LECTURE 9: assumptions and transformations

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.title[
# LECTURE 9: assumptions and transformations
]
.subtitle[
## FANR 6750 (Experimental design)
]
.author[
### Fall 2022
]

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# outline

1) Motivation

2) Assumptions

3) Transformations

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# assumptions

The key assumptions of ANOVA are that the residuals are **independent** and come from a **normal distribution** with mean 0 and **variance** `$\sigma^2$`

#### This implies the following:

1) Within each treatment group, the residuals are normally distributed

2) Treatment group variances are equal

3) There is no spatial, temporal, or other forms of correlation among the residuals - they are independent

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# normality assumption

### Assessment

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- Inspect histograms and boxplots of the data for each treatment group

- Inspect histograms and boxplots of the residuals

- Conduct a Shapiro-Wilk test of normality on the residuals

+ Goodness-of-fit test of normality 
 
 + Assumption of normality is rejected if p < 0.05

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# normality assumption

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# normality assumption

#### ANOVA is usually considered robust to violations of this assumption as long as they aren't severe and the sample size isn't too small.

#### However, some problems can arise

**Consequences of non-normality**

- Implausible confidence intervals

- Altered power if the data are heavily skewed (platykurtic or leptokurtic)

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# normality assumption

#### Cases where normality assumption might not hold

- Count data, especially when there are lots of zeros

- Proportion or percentage data

- Arbitrary scales, such as a 10-point taste test

- Weights of very small things

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# equal variance assumption

### Assessment

--
- Inspect histograms and boxplots of the data from each treatment group

- Plot group variances

- Conduct a Bartlett test of equality of variances

+ Assumption of homogeneity is rejected if p < 0.05

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# equal variance assumption

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# equal variance assumption

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# equal variance assumption

#### ANOVA is again considered robust to violations of this assumption, especially if sample sizes are similar among groups.

--
#### Consequences of heteroscedasticity

- Estimates of variance ( `$\sigma^2$` ) will be wrong

- Confidence intervals and F-tests will be affected

#### Cases where heteroscedasticity might be expected

- Count data, especially when there are lots of zeros

- Proportion or percentage data

- Arbitrary scales, such as a 10-point taste test

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# independence assumption

#### Residuals are independent if the value of one residual tells us nothing about the value of another residual

#### This assumption can be met by randomly selecting experimental units and randomly assigning them to treatment groups

#### It is unlikely to hold if multiple observations are made on the same experimental unit, or if the experimental units are highly clustered

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# independence assumption

### Assessment

- In general, this problem is hard to diagnose unless you know the details of the design

- Plot group variances against group means. There shouldn't be any pattern

- When temporal or spatial autocorrelation is a concern, auto-covariance plots can be used for diagnosis

### Consequences of non-independence

- Estimates of variance will generally be too small

- Power will be in inflated

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# what if assumptions don't hold?

### At least three options:

--
1) Use a more complicated model that relaxes some assumptions, such as:

+ Generalized linear models that don't assume normality

+ Linear models with multiple variance parameters

+ Time-series models or spatial models allowing for correlated residuals

2) Nonparametrics

--

3) Transform the data

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# transformations

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# transformations

### Key idea

> Transform the raw data ( `$u_{ij}$` ) so that the transformed data ( `$y_{ij}$` ) meet the assumptions of ANOVA

#### What kinds of transformations are valid?

- The transformation must maintain the rank order of the original data

- Common examples include:

+ Logarithmic transformations

+ Square-root transformations

+ Arcsine transformations  
    
    + Reciprocal transformations

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# logarithmic transformation

`$$\huge y = log(u + C)$$`

- The constant C is often 1, or 0 if there are no zeros in the data ( `$u$` )

- Useful when group variances are proportional to the means (count data)

--
- Could use `$log_{10}$` or any base, but the natural logarithm is preferred

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# logarithmic transformation

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# logarithmic transformation

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# square root transformation

`$$\huge y = \sqrt{u + C}$$`

- C is often 0.5 or some other small number

- Useful when group variances are proportional to the means (count data)

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# square root transformation

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# square root transformation

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# summary

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- The assumptions of ANOVA are made explicit by the linear model

- Colloquially, these assumptions are:

+ Within each treatment group, the data are normally distributed  
    
    + Treatment group variances are equal  
    
    + There is no spatial, temporal, or other forms of correlation among the residuals - they are independent  
    
--

- ANOVA is considered robust to violations of the first two assumptions

- Transformations, in some cases, can be used to meet all three assumptions

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- If transformations don't work, it is always possible to relax assumptions by adopting a more complicated model