Experimental Research Methods in Language

Learning

- Chapter 11
- Correlational Analysis

Correlational Analysis

- Can you think of real life examples of

correlations? - How do you express how much one thing is related

to another thing? - What is a positive correlation? What is a

negative one?

Correlational Analysis

- Correlation allows researchers to explore a

hypothetical relationship between variables. - Correlational analysis is typically used in

non-experimental research, such as survey

research or correlational research which aims to

examine whether there is an association between

variables of interest and if such an association

exists, to what extent the variables are

connected.

Correlational Analysis

- Correlation is fundamental to several advanced

statistical techniques, including factor

analysis, reliability analysis, and regression

analysis. - One of the best strategies to start learning

statistics for experimental research is via first

learning how correlations are analyzed.

The Size and Sign of a Correlation Coefficient

- There are five types of correlations introduced

in this chapter - Pearson Product Moment
- Point Biserial Correlation
- Spearmans Rho Correlation
- Kendalls Tau Correlation
- Phi Correlation.

The Size and Sign of a Correlation Coefficient

- A correlation coefficient is typically expressed

on a scale from 0 (i.e., 0, no relationship) to

1 (i.e., 100, perfect relationship). - If two variables are uncorrelated (i.e., 0),

there is no systematic relationship between them

and hence a prediction of one variable by the

other is not possible. - A positive () correlation two variables are

associated and move in the same direction in a

systematic way. - A negative (-) correlation suggests that the two

variables are associated with each other, but

move systematically in the opposite direction.

Curvilinear Relationship

- A curvilinear relationship At some level,

something can be positive, but when it exceeds a

certain level, it can become negative. - For example, we all know that some level of

anxiety/pressure is good for test performance

(because we will work harder and try to overcome

it), but too much anxiety is bad for test

performance because it takes control of our

emotions.

Hypothesis Testing in Correlation

- Researchers seek to test a hypothesis of whether

two variables are related. - The null hypothesis (H0) states that there is no

relationship between variable A and variable B

(i.e., correlation coefficient 0). - The non-directional alternative hypothesis (H1)

is that there is a relationship between variable

A and variable B (i.e., correlation coefficient ?

0). - A directional alternative hypothesis is there is

a positive relationship between variable A and

variable B (i.e., one-tailed). - The two-tailed test of significance is

recommended.

Hypothesis Testing in Correlation

- A typical p-value is taken to be 0.05. That is, a

5 chance is acknowledged to error in rejecting

the null hypothesis. - The degree of freedom (df) is determined by the

total number of cases (or sample size) minus 1

(i.e., N-1).

Effect Size and R-squared (R2)

- The sign of the correlation (i.e., or -) is not

related to the strength (or size) of the

correlation. - A negative correlation does not mean that the

finding is of no worth. - A correlation coefficient does not tell us how

much one variable accounts for the other. - In some research topics, fairly weak correlations

can be very important (i.e., they can indicate

theoretical or practical significance) and in

some cases even a shared variance of 10 can be

worth acting upon.

Shared Variance

Correction for Attenuation

- A correction for attenuation takes the

reliability coefficients of the two measures into

account in a Pearson correlation coefficient. - According to Hatch and Lazaraton (1991, p.444), a

correction for attenuation can be computed as - rAB vreliability of A x reliability

of B, - where rAB correlation coefficient.

Pearson Correlations

- The Pearson Product Moment correlation (Pearsons

r) is a parametric test for describing the

relationship between two continuous variables. - Known as a simple bivariate correlation.
- Pearson correlation can be used for numeric

variables on continuous scales such as interval

and ratio scales. - Two variables must be from the same participants.
- A data set must be normally distributed or close

to a normally distributed shape.

Five Statistical Assumptions for Pearson r

- Pairs of data are related (i.e., X and Y scores

are from the same person) - Continuous or interval-like data
- Normal distribution
- Linearity (i.e., the X-Y relationship that can be

represented as a straight line - A spread of score variability
- Assumptions 1 and 2 are easy to check.
- Assumptions 3 and 5 can be addressed by computing

descriptive statistics and creating a histogram. - Assumption 4 is checked through a scatterplot.

Point Biserial Correlations

- Point Biserial Correlation is a non-parametric

test - Is a special case of the Pearson correlation.
- Can be used to examine the relationship between a

dichotomous variable (e.g. male-female, and

yes-no) and a continuous variable (e.g., test

scores).

Spearmans Rho Correlations

- Spearmans Rho correlation (?) is a

non-parametric test. - Typically used for numeric variables on an

ordinal or ranked scale (e.g., ranked list of

test results, letter grades A-F, and steps on a

Likert scale). - Can calculate the correlation of an ordinal score

with an interval score. - Some information may be lost as continuous

variables are ranked.

Kendalls Tau b Correlation

- The Kendalls tau b correlation is a

non-parametric alternative to the Spearman

correlation. - Can be used to examine the level of agreement and

disagreement between two sources of data. - For example, if a number of judges are used to

score and rank candidates in order of performance

outcomes, we would like to see the extent to

which these judges agree with their ranking.

Phi Correlations

- Phi (ø) correlation is a non-parametric test.
- Is not used much in correlational studies.
- Is useful for examining the relationship between

two dichotomous variables (e.g., male or female,

living or dead, pass or fail, agree or disagree,

correct or wrong, homework or no homework, pair

work or individual work). - These variables can be assigned as 1 or 0, or 1

or 2, depending on how we code them in SPSS.

Factors Affecting Correlation Coefficients

- A correlation coefficient is affected by several

factors, some known and some unknown, including - Sample size
- Correlational test being used
- Outliers (extreme cases).
- Reliability of the research instruments.
- Restricted data or score range (e.g., truncated

data).

Discussion

- Why do you think it is not suitable to use

correlation to explain causation? - What is the difference between a correlation

coefficient and a shared variance? - Do you think whether setting a probability value

to be less than 0.05 is better or worse than

setting it at 0.01? Why do you think so?