Correlation analysis is undertaken to define the strength an direction of a linear relationship between two variables Two measurements are use to assess this relationship are: the Pearson’s product–moment correlation coefficient (r) The Spearman rank order correlation coefficient (rho)

Pearson’s product–moment correlation coefficient (r) Perfect Negative Correlation Perfect Positive Correlation The correlation coefficient ranges from -1 to +1. The sign shows the direction of the relationship The size in absolute value (without the sign) describe the strength of the relationship There are bivariate (between two variables) or “zero- order” correlations and partial

Bivariate (Zero – Order) correlations This correlation can be used for two continuous variables, i.e. interval (or scale in SPSS). For instance score on a measure of anxiety and self-efficacy. It can be used between one continuous variable and one dichotomous variable. For instance score on a measure of anxiety and gender (M/F). A fundamental assumption for correlation is linearity, the two variable should be linearly related and not curvilinear related

The linearity of the two variables is a sine qua non condition for any further correlation analysis

Generating a scatter plot in SPSS

Procedure for calculating the Pearson’s r in SPSS

SPSS output and Interpretation There is a strong negative correlation between mile per gallon and weight, r = – 0.80, n = 71. p < .001 There is a moderate positive correlation between price and weight, r = 0.54, n = 71. p < .001

Correlation matrix report with APA format Mean SD 1 2 3 1. Var A 4.23 0.59 - 2. Var B 3.82 1.04 .517** 3. Var C 4.87 0.75 -.284* .421*** Note: * Indicates p< .05; ** indicates p< .01; ***indicates p< .001 N = XXX Make sure to report all the abbreviated expressions in the “Note” part. Be it M for mean, SD for standard deviation, or any study variable

Partial Correlation Partial correlation are similar to PPMC as described above, to the exception that it allows to control for an incremental variable. Assume three variables A, B and C. You suspect C to influence the relationship between A and B. In other words, A and B may seems to be related, however it is due to a certain extend to C (the contaminating / confounding variable) Statistically removing (controlling for) C will enable you to remove its confounding or contaminating effect on the A – B relation. You will get a clearer and more accurate indication of the true relationship between A and B.

SPSS Procedure

Partial correlation Output and Interpretation

Partial correlation Output and Interpretation A partial correlation was used to determine the relationship between weight and VO2max (aerobic fitness) while controlling for age. There was a statistically significant, moderate, negative partial correlation between weight and VO2max (r=-.307, p=.002)*, indicating that heavy weight is negatively associated with lower aerobic fitness. The bottom half (red box) of the table repeats the same set of correlation analyses, but this time controlling for (taking out) the effects of your control variable (e.g. age). In this case the new partial correlation is –.314., p=.002 *A zero-order correlation

Partial correlation Output and Interpretation You should compare these two sets of correlation coefficients to see whether controlling for the additional variable had any impact on the relationship between your two variables. In this example there was only a small increase in the strength of the correlation (from –.307 to –.314). This suggests that the observed relationship between VO2max and weight is not due merely to the influence of age. *A zero-order correlation

BASIC ASSUMPTIONS Level of measurement – interval or ration (continuous), exception to dichotomous variables). Independence of observations – one observation should not affect another. Normality – each variable must be normally distributed (can be checked with histograms). Linearity – looking at a scatterplot, one should see (roughly) a “straight line” and not a curve. Outlier-free – r is influenced by extreme positive/negative value like the mean and SD. Correlation vs. Causality – two variables are associated to each other; one does not affect the other.

Spearman rank order correlation coefficient (rho) A non-parametric test as an alternative for the PPMCC – Pearson’s product – moment correlation coefficient (with no assumption of normally distribution of the underlying population). Ideal for nominal or ordinal scales, very small samples, and not normally distributed data.

Same procedure as the Pearson correlation Same procedure as the Pearson correlation. Rather unselect “Pearson” and select “Spearman” and then continue with the procedure.

Can be interpreted like the PPMCC

References Lund Research Ltd. (2013). Partial Correlation using SPSS Statistics. Retrieved from: https://statistics.laerd.com/spss- tutorials/partial-correlation-using-spss-statistics.php Pallant, J. (2013). SPSS survival manual. McGraw-Hill Education (UK). Van den Berg, R.G. (2014). SPSS Correlation Test. Retrieved from: https://www.spss-tutorials.com/spss-correlation-test/