Assumptions. Assumptions of linear models. Boxplot. Data exploration. Apply to response variable. Apply to error terms from linear model

Assumptions of linear models

Assumptions

• Apply to response variable

– within each group if predictor categorical

• Apply to error terms from linear model

• Normality

• Homogeneity of variance

• Independence

Data exploration

• Describe distribution of data

• Check assumptions of analysis

• Evaluate fit of model

• Find patterns in multivariate data

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Median

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Boxplot

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Outliers

1. SYMMETRICAL

EQUAL VARIANCES 2. SKEWED

4. UNEQUAL VARIANCES 3. OUTLIERS

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Scatterplots

• Plotting bivariate data

• Value of two variables recorded for each observation

• Each variable plotted on one axis (X or Y)

• Symbols represent each observation

• Assess relationship between two variables

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Model residuals

• Residual is difference between observed and predicted value of response variable

– regression model – ANOVA model

• Standardised (studentised) residuals

– follow a t-distribution

( y

− y

) ( y

− y $ )

Normality

Y normally distributed at each value of X:

– boxplots of Y, separate for each group if appropriate, should be symmetrical - watch out for outliers and skewness

– transformations of Y often help

– regression and ANOVA tests robust to this assumption

Homogeneity of variance

Variance (spread) of Y should be constant for each value of x

(homogeneity of variance):

– skewed populations or outliers produce unequal variances

– transformations that improve normality of Y will also usually make variance of Y more constant

Plots of residuals in regression

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Predicted y

Residual

x y

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Predicted y

Residual

x y

ANOVA checks

• Plot residuals (or variances) against group means

• Tests for equal variances – Bartlett’s, Cochran’s, Levene’s

tests

• ANOVA reliable if group n’s are equal and variances not too different:

• ratio of largest to smallest variance ≤ 3:1

Mean VarianceResiduals

Independence

Values of Y are independent of each other:

– no replicate used more than once

– observations independent within and between groups

– watch out for data which are a time series on same experimental or sampling units

– should be considered at design stage

Repeated measures analyses

– suitable for some non-independent designs

Linearity (regression)

True population relationship between Y and

– scatterplot of Y against X

– watch out for asymptotic or exponential patterns

– transformations of Y or Y and X often help

Transformations

• Transform variables to new scale – e.g. degrees Fahrenheit to degrees Celsius

• Statistical transformations

– non-linear (changes shape of distribution) – monotonic (retains rank order of values)

• If Y (therefore error terms) skewed:

– log or power transformation of Y – improves homogeneity of variance – can reduce influence of outliers

• If nonlinear relationship:

– linearise by transformation of Y and/or X

Data transformations

• Common transformations for biol data – log, square or 4^throot for skewed continuous

distributions

– arcsin√ for proportions and %

• Transformed variables must make biological sense

Transformation issues

• Zeros in skewed distributions

– log (y + constant) or power transformation

• Power transformations

– 4th root useful for abundance data with large range

• Base [10 or natural (e)] for log transformations – makes no difference to result

• Arcsin for % or proportions – little effect unless close to zero or 100

• Presentation of results

– back transformation of means and errors

• Generalised linear models – non-normal error distributions

Mussel clumps

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Other regression diagnostics

• Check assumptions

• Check fit of model

• Warn about influential observations and outliers

Anscombe (1973) data set

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R²= 0.667, y = 3.0 + 0.5*x, t = 4.24, P = 0.002

Outliers

• Unusual sample values very different from rest of sample

– detect using boxplots

• Sample values along way from fitted model – detect by analysing residuals from fitted model

• Solutions

– if impossible values, delete and adjust df – run analysis twice, outliers in and outliers omitted

• if result changes – problems!

Influence

• Cook’s D statistic:

– calculated for each observation – measures change in regression slope if

observation omitted

– observations with large D have large influence on estimated slope

• also large residual

• Observation 1 is X and Y outlier but not influential

• Observation 2 has large residual – outlier

• Observation 3 is very influential (large Cook’s D) - also outlier

Y 1

X 2

3

Assumptions not met - regression

• Transformations useful

• Non-parametric tests

• LAD, ranks – randomisation tests

• randomise observations or residuals

• Smoothing functions

Smoothers

• Nonparametric description of relationship between Y and X

– unconstrained by specific model structure

• Useful exploratory technique:

– is linear model appropriate?

– are particular observations influential?

• Used in generalized additive modeling (GAM)

Smoothers

• Each observation replaced by value reflecting neighbouring observations

– mean or median or predicted value of regression model through neighbouring observations

• Window size determines neighbouring observations – size of window (number of observations) determined by

smoothing parameter

• Adjacent windows overlap – resulting line is smooth

– smoothness controlled by smoothing parameter (size of windows)

• Any section of line robust to values in other windows

Types of smoothers

• Running (moving) means or averages:

– means or medians within each window

• Lo(w)ess:

– locally weighted regression scatterplot smoothing – observations within window

weighted differently – observations replaced by

predicted values from local

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Assumptions not met - ANOVA

• Robust if equal n

• Transformations useful

• Non-parametric tests

• Kruskal-Wallis for single factor designs

• ranks inappropriate for testing interaction terms – randomisation tests

• randomises observations or residuals

Generalized linear models

• Select distribution for response variable

– poisson, binomial, lognormal

• Logistic models

• Log-linear models

– count data in contingency tables

Outliers

• Observations further from fitted model than remaining observations

sample outliers in boxplots

• Large residual

outlier