variable and the criterion variable will improve the predictions made.

Using the Correlational Approach

• Correlational Analysis refers to the way data are analyzed and has nothing to do with the design of a study.– Experimental data can be subjected to

correlational analysis.– Relations that emerge can be associated with

cause and effect if the study is experimental, even if analysis is done with correlational statistics.

Using the Correlational Approach• Positive and Negative Correlations• With a positive (direct) correlation, when the value of

one variable increases, so does the value of the second.– Example: Perceived control over life is positively

correlated with life satisfaction: People who feel in control have greater life satisfaction.

• With a negative (indirect) correlation, when the value of one variable increases, the value of the second variable decreases.– Example: There is a negative correlation between how

many negative events happens in people’s lives and their feeling of control: People who go through many negative events have a feeling of less control in their lives.

Source: Lang, F. R., & Heckhausen, J. (2001). Perceived control over development and subject well-being: Differential benefits across adulthood. Journal of Personality and Social Psychology, 81, 509-523.

Using the Correlational Approach• Strength of Association

– The greater the absolute value of a correlation, the stronger the relation between variables (e.g., -.60 represents a stronger association between variables than +.50)

– By changing the way you score and label one variable, you can turn a negative correlation into a positive correlation, even though the data set does not change.

Using the Correlational Approach

Fictitious data showing the relation between (a) time spent studying for a test and percent correct [r = +.58] and (b) time spent studying for a test and percent wrong [r = -.58]. The scatter diagrams are based on the same data and tell the same story, but the correlation changes from positive in (a) to negative in (b) because the way the variable is scored (percent correct versus percent wrong) changes.

Using the Correlational ApproachFactors Affecting the Size of a Pearson Product-

Moment Correlation– There may be a relation between two variables,

but it may appear less strong than it really is– When the data show certain characteristics, the

correlation coefficient may diminish even when two variables are related

Using the Correlational ApproachFactors Affecting the Size of a Pearson Correlation Coefficient

Nonlinearity—When two variables do not have a linear relation, the Pearson r will be diminish relative to a situation when there is a linear relation.

Restricted Range—When measurements on one variable do not span a large range of values, the value of the Pearson r will be smaller.

Heterogeneous subsets—When the set of data consists of subgroups that show different patterns, the Pearson r might not reflect the pattern within those subsets.

Using the Correlational Approach

• The scatter diagram here shows a perfect relation between two variables, but the value of the Pearson r is 0 because the relation between variables is nonlinear.

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Nonlinear Relation Between Variables

Using the Correlational Approach

• The scatter diagram for the fictitious data here shows a positive relation between two variables, but the value of the Pearson r will be lower if the range of values for Verbal SAT score is restricted. For the full range of scores, r = .45 but for data in the restricted range (shown within the circle), the value of the correlation drops, r = .29

Restrictions in Range

Using the Correlational Approach

• The scatter diagram for the fictitious data here shows an overall positive relation between two variables, but each of the subsets has a negative correlation. When you combine subgroups into one overall data set, the correlation coefficient may not provide useful information.

Heterogeneous Subsets

Traditional Correlational TestsVarieties of the Correlation Coefficient

Pearson Product-Moment Correlation (Pearson r)–The most widely used correlation coefficient for bivariate correlations. It is typically cited for data that are on interval or ratio scales.

Spearman Correlation for Ranks (rs)–Correlation coefficient often used when data are ranked on an ordinal scale.

Point-Biserial Correlation–Correlation when one data for one variable are continuous and for the second variable are dichotomous (i.e., they can take on two values, like male-female or yes-no).

Phi Coefficient (Φ)–Correlation used when data for both variables are dichotomous.

Note: Although some sources present different formulas for these different types of correlation coefficients, they are all variations on the Pearson r, so you can simply use the regular formula for the Pearson r for all of them without bothering to learn different formulas (Howell, 2007) .

Linear Regression: Making PredictionsLinear Regression

– You can make predictions using a linear regression formula when you have one predictor variable that you use to estimate the value of a criterion variable.

– The formula is based on the simple formula for a straight line:

Y = a + bX,where

Y is the value of the criterion variable, a is a constant (the Y-intercept), b is the slope of

the line, and X is the value of the predictor variable

Linear Regression: Making PredictionsQuestion• Can you predict how funny a person will find

jokes if you know how good that person’s sense of humor is?

Method• Ask people to rate how funny they find a set

of jokes• Have them complete a sense of humor scale.

Linear Regression: Making Predictions

Result• Linear regression analysis showed that knowing

the sense of humor score helped generate reliable predictions of joke ratings: people with better sense of humor provided higher ratings.

• Predicted joke rating = Constant + Slope x Sense of Humor Score• a = constant (Y-intercept) = 1.84 • b = slope = 0.76

Linear Regression: Making PredictionsOne person had a sense of humor score of 4.A second person had a score of 2.Predicted rating = Intercept + (Slope x score)Predicted rating = 1.84 + (0.76 x 4) = 1.84 + 3.04 = 4.88Predicted rating = 1.84 + (0.76 x 2) = 1.84 + 1.52 = 3.36

So we would conclude that on the seven-point scale, a person with a sense of humor score of 4 would rate jokes a full point and a half higher than would a person with a sense of humor score of 2.

Source: Benfante, L., & Beins, B. C. (2007, October). Self-reflection and sense of humor: The big five personality characteristics and humor. Poster presentation at the annual convention of the New England Psychological Association, Danbury, CT.

Linear Regression: Making Predictions

Conclusion• If you know how much of a sense of humor a

person has based on a paper-and-pencil test, you can make reliable predictions about how that person will rate jokes.

• People with a better sense of humor seem to enjoy jokes more.

Correlations with Multiple Variables• Zero-order correlation: A correlational

analysis involving two variables.• Higher-order correlation: A correlational

analysis involving more than two variables– Multivariate statistics: Statistical approaches that

can accommodate simultaneous analysis of multiple variables

– Multiple regression: A correlational technique that employs more than one predictor variable to estimate the value of a criterion variable

Prediction with Multiple Variables• Question: Can we predict how funny a

person thinks he or she is based on how they rate jokes and on how extraverted they are?

• Method: Administer sense of humor and extraversion inventories and ask participants to rate a set of jokes.• Predictor variables: Extraversion score and

joke ratings• Criterion variable: Self-rating of funniness (1 =

not funny to 10 = very funny)

Prediction with Multiple Variables• Results: Multivariate analysis revealed that

knowing how extraverted people are and how they rate a set of jokes allows us to predict how funny they think they are.

Prediction with Multiple VariablesUse of Multiple Regression to Predict a Person’s Self-Rating of Funniness

Predictor Variable 1 = Extraversion score (Score of 1 to 5)Predictor Variable 2 = Average rating a person gave to the jokes (Score of 1 to 7)

Criterion Variable = Predicted self-rating of funniness

Constant (Y-intercept) = 3.48Variable 1 (Extraversion) Slope = 0.521Variable 2 (Joke rating) Slope = 0.348

Participant’s Extraversion Score = 3Participant’s Average Joke Rating = 2

Regression Equation:Y = Constant + (Slope1 x Score1) + (Slope2 x Score2) Y = 3.48 + (0.521 x 3) + (0.348 x 2) = 5.74

A person with an Extraversion score of 3 and an average joke rating of 2 would be predicted to give a self-rating of 5.74 for funniness on a scale of 1 to 10.

Source: Benfante, L., & Beins, B. C. (2007, October). Self-reflection and sense of humor: The big five personality characteristics and humor. Poster presentation at the annual convention of the New England Psychological Association, Danbury, CT.

Prediction with Multiple Variables• Conclusion: We can predict people’s self-

perceptions of how funny they are by assessing their behavior (i.e., how they rated jokes) and their personality (i.e., extraversion score). Knowing something about an individual’s personality and their actual enjoyment of jokes allows us to make reliable predictions about how they perceive themselves.

Controversy: The Mental Health of College Students

• College students generally have good mental health, but reported stress levels have been increasing.

• Can we predict suicidal ideation in order to minimize actual suicide attempts?

Controversy: The Mental Health of College Students

• Using multivariate statistics, Gutierrez et al. (2000) found • a positive relation between negative themes in a

student’s life and thoughts of suicide• a negative relation between types of social

relations and thoughts of suicide.

• These results could help a counselor spot potential problems before they become real problems.

Controversy: The Mental Health of College Students

Multiple regression has identified several factors associated with the likelihood of students seeking help

Gender (women are more likely)

Perceived stigma associated with seeking help (Less perceived stigma is associated with greater likelihood)

Level of emotional openness (greater openness is associated with greater likelihood)

Severity of symptoms (more severe symptoms are associated with greater likelihood)

Limitations of Multivariate Correlational Analysis

Limitations of Multivariate Correlational Analysis

Generalizability • Representative samples are necessary for confidence in generalizing beyond the sample• It is not always clear that results based on the measurements made in a given study will generalize to other measurements• Generalizations might not hold up over timeConfirmation Bias• Creating a model leads to only one possible model; researchers may fail to test alternate models