Exploring Parametric Statistics: Assessing Normality and Homogeneity of Variances, Study notes of Statistics

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Parametric Statistics:

Exploring Assumptions

• Reading - Field: Chapter

Exploring Assumptions

• Assumptions of parametric tests

based on the normal distribution

• Aim of this chapter:
• Quantify the assumption of normality o Graphical displays o Skew o Kurtosis o Normality tests
• Quantify the homogeneity of variances (when dealing with 2 or more samples): Levene’s test

Assessing Normality

• We do not have access to sample the entire

biological population, so we test the observed data

• 1. Central Limit Theorem
  • If N < 25, sampling distribution rarely normal
• 2. Graphical Displays
  • Histogram
  • Q-Q plot
• 3. Skewness / Kurtosis (point estimate +/- SE)
  • Do they overlap with 0? (normal distribution)

Assessing Normality - Graphically

Characteristics of Normal Distributions Unimodal, Symmetrical, Bell-shaped

Assessing Normality - Graphically

Comparing observations against a cumulative

normal distribution (same mean and S.D.)

Example: Festival Data Set

Biologist worried about potential health effects of music festivals. Measured hygiene of 810 concert-goers over the three days of a music festival. Hygiene measured using standardized index (from 0 to 4): 0 = you smell terribly 4 = you smell beautifully Import Download Festival Data (MusicFestival.xlsx) For ease of use, rename the Data Set “Festival”

Festival <- DownloadFestival

Explore Data Graphically: RCmdr

day1 day2 day histogram density

Graphs in Rcmdr – Quantiles

The solid red line is the expected pattern a normal distribution with the same mean and SD and the sampled data. Points outside of the dashed line envelope suggest significant deviations day

Graphs in Rcmdr – Quantiles

day 2 day 3

Note: The straight line represents the

expected pattern for a normal distribution

Explore Festival Data Set

We can also explore the summary statistics describing the three datasets (day1, day2, day3) using RCmdr: What statistics would you use to assess data normality? NOTE: multiple datasets can be analyzed at once

Explore Festival Data Set

Exploring the summary statistics describing the three datasets (day1, day2, day3) using RCmdr:

numSummary(Festival[,c("day1", "day2", "day3"), drop=FALSE], statistics=c("mean", "quantiles", "skewness", "kurtosis"), quantiles=c(.5), type="2") mean skewness kurtosis 50% n NA day1 1.7933580 8.865312 170.4502658 1.79 810 0 day2 0.9609091 1.095226 0.8222057 0.79 264 546 day3 0.9765041 1.032868 0.7315003 0.76 123 687

Further Explore Festival Data Set

Exploring additional datasets using other functions: stat.desc() function in psych package

stat.desc(Festival$day1, basic = FALSE, norm = TRUE) basic argument: Basic statistics included if TRUE (Note: FALSE is the default) norm argument: Statistics relating to normal distribution included if TRUE (Note: FALSE is the default)

Further Explore Festival Data Set

stat.desc(Festival$day1, basic = FALSE, norm = TRUE) median mean 1.790000e+00 1.793358e+ SE.mean C.I.mean.0. 3.318617e- 02 6.514115e- 02 var std.dev 8.920705e- 01 9.444949e- 01 coef.var

5.266627e- 01