Dear Six Sigma Practitioners,

We are aware the following underlying assumptions of One- Way ANOVA

• The k samples are normally distributed.
  
• The samples are independent of each other
  
• The k samples are all assumed to come from populations with the same variance
  
• Within each sample, the values are independent
  

In an ideal world, all assumptions are met however more often than not, assumptions of normality and homoscedasity are not met. So we use various transformations to transform data to meet assumptions and then carry out the hypothesis test. If data transformation doesn't work , then we move to Non-parametric tests.

I am listing down a few interesting situations that we may come across while carrying out One-Way ANOVA.

• The groups have unequal variance; the sample sizes are equal or almost equal. Can we carry out One-Way ANOVA?
  
• Let's say we did carry out One-Way ANOVA for groups with unequal variances. Can we still use comparisons like Tukey, Bonferroni etc?
  
• The group sample sizes are equal, symmetric, equal variances , the data is continuous but non-normal. What should we do?
  
• The groups have unequal variance; the sample sizes are significantly different. Can we carry out One-Way ANOVA?
  

Let's seek solutions keeping Transformations and Non-Parametric Tests aside.

Everybody is invited to suggest solutions

Regards,

shantanu kumar

Hello Shantanu,

Interesting questions there. It would have been interesting to view responses from members. Today, one participant mentioned to me that he is waiting for answers to these questions. I am sure there are many others who want to see answers. It may be a good idea for you to respond to these and then we can have more discussions on responses.

Regards,

VK

Solutions keeping Transformations and Non-Parametric Tests aside.

Question 1: The groups have unequal variance; the sample sizes are significantly different. Can we carry out One-Way ANOVA?

Solution: In this case, we can use Welch Statistic instead of ANOVA F-statistic. The Welch statistic is more powerful than the standard F when sample sizes and variances are unequal. Of course Brown-Forsythe statistic can be used in this situation however Welch Statistic is preferred over all other options.

Question 2: The groups have unequal variance; the sample sizes are equal or almost equal. Can we carry out One-Way ANOVA?

Solution: If the group sample size is equal, tend to follow normal distribution however groups have unequal variance, ANOVA is robust to this violation when the groups are of equal size. The F statistic is robust to unequal variances when sample sizes are equal or nearly equal. The sample size should be at least more than 5, and preferably more than 30.

Question 3: Let's say we did carry out One-Way ANOVA for groups with unequal variances. Can we still use comparisons like Tukey, Bonferroni etc?

Solution: When the groups have unequal variances, the multiple comparisons or post hoc test are different. Tamhanne's T2, Dunnett's T3, Games-Howell, Dunnett-C are the comparisons which don't assume equal variances. Please note that Post hoc test or comparisons results are valid to the extent that the standard F statistic is robust to violations of assumptions. If both variances and sample sizes are unequal, F-statistic is not robust.

Question 4:The group sample sizes are equal, symmetric, equal variances, the data is continuous but non-normal. What should we do?

Solution: This has been a debatable topic for years. In general, it should be borne in mind that when true statistical randomisation occurs, then violation of the assumption of normality is acceptable. The One-Way ANOVA F- test is fairly robust if the distribution is symmetrical. The one-way ANOVA's F test will not be much affected even if the population distributions are skewed, but the F test can be sensitive to population skewness if the sample sizes are seriously unbalanced. If the sample sizes are not unbalanced, the F test will not be seriously affected by light-tailedness or heavy-tailedness, unless the sample sizes are small (less than 5), or the departure from normality is extreme (kurtosis less than -1 or greater than 2).

Note: If the data aren't normally distributed, Bartlett test for variances, Hartley's test for variances are not preferred, as they are quite sensitive to non-normality. You can use Levene's Test.

Finally, we do have the option of Transformations and Non-Parametric Test, in case of violation of assumptions.

References

Brown, Morton B. and Forsythe, Alan B. (1974), Robust Tests for Equality of Variances, Journal of the American Statistical Association, 69, 364-367

Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proceedings of the Royal Statistical Society Series A 160, 268-282.

Levene, H. 1960. Robust Tests for the Equality of Variance. In: Contributions to Probability and Statistics, I. Olkin, eds. Palo Alto, Calif.: Stanford University Press.

Welch, B. L. 1951. On the Comparison of Several Mean Values: An Alternative Approach. Biometrika, 38:, 330-336.