Welch’s T-Test and Mann-Whitney U Test in SPSS

Welch’s T-Test and Mann-Whitney U Test in SPSS: What to Do When Assumptions Are Violated | Statistics Guide

Welch’s T-Test and Mann-Whitney U Test in SPSS: What to Do When Assumptions Are Violated (in English, With Example)

Checking assumptions for Independent Samples T-Test is essential. If any assumption is violated (such as homogeneity of variances or normality), alternatives like Welch’s T-Test or Mann-Whitney U Test can be used in SPSS. In this section, we’ll look step-by-step at how to perform these tests in SPSS when assumptions are violated, with an example, in English.

Example

Let’s take the same example used earlier, but with some modification so that assumptions are violated. Suppose we’re comparing test scores of two teaching methods (Method A and Method B):

  • Dependent Variable: Test Score (continuous variable).
  • Independent Variable: Teaching Method (categorical variable, 2 groups: Method A, Method B).

Data set (slightly modified):

  • Method A: 85, 90, 88, 92, 87 (same as before)
  • Method B: 75, 77, 80, 82, 84 (modified to have greater variance)

Now we’ll check whether assumptions are violated or not, and if they are violated, how to run Welch’s T-Test or Mann-Whitney U Test.

Step 1: Checking Assumptions

For Independent Samples T-Test, we check these assumptions:

  1. Dependent Variable Should Be Continuous: Test scores are continuous, so this assumption is met.
  2. Independent Variable Should Be Categorical: Teaching Method is categorical (2 groups), so this is met.
  3. Independence of Observations: Each student’s score is independent, so this is also met.
  4. No Significant Outliers:

    Create a boxplot in SPSS:

    Graphs > Chart Builder > Boxplot

    Drag Test_Score to Y-axis and Teaching_Method to X-axis.

    Check output. If there are no extreme outliers, this assumption is met.

  5. Normality:

    Run Shapiro-Wilk test to check normality:

    Analyze > Descriptive Statistics > Explore

    Add Test_Score to Dependent List and Teaching_Method to Factor List.

    Click Plots button, select Histogram and Normality plots with tests, and uncheck Stem-and-leaf.

    Click OK.

    Check Tests of Normality table in output:

    Teaching_Method Shapiro-Wilk Statistic df Sig.
    Method A 0.921 5 0.611
    Method B 0.896 5 0.487

    Interpretation: For both groups, Sig. > 0.05 (0.611 and 0.487), so data is approximately normal. Normality assumption is met.

  6. Homogeneity of Variances:

    This assumption is checked in Independent Samples T-Test output through Levene’s Test (we’ll see below). If Sig. < 0.05, variances are not equal, and assumption is violated.

Now we’ll run T-Test and check whether homogeneity of variances assumption is violated or not.

Step 2: Data Entry in SPSS

  1. Open SPSS: Create a new dataset.
  2. Define Variables:
    • Test_Score (Type: Numeric, Label: Test Score).
    • Teaching_Method (Type: Numeric, Label: Teaching Method, Values: 1 = Method A, 2 = Method B).
  3. Enter Data:
    Test_Score Teaching_Method
    85 1
    90 1
    88 1
    92 1
    87 1
    75 2
    77 2
    80 2
    82 2
    84 2

Step 3: Run Independent Samples T-Test (For Assumption Check)

  1. Go to Menu:

    Analyze > Compare Means > Independent-Samples T Test

  2. Select Variables:
    • Add Test_Score to Test Variable(s).
    • Add Teaching_Method to Grouping Variable.
  3. Define Groups:
    • Group 1: 1 (Method A), Group 2: 2 (Method B).
    • Click Continue.
  4. Click OK.

Check Output

  • Group Statistics:
    Teaching_Method N Mean Std. Deviation Std. Error Mean
    Method A 5 88.40 2.701 1.208
    Method B 5 79.60 3.782 1.691

    Interpretation: Mean for Method A = 88.40, Mean for Method B = 79.60.

    Standard deviation for Method B (3.782) is greater than Method A (2.701), indicating variance difference.

  • Independent Samples Test:
    Levene’s Test for Equality of Variances t-test for Equality of Means
    F = 2.614, Sig. = 0.040 t = 4.125, df = 8, Sig. (2-tailed) = 0.003, Mean Difference = 8.800, Std. Error Difference = 2.133
    t = 4.125, df = 7.071, Sig. (2-tailed) = 0.004, Mean Difference = 8.800, Std. Error Difference = 2.133

    Interpretation:

    • Levene’s Test: Sig. = 0.040 (< 0.05), meaning variances are not equal. Homogeneity of variances assumption is violated.
    • In this situation, we won’t use Equal variances assumed row. Instead, we’ll use Equal variances not assumed row, which gives Welch’s T-Test results.

Step 4: Interpret Welch’s T-Test Result

Interpret Welch’s T-Test results from Equal variances not assumed row:

  • t = 4.125, df = 7.071, Sig. (2-tailed) = 0.004.
  • p-value (0.004) < 0.05, so null hypothesis (that means of both groups are equal) is rejected.
  • Mean Difference = 8.800 (Method A scores are 8.8 points higher than Method B).

Conclusion: According to Welch’s T-Test, there is a statistically significant difference between test scores of Method A and Method B, t(7.071) = 4.125, p = 0.004.

APA Style Report:

A Welch’s T-Test revealed that test scores for Method A (M = 88.40, SD = 2.70) were statistically significantly higher than those for Method B (M = 79.60, SD = 3.78), t(7.071) = 4.125, p = 0.004.

Step 5: Mann-Whitney U Test (If Normality Assumption Is Also Violated)

If normality assumption is also violated (e.g., Sig. < 0.05 in Shapiro-Wilk test), then Welch's T-Test is also not appropriate because T-Test depends on normality. In this case, we use Mann-Whitney U Test (non-parametric alternative).

Modified Scenario: Suppose when checking normality, Shapiro-Wilk test result is like this:

Teaching_Method Shapiro-Wilk Statistic df Sig.
Method A 0.821 5 0.042
Method B 0.856 5 0.039

Here Sig. < 0.05, so normality assumption is violated. Therefore, we'll run Mann-Whitney U Test.

Mann-Whitney U Test in SPSS

  1. Go to Menu:

    Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples

  2. Select Variables:
    • Add Test_Score to Test Variable List.
    • Add Teaching_Method to Grouping Variable.
    • Define Groups: Group 1 = 1 (Method A), Group 2 = 2 (Method B).
  3. Select Test Type:
    • Check Mann-Whitney U.
    • Click Continue and OK.
  4. Check Output:

    Output will show Test Statistics table:

    Mann-Whitney U 2.000
    Wilcoxon W 17.000
    Z -2.611
    Asymp. Sig. (2-tailed) 0.009

    Interpretation:

    • Sig. (2-tailed) = 0.009 (< 0.05), so null hypothesis (that distributions of scores for both groups are same) is rejected.
    • Conclusion: There is a statistically significant difference between test scores of Method A and Method B.

APA Style Report:

A Mann-Whitney U Test revealed that test scores for Method A were statistically significantly higher than those for Method B, U = 2.000, p = 0.009.

Step 6: Visual Representation

To visually represent results of both tests:

  • Boxplot: Graphs > Chart Builder > Boxplot.
  • Bar Graph: Graphs > Chart Builder > Bar > Add Test_Score to Y-axis and Teaching_Method to X-axis.

These graphs help in visually understanding the results.

What to Use: Welch’s T-Test or Mann-Whitney U Test?

  • Welch’s T-Test:
    • Use when homogeneity of variances assumption is violated (Levene’s Test Sig. < 0.05), but normality assumption is met.
    • This is a robust version of T-Test that can handle unequal variances.
  • Mann-Whitney U Test:
    • Use when normality assumption is violated (Shapiro-Wilk Sig. < 0.05), or if data is non-parametric.
    • This is a non-parametric test that compares ranks instead of means.

Tips

  • Checking Assumptions Is Essential: Always check normality and homogeneity of variances before T-Test.
  • Data Size: Normality check can be tricky with small sample sizes, so prefer non-parametric tests.
  • Save Output: Save SPSS output in Word or PDF for future reference.
  • Practice: Practice Welch’s T-Test and Mann-Whitney U Test with different datasets.

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