Normality Testing, Distribution Fitting and Contingency Tables

Normality testing and goodness-of-fit testing both test against distributions

Several specific tests developed - conceptually not different as all test experimental data against theoretically expected distribution

Parametric tests require a normal distribution and equal variance

• Small sample size often reason for non-normal distributions

Compare data to normal distribution to determine what test to use:

• H1 - data differs from normal (non-parametric test)
• H0 - no difference between data and normal (parametric test)

Several (non-identical) tests to test normality:

• Kolmogorov-Smirnov - simplest and most commonly used
  • Compares two sets of data to determine whether they come from same distribution
  • One-sample - compares experimental data with expected distributions
  • Two-sample - compares two sets of experimental data to determine similarity
• Anderson-Darling - more powerful and gives more weight to tails
• Shapiro-Wilk

Which test to use depends on data = prior research necessary

Often good idea to use visual fit of data to confirm test results

Used when comparing homogeneity of variances of two samples

Levene's test most common:

• H1 - two variances different (non-parametric test)
• H0 - no difference between variables (parametric test)

Tests against any distribution - how expected differs from actual

Two main tests - chi-squared and G-test

• Always choose one and stick with it

G-test - use where observed frequencies of various categories and expected proportions for those categories weren't derived from data themselves

Chi-sqaured goodness-of-fit 

• Present data in table - observed and expected frequencies for various categories
• Ensure none of expected values <1
• Expected values derived from a distribution e.g. poisson, negative binomial, flat, ratio
• N0 - observed and expected frequencies are no different

Used when explanatory and response variables are categorical

In simplest scenario, explanatory and response variables both have two options

Common contingency table test are X2 and G-tests

If X2orG>P, then there is a significant difference

The p-value is more accurate the larger the sample size is (higher numbers). 

• No expected values <1 = categories should be combined
• No more than 20% expected values <5 = Fisher exact test

Fisher exact test - calculates p-value in 2x2 table directly = more accurate with smaller sample sizes (lower numbers)

• Uses factorials

Factorial - proudct of an integer and all integers below it e.g. 3! = 3x2x1