Interpreting SPSS Correlation Output - University of Dayton

Interpreting SPSS Correlation Output

Correlations estimate the strength of the linear relationship between two (and only two) variables. Correlation coefficients range from -1.0 (a perfect negative correlation) to positive 1.0 (a perfect positive correlation). The closer correlation coefficients get to -1.0 or 1.0, the stronger the correlation. The closer a correlation coefficient gets to zero, the weaker the correlation is between the two variables. Ordinal or ratio data (or a combination) must be used. The types of correlations we study do not use nominal data.

SPSS permits calculation of many correlations at a time and presents the results in a "correlation matrix." A sample correlation matrix is given below. The variables are:

Optimism: "Compared to now, I expect that my family will be better off financially a year from now. Life Satisfaction: Overall, life is good for me and my family right now. Entrepreneurial Interest:: I am interested in starting a business or investing in a business in the next six months.

All measures were recorded on five point Likert scales anchored by Strongly Disagree (1) to Strongly Agree (5).

Family's income will improve Pearson Correlation Sig. (2-tailed)

Life is good right now.

N Pearson Correlation Sig. (2-tailed)

Consider business

N Pearson Correlation Sig. (2-tailed)

N

Family's income will improve 1

491 .494**

.000 488 .248** .000

469

Life in is good right now. .494** .000

488 1

501 .228**

.000

475

Consider business

.248** .000

469 .228**

.000 475

1

479

The correlation coefficient for Optimism and Satisfaction is 0.494. For survey scale type data this is pretty large. The number of respondents in the sample answering both items is 488. p-value for this correlation coefficient is .000. It's not technically zero. SPSS does not give p-values to more than three decimal places

The statistical hypothesis test for this p-value is:

Notice the diagonal of ones. These are perfect correlations between variables and themselves. The matrix is symmetrical on either side of the diagonal, meaning all correlations are given twice.

H0: There is no significant relationship between Optimism and Life Satisfaction. Ha: There is a statistically significant relationship between Optimism and Life Satisfaction.

Because p < .05, reject the null of no relationship and conclude that the relationship is statistically significant.

Interpreting SPSS ANOVA Output

Analysis of Variance (ANOVA) tests for differences in the mean of a variable across two or more groups. The dependent (Y) variable is always ordinal or ratio data while the independent (X) variable is always nominal data (or other data that's converted to be nominal). With ANOVA, the independent variable can have as many levels as desired.

A sample of SPSS ANOVA output is below and on the following page. The variables in this example are:

Entrepreneurial Interest (Y):: I am interested in starting a business or investing in a business in the next six months. State of Residence(X): Florida, Nevada, Texas

Descriptive Statistics

Dependent Variable: Consider starting your own business

data collection location

Mean Std. Deviation

Nevada Florida

1.24

2.474

2.61

1.057

Texas Total

2.40

1.069

2.22

1.582

N 29 49 50

128

This table simply provides the means, standard deviations, and group sizes for the dependent variable for all three levels of the independent variable.

Tests of Between-Subjects Effects Dependent Variable: Consider starting your own business

Source Corrected Model Intercept location Error Total Corrected Total

Type III Sum of Squares 36.932a 522.197 36.932 280.943 948.000 317.875

df

Mean Square

2

18.466

1

522.197

2

18.466

125

2.248

128

127

F 8.216 232.341 8.216

Sig. .000 .000 .000

This table gives the main effects of the AVOVA test. This is where you look first to see if any significant differences exist in the dependent variable between levels of the independent variable.

For this ANOVA application, the only Source variable we're interested in is our independent variable, location. If you follow across to the right to the "Sig." column, you find the p-value for this hypothesis test. That test is:

H0: There is no difference between the three states in respondent interest in starting a business. Ha: At least one state will differ from the others in interest in starting a business. The alternative hypothesis is worded this way because there are three levels of the independent. The pvalue indicates that the null should be rejected, but it does not say how the states differ. We use "multiple group comparisons" to that (even though in this case it's pretty obvious).

There are many ways to calculate multiple difference tests. The one used below is called the Least Significant Differences test (LSD). It compares all possible pairs of levels of the independent variable and tests each for significance in a way that controls what's referred to as the experiment-wide error rate. Note that all information is given twice. That is, Texas is compared to Florida and then Florida is compared to Texas. It can get confusing.

Consider starting your own business LSD

(I) data collection location

(J) data collection location

Nevada

Florida

Florida Texas

Texas Nevada Texas Nevada

Florida

Multiple Comparisons

Mean Difference (I-J) -1.37* -1.16* 1.37* .21 1.16*

-.21

Std. Error .351 .350 .351 .301 .350 .301

95% Conf. Interval

Sig. Lower Bound Upper Bound

.000

-2.07

-.68

.001

-1.85

-.47

.000

.68

2.07

.483

-.38

.81

.001

.47

1.85

.483

-.81

.38

The hypothesis test for each pairwise comparison is the same: H0: There is no difference between the two states being compared. Ha: There is a difference between the two states being compared.

There are three possible comparisons: Nevada with Florida. The p-value is .000, meaning reject the null and conclude the state differ. Nevada with Texas. The p-value is .001, meaning reject the null and conclude the state differ. Florida with Texas. The p-value is .483, meaning do not reject the null. Conclude the states don't differ.

Thus, respondents in Texas and Florida don't significantly differ in their interests in starting a business. However, these two states significantly differ from Nevada. Check the descriptive statistics and you'll see that the mean values show how. There's significantly less interest among Nevada respondents in starting a business than respondents from Florida and Texas, who are equally interested.

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