Correlations and linear regression Flashcards by Mia Philip

Define

Correlations

Statistical technique for measuring the extent to which two variables are associated Measures the pattern of responses across variables

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Define

Linear regression

a linear model to predict the value of one variable from another

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One-tailed

a statistical test in which the critical area of a distribution is one-sided so that it is either greater than or less than a certain value, but not both

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Define

Two-tailed

a method in which the critical area of a distribution is two-sided and tests whether a sample is greater than or less than a certain range of values

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Variance

tells us how much scores deviate from the mean

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Covariance

similar to the variance, but tells us how much on two variables differ from their means

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Correlation coefficient

The standardised version of covariance

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Pearson correlation coefficient

a parametric statistic that measures linear correlation between two variables X and Y. It has a value between +1 and −1.

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Correlation matrix

a table showing correlation coefficients between variables

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Post hoc

statistical analyses that were specified after the data were seen

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Spearman correlation

a nonparametric measure of rank correlation. It assesses how well the relationship between two variables can be described using a monotonic function

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Monotonic

relationships that are consistently one-directional

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Coefficient of determination (r^2)

the proportion of the variance in the dependent variable that is predictable from the independent variable

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Shared variance

the extent to which two variables vary together

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Partial correlation

Measures the relationship between two variables, controlling for the effect that a third variable has on them both

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Semi-partial (part) correlation

Measures the relationship between two variables, controlling for the effect that a third variable has on one of the others

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Zero order correlation

the correlation between two variables when you do not control for anything

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Directionality problem

it is not possible to determine which variable is the cause, and which is the effect

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Residual

The difference between the observed value of the dependent variable (y) and the predicted value (ŷ)

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Definition

Statistical technique for measuring the extent to which two variables are associated Measures the pattern of responses across variables

Correlations

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Definition

a linear model to predict the value of one variable from another

Linear regression

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Definition

a statistical test in which the critical area of a distribution is one-sided so that it is either greater than or less than a certain value, but not both

One-tailed

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Definition

a method in which the critical area of a distribution is two-sided and tests whether a sample is greater than or less than a certain range of values

Two-tailed

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Definition

tells us how much scores deviate from the mean

Variance

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# Definition similar to the variance, but tells us how much on two variables differ from their means

Covariance

# Definition The standardised version of covariance

Correlation coefficient

# Definition a parametric statistic that measures linear correlation between two variables X and Y. It has a value between +1 and −1.

Pearson correlation coefficient

# Definition a table showing correlation coefficients between variables

Correlation matrix

# Definition statistical analyses that were specified after the data were seen

Post hoc

# Definition a nonparametric measure of rank correlation. It assesses how well the relationship between two variables can be described using a monotonic function

Spearman correlation

# Definition relationships that are consistently one-directional

Monotonic

# Definition the proportion of the variance in the dependent variable that is predictable from the independent variable

Coefficient of determination (r^2)

# Definition the extent to which two variables vary together

Shared variance

# Definition Measures the relationship between two variables, controlling for the effect that a third variable has on them both

Partial correlation

# Definition Measures the relationship between two variables, controlling for the effect that a third variable has on one of the others

Semi-partial (part) correlation

# Definition the correlation between two variables when you do not control for anything

Zero order correlation

# Definition it is not possible to determine which variable is the cause, and which is the effect

Directionality problem

# Definition The difference between the observed value of the dependent variable (y) and the predicted value (ŷ)

Residual

\_\_\_\_\_\_\_\_\_\_\_\_exists when changes in one variable tend to be accompanied by persistent and predictable changes in the other variable

**Association/Relationship** exists when changes in one variable tend to be accompanied by persistent and predictable changes in the other variable

What is the range of correlation values?

-1 to +1

What does a correlation of 0 indicate?

No association (e.g. null hypothesis)

What does the significance of a correlation depend on?

Sample size (n) Alpha value (one vs two-tailed)

Size of correlation (ignoring direction), must be ___ critical value given for that df

Size of correlation (ignoring direction), must be **≥** critical value given for that df

The \_\_\_\_\_\_\_tells us how much scores deviate from the mean

The **variance** tells us how much scores deviate from the mean

The \_\_\_\_\_\_\_\_\_\_\_of the two variables is similar to the variance, but tells us how much on two variables differ from their means

The **covariance** of the two variables is similar to the variance, but tells us how much on two variables differ from their means

What is the variance equation?

What is the covariance equation?

What are some of the issues with covariance?

It depends upon the units of measurement e.g., The covariance of two variables measured in miles might be 4.25, but if the same scores are converted to kilometres, the covariance is 11

How do you get around the issue of units in covariance?

Standardise it | (divide the SD of both variables)

The standardised version of covariance is known as the \_\_\_\_\_\_\_\_\_\_\_\_\_\_

The standardised version of covariance is known as the **correlation coefficient**

What happens to the range when you standardise the covariance?

It goes from being unrestricted to between -1 and +1

What is the parametric correlation coefficient?

Pearson

What is the formula for the Pearson/Spearman correlation coefficient?

Why is it not necessarily a good idea to run a correlation matrix for all your variables?

The more tests you run, the more likely a false positive will occur

What are the assumptions of a Pearson correlation?

Both variables measured on an interval or ratio scale Both variables should be normally distributed (For statistical inference) Relationship between the variables must be linear

What happens if normality is violated when you want to run a Pearson correlation?

If N \> 30, can use central limit theorem to justify proceeding, despite violation Otherwise, use a Spearman correlation

What happens if linearity is violated when you want to run a Pearson correlation?

If relationship is monotonic, use a Spearman correlation Otherwise, try transforming the data to achieve linearity

What does a Spearman correlation measure?

It measures the association between two ordinal variables; that is, X and Y both consist of ranks It measures the consistency of direction of the association between two interval/ratio variables

Why is a Spearman correlation less powerful than a Pearsons?

The continuous data must be converted into ranks before conducting the correlation

True or False: You could conduct a Spearman correlation on this data

False It is nonmonotonic

What statistic do you use to measure the relationship strength of a correlation?

Coefficient of determination (r²)

If r = .411, what is the coefficent of determination?

r² = .169 Therefore, 16.9% shared variability

a ____________ is the correlation between two variables when you control (i.e., hold constant) the effects of a third variable on both of the other variables

a **partial correlation** is the correlation between two variables when you control (i.e., hold constant) the effects of a third variable on both of the other variables

What does this equation show?

Shows the association between and Y and X1 after removing the overlap of X2 with X1, and with Y

Why is semi-partial correlation used in multiple regression?

Used in multiple regression because if you square the semi-partial correlation, it tells you the variability in the outcome uniquely accounted for by one specific predictor variable i.e., controls for the relationships between predictors, so the outcome variable’s relationship with other predictors is still taken into account

What does this equation show?

How much of the total variability in Y is uniquely explained by X1

What is the difference between partial correlation and semi-partial correlation?

**Partial correlation:** Measures the relationship between two variables, controlling for the effect that a third variable has on them _both_ **Semi-partial correlation:** Measures the relationship between two variables, controlling for the effect that a third variable has on _only one of the others_

What is a 1st and 2nd order correlation?

1st order correlation = partial correlation that controls for 1st variable 2nd order correlation = partial correlation that controls for 2 variables

If I find that the Pearson correlation between exam performance and revision time is .384 for males and .442 for females, can I test whether the correlation is ‘significantly’ stronger for females than males?

The best way to test whether the association between two variables differs by group is to use multiple regression with interactions, which we will discuss later

Can partial correlations be performed on non-parametric data? (i.e., is there a way to do Spearman partial correlations?)

Yes! Use Spearman’s partial rank order correlation

What are the two options for dealing with missing data?

Exclude cases pairwise Exclude cases listwise

What happens if we exclude cases pairwise?

For EACH correlation, exclude participants who do not have a score for both variables

What happens if we exclude cases listwise?

Across ALL correlations, exclude participants who do not have a score for every variable

What is the linear regression equation?

Yi = b₀ + b₁X₁ + εi

What do the dots and the lines represent?

Dots = actual scores lines = difference between actual scores and predicted scores (residuals)

What is the regression line equation of this data?

Yi = b0 + b1X1 + εi Yi = 43.9 + 0.65X + εi

What does this value tell us?

Here .411 means that a 1 SD increase in revision time is expected to relate to a .411 SD increase in exam performance