week 4

• Covariation

  • Correlation underlies many different statistical analyses we use in this unit, e.g. internal consistency, regression, factor analysis
• Total error = sum of deviance =  (xi - X̄)
• Sum of squared errors (SS) =  (xi - X̄) (xi - X̄)
• Variance (s2) = ss/(N-1) = ( (xi- X ̄)2)/(N-1)
• Standard deviation (s) = √ ((xi- X ̄2)/(N-1)
• Linear correlation

  • To what extent do two variables have a linear relationship?
  • A simple/straight-line/constant relationship
  • To interpret a linear correlation, look at the: Direction, Strength, Statistical significance
• Types of correlation

  • No relationship (r is close to 0)
  • Linear relationship
  • Non-linear relationship
• How do you choose which type of correlation to use?

  It depends on the levels of measurement of your variables
• Nominal by nominal

  • Contingency (crosstabs) table of observed and expected frequencies, Row and/or column percentages, Marginal totals, Clustered bar chart, Chi-square, Phi (Φ) or Cramer’s V
• Chi-square


  To calculate: χ2 = Σ((O-E) ^2/E), χ2 = the test statistic that approaches a χ2 distribution, O = frequencies observed, E = frequencies expected (by the null hypothesis), χ2 = sum of: (observed – expected) ^2 / expected, Expected counts are the cell frequencies that should occur if the variables are not correlated, Chi-square is based on the squared differences between the actual and expected cell counts
• Effect size: χ2(1) = 292, p < .001, Φ = .18, 2x2 -> use the Phi effect size
• PHI (Φ) & CRAMER’S V


  These are non-parametric measures of correlation, Phi is used for 2x2, 2x3, or 3x2 analyses, Cramer’s V is used for 3x3 and greater analyses
• Write up example: 'A Chi-squared analysis revealed a significant, small association (χ2(1) = 292, p < .001, Φ = .18). Returning to the cross-tabulation results, women were more likely to say it is true that the father’s gene determines the sex of a baby, while men are more likely to say this is false'
• Ordinal by ordinal


  • Spearman’s rho (rs), Kendall tau (τ)
• Graphing: Ordinal by ordinal data is difficult to visualize – it is non-parametric, but it can have many points. You can use non-parametric approaches (e.g. clustered bar chart), Parametric approaches (e.g. scatterplot with line of best fit)
• Spearman’s rho


  Spearman’s rho is also called Spearman’s rank order correlation. Use this when your data is ranked (ordinal) LOM. The formula is the same as the Pearson’s product-moment correlation. Be careful with your interpretation so you are considering the underlying ranked scales
• Kendall’s tau

  Three versions: Tau a – does not take joint ranks into account (e.g. tied for 2nd place), Tau b – takes joint ranks into account, works for square tables (i.e. 2x2, 3x3, 4x4...), Tau c – takes joint ranks into account, works for rectangular tables (2x3, 4x2, etc.)
• Response options

  • Very unsure, Fairly unsure, Fairly sure, Very sure
• Clustered bar chart


  Have to first change the format of one variable to categorical in jamovi
• Scatterplot


  For this option, both variables in jamovi should be at least ordinal
• Interpretation: There is a significant effect where people who are very confident in their first answer also tend to be very confident in their second answer. τb = .32, p < .001
• Dichotomous by interval/Ratio

  • Correlational approaches: You have one dichotomous and one interval/ratio variable
• There is a significant effect where people who are very confident in their first answer also tend to be very confident in their second answer
• Antibiotics kill viruses as well as bacteria
• Interpretation of positive relationship

  • The higher one scores on the interval measure, the more likely they are to choose 'false'
• Interpretation of negative relationship

  • The higher one scores on the interval measure, the more likely they are to choose 'true'
• There was a small, significant relationship

  rpb = .12, p < .001
• Those who believe science is more credible

  Are more likely to report that antibiotics do NOT kill viruses as well as bacteria
• Those who are more sceptical about science

  Are more likely to say it is true that antibiotics kill viruses as well as bacteria
• Correlation is a measure of the standardized covariance
• The relationship between X and Y is the covariance of X and Y divided by the product of the standard deviations of X and Y
• The covariance between two variables is the variance that they share
• Covariance reflects the direction of the relationship (+/-)
• Covariance is unstandardized – depends on two variables having the same scale of measurement
• When you standardize the covariance (by dividing it by the cross-product of the SDs), you get the correlation
• Correlation is an effect size – a standardized measure of the strength of the relationship
• It’s rare to get correlations of 0 (no relationship at all)
• How do we decide if we have a ‘real’ relationship between our variables?
• What is the likelihood that an observed relationship is due to chance?
• What is the likelihood that it reflects a true relationship?