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FONTS

Correlation and Partial correlation

Interpretation of output from scatterplot

Examples of Correlation Coefficients

Step 1: Checking for outliers-

Step 2: Inspecting the distribution of data points

PRELIMINARY ANALYSES FOR CORRELATION

Two types of correlation

Step 3: Determining the direction of the relationship between the variables

The syntax generated from this procedure is:

CORRELATIONS

/VARIABLES=tpstress tpcoiss

/PRINT=TWOTAIL NOSIG

/MISSING=PAIRWISE .

NONPAR CORR

/VARIABLES=tpstress tpcoiss

/PRINT=SPEARMAN TWOTAIL NOSIG

/MISSING=PAIRWISE

3. A positive correlation coefficient occurs when the values of both variables increase together.

(ex, between studying hard and high grades in school.)

4. A negative correlation occurs when the increase of one variable corresponds with the decrease of another.

(ex, seen in less stage fright when more time was spent practicing lines in a play.)

1. A correlation coefficient close to zero indicates a weak linear relationship between two variables.

(ex, between one's type of pet and one's personality.)

1. Simple bi variate correlation (zero order correlation)- between two variables.

2.Partial correlation- the relationship between two variables while controlling for another variable.

5. To run this new syntax, you need to highlight the text from CORRELATIONS, down to and including the full stop at the end. It is very important that you include the full stop in the highlighted section. Alternatively, you can click on Correlations on the left-hand side of the screen.

6. With this text highlighted, click on the green triangle or arrow-shaped icon (>), or alternatively click on Run from the Menu, and then Selection from the drop-down menu that appears. This tells SPSS that you wish to

run this procedure. The output generated from this procedure is shown as follows

2. A correlation coefficient of zero would indicate that there is no correlation, or relationship, between two variables.

(ex, between shoe size and number of books read)

Procedure for requesting Pearson r or Spearman rho

1. From the menu at the top of the screen, click on Analyze, and then select Correlate, then Bivariate.

2. Select your two variables and move them into the box marked Variables (e.g. Total perceived stress: tpstress, Total PCOISS: tpcoiss). If you wish you can list a whole range of variables here, not just two. In the resulting matrix, the correlation between all possible pairs of variables will be listed. This can be quite large if you list more than just a few variables.

3. In the Correlation Coefficients section, the Pearson box is the default option. If you wish to request the Spearman rho (the non-parametric alternative), tick this box instead (or as well).

4. Click on the Options button. For Missing Values, click on the Exclude cases pairwise box. Under Options, you can also obtain means and standard deviations if you wish.

5. Click on Continue and then on OK (or on Paste to save to Syntax Editor).

Procedure for generating a scatter plot

1. From the menu at the top of the screen, click on Graphs, then select Legacy Dialogs.

2. Click on Scatter/Plot and then Simple Scatter. Click Define.

3. Click on the first variable and move it into the Y-axis box (this will run vertically). By convention, the dependent variable is usually placed along the Y-axis (e.g. Total perceived stress: tpstress)

4. Click on the second variable and move to the X-axis box (this will run across the page). This usually the independent variable (e.g. Total PCOISS: tpcoiss)

5. In the Label Cases by: box, you can put your ID variable so that outlying points can be identified.

6. Click on OK (or on paste to save to Syntax Editor)

The output generated from this procedure is shown as follows.

Chapter 11: Correlation

The output generated from this procedure (showing both Pearson and Spearman results) is presented below.

• What it does: Correlation describes the relationship between two continuous variables, in terms of both the strength of the relationship of the direction.

Details of Example

In survey4ED.sav designed to explore the factors that affect respondent’s psychological adjustment and well being. The author was interested in assessing the correlation between respondent’s feelings of control and their level of perceived stress. Details of the two variables are provided below.

-A measurement used to show the strength between two variables using a value between -1 and 1; it is often symbolized in mathematical formulas as the letter r.

• Examples 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?

• Assumptions: assumptions common to all techniques.

1. Level of measurement- ( Nominal, ordinal, and interval)

2. Related pairs- each subject must provide a score on both variable X and Y (related pairs)

3. Independence of observations- The observations that make up your data must be independent of one another.

• Non- parametric alternative: Spearman Rank order Correlation. (rho)

  • The correlation coefficient shows the strength of a relationship; the closer the correlation coefficient is to positive or negative 1, the stronger the relationship, whether it be negative or positive, between the two variables.
  • The correlation coefficient cannot be less than -1 or greater than 1.

• What you need: Two variables: both continuous, or one continuous and the other dichotomous (two values)

INTERPRETATION OF OUTPUT FROM CORRELATION

COMPARING THE CORRELATION COEFFICIENTS FOR TWO GROUPS

Step 1: Checking the information about the sample

Step 2: Determining the direction of the relationship

Step 3: Determining the strength of the relationship

Step 4: Calculating the coefficient of determination

Step 5: Assessing the significance level.

The procedure for obtaining and interpreting a Pearson product- moment correlation coefficient (r) is presented along with Spearman Rank Order Correlation (rho)

Pearson r is designed for....

1. Interval level (continuous) variables. (e.g. scores on a measure of self esteem)

2. One dichotomous variable (e.g. sex: M/F)

Spearman rho is designed for use with ordinal level or ranked data and is particularly useful when your data does not meet the criteria for Pearson correlation.

Procedure for comparing correlation coefficients for two groups of Participants

Step 1: Split the sample

1. From the menu at the top of the screen, click on Data, then select Split File.

2. Click on Compare Groups.

3. Move the grouping variable (e.g. sex) into the box labelled Groups based

on. Click on OK (or on Paste to save to Syntax Editor). If you use syntax, remember to run the procedure by highlighting the command and clicking on Run.

4. This will split the sample by sex and repeat any analyses that follow for these two groups separately.

Procedure for obtaining correlation coefficients between one group of variables and another group of variables

PRESENTING THE RESULTS FROM CORRELATION

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 < .0005, with high levels of perceived control associated with lower levels of perceived stress.

TESTING THE STATISTICAL SIGNIFICANCE OF THE DIFFERENCE BETWEEN CORRELATION COEFFICIENTS

Step 2: Correlation

1. Follow the steps in the earlier section of this chapter to request the correlation between your two variables of interest (e.g. Total optimism: toptim, Total negative affect: tnegaff). The results will be reported separately for the two groups.

The syntax for this command is:

CORRELATIONS

/VARIABLES=toptim tnegaff

/PRINT=TWOTAIL NOSIG

/MISSING=PAIRWISE .

3. Click on Paste. This opens the Syntax Editor window.

4. Put your cursor between the first group of variables (e.g. tposaff, tnegaff, tlifesat) and the other variables (e.g. tpcoiss and tmast). Type in the word WITH (tposaff tnegaff tlifesat with tpcoiss tmast). This will ask SPSS to calculate correlation coeffi cients between tmast and tpcoiss and each of the other variables listed. The fi nal syntax should be:

CORRELATIONS

/VARIABLES=tposaff tnegaff tlifesat with tpcoiss tmast

/PRINT=TWOTAIL NOSIG

/MISSING=PAIRWISE.

1. From the menu at the top of the screen, click on Analyze, then select Correlate, then Bivariate.

2. Move the variables of interest into the Variables box. Select the first group of variables (e.g. Total Positive Affect: tposaff, total negative affect: tnegaff, total life satisfaction: tlifesat), followed by the second group (e.g. Total PCOISS: tpcoiss, Total Mastery: tmast). In the output that is generated, the first group of variables will appear down the side of the table as rows and the second group will appear across the table as columns. Put your longer list first; this stops your table being too wide to appear on one page

DETAILS OF EXAMPLE

INTERPRETATION OF OUTPUT FROM PARTIAL CORRELATION

Correlation relationship among a group of variables

Chapter 12: Partial Correlation

What it does: Partial correlation allows you to explore the relationship between

two variables, while statistically controlling for (getting rid of) the effect of another variable that you think might be contaminating or infl uencing the relationship.

Assumptions: For full details of the assumptions for correlation, see the introduction to Part Four. Before you start the following procedure, choose Edit from the menu, select Options, and make sure there is a tick in the box No scientific notation for small numbers in tables.

Example of research question: After controlling for participants’ tendency to present themselves in a positive light on self-report scales, is there still a significant relationship between perceived control of internal states (PCOISS) and levels of perceived stress?

What you need: Three continuous variables:

• two variables that you wish to explore the relationship between (e.g. Total PCOISS,

Total perceived stress)

• one variable that you wish to control for (e.g. total social desirability: tmarlow).

ADDITIONAL EXERCISES

Health

Data fi le: sleep4ED.sav. See Appendix for details of the data fi le.

1. Check the strength of the correlation between scores on the Sleepiness and Associated

Sensations Scale (totSAS) and the Epworth Sleepiness Scale (ess).

2. Use Syntax to assess the correlations between the Epworth Sleepiness Scale (ess)

and each of the individual items that make up the Sleepiness and Associated

Sensations Scale (fatigue, lethargy, tired, sleepy, energy).

The output provides you with a table made up of two sections:

1. In the top half of the table is the normal Pearson product-moment correlation matrix between your two variables of interest (e.g. perceived control and perceived stress), not controlling for your other variable. In this case, the correlation is –.581.

The word ‘none’ in the left-hand column indicates that no control variable is in operation.

2. The bottom half 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. social desirability). In this case, the new partial correlation is –.552.

In this case, A and B may look as if they are related, but in fact their apparent relationship is due, to a large extent, to the influence of C. If you were to statistically control for the variable C then the correlation between A and B is likely to be reduced, resulting in a smaller correlation coefficient.

Partial correlation is similar to Pearson product-moment correlation except that it allows you to control for an additional variable. This is usually a variable that you suspect might be influencing your two variables of interest. By statistically removing the influence of this confounding variable, you can get a clearer and more accurate indication of the relationship between your two variables.

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