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Table of contents

1. Types of response variable distributions

In statistic we can find four main distributions:

1.1 Gaussian or Normal distribution

distribution

Is a type of continuous probability distribution. Is defined by two statistics:

  • Mean ($\mu$) = 0

  • Standard deviation ($\sigma$) = 1. It is a measure how spread out our date are.

The parameter $\mu$ is the mean of the distribution, while the parameter $\sigma$ is its standard deviation. The variance of the distribution is $\sigma^2$. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.

Key points:

  • A normal distribution is the proper term for a probability bell curve.

  • Normal distributions are symmetrical, but not all symmetrical distributions are normal.

  • In normal distribution:

    $\mu$ = median = mode

1.2 Poisson distribution

This distriburtion has the following characteristics:

  • Created by integers (usually counts)

  • It can not have negative values

  • Its distribution is described by only one parameter, lambda ($\lambda$), so that the mean is equal to the variance of the distribution

    $\lambda$ --> $\mu$ = $\sigma^2$

⚠️ Poisson distributions with high $\lambda$ (i.e. far from zero) can resemble a Gaussian

1.3. Binomial Distribution

1.4 Negative Binomial distribution

More spread out distributions than a Poisson (variance > mean). Their characteristics are:

  • Distribution of integers (usually counts)
  • It can not have negative values
  • Are described by two parameters:
    • Mean ($\mu$)
    • Inflated variance (called size in R)

The variance is defines by:

$$ \mu \ + \mu^2 / size $$

When size is very large 1/size tend to 0 and the variance ends up looking by $\mu$

Depending on the nature of our response variable, we will work with different types of models:

Linear General Models Linear Generalised Linear Models
Gaussian Distribution Poisson Distribution
Binomial Distribution
Negative Binomial Distribution

⚠️ If the distribution of the response variable is neither Poisson, Negative Binomial or Binomial, but it does not exhibit Gaussian characteristics eirther then:

TRANSFORM THE RESPONSE VARAIBALE AND APPLY GENERAL LINEAR MODELS

2. Types predictors

2.1 Continous factors

2.2 Offsets

2.3 Within- and between-groups factors

2.4 Fixed and random factors

2.5 Nested factors

3. Basic n-way ANOVA models

4. Tables and types of contrasts

5. Check the assumptions for the "good" model

5.1 Normality (residuals)

In statistics, normality tests are used to determine if a data set is well-modeled by a normal distribution and to compute how likely it is for a random variable underlying the data set to be normally distributed.

The residuals of the model should fit a normal distribution. Should look something like this:

METER UNA FOTO DE LA DISTRIBUCION NORMLA

How we can explore the normality in our residuals?

  • Visually:

    • Histogram

    • Q-Q plot (normal probability plot)

  • Analytically / Statistically:

    • Test the Shapiro-Wilk

      The null hypothesis of this test is that the population is normally distributed. Thus:

      • If p-value < 0.05 ==> Data NOT NORMALLY distributed
      • If p-value > 0.05 ==> Data NORMALLY distributed

      More info about Shapiro test here

    • Kurtosis (K)

      Kurtosis tell us the height and sharpness of the central peak, relative to that of a standard bell curve. The null hypothesis of this test is that the population is normally distributed.

      • If K > 0 (leptokurtosis) ==> greater type II error, accept null hypothesis when is false
      • If K < 0 (platikurtosis) ==> greated type I error,

    • Skew

5.2 Homocedasticity (residuals)

  • The variance of the residuals should be similar across model predictions.

  • A random scatter pattern of points should appear, without drawing any geometric pattern (e.g., like many randomly distributed balls on a billard table)

⚠️ SITUATION WE SHOULD NOT HAVE ⚠️ violation of the homoscedasticity assumption as there is a triangular pattern indicating that tehre is heterogeneity in the variance of the residuals across the model predictions. There is greater variance at higer preficted values.

5.3. Influent and lost points

5.4. Variance homogeneity

5.5. Independence between predictor variables-factors - collinearity

The predictor variables were for a long time called independent. This was an explicit recognition that they shoud be uncorrelated with each other.

If there is dependence between the predictor variables, we have COLLINEARITY

The collinearity could be the result of:

  • By definition the predictor variables are correlated ( e.g. altitude and temperature)

  • Because there is not homogeneity in the sample size along the different factors levels.

The problems that we can find when the variables are correlated:

  • Variables cancel each other out
  • significance estimates are altered
  • Effect sizes are altered
  • No convergence between type I and type II sum of squares (SS) results.

Exploring collinearity between predictor variables:

  • Correlation between variables.

  • VIF index

    AFTER ALL THESE EXPLORATION OF THE CANONICAL ASSUMPTIONS OF THE MODELS...

    WE CAN NOW PROCEED TO ASSESS THEIR RESULTS

How to interpret the model results here