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7.Correlation.pptxCorrelation
Correlation analysis is used to describe the strength and direction of the linear relationship between two variables. There are a number of different statistics available from IBM SPSS, depending on the level of measurement and the nature of your data.
The procedure for obtaining and interpreting a Pearson product-moment correlation coefficient (r) is presented along with Spearman Rank Order Correlation (rho).
Correlation
تخدم تحليل الارتباط لوصف قوة واتجاه العلاقة الخطية بين متغيرين. هناك عدد من الإحصائيات المختلفة المتاحة من IBM SPSS ، اعتمادًا على مستوى القياس وطبيعة البيانات الخاصة بك. يتم تقديم الإجراء الخاص بالحصول على معامل ارتباط بيرسون اللحظي للمنتج وتفسيره (r) جنبًا إلى جنب مع ارتباط ترتيب رتبة سبيرمان (rho).
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Pearson r is designed for interval level (continuous) variables. It can also be used if you have one continuous variable (e.g. scores on a measure of self-esteem) and one dichotomous variable (e.g. sex: M/F). Spearman rho is designed for use with ordinal level or ranked data
Pearson correlation coefficients (r) can only take on values from –1 to +1. The size of the absolute value (ignoring the sign) provides an indication of the strength of the relationship.
A scatterplot of this relationship would show a straight line.
Correlation
A correlation of 0 indicates no relationship between the two variables. Knowing the value on one of the variables provides no assistance in predicting the value on the second variable. A scatterplot would show a circle of points, with no pattern evident.
DETAILS OF EXAMPLE
To demonstrate the use of correlation, I will explore the interrelationships among some of the variables included in the survey.sav data file
In this example, we interested in assessing the correlation between respondents’ feelings of control and their level of perceived stress.
Example of research question: Is there a relationship between the amount of control people have over their internal states and their levels of perceived stress?
Do people with high levels of perceived control experience lower levels of perceived stress?
DETAILS OF EXAMPLE
What you need: Two variables: both continuous, or one continuous and the other dichotomous (two values).
What it does: Correlation describes the relationship between two continuous variables, in terms of both the strength of the relationship and the direction.
PRELIMINARY ANALYSES FOR CORRELATION
Before performing a correlation analysis, it is a good idea to generate a scatterplot.
Interpretation of output from scatterplot
The scatterplot can be used to check a number of aspects of the distribution of these two variables.
Step 1: Checking for outliers
Check your scatterplot for outliers—that is, data points that are out on their own, either very high or very low, or away from the main cluster of points.
The distribution of data points can tell you a number of things:
• Are the data points spread all over the place? This suggests a very low correlation.
• Are all the points neatly arranged in a narrow cigar shape? This suggests quite a strong correlation.
Step 2: Inspecting the distribution of data points
Step 2: Inspecting the distribution of data points
• Could you draw a straight line through the main cluster of points, or would a curved line better represent the points? If a curved line is evident (suggesting a curvilinear relationship) Pearson correlation should not be used, as it assumes a linear relationship.
• What is the shape of the cluster? Is it even from one end to the other? Or does it start off narrow and then get fatter? If this is the case, your data may be violating the assumption of homoscedasticity.
Step 3: Determining the direction of the relationship between the variables
The scatterplot can tell you whether the relationship between your two variables is positive or negative.
If a line were drawn through the points, what direction would it point—from left to right; upward or downward?
An upward trend indicates a positive relationship; high scores on X associated with high scores on Y.
A downward line suggests a negative correlation; low scores on X associated with high scores on Y.
In this example, we appear to have a negative correlation, of moderate strength.
Once you have explored the distribution of scores on the scatterplot and established that the relationship between the variables is roughly linear and that the scores are evenly spread in a cigar shape, you can proceed with calculating Pearson or Spearman correlation coefficients.
Before you start the following procedure, choose Edit from the menu, select Options, and on the General tab make sure there is a tick in the box No scientific notation for small numbers in tables in the Output section.
Calculating Pearson or Spearman correlation coefficients.
Calculating Pearson or Spearman correlation coefficients.
The output generated from this procedure (showing both Pearson and Spearman results) is presented below.
Calculating Pearson or Spearman correlation coefficients.
INTERPRETATION OF OUTPUT FROM CORRELATION
Step 1: Checking the information about the sample
The first thing to look at in the table labelled Correlations is the N (number of cases). Is this correct? If there are a lot of missing data, you need to find out why.
In the above example we have 426 cases that had scores on both of the scales used in this analysis
Step 2: Determining the direction of the relationship
INTERPRETATION OF OUTPUT FROM CORRELATION
In the example given here, the Pearson correlation coefficient (–.58) and between Spearman rho value (–.56) are negative, indicating a negative correlation perceived control and stress. The more control people feel they have, the less stress they experience.
Step 3: Determining the strength of the relationship
The second thing to consider is the direction of the relationship between the variables.
The third thing to consider in the output is the size of the value of the correlation coefficient.
This can range from –1 to 1. This value will indicate the strength of the relationship between your two variables. A correlation of 0 indicates no relationship at all, a correlation of 1 indicates a perfect positive correlation, and a value of –1 indicates a perfect negative correlation.
Step 3: Determining the strength of the relationship
INTERPRETATION OF OUTPUT FROM CORRELATION
Interpret values between 0 and 1? The following guidelines:
small r=.10 to .29
medium r=.30 to .49
large r=.50 to 1.0
In the example presented above, there is a large correlation between the two variables (above .5), suggesting quite a strong relationship between perceived control and stress.
Step 4: Calculating the coefficient of determination
INTERPRETATION OF OUTPUT FROM CORRELATION
To get an idea of how much variance your two variables share, you can also calculate what is referred to as the ‘coefficient of determination’.
Square r value (multiply it by itself). To convert this to ‘percentage of variance’, just multiply by 100 (shift the decimal place two columns to the right).
For example, two variables that correlate r=.2 share only .2 × .2 = .04 = 4 per cent of their variance.
In our example the Pearson correlation is .581, which, when squared, indicates 33.76 per cent shared variance. Perceived control helps to explain nearly 34 per cent of the variance in respondents’ scores on the Perceived Stress Scale.
Step 5: Assessing the significance level
INTERPRETATION OF OUTPUT FROM CORRELATION
The next thing to consider is the significance level (listed as Sig. 2 tailed).
The level of statistical significance does not indicate how strongly the two variables are associated (this is given by r or rho), but instead it indicates how much confidence we should have in the results obtained.
The significance of r or rho is strongly influenced by the size of the sample.
In a small sample (e.g. n=30), you may have moderate correlations that do not reach statistical significance at the traditional p<.05 level. In large samples (N=100+), however, very small correlations (e.g. r=.2) may reach statistical significance.
PRESENTING THE RESULTS FROM CORRELATION
The results of the above example using Pearson correlation could be presented in a research report as follows.
The relationship between perceived control of internal states (as measured by the PCOISS) and perceived stress (as measured by the Perceived Stress Scale) was investigated using Pearson product-moment correlation coefficient. Preliminary analyses were performed to ensure no violation of the assumptions of normality, linearity and homoscedasticity. There was a strong, negative correlation between the two variables, r = –.58, n = 426, p < .001, with high levels of perceived control associated with lower levels of perceived stress.
In this case, it would be awkward to report all the individual correlation coefficients in a paragraph; it would be better to present them in a table. One way this could be done is as follows:
PRESENTING THE RESULTS FROM CORRELATION
COMPARING THE CORRELATION COEFFICIENTS FOR TWO GROUPS
Sometimes when doing correlational research you may want to compare the strength of the correlation coefficients for two separate groups. For example, you may want to look at the relationship between optimism and negative affect for males and females separately.
The output generated from the correlation procedure is shown below.
Interpretation of output from correlation for two groups
From the output given above, the correlation between Total optimism and Total negative affect for males was r=–.22, while for females it was slightly higher, r=–.39.
It is important to note that this process is different from testing the statistical significance of the correlation coefficients reported in the output table above. The significance levels reported above (for males: Sig. = .003; for females: Sig. = .000).
