13 Violations of assumptions

# 13 Violations of assumptions - Assumptions of t-tests...

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Assumptions of t -tests ! Random sample(s) ! Populations are normally distributed ! ( for 2-sample t ) Populations have equal variances Detecting deviations from normality ! Previous data/ theory ! Histograms ! Quantile plots ! Shapiro-Wilk test Detecting deviations from normality: by histogram Biomass ratio Frequency Detecting deviations from normality: by quantile plot

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Detecting deviations from normality: by quantile plot Normal data Detecting differences from normality: Shapiro-Wilk test A Shapiro-Wilk test is used to test statistically whether a set of data comes from a normal distribution. What to do when the assumptions are not true ! If the sample sizes are large, sometimes the parametric tests work OK anyway ! Transformations ! Non-parametric tests ! Randomization and resampling The normal approximation ! Means of large samples are normally distributed ! So, the parametric tests on large samples work relatively well, even for non-normal data. ! Rule of thumb- if n > ~50, the normal approximations may work
Parametric tests - Unequal variance ! Welch’s t- test would work ! If sample sizes are equal and large, then even a ten-fold difference in variance is approximately OK Data transformations A data transformation changes each data point by some simple mathematical formula. Log-transformation ! Y = ln Y [ ] Y Y' = ln[Y] Frequency biomass ratio ln[biomass ratio] Carry out the test on the transformed data! Biomass ratio ln[Biomass Ratio] 1.34 0.30 1.96 0.67 2.49 0.91 1.27 0.24 1.19 0.18 1.15 0.14 1.29 0.26

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The log transformation is often useful when: ! the variable is likely to be the result of multiplication of various components. ! the frequency distribution of the data is skewed to the right ! the variance seems to increase as the mean gets larger ( in comparisons across groups).
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