Probability distributions/transformations

P = probability

Probability of null hypothesis being true

Null hypothesis for Shapro-Wilk: normal distribution

Let's set alpha = 0.05 (cut off for significantly unlikely results)

The p > 0.05, we do not have sufficient evidence to reject null hypothesis (that distribution is normal)

This does not mean that we are certain that it is normal.

Many different types

Important to estimate

 - Event probabilities - p values, outliers

 - Uncertainty - p values, confidence intervals.

Common types in psychology

 - Normal; lognormal; binomial; Poisson

- Where logrithm of data is normally distributed

- Assymmetrical with right skew (long tails)

 - Parameters: mean, SD

 - High chance of high values than normal

 - Common in biology, finance, natural events

- Log-transformation yields normal data.

- Only two values possible, usually recorded as 0 and 1.

Key parameter: probability of success (p)

        - Often analysed as normal percentage data

Problem: largest variance around p = 0.5, little variance as we go towards p=0 or p=1, hence skewed distribution.

- For binary data

 - Parameter: P = success probability

                    N = number of trials

- Assymmetrical is p = 0.5

- Assymmetry increases towards the extremes (p=0, or p=1)

- Count of events in fixed period of time or space.

 - Only whole numbers are possible.

Key parameters: Number of events counted. Often analysed as normal data.

Problems: Assymmetrical distribution when average count is very low (0.5) that we cannot assume its normal.

Probability that a given number of events occur independently in a fixed/space interval.

Single parameter 'lambda': expected number of events, based on average rate and interval size.

Assymmetrical

Related to binomial

For event counts and single detection theory in psychology.

- Statistical tests to calculate probalities based on known probability desnsity function (PDF)

PDF - a function that necessary to understanding whether a value of a variable you're measuring lies within the same unterval so area that that function runs through.

- Hence, accurate statistical results depend entirely on using the most appropriate PDF to describe your data.

- Selection of appropraite PDF guided by:

          - Type of data (e.g. binomial for binary data)

          - Match to shape of data distribution (e.g. histogram)

- Specific stats tests for group comparisons

        - binomial test equivalent to t-test for binomial data

        - exact rate ratio test for Poisson data

- More generic non-parametric methods largely rely on ranked data

- Generalised linear models are multiple regression models where you can specify the distributions type of your outcome data, no transformation need.

- Symmetrical of normal distribution

- 95% confidence interval often used especially in Psychology.

       - Based on 2SD (1.96) distance from the mean

       - 2 sigma rule (SD)

- 3 sigma (99.7% confidence) usually considered near-certain.

Depend of definition

 - Very strict definition for normal data can be baswed on 2SD from mean 95% confidence interval)

      - If you recorded 100 values, how many would be falsly considered as outliers? 

      - What effect would the 2SD have on the SD of your data?

      - What is the likely effect on significance test?

 - 3SD (inner 99.7% of values) are often used.

 - Often better to transform data and not exclude outliers or use both strategies.

- Run analyses with and without outliers to check if results are robust or driven by potential outliers.

- Can scale data without 'affecting distribution shapes e.g. z-score transformations.

- Change shape of your data distribtion, e.g. log transformation.

- Should not re-order values (larger must remain larger)

- Transformation data may be better suited for analysis.

      - if better fit to a known pdf.

- When data has right-hand skew and could be log-normal.

- Reduces larger values more than smaller values, hence reduces skew.

To normalise scale of distribution to mean =0, SD = 1.

Does not affect shape of the distribution

- Assign 1 to lowest value, 2 to the next etc.

- Same scores for same values or tie breaking methods.

- Generic way to deal with heavily skewed or difficult data.

- E.g. identical rank scoresfor log-normal and normal from examples above.

Always check your data before analysis

First make sure you have resonable values - e.g. height in cm should not contain 0 or 354. Unreasonable numbers should be deleted with no number left in cell, not in a 0.

Then check for floor/cieling effects - were data points bunch up around lowest/highest possible values.

Then check distribution shape -apply transformations if necessary. Deal with outliers - filter or delete. Possibly run analyses with and without outliers to determine their effect on overall result, robust effect should not change.