Fundamental Statistics in
Applied Linguistics Research
Spring 2010
Weekend MA Program on Applied English
Dr. Da-Fu Huang
5.Finding relationships using correlation
5.1 Scatterplots : Visual inspection of your data
 Examining the linearity assumption
 Graphs > Legacy Dialogs > Scatter / Dot
> Simple Scatter (for two-variable SP) > Define
> one variable in the x-axis, and another in the y-axis,
press OK
 Adding a regression line (fit line) or a Loess line to a
SP
 Open the Chart Editor by double-clicking the created
SP > Elements > Fine Line at Total
> Properties > Linear (for a straight regression line)
OR Loess (for a line fitting the data more closely)
5.Finding relationships using correlation
Example datasets: DeKeyser (2000)
Viewing simple SP data by categories
Simple SP > Set Markers By, adding a
categorical variable > customize the graph by
adding fit lines, changing labels, or changing
properties of the plotting characters from the
Chart Editor
Application activities: Q’s 1-5, PP156-157
5.Finding relationships using correlation
5.2 Multiple Scatterplots
Graphs > Legacy Dialogs > Scatter / Dot
> Matrix Scatter (for more than two variables)
> Define
5.Finding relationships using correlation
5.3 Assumptions of parametric correlation
(Pearson’s r) (cf. Table 6.1, P160)
Linearity between each pair of variables
Independence of observations
Normal distribution of variables
Homoscedasticity (constant variance)
(the variance of the residuals for every pair of
points on the independent variable is equal)
5.Finding relationships using correlation
5.4 Effect size for correlation
R2 as a measure of how much of the variance in
one variable is accounted for by the other variable
R2 as a measurement of how tightly the points in a
scatterplot fit the regression line.
R2 is a percentage of variance (PV) effect size,
from the r family of effect sizes.
Cohen (1992)’s definition of effect size for R2 :
 R2 = .01 (small)
 R2 = .09 (medium)
 R2 = .25 (large)
Effect size for correlation (R2)
5.Finding relationships using correlation
5.5 Calculating correlation coefficients
Analyze > Correlate > Bivariate
> Move variables on the left to the Variables
on the right > Choose correlation coefficient
type
 Application activities: Q’s 1-4, P165
5.Finding relationships using correlation
5.6 Output and reporting of a correlation
 4 pieces of info desired in the output
Correlation coefficient (Pearson’s r, Spearman’s
rho, etc)
95% CI
Sample size (N) involved in the correlation
p-value
Calculation of the CI (by typing in r and N) at
the http://glass.ed.asu.edu/stats/analysis/rci.html
Double-click on the table > SPSS Pivot Table >
Format > Table Looks
Output of a correlation
Correlations
Total score on
gjtscore
gjtscore
Pearson Correlation
1
Sig. (2-tailed)
N
Total score on aptitude test Pearson Correlation
totalhrs
totalhrs
aptitude test
.079
.184
**
.267
.009
200
200
200
.079
1
.075
Sig. (2-tailed)
.267
N
200
200
200
.184**
.075
1
.009
.293
200
200
Pearson Correlation
Sig. (2-tailed)
N
**. Correlation is significant at the 0.01 level (2-tailed).
.293
200
5.Finding relationships using correlation
5.7 Sample of reporting a correlation
 Larson-Hall (2010, PP165-166)
Written and tabular forms
5.Finding relationships using correlation
5.8 Partial correlation
 Subtract the effects of a variable from the correlations
we are concerned with
 LOR -Cor Production Accuracy
 LOR - Cor Aptitude Test Score
 Age -Cor Production Accuracy
 Age - Cor Aptitude Test Score
 Age +Cor LOR
Analyze > Correlate > Partial
 Put the variable you want to control for in the Controlling For
box
 Put the other variables in the Variables box
Reporting results of partial correlation (P168)
5.Finding relationships using correlation
5.9 Point-Biserial correlations (rpb) & Test
Analysis
 Correlation between a dichotomous variable (only two
choices) and a continuous variable
 One way to determine item discrimination in classical
test theory is to conduct a corrected point-biserial
correlation, scores for the item crosses with scores for
the entire test, minus that particular item
Analyze > Scale > Reliability Analysis
Put the score for the total test and the individual
items in the “Items” box. Open the Statistics and
tick “Scale if item deleted.”
5.Finding relationships using correlation
5.10 Inter-rater Reliability
Inter-rater reliability or the measurement of
Cronbach’s alpha as intraclass correlation for cases
of judges rating persons
Problem with using the average inter-item
correlation as a measurement of reliability between
judges is that we are not sure whether the judges
rated the same people the same way, or just if the
trend of higher and lower scores for the same
participant was followed
5.Finding relationships using correlation
5.10 Inter-rater Reliability
Analyze > Scale > Reliability Analysis
Put the items which contain judges’ ratings of the
participants in the “Items” box. Open the
Statistics and tick “intraclass correlation
coefficient” box
Choose Two-Way Random. Also tick “Scale if
item deleted” and “Correlations.”
Look for Cronbach’s alpha in the output
For overall test reliability, put all of dichotomous test
items into the “Items” box in the Reliability analysis and
obtain Cronbach’s alpha, also known as the KR-20
measure of reliability