SPSS Statistical Software

While descriptive statistics summarize the characteristics of a dataset, inferential statistics allow you to make conclusions and predictions based on the data. Descriptive statistics allow you to describe a dataset, while inferential statistics allow you to make inferences based on a dataset.

When you have collected data from a sample, you can use inferential statistics to understand the larger population from which the sample is taken. Generally with research, it is not possible for us to collect data from every individual in the world, so we collect data from a sample of individuals and then use our results from our sample to make predictions about what we would expect to see for all individuals based on our results. However, unlike this oversimplified explanation, you likely have specific groups of individuals (i.e., specific populations) you are interested in researching (for example: all women over the age of 35, all employed adults in the US, all Christian families in Texas, all pediatric doctors in Mexico, etc.) so your population would be all individuals who fit your criteria of interest, not just all individuals in general in the entire world. 

Inferential statistics have 2 main uses:

1. Making estimates about populations of interest (for example, using the average SAT score of a sample of 11th graders to infer what the average SAT score of all 11th graders in the US might be).
2. Testing hypotheses to draw conclusions about populations (for example, the relationship between SAT scores and family income).

(Information adapted from Bhandari, P. (2023). Inferential Statistics | An Easy Introduction & Examples. Scribbr).

Correlations are a simple statistical analysis that are great for helping you predict values of one variable based on another. 

Correlations are a measure of how strongly 2 variables are related to each other. The number you will see in a correlation analysis will represent the strength of the relationship between the 2 variables. Correlations range from -1 to +1, and values closer to either -1 or +1 signify stronger relationships. A correlation of 0 means no relationship.

Positive correlations mean as one of the variables increases, so does the other. Negative correlations means as one variable increases, the other decreases.

Let’s run a correlation!

Background Info on the Pearson Correlation

As a note, the Pearson bivariate correlation (bivariate just means 2 variables) is the most common type of correlation you will come across in research, though you will often just see it simply referred to as a "correlation" or "correlational analysis." There are also other types of correlations, such as the Spearman rank-order correlation, however, for most intents and purposes with quantitative data, the Pearson correlation is the one you will likely use. This is because the Pearson correlational analysis is for Scale data, whereas the Spearman rank order is for Ordinal (rank-ordered) data. It is not advised to run correlations on Nominal data, it will not give meaningful results as the numeric values of Nominal variables just represent categories.

As another note, correlations only measure linear relationships, not parabolic, cubic, or other non-linear relationships. So even though your correlational analysis may not be statistically significant, it is possible that the variables you are looking at relate to each other in some other way. (Two variables can be perfectly related, but if the relationship is not linear, a correlation coefficient is not an appropriate statistic for measuring their association).

Running a Pearson Correlation

1. Click on Analyze on the menubar, then select Correlate, then select Bivariate.
    
2. Let's use the the Age and Salary variables for this example. In the popup window, move Age and Salary over to the Variables box. (You can select more than two variables, but we will just use two for now). You'll notice that Pearson is selected by default for the Correlation Coefficient, this is what we want. Leave the Test of Significance as Two-Tailed (I'll explain down below the difference). Optional: you can check the box for Show only the lower triangle if you don't want the mirrored-image upper triangle to display. (I'll show the difference down below). Click OK to run the analysis.
    
3. Now you will see the Correlational analysis appear in your Output window. You should see the header Correlations and one table below it also labelled Correlations. It should look like this: 
    
4. How to interpret the Correlation table:
   1. You will see each variable you selected listed on the left side of the table and at the top. This is because we interpret correlations by looking at the intersection of separate variables in the table. 
   2. Look at the Age column and its intersection with the Salary row. We see 3 small rows labelled Pearson Correlation, Sig. (2-tailed), and N. 
      1. The Pearson Correlation row shows us the Pearson correlation coefficient for the linear relationship between Age and Salary. Correlations are reported with the letter r. There is a moderately strong, positive correlation between Age and Salary  (r = .762). (Remember, the closer a correlation coefficient is to -1 or +1, the stronger the relationship between the 2 variables).
      2. The Sig. (2-tailed) row shows us the p-value (or significance value) of this correlation coefficient. For Age and Salary, we have a p-value of less than .001. We report this as: p < .001. Altogether with correlation coefficient, you would report it as: There is a moderately strong, positive correlation between Age and Salary  (r = .762, p < .001). Note: p-values less than .05 are generally considered statistically significant, meaning that we are fairly certain that the 2 variables are actually related and this result wasn't just due to random chance. A p-value of .05 roughly translates to there being a 5% chance that the results are due to random chance; you want lower p-values because lower p-values indicate that there is an even smaller chance that the results are due to random chance. For example a p-value of .01 translates to there being a 1% chance that the results are due to random chance.
      3. The N row shows us the sample size for the two variables in this correlational analysis. In our example, we see that 50 individuals provided data for both Age and Salary. 

Another Pearson Correlation Example

As another example, let's run a correlation with 4 variables: Age, Salary, Years Employed, and Anxiety 1. Follow the steps above, but this time in Step 2, move Age, Salary, Years Employed, and Anxiety 1 over to the Variables box. Then click OK to run the analysis. See the resulting output table below. (You can add as many variables as you want to a correlational analysis, but keep in mind that the resulting correlation table will get increasingly larger with the more variables you add, and it may consequently become more difficult for you to read the table accurately).

Additional Options: Show Only the Lower Triangle

When you check the box for Show only the lower triangle, the resulting correlation table will not display the mirrored-image, identical values in the upper portion of the table. You can see what this looks like for our Age and Salary correlation table below. Now we see blank cells directly underneath the Salary column where it intersects with the Age row because these cells would contain the exact same values as the cells in the Salary row where it intersects with the Age column. These repeated/mirrored values in the upper "triangle" of the table are not shown. 

The effect of showing only the lower triangle becomes even more apparent in larger correlation tables with more variables; take a look at the correlation table below with Age, Salary, Years Employed, and Anxiety 1. (The top table does not have the box checked for showing only the lower triangle. The bottom table does have the box checked).
 

Notice how much easier it is to read the table when we check the box for Show only the lower triangle. As the upper triangle just repeats/mirrors the values in the lower triangle, you do not need to have the upper triangle visible to be able to interpret your correlation results.

Additional Options: One-Tailed vs Two-Tailed

Generally, you’ll want to use the Two-Tailed test of significance (also called a two-tailed p-value) because a two-tailed test will test for any relationship between the 2 variables. A One-Tailed test only tests for one specific direction (either positive or negative, but not both), and you would have had to make a hypothesis about the specific direction you expected to see prior to running the analysis in order to use a one-tailed test of significance (i.e., a one-tailed p-value). A two-tailed test tests for both positive or negative relationships, so if you don’t know how the variables may relate and just want to know if they relate, use the two-tailed test.

When in doubt, it is almost always more appropriate to use a two-tailed test. A one-tailed test is only justified if you have a specific prediction (hypothesis) about the direction of the difference (e.g., Age being positively correlated with Salary), and you are completely uninterested in the possibility that the opposite outcome could be true (e.g., Age being negatively correlated with Salary).

Additional Options: Style - Highlighting Significant Correlations

Another useful option/setting that you can play around with is the Style settings of the correlation table.

1. Click on the Style button within the Correlations popup window to open the Style settings window. 
2. This brings up the Table Style editor window. Click in the cell under Value, then click the dropdown arrow that appears. Select Significance.
    
3. Once you select Significance, information will populate in the Dimension, Condition, and Format columns. The information that populates by default specifies that for significance values (p-values) less than or equal to .05, SPSS will format those cells to be bright yellow. That is, all significant correlations will be highlighted in yellow. (You can adjust the Condition to other p-values if you would like; for example you could change it to be .01 so only correlations with significance values less than or equal to .01 are highlighted. You can also adjust the Format to other colors instead of yellow). For our purposes, the default conditions of .05 and yellow are fine, so let's just click Continue to save these Style settings. 
    
4. Then click OK on the initial Correlation popup window to run the analysis. Now you'll see the following output with our significant correlations highlighted. (Note: I checked the box for Show only the lower triangle).
    

Note: We selected Significance when we were adjusting the Style settings, but you could instead select Correlation and set a specific value of correlation coefficient for SPSS to then highlight in your table. You can set the condition to specify highlighting the cells that have a correlation coefficient equal to or higher than your specified value. Or you could have both a Significance condition and a Correlation condition - if you click Add in the Style settings window, you can add multiple conditions. 

The Chi-Square Test of Independence (a.k.a., Chi-Square Test of Association) determines whether there is an association/relationship between nominal or ordinal variables. (This is different from a Pearson correlation because a Pearson correlation is meant for testing associations/relationships between scale variables). The Chi-Square Test is an extension of the Crosstabs analysis in SPSS. While Crosstabs can show you how two nominal (or ordinal) variables compare, the Chi-Square Test can assess if these variables are significantly related or not. 

Examples of when to use a Chi-Square Test of Independence:

• You want to see if Level of Education (e.g., High School Diploma, Associate's Degree, Bachelor's Degree, Master's Degree) is related to Marital Status.
• You want to see if Gender is related to Smoking Status (i.e., whether someone smokes cigarettes or not).
• You want to see if Ethnicity is related to Political Party Affiliation.
• You want to see if the effectiveness of a public health intervention, like a flyer vs phone call, is related to an outcome, such as the rate of recycling.
• You want to see if if someone's age group is associated with their preference for social media platform.

(Not to be confused with the Chi-Square Goodness-of-Fit test, which is an entirely different test. The Chi-Square Goodness-of-Fit test is for when you want to determine if a the distribution of a single categorical variable matches a specific, known distribution, like the normal distribution. The Chi-Square Test of Independence is for testing whether two categorical variables are statistically associated, and it uses a crosstabs/contingency table to compare the two variables).

Extra note: The chi-square test of independence is a nonparametric test. A non-parametric test is a statistical test that does not assume that the data comes from a population with a specific distribution, such as the normal distribution. These tests, also known as "distribution-free" tests, are useful for data that is non-normally distributed, has small sample sizes, contains outliers, or is measured at the ordinal or nominal level. Because the chi-square test of independence is for assessing nominal and/or ordinal data, it is considered a non-parametric test.

Statistical Assumptions of the Chi-Square Test of Independence

1. Two categorical variables (can be nominal and/or ordinal)
2. Each variable has 2 or more categories/groups
3. Independence of observations
   1. All of your participants/data-points should be independent of each other. This means each participant is unique, in other words, nobody took your survey or submitted data more than once. Each data point is unique.
   2. The categorical variables are not "paired" in any way (e.g., the variables you are using in this analysis are not paired pre-test and post-test observations).
4. Relatively large sample size.
   1. Expected frequencies for each cell are at least 1.
   2. Expected frequencies should be at least 5 for the majority (80%) of the cells. (In other words, no more than 20% of your cells should have expected counts less than 5. SPSS notes this percentage at the bottom of the relevant tables when you run the analysis).

For further info on these assumptions, check out these resources: Laerd Statistics Chi-Square Test for Association Guide, Scribbr Chi-Square Test of Independence Guide, Kent State Chi-Square Test of Independence Guide. 

Running a Chi-Square Test of Independence

1. Click on Analyze on the menubar, then select Descriptives, then select Crosstabs. (Note: You'll notice there is an option at the bottom of the list called ChiSquare, which also allows you to run a Chi-Square Test of Independence. However, I recommend using the Crosstabs menu to run the Chi-Square Test of Independence because it allows you to specify/adjust useful parameters and options for the analysis that are not available in the stand-alone ChiSquare menu).
    
2. In the popup window, select one nominal/ordinal variable and move it to the Row(s) box and then select another nominal/ordinal variable and move it to the Column(s) box. For example purposes (using the example dataset), let's select Gender and move it over to the Row(s) box, and then select Work Field and move it over to the Column(s) box. 
   (Note: it doesn't matter which one you put in the Row(s) box and which in the Column(s) box, it just changes the orientation of the table. I suggest putting the variable with more categories in the Row(s) box and the variable with less categories in the Column(s) box so you get a tall, narrow table instead of a short, wide table that you may need to scroll left/right to fully see). 
    
3. Click on the Statistics button on the right. In the little window that appears, check the box for Chi-Square. Also check the box for Phi and Cramer's V. Click Continue. (Note: if you forget to check the box for Chi-Square, then SPSS will just run a regular crosstabs analysis).
   1. Phi and Cramer's V are effect sizes for the Chi-Square analysis. A chi-square analysis just tells you if the 2 variables are significantly related, not the strength of that relationship. The effect size (Phi or Cramer's V) can tell you the strength of the relationship. The analysis outputs both effect size values, but which one you need to use depends on how many categories are in each of the two variables you are analyzing. If each variable only has 2 categories (e.g., Gender: Male or Female, and Smoking Status: Yes smoker or Not smoker), you will use Phi. If one (or both) of the variables has more than 2 categories (e.g., Work Field has 5 categories), then you will use Cramer's V.
       
4. Click on the Cells button on the right. In the little window that appears, check the box under Counts for Expected (also make sure Observed is checked too), check the box under Percentages for Row, check the box under Residuals for Adjusted Standardized. Click Continue.
    
5. Optional: After clicking Continue and returning to the main Crosstabs window, you can check box for Display Clustered Bar Charts if you would like SPSS to include in the output a bar chart with the frequency counts of how many (in our example) men and women work in each Work Field category.
6. Click OK.
7. Now you should see the Crosstabs with Chi-Square analysis in your output window. 
    
    
    
8. How to interpret the output:
   1. Case Processing Summary table: This table just shows you the number of individuals in your sample that provided data for both of your specified variables and if there is any missing data.
   2. Crosstabulation table: The Crosstabulation table (labelled with the names of the 2 variables you selected - in our example, Gender * Work Field Crosstabulation) is the crosstabs results table. Each row and column will correspond to the variables you selected for the analysis.
      1. We see Work Field on the top of the table with each specific Work Field category listed in the columns. Gender is listed at the left side of the table with the rows representing each category of the Gender variable. 
      2. The cells in the intersections of each column and row represent the number of individuals in our sample who fell under both of the corresponding categories. Because we checked the boxes for Count, Expected Count, Row Percentages, and Adjusted Standardized Residuals, we see cells for each of those specifications. For example, the cell that intersects BUS and Male has a value of 6 in it, indicating that 6 individuals in our sample responded that they are Male and that they work in the Business field. If we look at Female and HEALTH, we see that 8 women in our sample work in the Health field.
      3.  
      4. The Total column lists the total number of individuals in our sample who work in each Work Field category, regardless of their Gender. Each total value is the sum of the cells to the left of it. We see we have 10 individuals who work in the Business field, 9 who work in the Education field, 9 who work in the Government field, and so on. (This column should show us the same numbers as if we ran Frequencies on just the Work Field variable)
         1. The Total row lists the total number of individuals in our sample who fall under each Gender category, regardless of their Work Field. Each total value is the sum of the cells above it. We see we have 24 Males in our sample and 26 Females. (This row should show us the same numbers as if we ran Frequencies on just the Gender variable).

Multivariate analyses consist of analyzing more than one dependent variable at once. This is in contrast to all of the above Univariate analyses which only include one dependent variable per analysis. 

To conduct multivariate analyses in SPSS, you will often need to use the General Linear Model function.

Coming Soon!