IQ alone might not predict examination success very well. Motivation, on its own, might not predict examination success either. However, together these two variables might predict exami- nation success much better. In life, it is rare to find simple relationships where one variable predicts another, without the influence of anything else. It is more realistic to use multiple vari- ables in statistical analyses.
Example from the literature
Association between depression and aggression in rural women
Meyrueix and colleagues (2015) carried out research on the potential role of aggression in fifty-four depressed rural women. The women completed questionnaires on depression and aggression. The authors state that: ‘Linear regression indicated that aggression was significantly associated with depression, with aggression explaining 16% of variance in depression (β=.399, r2=.159, p=0.003).’ (p. 139)
CHAPTER 12 Regression analysis 399
Let’s say you have collected information on examination success – percentage marks, for instance. You have also measured IQ and given a questionnaire on motivation levels. You could have more than two explanatory variables, of course, but for the purposes of explanation, it is easier to keep to two explanatory variables. How well does the combination of these two vari- ables predict examination success? Multiple regression not only enables us to answer this ques- tion, it also allows us to discover the relative contribution of each separate variable.
So multiple regression shows us the cumulative effects of a set of explanatory variables (x1, x2, etc.) on a dependent variable (called y), and also the separate effects of these explanatory variables.
Let’s say we wanted to do a simple linear regression, predicting exam success from IQ. We would obtain a scattergram, putting exam success on the y-axis, as it is the criterion variable, and IQ on the x-axis, as it is the explanatory variable. The scattergram, with line of best fit, might look like Figure 12.11.
The scattergram of motivation and examination, with line of best fit, might look like F igure 12.12. Both might predict examination success separately, but the prediction may be even better using both together.
In Figure 12.13, both IQ and motivation are shown (in 3D form) relating to examination success (on the y-axis). In this case, instead of a line of best fit, we have a plane of best fit. We can imagine this as a sheet of Perspex, cutting through the cube. The best-fitting plane is one that has the dots closest to the Perspex. It is not possible to imagine, or draw, in more than three dimensions, but SPSS has no problem in analysing data using many explanatory variables.
Multiple regression analysis is, not surprisingly, similar to the regression analysis using one explanatory variable. The following are the statistics that result from a multiple regression analysis.
Figure 12.11 Plot of examination success against IQ IQ
Exam
90 100 110 120 130 140 150 160
10 20 30 40 50 60 70 80 90
Figure 12.12 Plot of examination success against motivation Motivation
Exam
20 30 40 50 60 70 80 90
10 20 30 40 50 60 70 80 90
Statistics without maths for psychology 400
Important parts of the output
The first section confirms that both IQ and Motivation are entered, and that the criterion vari- able is EXAM. The method ‘enter’ means that both IQ and Motivation were entered together in order to predict exam success.
Figure 12.13 Plot of examination success against motivation and IQ 20
40 60 80 100
100 110 120 130 140 150 160 40 30
60 50 80 70 90
Motivation IQ
Exam
Method Variables Entered/Removedb
Enter Variables Entered
IQ, MOTIVa
Variables Removed Model
1
a. All requested variables entered b. Dependent Variable: EXAM
Model summary
The R-value (0.762) is the correlation between EXAM and both of the explanatory variables.
The R2 (0.579) has been adjusted downwards to 0.52.
Std. Error of the Estimate Model Summary
11.906 R Square
.579
Adjusted R Square .515 R
.762a Model
1
a. Predictors: (Constant), IQ, MOTIV
In section 12.1.6, you had only one variable listed in the model summary. Remember that r=b in that case. Here, however, we have two explanatory variables.
Remember that in linear regression this was simply Pearson’s r, the correlation between x and y. In multiple regression, however, r becomes R, and it is the correlation between all the xs and y.5 So in this case, it is the correlation between examination success, and IQ and motivation, y. In this study, R is 0.76.
5 Multiple R actually represents the correlation between the actual y scores and the predicted y scores.
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R2
If you square 0.76 you will obtain 0.58. R2 represents the correlation between all the exploratory variables together with the criterion variable. This means all the variance (both shared and unique) of the exploratory variables in relation to the criterion variable. In our particular sample, 58% of the variance in exam success can be accounted for by IQ and motivation. However, SPSS adjusts this figure downwards to give an estimate of the population R2, otherwise our r is too optimistic. This is because the sample regression line will always fit the sample better than it will the population (since it is the best-fitting line for the sample). So we adjust down- wards. The formula for this takes into account the number of participants and variables. Thus we can say we have accounted for 52% of the variance in y, by our explanatory variables.