Lecture 3 - Quantitative Methods

Question 1

Difference b/w chi-square and regression analysis

Answer 1

Þ Chi-square analysis looks at the association between two categorical
variables.
Þ Regression analysis looks at predicting one continuous variable from another

Question 2

What is a correlation?

Answer 2

the association b/w 2 continuous variables

Question 3

Correlation is a form of. . .

Answer 3

BIVARIATE analysis (measure of relationship b/w 2 variables)

Correlation quantifies the relationship (typically linear) between two variables X & Y
in terms of direction, degree.

Question 4

An association b/w 2 variables can be. . .

Answer 4

LINEAR or NON-LINEAR

Question 5

What does the Pearson correlation coefficient (r)?

Answer 5

measures the linear association between two
continuous variables.

It compares how much the two variables vary together to how much they vary separately.

It ranges from -1 to 1. A r value of 0 indicates no association between 2 variables.

Question 6

R squared (r^2) is. . .

Answer 6

the percentage of variance accounted for (i.e. how much of the variation in the outcome variable is explained by the predictor)

Question 7

What is variability?

Answer 7

how much a given variable varies from observation to observation

Question 8

What is covariability?

Answer 8

how much two variables vary together

Question 9

Degree of relationship

Answer 9

® Small effect: 0.1 < r < 0.3, or -0.3 > r > -0.1

® Medium effect: 0.3 < r < 0.5, or -.3 > r > -0.5

® Large effect: 0.5 < r < 0.7, or -0.5 > r > -0.7

Question 10

What can greatly influence the value of a correlation?

Answer 10

extreme scores / outliers

Question 11

Pearson’s r is. . .

Answer 11

BIDIRECTIONAL

• meaning, the correlation b/w Variable B and Variable A is the same as that b/w Variable A & B

Question 12

Regression toward the mean:

Answer 12

® With imperfect correlation, an extreme score on one measure tends to be followed
by a less extreme score on the other measure.

® This is because extreme scores are often due to chance. And if it’s due to chance, it’s
extremely unlikely that the other value will also be extreme.

Question 13

Null hypothesis testing

Answer 13

we assume there is no effect (i.e. no association)

Question 14

The null hypothesis –

Answer 14

ρ = 0. Rho (ρ) is the correlation in the population

® If the probability of finding an r this big if the real association in the population (ρ) is small (p < .05), we reject the null hypothesis. So, we infer that the correlation value for the population is NOT zero. There is significant association between the two
variables.

Question 15

ρ will depend on. . .

Answer 15

the size of the sample (n)

it will also depend on
the degrees of freedom (df).

® For correlation, df = n – 2.
® For a small correlation to be significant, a high df (lots of participants/data) is
required.

Question 16

When is Spearman’s correlation coefficient used?

Answer 16

• when the data is ORDINAL (ranked) & when the data is one-directional but NON-LINEAR
• Convert the data to ranks before calculating correlations. Converting to ranks can linearize non-linear data.
• Useful for non-linear data and data sets with outliers.

Question 17

What is regression used for?

Answer 17

• predicting one variable from another
• However, unless the correlation is perfect (r = +1 or r = -1), the prediction will not be
  exact.
• Looks for the line of best fit (regression line)

Question 18

Errors in regression are assumed to be

Answer 18

• independent
• normally distributed (w a mean of 0)
• homoscedastic (equal error variance for levels of predicted Y)

Question 19

How is the assumption of the independence of errors assessed in regression?

Answer 19

• by reflecting on the sampling procedure for the study (and NOT by looking at plots of residuals)

Question 20

What does the least squares parameter estimate?

Answer 20

estimate parameters by minimizing the total squared error (regression coefficients)

Question 21

What does the ANOVA (F test) for aggression tell us?

Answer 21

• F test tells us whether the variance explained is significantly different from zero
• if the F test is NOT significant, the regression is worthless. The predictor does not explain the outcome variable at all

® Null hypothesis: r^2 = 0.
® The F test has 2 degrees of freedom. For simple regression, the first is always 1, the
second df is the same as for the earlier t-test of regression coefficients.

Question 22

Regression parameters (a, b)

Answer 22

® The intercept (a) is the estimated value of Y when X = 0.
® Slope (b) (gradient) (the standardized/regression coefficient in JASP) indicates
whether there is a relationship between X & Y, whether that relationship is positive
or negative, and the estimated change in Y when X increases by 1. For H0: b = 0
® Slope can be transformed into standardized form (covert X & Y into z-scores, and
then do the regression). This is called a standardized regression coefficient (called
beta in SPSS/JASP), and is the same as the correlation coefficient for bivariate data.

Question 23

Regression diagnostics

Answer 23

a) Histogram of residuals – normality: we want the errors to be approximately
normally distributed

b) Residuals plot – homoscedasticity: a scatterplot of residuals against predicted values
to check for heteroscedasticity. Absence of any systemic pattern supports the
assumption of homoscedasticity (where the variance of the residual/error term in a
regression model is constant).