Chapter 8 - bivariate correlational research

Bivariate correlations

• Associations or relationships between exactly two variables

  • Association claim - two variables 

  • A study that reports a bivariate correlation may have measured more than two variables

Special case: categorical data 

• Scatterplots work well with quantitative data

  • Can have issues with ordinal data

• But scatterplots are often less clear for categorical data → ex. If you only have two categories (cant make helpful scatterplot)

  • Bar graphs can be used instead

    • Allows visual comparison of group means

    • Can use r for categorical data, but its more common to estimate the magnitude of difference between group averages 

T-test

• statistic to test the differences between two group averages 

• Peaks are averages, the higher the peak, the more significant differences (lower p values) and vice versa 

• Not exclusive to experiments or association claims

Statistical validity 

1. How strong is the relationship?

2. How precise is the estimate?

3. Has it been replicated?

4. Are outliers affecting the association?

5. Is there a restriction of range?

6. Is the association curvilinear? 

Effect size

measure of the strength of the relationship between two variables in a population

• All else being equal, larger effect size are more meaningful (stronger relationship)

• Small effect sizes are important too

Effect size magnitude 

• Useful for predicting where someone’s data point will land on the prediction line in a scatter plot 

• When everything is equal, larger effect sizes are usually considered more important

  • Some exceptions

How precise is the estimate?

• 95% confidence interval (95% CI)

  • Range will contain the true population value 95% of the time 

  • Better that CI does not contain zero in the range

  • CI that does contain zero? → possibility that there is no association between the two variables 

 

• Smaller CI is better 

• Error bars often represent one standard deviation of uncertainty, one standard error, or a particular CI → the narrower error bars = the more precise it is (better)

• Estimates based on smaller samples are less stable → wider and less precise CIs

Has it been replicated?

• Replication allows for more accurate findings 

• Others doing smth similar and found similar results 

• True variations and different variations 

• More generalization = better external validity 

Are outliers affecting the association?

Outliers: extreme scores that lies far away from the rest of the scores 

• Outlier in larger sample → smaller effect on the overall sample

• Outlier in smaller sample → larger effect on the overall sample 

Is there a restriction of range?

• restriction of range that prevent us from getting the full sample

• Can influence conclusions 

  • Ex. SAT scores cut off for college is 1200 → not the full sample

Is the association curvilinear? 

• Do the data points make a curve/u-shape on the scatter plot?

• Nonlinear → not meant to be captured by correlations

Is it statistically significant?

Statistical significance: conclusions drawn about how probable it is that a correlation that size would come from a population with no correlation

• Tldr: are these results due to chance?

Logic behind statistical inference

• Researches collect data from a sample to make conclusions about a larger population

• If there is association in the sample, we assume association exists in the population

• If no association in population, association in sample may have been chance

Significance in journal articles

• Usually tell us

• P = 0.05 is usually what to expect

  • Can be recorded in different ways - may make it difficult to read through methods

Internal validity 

• Can we make a causal inference from this association?

  • 3 causal criteria:

    • Covariance: do results show that the variables are correlated?

    • Temporal precedence: does method establish which variable came first in time?

    • Interval validity (third-variable problem): is there a third variable that is associated with the other two variables independently? 

• When a 3rd variable is a problem

  • Ex. hair length and height → shorter = longer hair length

    • third variable is gender of the individuals

  • weight and height → taller = heavier

    • gender of the individuals is a factor but is not a third variable 

Moderating variables

• When the relationship between two variables changes depending on the level of another variable. 

• The other variable is called a moderator 

• Ex. association between team success and game attendance

  • Place A has higher association

  • Place B has lowe association

    • But moderator is residential mobility (transience - do people move there and live there for a long time? 

    • High transience/residential mobility = higher association

    • Low transience/residential mobility = lower association