Experimental Research Methods in Language Learning

1
Experimental Research Methods in Language
Learning

• Chapter 11
• Correlational Analysis

2
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?

3
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.

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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.

5
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.

6
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.

7
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.

8
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.

9
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).

10
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.

11
Shared Variance
12
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.

13
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.

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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.

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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).

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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.

17
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.

18
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.

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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).

20
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?