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Session 5: Exploring Assumptions Normality and Homogeneity of Variance

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Outliers Impact

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Assumptions Parametric tests based on the normal distribution assume: Additivity and linearity Normality something or other Homogeneity of Variance Independence

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Additivity and Linearity The outcome variable is, in reality, linearly related to any predictors. If you have several predictors then their combined effect is best described by adding their effects together. If this assumption is not met then your model is invalid.

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Normality Something or Other The normal distribution is relevant to: Parameters Confidence intervals around a parameter Null hypothesis significance testing This assumption tends to get incorrectly translated as ‘your data need to be normally distributed’.

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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.

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The P-P Plot

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Assessing Skew and Kurtosis

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Homoscedasticity/ Homogeneity of Variance When testing several groups of participants, samples should come from populations with the same variance. In correlational designs, the variance of the outcome variable should be stable at all levels of the predictor variable. Can affect the two main things that we might do when we fit models to data: – Parameters – Null Hypothesis significance testing

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Assessing Homoscedasticity/ Homogeneity of Variance Graphs (see lectures on regression) Levene’s Tests Tests if variances in different groups are the same. Significant = Variances not equal Non-Significant = Variances are equal Variance Ratio With 2 or more groups VR = Largest variance/Smallest variance If VR < 2, homogeneity can be assumed.

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Independence The errors in your model should not be related to each other. If this assumption is violated: Confidence intervals and significance tests will be invalid.

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Reducing Bias Trim the data: Delete a certain amount of scores from the extremes. Windsorizing: Substitute outliers with the highest value that isn’t an outlier Analyze with Robust Methods: Bootstrapping Transform the data: By applying a mathematical function to scores

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Trimming the Data

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Robust Methods

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Transforming Data Log Transformation (log()): Reduce positive skew. Square Root Transformation (): Also reduces positive skew. Can also be useful for stabilizing variance. Reciprocal Transformation (1/): Dividing 1 by each score also reduces the impact of large scores. This transformation reverses the scores, you can avoid this by reversing the scores before the transformation, 1/(-).

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Log Transformation

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Square Root Transformation

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Reciprocal Transformation

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But …

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To Transform … Or Not Transforming the data helps as often as it hinders the accuracy of F (Games & Lucas, 1966). Games (1984): The central limit theorem: sampling distribution will be normal in samples > 40 anyway. Transforming the data changes the hypothesis being tested E.g. when using a log transformation and comparing means you change from comparing arithmetic means to comparing geometric means In small samples it is tricky to determine normality one way or another. The consequences for the statistical model of applying the ‘wrong’ transformation could be worse than the consequences of analysing the untransformed scores.