Bivariate Correlational Research Notes

Bivariate Correlational Research

Key Terms

• Mean: The average of a set of numbers.
• Effect sizes: A quantitative measure of the magnitude of a phenomenon.
• Statistically significant: The likelihood that a relationship between two or more variables is caused by something other than random chance.
• Outlier: A data point that differs significantly from other data points in a sample.
• Restriction of range: Occurs when there is a limited range of scores on one of the variables in a correlational study, which can underestimate the true association between the variables.
• Curvilinear association: A relationship between two variables that is not linear; it curves.
• Directionality problem: In correlational research, the difficulty in determining which variable causes the other.
• Moderator: A variable that affects the relationship between two other variables.

Associations Between Variables

• Associations describe relationships between variables.
• Bivariate association: An association that involves exactly two variables.
  • Example: Marital satisfaction vs. Meeting Context (Online vs. Offline).
  • Cacioppo et al., 2013:
    • Asked 19,000 responders about whether they met their spouse online or offline.
    • Asked them to indicate their degree of happiness in their marriage using a 7-point rating scale (1 = extremely unhappy, 7 = perfect).
    • People who met online scored a little higher on the CSI (Funk and Rogge, 2017).
• Dispositional greed and satisfaction with life:
  • Hoyer et al., 2022 recruited 2,367 Dutch adults aged 16 to 95.
  • Used Pavot & Diener’s 1993 subjective well-being scale (LiSS).
  • $r = -.14$ between greed and well-being.

Describing Associations Between Two Quantitative Variables

• Using scatterplots and the correlation coefficient $r$. E.g.,
  • Relationship between deep talk and well-being, $r = .26$.
  • Correlation between greed and well-being.
• Correlation $r$ has two qualities: direction and strength.
• Correlation $r$ is designed to describe the slope of the best-fitting straight line through the scatterplot.

Describing Associations with Categorical Variables/Data

• Example: Marital satisfaction and online/offline meeting of spouse.
• It is more common to estimate the magnitude of difference between means (group averages).
• Scatter plots might not be the primary choice for categorical data; bar plots might be more suitable.
• Researchers can present the mean of the distribution of the data or show the individual data points.

Assessing Validities of Correlational Research

Construct Validity

• How well was each (of the two) variables measured?
• Evaluate the researchers’ operationalizations of variables.
• Does the measure have good reliability?

Statistical Validity

• How well do the data support the conclusion?
  • How strong is the relationship? (Effect size!).
  • How precise is the estimate? (Confidence interval!).
  • Has the finding been replicated?
  • Could outliers affect the association?
  • Is there restriction of range? (Similar to floor or ceiling effects).
  • Is the association curvilinear?

Internal and External Validity

• Internal validity is more applicable to experiments.
• Internal validity: Can we make a causal inference from an association? Applying the three causal criteria.
• External validity: To whom can the association be generalized? How important is external validity? Consider the role of moderating variables.

Statistical Validity in Detail

• How strong is the relationship? What is the effect size?
  • Effect size describes the strength of a relationship between the variables.
  • Tiny effects can add up (e.g., agreeableness and enjoying social interactions; $r = 0.07$; Ozer & Benet-Martínez, 2006).
  • “Explaining behavior is like a complicated recipe, in which each ingredient plays a small role in the overall flavor.”
• How precise is the estimate? What is the confidence interval?
  • Usually reported in square brackets.
  • Tells you the estimate of the mean if you were to run the same study numerous times. E.g., greed and well-being $r = .14$ while its 95% CI was $[-.18, -.10]$. What does this mean?
  • Sample size shrinks the standard error of the mean, which also means CI has a narrow range.
  • If CI includes zero (0) that means the association is not statistically significant. Meaning, we cannot rule out the possibility that the true association is zero.

Replication and Outliers

• Has it been replicated?
• Could outliers be affecting the association?
  • It can! Especially when sample size is relatively smaller for topic of interest.
  • It can make a medium sized correlation appear stronger, or a strong one appear weak or even not existing.

Restriction of Range

• Is there restriction of range?
  • Especially relevant for cases where one variable has very little variance.
  • Similar to floor or ceiling effects.
  • Not having the full range can lead to underestimation of any true association.
  • There is a statistical technique called correction for restriction of range.

Curvilinear Association

• Is there restriction of range?
• Is the association curvilinear?
  • Correlation between one variable and the square root of the other variable can hint a curvilinear relationship.

Causal Criteria

• Case study: Meeting context and marital satisfaction
  1. Covariance of cause and effect: yes, there is relationship here.
  2. Temporal precedence: yes, meeting comes before than marriage.
  3. Internal validity: not really satisfied because a third variable problem.
     • Relationship motivation could be the third variable, where relationship-motivated individuals may be more likely to use these online apps, and feel greater marital satisfaction.

Third Variables and Moderators

• The relationship can be explained by a third variable: gender in the relationship between hair length and height.
• The relationship is not mediated by gender in the relationship between weight and height.
• Moderating variables: when the relationship between the two changes depending on the level of another variable.
  • Oishi et al., 2007: city’s level of residential mobility determines if attendance is increased with the city’s baseball team’s success.
  • “Degree of residential mobility moderates the relationship between success and attendance”
• Example of a moderator variable not affecting the observed association between the two variables.

Review Questions

• Suppose you hear that conscientious people are more likely to get regular health checkups. Which of the following correlations between conscientiousness and getting checkups would probably support this claim?
  • B. $r = .25$
• A study found that people who like spicy foods are generally risk takers. Which of the following questions interrogates the construct validity of this correlation?
  • D. How well did they measure each variable: risk taking and liking spicy foods?
• Which of the following sentences describes a moderator for the relationship between risk taking and liking spicy foods?
  • A. There is a positive relationship between liking spicy foods and risk taking for men, but no relationship for women.

Case Study: Anxiety and Negativity Bias

• The heightened negativity bias in visual perception among individuals with clinical levels of anxiety results from an overactive threat detection system.
• This bias stems from dysregulated attentional and neural mechanisms, such as hyperactivity in the amygdala and altered top-down modulation by the prefrontal cortex, which lead to an increased sensitivity to negative stimuli.
• Interpretation bias is a core diagnostic feature of anxiety.
• Is visual anger bias amplified for people with clinical levels of anxiety?
• Does bias in visual judgments relate to threat bias in other linguistic/cognitive domains?
• How do linguistic/visual biases contribute to maintenance of anxiety?

Studies and Designs

Participants

• Youth & emerging adults.
• Anxiety across the samples.
• Youth (n=40), Emerging Adults (n=44), Full Sample (N=84).
• Anxious (n=22), Control (n=22), Full Sample (N=44).

Design

• Visual Bias Task
• Linguistic Bias Task

Results

Results

• Linguistic Bias Task: Pos. corr. w anxiety, $r$ = .62, $p$ < .001
• Visual Bias Task: corr. w anxiety, $r$ = -.26, $p$ = .019
• $t(82) = 5.56$, $p$ < .001, $d$ = 1.21
• $t(82) = 2.48$, $p$ < .015, $d$ = .54
• $r$ = -.28, $p$ = .009
• Linear Regression
  1. Accounts for 39% of anxiety symptoms.
  2. VB does not have explanatory power beyond LB.
  3. Correctly classifies 73% of anxiety cases.
• Rozenman et al., 2024

Further Study

• Study n = 114
• 2 alternative forced choice paradigm
• 240 trials
• Mouth curvature varied
• Participants then provided self-reported anxiety scores
• Association between negativity bias and anxiety is assessed
• Screen for Adult Anxiety Related Disorders (SCAARED)

Hypothesis and Results

• Hypothesis: The negativity bias in visual perception will correlate with the and clinical levels of anxiety.
• $r= .14$
• $r = .08$
• Hypothesis: There is a positive association between negativity bias in visual perception and clinical levels of anxiety
• $r= .12$
• $r = .08$

Next Steps

• Multivariate Correlational Research (chapter 9).
• Gets closer to making a causal claim.
• Multiple regression analysis.
• Longitudinal designs.