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, PP

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 R 2 as a measure of how much of the variance in one variable is accounted for by the other variable R 2 as a measurement of how tightly the points in a scatterplot fit the regression line. R 2 is a percentage of variance (PV) effect size, from the r family of effect sizes. Cohen (1992)’s definition of effect size for R 2 : R 2 =.01 (small) R 2 =.09 (medium) R 2 =.25 (large)

Effect size for correlation (R 2 )

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 Double-click on the table > SPSS Pivot Table > Format > Table Looks

Output of a correlation

5.Finding relationships using correlation 5.7 Sample of reporting a correlation Larson-Hall (2010, PP ) 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 (r pb ) & 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