-Learn as you go philosophy

-Data exploration

-Linear regression modelling

-Linear regression assumptions

Nereis data

here I think:
concentration (numeric) nutrient concentration
biomass (numeric) polychaete biomass
nutrient (factor) nutrient type - reads in as numeric but is actually a cetegorical factor

##   concentration biomass nutrient
## 1         0.050     0.0        1
## 2         0.105     0.0        1
## 3         0.105     0.0        1
## 4         0.790     0.5        1
## 5         0.210     0.5        1
## 6         2.100     0.5        1

par(mfrow = c(1,2))
Nereis <- read.table(file = "Nereis.txt", header = T)
dotchart(Nereis$concentration,   groups = factor(Nereis$nutrient),
  ylab = "Nutrient",xlab = "Concentration",   
  main = "Clevelanddotplot",  pch = Nereis$nutrient)

plot(jitter(nutrient) ~ concentration, data = Nereis,
     pch = Nereis$nutrient)

# NB possible effect (bigger mean Nutrient 2), 
# Also variance seems lower on nutrient level 3
boxplot(concentration ~ factor(nutrient),
        data = Nereis,
        ylab = "Concentration",
        xlab = "Nutrient")

\(Y_i = \alpha + \beta \times X_i +\epsilon _i\)

\(Y_i\) = dependent var
\(X_i\) = explanatory var
\(\alpha\) and \(\beta\) = intercept and slope
\(\epsilon _i\) = residual error

\(\epsilon _i \sim N(0, \sigma ^2)\)

Assumption the residual error is Gaussian
with expected value = 0, variance = \(\sigma ^2\)

-assume Gaussian residual error
-homoscedasity of error
-no weird values
RIKZ data, R = spp richness, NAP = tide height
P = probability density

Vanilla linear model full assumptions

-Gaussian residuals

-Homogeneous variance

-“fixed” X (discuss briefly)

-Independence

-Correct model specification…

“the underlying concept of normality is grossly misunderstood by many researchers. The linear regression model requires normality of the data, and therefore of the residuals at each X value”

Important but is it black and white? (Sokal and Rohlf, 1995; Zar, 1999)

“heterogeneity (violation of homogeneity), also called heteroscedasticy, happens if the spread of the data is not the same at each X value, and this can be checked by comparing the spread of the residuals for the different X values”

“heterogeneity (violation of homogeneity), also called heteroscedasticy, happens if the spread of the data is not the same at each X value, and this can be checked by comparing the spread of the residuals for the different X values”

-Concept of fixed versus random “effects”

-Explanatory variables are fixed if:
1) experimentally assigned
2) low error in sample estimate relative to pop’n

-Can be serious (ref to Faraway 2005)

This is the most serious of violated assumptions in linear models and is very, very common too.

2 related causes:

-Dependence structure inherent in the model
(e.g. multiple samples in a plot)

-Other dependence in the data
(e.g. measuring growth at multiple points in time)

NB we fix this with a mixed effects model…

Clams <- read.table("Clams.txt", header = T)
str(Clams)

## 'data.frame':    398 obs. of  5 variables:
##  $ MONTH   : num  11 11 11 11 11 11 11 11 11 11 ...
##  $ LENGTH  : num  28.4 16.6 13.7 17.4 11.8 ...
##  $ AFD     : num  0.248 0.052 0.028 0.07 0.022 0.187 0.361 0.05 0.087 0.128 ...
##  $ LNLENGTH: num  3.35 2.81 2.62 2.86 2.47 ...
##  $ LNAFD   : num  -1.39 -2.96 -3.57 -2.65 -3.83 ...

Month - month of measurement
Length - length (mm?)
AFD - weight
LNLENGTH - log(Length)
LMAFD - log(AFD)

models: LNAFN ~ LNLENGTH + MONTH LNAFN ~ LNLENGTH * MONTH

Clams$MONTH <- factor(Clams$MONTH)
M1 <-lm(LNAFD ~ LNLENGTH * MONTH, data = Clams)
drop1(M1, test = "F")

## Single term deletions
## 
## Model:
## LNAFD ~ LNLENGTH * MONTH
##                Df Sum of Sq    RSS     AIC F value  Pr(>F)  
## <none>                      6.4490 -1616.8                  
## LNLENGTH:MONTH  5   0.20328 6.6523 -1614.4  2.4334 0.03444 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1