FND-STA-Significance

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Significance

(Probability and p-values; significance as a threshold of p-values; deriving probability distributions from simulation and interpreting in terms of significance.)


 


Abstract:

The probability of an event is the chance of it occurring, but how do we relate that to the question whether an observation is significant? In this context we talk about p-values and the meaning of a p-value is not the same as the probability of an observation. The p-value of an observation is the probability that - assuming a null hypothesis is true - an event as extreme or more extreme is observed. This unit contains R code to study this concept.

Objectives:

  • Introduce the difference between p-values and event probability;
  • Show how we interpret p-values in terms of "significance";
  • Illustrate this with an example;
  • Present a permutation example, a strategy that can be used as an alternative to the integration of probability density functions.
  • Discuss a common error that is made when establishing the significance of an observation in the biomedical sciences.

Outcomes:

After working through this unit you should...
  • Be able to define a "p-value";
  • Be able to set up a permutation test or a sampling simulation to estimate a probability density;
  • Be able to interpret the frequency of values in that probability density in terms of a p-value;
  • Be able to critically assess whether an observation should be considered "significant" in that context;
  • Be able to identify a common error that is made in the literature when two effects are compared.

Deliverables:

  • Time management: Before you begin, estimate how long it will take you to complete this unit. Then, record in your course journal: the number of hours you estimated, the number of hours you worked on the unit, and the amount of time that passed between start and completion of this unit.
  • Journal: Document your progress in your Course Journal. Some tasks may ask you to include specific items in your journal. Don't overlook these.
  • Insights: If you find something particularly noteworthy about this unit, make a note in your insights! page.

    • Prerequisites:
    This unit builds on material covered in the following prerequisite units:


     


    This page is tagged for revision; expect changes and proceed with caution.


     



     


    Contents

    "Significance" concepts in practice

    Here we discuss the idea of a p-value, in particular how to compute "empirical p-values". These are very easy to compute and simulate in R. There is just one thing to be aware of: while we normally approximate a p-value from observed events r divided by the number of observations N as r / N, if we use this approach to evaluate significance, i.e. we are asking whetehr our r events are taken from the same distribution as the N observations, we need to apply a correction factor: (r + 1) / (N + 1)[1].

     
    Empirical p-value
    (r + 1) / (N + 1)
    for r events of interest in N observations.
     

    Task:

     
    • Open RStudio and load the ABC-units R project. If you have loaded it before, choose File → Recent projects → ABC-Units. If you have not loaded it before, follow the instructions in the RPR-Introduction unit.
    • Choose Tools → Version Control → Pull Branches to fetch the most recent version of the project from its GitHub repository with all changes and bug fixes included.
    • Type init() if requested.
    • Open the file FND-STA-Significance.R and follow the instructions.


     

    Note: take care that you understand all of the code in the script. Evaluation in this course is cumulative and you may be asked to explain any part of code.


     


     

    Controversies

    Task:
    Examine the papers below that introduce difficulties with P-values and statistical significance. Rephrase the issues in your own words to make sure that you understand what the discussion is about.

    Baker (2016) Statisticians issue warning over misuse of P values. Nature 531:151. (pmid: 26961635)

    [ PubMed ] [ DOI ]

    Nieuwenhuis et al. (2011) Erroneous analyses of interactions in neuroscience: a problem of significance. Nat Neurosci 14:1105-7. (pmid: 21878926)

    [ PubMed ] [ DOI ] In theory, a comparison of two experimental effects requires a statistical test on their difference. In practice, this comparison is often based on an incorrect procedure involving two separate tests in which researchers conclude that effects differ when one effect is significant (P < 0.05) but the other is not (P > 0.05). We reviewed 513 behavioral, systems and cognitive neuroscience articles in five top-ranking journals (Science, Nature, Nature Neuroscience, Neuron and The Journal of Neuroscience) and found that 78 used the correct procedure and 79 used the incorrect procedure. An additional analysis suggests that incorrect analyses of interactions are even more common in cellular and molecular neuroscience. We discuss scenarios in which the erroneous procedure is particularly beguiling.


    Self-evaluation

    Notes

    1. ↑
      North et al. (2003) A note on calculation of empirical P values from Monte Carlo procedure. Am J Hum Genet 72:498-9. (pmid: 12596795)

      [ PubMed ] [ DOI ]

    Further reading, links and resources

    Duncan J Murdoch, Yu-Ling Tsai & James Adcock (2008) P-Values are Random Variables. The American Statistician 62:3:242-245. (pmid: None)
    [ DOI ] P-values are taught in introductory statistics classes in a way that confuses many of the students, leading to common misconceptions about their meaning. In this article, we argue that p-values should be taught through simulation, emphasizing that p-values are random variables. By means of elementary examples we illustrate how to teach students valid interpretations of p-values and give them a deeper understanding of hypothesis testing.


     



     

    If in doubt, ask! If anything about this learning unit is not clear to you, do not proceed blindly but ask for clarification. Post your question on the course mailing list: others are likely to have similar problems. Or send an email to your instructor.


     

    About ...
     
    Author:

    Boris Steipe <boris.steipe@utoronto.ca>

    Created:

    2017-08-05

    Modified:

    2017-11-01

    Version:

    1.1

    Version history:

    • 1.1 Added definition of empirical p-value
    • 1.0 First live
    • 0.1 First stub

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