# Guide to Essential Biostatistics XVII: Inferential Statistics – post ANOVA tests

**In the previous articles in this series, we explored the ****Scientific Method****, ****Proposing Hypotheses**** and ****Type-I and Type-II**** errors, Designing and implementing experiments (****Significance, Power, Effect****, ****Variance****, ****Replication****, ****Experimental Degrees of Freedom**** and Randomization), Critically evaluating experimental data (****Q-test****; ****SD, SE, and 95%CI****) as well as ****Two-Sample Means Comparisons (the t-test)****. **

Post-ANOVA tests are typically performed after the f-ratio has indicated that not all means are equal.

Depending on our requirements, different post-ANOVA tests allow different comparisons, for example: to compare every mean to every other mean (the most common application), to compare each mean to a control mean, or to compare selected means.

Among the post-ANOVA tests to compare every mean to every other mean, we are provided with a number of methods, the most common of which (at least in the biological sciences) are the Tukey’s test and the Newman-Keuls test (or Student-Newman-Keuls test; SNK).

The Newman-Keuls post-ANOVA test is considered less stringent (has less power – see previous article on Power) than Tukey’s test, and may indicate that a difference in means is statistically significant, whereas the more stringent Tukey’s test might indicate that the difference is not statistically significant.

As the risk of a Type-I error (where there really is no difference) is greater for the Newman-Keuls test than for the Tukey test, statisticians tend to recommend Tukey’s test.

However, for Tukey’s test the chance of a Type-II error (missing a real difference) is greater. This is the “one that got away” type error, also termed “false negative”, where you miss out on something that really is extraordinary.

*Figure 1: Summary of post-ANOVA tests most commonly used to identify differences between all group means.*

Taking the strategic research objectives into consideration will facilitate a reasoned choice of test method. Thus, if the cost of a false positive is high (for example if it leads to the initiation of a costly development process) a more stringent method to determine the significance of differences between treatment means such as Tukey’s test may be appropriate.

Conversely, if the cost of a false negative is high (leading to you missing out on a valuable discovery) a less stringent method such as Newman-Keuls test may be appropriate.

Newman-Keuls test may thus be appropriate for early-stage discovery processes, while Tukey’s may be more appropriate as a decision-making tool as discovery candidates approach development phases.

The means and results of post-ANOVA tests are typically presented as bar graphs (with error bars correctly defined), where different letters above bars (means) indicate significant differences (p<0.05). Conversely, bars labeled with the same letter are not significantly different from each other.

For the example dataset above, performing Tukey’s test indicates that the means of treatment A and Treatment B are significantly different from the untreated control (p<0.001) while the difference between the means of Treatment A and Treatment B __is not__ significantly different (p>0.05).

*Figure 2: Mean plant heights for untreated control and two herbicide treatments. Different letters above bars indicate significant differences (p<0.05; Tukey’s post-ANOVA test).*

In contrast, applying the SNK test to the example dataset confirms that the means of treatment A and Treatment B are significantly different than that of the untreated control (p<0.001) while the lower stringency of this tests indicates that the difference between the means of Treatment A and Treatment B __is__ significantly different (p<0.05):

*Figure 3: Mean plant heights for untreated control and two herbicide treatments. Different letters above bars indicate significant differences (p<0.05; SNK post-ANOVA test).*

A number of additional post-ANOVA tests are available for specific comparisons (to compare each mean to a control mean, or to compare selected means, e.g. Bonferroni’s test), or for situations for which the assumption of equal variances may (e.g. Duncan’s test – a more stringent modification of the Newman-Keuls test i.e. with more power) or may not (e.g. Dunnett’s test) be required. These necessitate an in-depth understanding of statistical analysis and are thus outside the scope of this book – the reader is encouraged to refer to a statistician.

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20 years ago, I worked at Copenhagen University and the University of Adelaide on plant responses to biotic and abiotic stress in crops.

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