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Correlation practical
In this practical we will investigate whether there is a relationship between two variables by looking how correlated they are.
Two of the key predictors of academic achievement proposed by theories of student learning are self-efficacy and instrumental motivation. Self-efficacy refers to an individual’s confidence in their ability to perform a task well, and instrumental motivation to the belief that learning will be useful for one’s later career. In this practical we address the question of whether these two constructs are correlated, or specifically whether students who are more confident in their abilities in science are systematically more (or less) likely to view science learning as important for their future prospects. The PISA measure of science self-efficacy (SCIEEFF) was derived from students’ responses to questions on how easy they would find it to perform eight science tasks on their own, such as “Identify the better of two explanations for the formation of acid rain”. The measure of instrumental motivation (INTMOVSCI) was derived from four items in which students rated their agreement with statements like “Many things I learn in my school science subjects will help me to get a job” (see PISA datafile description for further details).
Correlations in SPSS (Practical)
To do this we will begin by simply plotting the two variables in SPSS:
Select Scatter/Dot from the Legacy diagnostics available from the Graphs menu. Select Simple Scatter and click on Define to bring up the Simple Scatterplot window. Copy the Science self-efficacy score[SCIEEFF] variable into the Y Axis box. Copy the Science instrumental motivation score[INSMOVSCI] variable into the X Axis box. Click on the OK button.
SPSS will then draw a scatterplot of the two variables which can be seen below:
Looking at the scatterplot there appears to be a positive correlation between the variables with larger values of SCIEEFF associated with larger values of INSMOVSCI (an upward sloping relationship) but this relationship is not that strong with possibly a few more points in the bottom-left and top-right quarters of the plot. We want to test whether any correlation we observe in the scatterplot is significant but there are several different correlation coefficients for different situations. The first correlation coefficient that we will look at is the Pearson correlation coefficient. This correlation requires the variables to be continuous and, in smaller samples, to be normally distributed so we will firstly look at whether a normal distribution is suitable.
To do this we need to the following in SPSS:
Select Descriptive Statistics from the Analyze menu. Select Explore from the Descriptive Statistics sub-menu. Click on the Reset button. Copy the Science self-efficacy score[SCIEEFF] and Science instrumental motivation score[INSMOVSCI] variables into the Dependent List: box. Click on the Plots... button. On the screen that appears select the Histogram tick box. Unselect the Stem and leaf button.
Select the Normality plots with tests button. Click on the Continue button. Click on the OK button.
This set of instructions will create a whole list of outputs - both tables and figures - in SPSS. We will focus on two figures each for our two variables and then one table.
We will first look at a histogram of the variable, SCIEEFF. This can be found in amongst the set of output objects and looks as follows:
Ideally for a normal distribution this histogram should look symmetric around the mean of the distribution, in this case .3688. This distribution appears to be reasonably symmetric.
We will next look at a histogram of the variable, INSMOVSCI. This can also be found in amongst the set of output objects and looks as follows:
Again for a normal distribution this histogram should look symmetric around the mean of the distribution, in this case .3827. This distribution appears to be significantly skewed to the left (negatively skewed).
We will next look at statistical tests for the two variables to see if they back up our visual impressions from the histograms.
The Kolmogorov-Smirnov test is used to test the null hypothesis that a set of data comes from a normal distribution. An alternative test derived by Shapiro and Wilks is sometimes also available in SPSS but will not be described here. The available test statistics are presented in the table below that will be amongst the outputs from the Explore command:
As we found a significant effect in the Kolmogorov Smirnov test for INSMOVSCI we should see the points diverging from the line in the plot above with either some outlying values lying away from the line or even the shape of the points forming a non-linear pattern. We will now finally turn our attention to the main topic of this practical which is the calculation of the correlation between our two variables. SPSS offers several correlation coefficients and we will consider these here in turn. All three are available through the Analyse-
Correlate->Bivariate option in SPSS.
Select Bivariate... from the Correlate option available from the Analyse menu. Copy the Science self-efficacy score[SCIEEFF] and the Science instrumental motivation score[INSMOVSCI] variables into the Variables box. Click on the Options button and Select the Means and Standard deviations tick box. Click on the Continue button to return to main window. Click on the OK button.
The correlation command will produce two output tables. The first table which we show below simply gives means and standard deviations for the two variables we are comparing.
Descriptive Statistics Mean Std. Deviation N Science self-efficacy score .3671 1.19427 4726 Science instrumental motivation score .3819 .96666 4791
In the next table we see the correlation matrix for the variables we are considering:
Correlations Science self-efficacy score Science instrumental motivation score Science self-efficacy score Pearson Correlation 1. Sig. (2-tailed). N 4726 4694 Science instrumental motivation score Pearson Correlation .327 1 Sig. (2-tailed). N 4694 4791 **. Correlation is significant at the 0.01 level (2-tailed).
The Correlate option can be used for more than two variables simultaneously and will then give all correlations hence the output table is in this matrix format. The table contains three numbers for each possible correlation (including the correlations of variables with themselves which always takes the value 1). For each correlation there is an estimate of the correlation, an accompanying p value and a sample size on which the correlation has been calculated. Here we are interested in the Pearson correlation between SCIEEFF and INSMOVSCI which can be found in two places in the table - either in the row for SCIEEFF and column for INSMOVSCI or the row for INSMOVSCI and column for SCIEEFF. Note that the SPSS table repeats exactly the same information twice, but in the write-up of results it should only be reported once!
In this case the correlation (reported as the statistic r ) takes value .327. The widely-used rules specified by Cohen regard a correlation of r =.1 as small, r =.3 as moderate, and r =.5 as large. Here, then, we see a moderate positive correlation. The correlation is given in the table, along with a significance value and a sample size which in this case is 4694. This is the number of observations in which both SCIEEFF and INSMOVSCI where observed.
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We can test if this correlation is significantly different from zero which will depend on (i) the magnitude of the correlation and (ii) the number of observations on which the correlation is based.
The p value (quoted under Sig. (2-tailed)) is .000 (reported as p < .001) which is less than 0.05. We therefore have significant evidence to reject the null hypothesis that the correlation is 0.
We would report the result as follows: The variables SCIEEFF and INSMOVSCI were significantly and moderately positively correlated r = .327, N = 4694, p < .001. Note there is no need for a table when reporting a single correlation.
The Pearson correlation coefficient is appropriate to use when both variables can be assumed to follow a normal distribution or when samples are very large. If this is not the case then an alternative is the Spearman rank correlation. This correlation works in much the same way as the Pearson coefficient but is calculated on the ranks of the data points rather than the points themselves. To calculate the Spearman correlation we need to return to the Bivariate screen and do the following:
Select Bivariate... from the Correlate option available from the Analyse menu. Check that the Science self-efficacy score[SCIEEFF] and the Science instrumental motivation score[INSMOVSCI] variables are still in the Variables box. Deselect the Pearson tick box. Select the Spearman tick box. Click on the OK button.
In the table produced we see the correlation matrix for the variables we are considering:
Correlations Science self-efficacy score Science instrumental motivation score Spearman's rho Science self-efficacy score Correlation Coefficient 1.000. Sig. (2-tailed).. N 4726 4694 Science instrumental motivation score Correlation Coefficient .333 1. Sig. (2-tailed).. N 4694 4791 **. Correlation is significant at the 0.01 level (2-tailed).
For each correlation there is once again an estimate of the correlation, an accompanying p value and a sample size on which the correlation has been calculated. Here we are interested in the Spearman correlation between SCIEEFF and INSMOVSCI is repeated in two places in the table - either in the row for SCIEEFF and column for INSMOVSCI or the row for INSMOVSCI and column for SCIEEFF.
In this case the correlation (reported as the statistics rho) takes value .333. This represents a moderate positive correlation. The correlation is given in the table, along with a significance value and a sample size which in this case is 4694. This is the number of observations in which both SCIEEFF and INSMOVSCI where observed.
We can test if this correlation is significantly different from zero which will depend on (i) the magnitude of the correlation and (ii) the number of observations on which the correlation is based.
The p value (quoted under Sig. (2-tailed)) is .000 (reported as p < .001) which is less than 0.05. We therefore have significant evidence to reject the null hypothesis that the correlation is 0.
We would report the result as follows: The variables SCIEEFF and INSMOVSCI were significantly and moderatly positively correlated r = .333, N = 4694, p < .001.
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The third possible correlation is known as Kendalls Tau-b and has desirable properties when the variables take values that are ordered categories (i.e. ordinal variables). To calculate the Kendalls Tau-b we need to return to the Bivariate screen and do the following:
Select Bivariate... from the Correlate option available from the Analyse menu. Check that the Science self-efficacy score[SCIEEFF] and the Science instrumental motivation score[INSMOVSCI] variables are still in the Variables box. Deselect the Spearman tick box. Select the Kendall tau-b tick box. Click on the OK button.
In the table produced we see the correlation matrix for the variables we are considering:
