Chapter 8 - Correlations Mmlecture
notes
Modeling Relationships
Starting point of correlational analysis is looking at the scatter plots
Chapter 6 - Overview of assumptions
Assumptions: A conditions that ensures that what you're trying to do works -> Can only use particular tests if certain
assumptions are met
Additivity and Linearity
Normally distributed something or other
Homoscedasticity/Homogeneity of variance
Independence
outcome modelterror
Recap on original equation presented -> means that the data we observed can be predicted from the model we choose to fit
the data plus some error in prediction
• This model can depend on the design of the study, type of data and what one is trying to achieve
• E.g., if one wants to model a relationship between variability then one is trying to predict and outcome variable from a
predictor variable -> Therefore the predictor variable should be included in the model (predictor variables are given the
letter X) outcome
bixilterror
• This equation means the 'outcome for an entity is predicted from their score on the predictor variable plus error'
• When one work with raw data, one needs to know the outcome without the predictor present (if there was no predictor
variable, what would the outcome be?) - We add this value into the model as constant b0 (known as the intercept)
Covariance
To see if two variables are associated is to look at whether they covary positivecovariance
Variance: The average amount of data that varies from the mean.
Covariance: If two variables are related and one variable deviates from the mean, one would expect the other variable to
deviate from the mean in a similar way (See figure on page 514)
How does one calculate the exact similarity between the patterns of differences of the two variables?
• Cannot calculate the total amount of deviation as one would cancel another out
• In a single-variable case, we squared the deviation to eliminate the problem
• However, when there are two variables, rather than squaring, one can multiple the deviation for one variable by the
corresponding deviation for the second variable - This is getting the cross-produce deviations
• Positive covariance -> means that as one variable deviates from the mean, the other variable deviates in the same
direction
• Negative covariance -> as one variable deviates from the mean, the other variable deviates in the opposite direction
• Important to note that covariance depends on the scales of measurement used (cannot compare covariances in an
objective way)
,Standardization and the correlation coefficient
To overcome the problem of dependence on the measurement scale -> covariance needs to be converted into a standard set
of units (standardization)
The standardized covariance is known as a correlation coefficient (Pearson's correlation coefficient - how to do it)
By Standardizing the covariance we end up with a value that lies between -1 and +1 (correlation)
Pearsons correlation coefficient is also often used as an effect size measure
• Bivariate correlation: Is the correlation between two variables
The significance of the correlation coefficient
Pearson's r is known to not be normally distributed (meaning cannot use z-scores)
Thanks to Fisher (1921) we can adjust r into a z-score (zr) so that its sampling distribution is normal.
Confidence Intervals for r
To compute confidence intervals for r, we use what we have learnt about converting r to zr (to make the sampling
distribution normal) and use the associated standard errors. By using zr we can construct a confidence interval the normal
way ->
Zr I 1.96 5Ear
Data entry for correlation analysis
When looking at the relationship between variables, each variable should be entered in a separate column, with each
participants score in the rows.
Bivariate Correlation
When computing bivariate correlation, one should check for sources of biases first (linearity and normality) - Since we are
fitting a linear model to the data, if the relationship between the variables is not linear, the model is invalid. (a
transformation can help deal with this)
General procedure for correlations using SPSS
Analyze -> Correlate -> Bivariate. Here you click the variables box and transfer them.
SPSS will create a correlation matrix of correlation coefficients for all combinations of specific variables
Having selected the variables, you can choose between 3 correlation coefficients
(Pearson's: There is a need for a linear relationship between two variables.
Spearman rho (also known as spearman's rank order correlation): applies in cases that lack normal distribution
, Kendall's tau: Use when your variables of interest are continuous and contain outliers or ordinal. And when you only have
two variables.
Can also specify whether the test is one (directional) or two-tailed (non-directional)
Pearson's Correlation Coefficient using SPSS
Run through the general procedure listed above and click Pearson (the default)
Important to note is that if the confidence interval does not cross over 0, then it is very plausible that there is an effect (as
in if the confidence interval crosses 0, it is possible that the population value could be 0 (no effect at all)
Coefficient of determination: Known as the correlation coefficient squared (the measure of the amount of variability in
one variable that is shared by the other) E.g., exam performance can vary, reflecting that peoples scores are not identical,
they will have varied for a number of reasons (different ability, different levels of preparation)
• Adding up this individual variability gives us an estimate of how much variability in exam performance there was in
total
Spearman's Correlation Coefficient
Denoted as rs, is a non-parametric statistic that is useful to minimize the effects of extreme scores or the effects of the
violations of assumptions discussed in chapter 6.
When ranking individuals performance on lie test, (since it is a meaningful ranking/ordinal) spearman's is used
Kendall's Tau (non-parametric)
Should be used instead of Spearman's when you have a small data set with a large number of tied ranks
Biserial and Point-Biserial Correlations
Investigating the relationship between two variables, when one variable is dichotomous (0,1) (e.g., being pregnant or not
pregnant, being dead or alive)
The biserial and point-biserial correlations should be used in these situations
Difference between the use of biserial vs point-biserial correlations
• Point-biserial correlation is used: when one variable is a discrete dichotomy (e.g., pregnancy)
• Biserial correlation coefficient: is used when one variable is a continuous dichotomy (e.g., passing or failing the exam
- someone can just fail the exam vs someone can largely (with a great margin) fail the exam)
Partial and Semi-partial correlation
Semi-partial correlation
• A type of correlation that allows one to look at the relationship between 2 variables, accounting for the effect of a third
variable. (E.g., exam performance is negatively related to exam anxiety, but positively related to revision time, and
revision time itself was negatively related to exam anxiety.
• If revision time is related to both exam anxiety and exam performance then to get a measure of the unique relationship
between exam anxiety and exam performance, we need to account for revision time.
notes
Modeling Relationships
Starting point of correlational analysis is looking at the scatter plots
Chapter 6 - Overview of assumptions
Assumptions: A conditions that ensures that what you're trying to do works -> Can only use particular tests if certain
assumptions are met
Additivity and Linearity
Normally distributed something or other
Homoscedasticity/Homogeneity of variance
Independence
outcome modelterror
Recap on original equation presented -> means that the data we observed can be predicted from the model we choose to fit
the data plus some error in prediction
• This model can depend on the design of the study, type of data and what one is trying to achieve
• E.g., if one wants to model a relationship between variability then one is trying to predict and outcome variable from a
predictor variable -> Therefore the predictor variable should be included in the model (predictor variables are given the
letter X) outcome
bixilterror
• This equation means the 'outcome for an entity is predicted from their score on the predictor variable plus error'
• When one work with raw data, one needs to know the outcome without the predictor present (if there was no predictor
variable, what would the outcome be?) - We add this value into the model as constant b0 (known as the intercept)
Covariance
To see if two variables are associated is to look at whether they covary positivecovariance
Variance: The average amount of data that varies from the mean.
Covariance: If two variables are related and one variable deviates from the mean, one would expect the other variable to
deviate from the mean in a similar way (See figure on page 514)
How does one calculate the exact similarity between the patterns of differences of the two variables?
• Cannot calculate the total amount of deviation as one would cancel another out
• In a single-variable case, we squared the deviation to eliminate the problem
• However, when there are two variables, rather than squaring, one can multiple the deviation for one variable by the
corresponding deviation for the second variable - This is getting the cross-produce deviations
• Positive covariance -> means that as one variable deviates from the mean, the other variable deviates in the same
direction
• Negative covariance -> as one variable deviates from the mean, the other variable deviates in the opposite direction
• Important to note that covariance depends on the scales of measurement used (cannot compare covariances in an
objective way)
,Standardization and the correlation coefficient
To overcome the problem of dependence on the measurement scale -> covariance needs to be converted into a standard set
of units (standardization)
The standardized covariance is known as a correlation coefficient (Pearson's correlation coefficient - how to do it)
By Standardizing the covariance we end up with a value that lies between -1 and +1 (correlation)
Pearsons correlation coefficient is also often used as an effect size measure
• Bivariate correlation: Is the correlation between two variables
The significance of the correlation coefficient
Pearson's r is known to not be normally distributed (meaning cannot use z-scores)
Thanks to Fisher (1921) we can adjust r into a z-score (zr) so that its sampling distribution is normal.
Confidence Intervals for r
To compute confidence intervals for r, we use what we have learnt about converting r to zr (to make the sampling
distribution normal) and use the associated standard errors. By using zr we can construct a confidence interval the normal
way ->
Zr I 1.96 5Ear
Data entry for correlation analysis
When looking at the relationship between variables, each variable should be entered in a separate column, with each
participants score in the rows.
Bivariate Correlation
When computing bivariate correlation, one should check for sources of biases first (linearity and normality) - Since we are
fitting a linear model to the data, if the relationship between the variables is not linear, the model is invalid. (a
transformation can help deal with this)
General procedure for correlations using SPSS
Analyze -> Correlate -> Bivariate. Here you click the variables box and transfer them.
SPSS will create a correlation matrix of correlation coefficients for all combinations of specific variables
Having selected the variables, you can choose between 3 correlation coefficients
(Pearson's: There is a need for a linear relationship between two variables.
Spearman rho (also known as spearman's rank order correlation): applies in cases that lack normal distribution
, Kendall's tau: Use when your variables of interest are continuous and contain outliers or ordinal. And when you only have
two variables.
Can also specify whether the test is one (directional) or two-tailed (non-directional)
Pearson's Correlation Coefficient using SPSS
Run through the general procedure listed above and click Pearson (the default)
Important to note is that if the confidence interval does not cross over 0, then it is very plausible that there is an effect (as
in if the confidence interval crosses 0, it is possible that the population value could be 0 (no effect at all)
Coefficient of determination: Known as the correlation coefficient squared (the measure of the amount of variability in
one variable that is shared by the other) E.g., exam performance can vary, reflecting that peoples scores are not identical,
they will have varied for a number of reasons (different ability, different levels of preparation)
• Adding up this individual variability gives us an estimate of how much variability in exam performance there was in
total
Spearman's Correlation Coefficient
Denoted as rs, is a non-parametric statistic that is useful to minimize the effects of extreme scores or the effects of the
violations of assumptions discussed in chapter 6.
When ranking individuals performance on lie test, (since it is a meaningful ranking/ordinal) spearman's is used
Kendall's Tau (non-parametric)
Should be used instead of Spearman's when you have a small data set with a large number of tied ranks
Biserial and Point-Biserial Correlations
Investigating the relationship between two variables, when one variable is dichotomous (0,1) (e.g., being pregnant or not
pregnant, being dead or alive)
The biserial and point-biserial correlations should be used in these situations
Difference between the use of biserial vs point-biserial correlations
• Point-biserial correlation is used: when one variable is a discrete dichotomy (e.g., pregnancy)
• Biserial correlation coefficient: is used when one variable is a continuous dichotomy (e.g., passing or failing the exam
- someone can just fail the exam vs someone can largely (with a great margin) fail the exam)
Partial and Semi-partial correlation
Semi-partial correlation
• A type of correlation that allows one to look at the relationship between 2 variables, accounting for the effect of a third
variable. (E.g., exam performance is negatively related to exam anxiety, but positively related to revision time, and
revision time itself was negatively related to exam anxiety.
• If revision time is related to both exam anxiety and exam performance then to get a measure of the unique relationship
between exam anxiety and exam performance, we need to account for revision time.
