• Residuals - As the residuals are the differences between the observed and predicted values along a vertical plane, they provide a measure of how much of an outlier each point is in y-space (on y-axis). The patterns of residuals against predicted y values (residual plot) are also useful diagnostic tools for investigating linearity and homogeneity of variance assumptions

• Leverage - a measure of how much of an outlier each point is in x-space (on x-axis) and thus only applies to the predictor variable. (Values > 2*(2/n) for simple regression are cause for concern)

• Cook’s D statistic is a measure of the influence of each point on the fitted model (estimated slope) and incorporates both leverage and residuals. Values ≥ 1 (or even approaching 1) correspond to highly influential observations.

• running medians (or less robust running means) generate predicted values that are the medians of the responses in the bands surrounding each observation.
• loess and lowesse (locally weighted scatterplot smoothing) - fit least squares regression lines to successive subsets of the observations weighted according to their distance from the target observation and thus depict changes in the trends throughout the data cloud.
• kernel smoothers - new smoothed y-values are computed as the weighted averages of points within a defined window (bandwidth) or neighbourhood of the original x-values. Hence the bandwidth depends on the scale of the x-axis. Weightings are determined by the type of kernel smoother specified, and for. Nevertheless, the larger the window, the greater the degree of smoothing.
• splines - join together a series of polynomial fits that have been generated after the entire data cloud is split up into a number of smaller windows, the widths of which determine the smoothness of the resulting piecewise polynomial.

x <- zfish$SL
y <- zfish$Weight

size_lm <- lm(y~x)
summary(size_lm)

plot(y~x, col = "blue")
abline(size_lm, col = "red")

Note that when we run the lm() function we use Model I regression. Is this appropriate?

install.packages(“lmodel2”)
library(lmodel2)

size_lm_ModII <- lmodel2(y ~ x)
size_lm_ModII

abline(a=0.1912797, b=0.1097880, lty=2)

You should have generated OLS, MA, and SMA regression models, and the last plots SMA line from parameter estimates

• Say you measured green fluorescent protein abundance in cell culture across several key, carefully controlled temperatures.

• You ultimately want to know whether protein expression changes as a function of temperature in your experiment.

• Read in the gfp_temp.tsv data and perform a regression analysis.

• Address the following questions
  • Which is X and which is Y?
  • Are residual assumptions met?
  • What model of regression should we use in this case and why?
  • What is our decision regarding our null hypothesis of no relationship?

• power
• exponential
• polynomial

• Read in the RNAseq_lip.tsv and examine the continuous variables.
• We are interested in whether Gene01 and/or Gene02 expression levels influence lipid content.
• First plot \(Y\) vs \(Gene01\), \(Y\) vs \(Gene02\), then set up and test a multiplicative model.
• What are the parameter estimates of interest in this case?
• What are the outcomes from our hypothesis tests?

RNAseq_Data <- read.table(‘RNAseq_lip.tsv', header=T, sep=‘\t')

y <- RNAseq_Data$Lipid_Conc
g1 <- RNAseq_Data$Gene01
g2 <- RNAseq_Data$Gene02
g3 <- RNAseq_Data$Gene03
g4 <- RNAseq_Data$Gene04

Mult_lm <- lm(y ~ g1*g2)
summary(Mult_lm)

• Now get rid of the interaction term, and set up a purely additive model
• Did any of our estimates change? Why?
• Did the degrees of freedom change? Why?

Add_lm <- lm(y ~ g1+g2)
summary(Add_lm)

• Finally try different combinations of genes

RNAseq_Data <- read.table(‘RNAseq_lip.tsv', header=T, sep=‘\t')

RNAseq_lm_linear <- lm(y ~ g4)
summary (RNAseq_lm_linear)
influence.measures(RNAseq_lm_linear)

RNAseq_lm_poly <- lm(y ~ poly(g4, 2))
summary (RNAseq_lm_poly)

lines(y, fitted(lm(y ~ poly(g4, 2))), type="l", col="blue")
influence.measures(RNAseq_lm_poly)

• Try this for different genes
• Try this for different degree polynomials

• MSresiduals- represents the mean amount of variation unexplained by the model, and therefore the lowest value indicates the best fit.

• Adjusted \(r^2\) - (the proportion of mean amount of variation in response variable explained by the model) is calculated as adj. r2 which is adjusted for both sample size and the number of terms. Larger values indicate better fit.

• Mallow’s Cp - is an index resulting from the comparison of the specific model to a model that contain all the possible terms. Models with the lowest value and/or values closest to their respective p (the number of model terms, including the y-intercept) indicate best fit.

• Akaike Information Criteria (AIC) - there are several different versions of AIC, each of which adds a different constant designed to penalize according to the number of parameters and sample size to a likelihood function to produce a relative measure of the information content of a model. Smaller values indicate more parsimonious models. As a rule of thumb, if the difference between two AIC values (delta AIC) is greater than 2, the lower AIC is a significant improvement in parsimony.

• Schwarz or Bayesian Information Criteria (BIC or SIC) - is outwardly similar to AIC. The constant added to the likelihood function penalizes models with more predictor terms more heavily (and thus select more simple models) than AIC. It is for this reason that BIC is favored by many researchers.