Testing Assumptions: The Shapiro-Wilk Test and the Levene Test

Testing Assumptions: The Shapiro-Wilk Test and the Levene Test

Recall from Unit 7 that two assumptions of the t test are that:

1. The outcome variable Y is normally distributed.
2. The variance of Y scores is approximately equal across groups (homogeneity of variance assumption).

The Shapiro-Wilk Test

In addition to a visual inspection of histograms and calculation of skewness and kurtosis values, SPSS provides a formal statistical test of normality referred to as the Shapiro-Wilk test. A perfect normal distribution will have a Shapiro-Wilk value of 1.00. Values less than 1.00 indicate an increasing departure from a perfect normal shape. The null hypothesis of the Shapiro-Wilk test is that the distribution is normal. When the Shapiro-Wilk test indicates a p value less than .05, the normality assumption may be violated, which can be problematic.To obtain the Shapiro-Wilk test in SPSS, follow the step-by-step guide for t tests that is provided in the Unit 8 assignment. SPSS provides the Shapiro-Wilk test output for interpretation. A significant Shapiro-Wilk test ( p < .05) suggests that the distribution is not normal and interpretations may be affected. However, the t test is fairly robust to violations of this assumption when sample sizes are sufficiently large (that is, greater than 100 members).

The Levene Test

The homogeneity of variance assumption is tested with the Levene test. The Levene test is automatically generated in SPSS when an independent samples t test is conducted. The null hypothesis for the Levene test is that group variances are equal. A significant Levene test ( p < .05) indicates that the homogeneity of variance assumption is violated. In this case, report the “Equal variances not assumed” row of the t-test output from SPSS. This version of the t test uses a more conservative adjusted degrees of freedom ( df) that compensates for the homogeneity violation. The adjusted df can often result in a decimal number (such as df = 13.4), which is commonly rounded to a whole number in reporting ( df = 13). If the Levene test is not significant (that is, homogeneity is assumed), report the “Equal variances assumed” row of the t-test output from SPSS.

Proper Reporting of the Independent Samples t Test

Reporting a t test in proper APA style requires an understanding of several elements, including the statistical notation for an independent samples t test ( t), the degrees of freedom in parentheses, the t value, the probability value, and the effect size. To provide context, provide the means and standard deviations for each group. Warner (2013) also recommends reporting the 95% confidence interval (CI) for the difference in sample means. Consider the following example from Warner (2013, p. 213):

The mean HRs differed significantly, t(18) = −2.75, p = .013 (two-tailed). Mean HR for the no-caffeine group ( M = 57.8, SD = 7.2) was about 10 bpm lower than mean HR for the caffeine group ( M = 67.9, SD = 9.1). The effect size, as indexed by η ², was .30; this is a very large effect. The 95% CI for the difference between sample means, M ₁ − M ₂, had a lower bound of −17.81 and an upper bound of −2.39.

t, Degrees of Freedom, and t Value

The statistical notation for an independent samples t test is t, and following it is the degrees of freedom for this statistical test. The degrees of freedom for t is n₁ + n₂ − 2, where n₁ equals the number of participants in Group 1 and n₂ equals the number of participants in Group 2. In the example above, there are 10 people in each group: N = 20 ( n₁ = 10; n₂ = 10), so the df = 18 ( n₁ + n₂− 2). Warner (2013) recommends that the t test should not be conducted with groups of fewer than 10 members. The t value is a ratio of the difference in group means divided by the standard error of the difference in sample means. The t value can be either positive or negative.

Probability Value

Appendix B (pp. 1056–1057) of the Warner text provides critical values of t for rejecting the null hypothesis. In the example above, with 18 degrees of freedom and alpha level set to .05 (two-tailed), the table indicates a critical value of ±2.101 to reject the null hypothesis. The obtained t value above is −2.75, which exceeds the critical value required to reject the null hypothesis. SPSS determined the exact p value to be .013. This p value is less than .05, which indicates that the null hypothesis should be rejected for the alternative hypothesis—that is, the two groups are significantly different in mean heart rate.

Effect Size

A common index of effect size for the independent samples t test is eta squared (η²). SPSS does not provide this output for the independent samples t test, but it is easily calculated by hand with the following formula: t² ÷ ( t² + df). In the example above, the calculation is (−2.75)² ÷ [(−2.75)² + 18] = 7.56 ÷ [(7.56 + 18)] = 7.56 ÷ 25.56 = .30. The effect size is interpreted using Table 5.2 in the Warner text (p. 208).

Confidence Intervals

Recall that confidence intervals (CIs) were introduced in Unit 2. Standard APA reporting of the independent samples t test should include the 95% confidence interval for the difference in sample means, which is provided in the SPSS output for the t test.The Warner text provides a “Results” example at the end of each chapter for all statistics studied in this course. You are encouraged to review these examples and follow their structure when writing up Section 4, “Interpretation,” of the DAA Template.

Reference

Warner, R. M. (2013). Applied statistics: From bivariate through multivariate techniques (2nd ed.). Thousand Oaks, CA: Sage.

OBJECTIVES

To successfully complete this learning unit, you will be expected to:

1. Interpret SPSS t-test output and draw conclusions.
2. Articulate a research question, null hypothesis, and alternative hypothesis.
3. Analyze the application of t tests in your career.

[u08s1] Unit 8 Study 1

Studies

Readings

Use your IBM SPSS Statistics Step by Step text to complete the following:

• Read Chapter 11, “The t Test Procedure.” This reading addresses the following topics:
  • Independent samples t test.
  • Paired samples t test.
  • One-sample t test.
  • Significance testing.
  • SPSS commands.
  • Reporting and interpreting SPSS output.

PSY Learners – Additional Required Readings

In addition to the other required study activities for this unit, PSY learners are required to read the following:Delphin-Rittmon, M. E., Flanagan, E. H., Bellamy, C. D., Diaz, A., Johnson, K., Molta, V., & . . . Ortiz, J. (2015). Learning from those we serve: Piloting a culture competence intervention co-developed by university faculty and persons in recovery. Psychiatric Rehabilitation Journal. doi:10.1037/prj0000155

[u08a1] Unit 8 Assignment 1

t Tests

Resources

•  t Tests Scoring Guide.
•  DAA Template.
•  SPSS Data Analysis Report Guidelines.
•  IBM SPSS Step-by-Step Guide: t Tests.
•  Copy/Export Output Instructions.
•  APA Style and Format.

See the Resources area for links to resources that you will use for this assignment:

• You will complete this assignment using the DAA Template.
• Read the SPSS Data Analysis Report Guidelines for a more complete understanding of the DAA Template and how to format and organize your assignment.
• Refer to the IBM SPSS Step-By-Step Guide: t Tests for additional information on using SPSS for this assignment.
• If necessary, review the Copy/Export Output Instructions to refresh your memory on how to perform these tasks. As with your previous assignments, your submission should be in narrative format with supporting statistical output (table and graphs) integrated into the narrative in the appropriate places (not all at the end of the document).

You will analyze the following variables in the grades.sav data set:

• gender
• gpa

Step 1: Write Section 1 of the DAA.

• Provide the context of the grades.sav data set.
• Include a definition of the specified variables (predictor, outcome) and corresponding scales of measurement.
• Specify the sample size of the data set.

Step 2: Write Section 2 of the DAA.

• Analyze the assumptions of the t test.
• Paste the SPSS histogram output for gpa and discuss your visual interpretations.
• Paste SPSS descriptives output showing skewness and kurtosis values for gpa and interpret them.
• Paste SPSS output for the Shapiro-Wilk test of gpa and interpret it.
• Report the results of the Levene test and interpret it.
• Summarize whether or not the assumptions of the t test are met.

Step 3: Write Section 3 of the DAA.

• Specify a research question related to gender and gpa.
• Articulate the null hypothesis and alternative hypothesis.
• Specify the alpha level.

Step 4: Write Section 4 of the DAA.

• Paste the SPSS output of the t test.
• Report the results of the SPSS output using proper APA guidelines (refer to the Unit 8 Introduction and the “Results” example from the Warner text in Chapter 5). Include the following:
  • t.
  • Degrees of freedom.
  • p value.
  • Effect size.
  • Interpretation of effect size.
  • Means and standard deviations for each group.
  • Mean difference.
  • 95% confidence interval of the difference of sample means.
• Interpret the results against the null hypothesis.

Step 5: Write Section 5 of the DAA.

• Discuss the implications of this t test as it relates to the research question.
• Conclude with an analysis of the strengths and limitations of the t test.

Submit your DAA Template as an attached Word document in the assignment area.