Assumptions of Parametric Tests

Assumptions of Tests Based on Normal Distribution

• additivity

• normality of something or other

• homogeneity of variance/homoschedasticity

• independence

Additivity

• outcome = model + error

  • model used to predict variability

• the outcome variable (DV) is linearly related to any predictors (IV)

• if you have several predictors (IVs) then their combined effect is best described by adding their effects together

• if this assumption is met then your model is valid

Normally Distributed Something or Other

• the normal distribution is relevant to:

  • parameter estimates

  • confidence intervals around a parameter

  • null hypothesis significance testing

When does the Assumption of Normality Matter?

• in small samples

  • the central limit theorem allows us to forget about this assumption in larger samples

• in practical terms, as long as your sample is fairly large, outliers are a much more pressing concern than normality

Spotting Normality

• we don’t have access to the sampling distributions so we usually test the observed data

• central limit theorem

• graphical displays

Spotting Normality with the Central Limit Theorem

• if N > 30, the sampling distribution is normal anyway

Spotting Normality with graphical displays

• can use histograms or P-P Plots

P-P Plots

(probability-probability)

• when it sags: kurtosis is an issue

• when it forms an “S”: skewness is an issue

Values of Skew/Kurtosis

• 0 in a normal distribution

• convert to z (by dividing value by SE)

  • value greater than 1.96 = significantly different from normal

  • should be used for smaller sample sizes, if at all

Kolmogorov-Smirnov Test

• tests if data differ from a normal distribution

  • significant = non-normal data

  • non-significant = normal data

*for large sample sizes

Shapiro-Wilk Test

• tests if data differ from a normal distribution

  • significant = non-normal data

  • non-significant = normal data

*for small sample sizes

Homoschedasticity

• measuring variance of errors

  • variance of outcome variable should be stable across all conditions

Homogeneous

• uniform error rate across categories

Heterogeneous

• difference in error rate across categories

  • violation of homoschedasticity

Independence

• observations are completely independent from each other

  • violation example — 2 participants talk and share notes between the first and second parts of a test so their scores are no longer independent and now have a correlation

Violation of Assumptions of Independence

• grouped data

Violation of Assumptions of Normality

• robustness of test

• transformations

Violation of Assumptions of Homogeneity

• robustness and unequal sample sizes

• transformations

  • if you have not normal data, one way to make data normal is to conduct a transformation

Types of Transformations

• logarithmic transformation

• square root transformation

• reciprocal transformation

• arcsine transformation

• trimmed samples

• windsorized sample

Transformations

• always examine and understand data prior to performing analyses

• know the requirements of the data analysis technique to be used

• utilize data transformation with care and never use unless there is a clear reason

Square Root Transformation

• help decrease skewness and stabilize variances (homoschedasticity)

Reciprocal Transformation

• reduces influence of extreme values (outliers)

Arcsine Transformation

• elongates tails (good for leptokurtic distributions)

Trimmed Samples

• not really a transformation

• fixed value of extreme values you cut off

Windsorized Samples

• similar to trimmed samples

• extreme values replaced by values that occur at 5% in the tails