“linear model” “general linear model” “linear predictor” “regression” “multiple regression” “multiple linear regression” “multiple linear regression model”

Learn to predict a response variable as

• a straight line relationship with a predictor variable
• or more than one predictor variable
• actually it doesn’t have to be a straight line
• some of the predictors could be categorical (ANOVA)
• and there could be interactions

• test which variables and interactions the data supports as predictors
• give a confidence interval on any estimates

• meet many ideas that form the basis of machine learning and statistics

Many features of the S language (predecessor to R) were created to support working with linear models and their generalizations:

• data.frame type introduced to hold data for modelling.

• factor type introduced to hold categorical data.

• y ~ x1+x2+x3 formula syntax specify terms in models.

• Manipulation of “S3” objects holding fitted models.

• Rich set of diagnostic visualization functions.

Primary reference is “Statistical models in S”.

We will be using vectors and matrices extensively today.

In mathematics, we usually treat a vector as a matrix with a single column. In R, they are two different types. * R also makes a distinction between matrix and data.frame types. There is a good chance you have used data frames but not matrices before now.

Matrices contain all the same type of value, typically numeric, whereas data frames can have different types in different columns.

A matrix can be created from a vector using matrix, or from multiple vectors with cbind (bind columns) or rbind (bind rows), or from a data frame with as.matrix.

Matrix transpose exchanges rows and columns. In maths it is indicated with a small t, eg \(X^\top\). In R use the t function, eg t(X).

\[ X = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix} \quad \quad X^\top = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \]

“dot product” “weighted sum” “linear combination” “linear function” …

Taking the dot product of two vectors we multiply corresponding elements together and add the results to obtain a total.

In mathematical notation:

\[ a^\top b = a_1 b_1 + a_2 b_2 + \dots + a_n b_n \]

In R:

sum(a*b)

Lengths

A vector can be thought of as an arrow in a space.

The dot product of a vector with itself \(a^\top a\) is the square of its length.
= Pythagoras, but in as many dimensions as we like.
= Euclidean distance (squared).

Right angles

Two vectors at right angles have a dot product of zero. They are orthogonal.

Taking the product of a matrix \(X\) and a vector \(a\) with length matching the number of columns, the result is a vector containing the dot product of each row of the matrix \(X\) with the vector \(a\):

\[ Xa = \begin{bmatrix} x_{1,1} a_1 + x_{1,2} a_2 + \dots \\ x_{2,1} a_1 + x_{2,2} a_2 + \dots \\ \vdots \end{bmatrix} \]

Equivalently, it’s a weighted sum of the columns of \(X\):

\[ Xa = a_1 \begin{bmatrix} x_{1,1} \\ x_{2,1} \\ \vdots \end{bmatrix} + a_2 \begin{bmatrix} x_{1,2} \\ x_{2,2} \\ \vdots \end{bmatrix} + \dots \]

In R:

X %*% a

(Can also multiply two matrices with %*%. We won’t need this today.)

• A line in two or more dimensions, passing through the origin.
• A plane in three or more dimensions, passing through the origin.
• A point at the origin.

These are all examples of subspaces.

Think of all the vectors that could result from multiplying a matrix \(X\) with some arbitrary vector.

If the matrix \(X\) has \(n\) rows and \(p\) columns, we obtain an (at most) \(p\)-dimensional subspace within an \(n\)-dimensional space.

A subspace has an orthogonal subspace with \(n-p\) dimensions. All the vectors in a subspace are orthogonal to all the vectors in its orthogonal subspace.

Do section: Vectors and matrices

A model can be used to predict a response variable based on a set of predictor variables. * Alternative terms: depedent variable, independent variables.

We are using causal language, but really only describing an association.

The prediction will usually be imperfect, due to random noise.

A response \(y\) is produced based on \(p\) predictors \(x_j\) plus noise \(\varepsilon\) (“epsilon”):

\[ y = \beta_1 x_1 + \beta_2 x_2 + \dots + \beta_p x_p + \varepsilon \]

The model has \(p\) terms, plus a noise term. The model is specified by the choice of coefficients \(\beta\) (“beta”).

This can also be written as a dot product:

\[ y = \beta^\top x + \varepsilon \]

The noise is assumed to be normally distributed with standard deviation \(\sigma\) (“sigma”) (i.e. variance \(\sigma^2\)):

\[ \varepsilon \sim \mathcal{N}(0,\sigma^2) \]

Typically \(x_1\) will always be 1, so \(\beta_1\) is a constant term in the model. We still count it as one of the \(p\) predictors. * This matches what R does, but may differ from other presentations!

For vector of coefficients beta and vector of predictors in some particular case x, the most probable outcome is:

y_predicted <- sum(beta*x)

A simulated possible outcome can be generated by adding random noise:

y_simulated <- sum(beta*x) + rnorm(1, mean=0, sd=sigma)

But where do the coefficients \(\beta\) come from?

Say we have observed \(n\) responses \(y_i\) with corresponding vectors of predictors \(x_i\):

\[ \begin{align} y_1 &= \beta_1 x_{1,1} + \beta_2 x_{1,2} + \dots + \beta_p x_{1,p} + \varepsilon_1 \\ y_2 &= \beta_1 x_{2,1} + \beta_2 x_{2,2} + \dots + \beta_p x_{2,p} + \varepsilon_2 \\ & \dots \\ y_n &= \beta_1 x_{n,1} + \beta_2 x_{n,2} + \dots + \beta_p x_{n,p} + \varepsilon_n \end{align} \]

This is conveniently written in terms of a vector of responses \(y\) and matrix of predictors \(X\):

\[ y = X \beta + \varepsilon \]

Each response is assumed to contain the same amount of noise:

\[ \varepsilon_i \sim \mathcal{N}(0,\sigma^2) \]

\[ y = X \beta + \varepsilon \] \[ \varepsilon_i \sim \mathcal{N}(0,\sigma^2) \]

Maximum Likelihood* “Likelihood” has a technical meaning in statistics: it is the probability of the data given the model parameters. This is backwards from normal English usage. estimate:
We choose \(\hat\beta\) to maximize the likelihood of \(y\).

A very nice consequence of assuming normally distributed noise is that the vector \(\varepsilon\) has a spherical distribution, so the choice of \(\hat\beta\) making \(y\) most likely is the one that places \(X \hat\beta\) nearest to \(y\).

\[ \hat \varepsilon = y - X \hat \beta \]

The length of \(\hat \varepsilon\), which we seek to minimize, is the square root of the sum of squares of its elements \(\hat \varepsilon^\top \hat \varepsilon\). So this is also called a least squares estimate.

Our starting point for modelling is usually a data frame, but the process of getting \(y\) and \(X\) from the data frame has some complications, and a special “formula” syntax.

First look at examples then discuss process in detail.

Do section: Single numerical predictor

We’ve just seen the formula

height ~ age

• Left hand side specifies response \(y\).
• Right hand side specifies model matrix \(X\).

Terms refer to columns of a data frame or variables in the environment.

There are some rules to learn about the construction of \(X\).

From the formula height ~ age, X had two columns. What gives?

Intercept term 1 is implicit (predictor consisting of all 1s).

• height ~ 1 + age to be explicit.

Omit with -1 or 0.

• height ~ 0 + age to force a model where newborns are 0cm tall.

Factors in R represent categorical data, such as treatment groups or batches in an experiment.

R converts a factor into \(n_\text{levels}-1\) columns of “indicator variables”, aka “one-hot” encoding. * The fundamental issue here is that we can’t estimate the coefficients in a model that has both a baseline effect and effects of each level of the factor. R’s solution is to treat the first level as the baseline. This won’t affect the predictions the model makes, but we need to be careful interpreting the coefficients.

The encoding of factors can be adjusted using the C() function within a formula. The default differs between S and R languages!

f <- factor(c("a","a","b","b","c","c"))

model.matrix(~ f)

##   (Intercept) fb fc
## 1           1  0  0
## 2           1  0  0
## 3           1  1  0
## 4           1  1  0
## 5           1  0  1
## 6           1  0  1
## attr(,"assign")
## [1] 0 1 1
## attr(,"contrasts")
## attr(,"contrasts")$f
## [1] "contr.treatment"

Need to be careful interpreting the meaning of coefficients for factors.

• (Intercept) is the mean for “a”.
• fb is the step from “a” to “b”.
• fc is the step from “a” to “c”.

f <- factor(c("a","a","b","b","c","c"))

model.matrix(~ f)

##   (Intercept) fb fc
## 1           1  0  0
## 2           1  0  0
## 3           1  1  0
## 4           1  1  0
## 5           1  0  1
## 6           1  0  1
## attr(,"assign")
## [1] 0 1 1
## attr(,"contrasts")
## attr(,"contrasts")$f
## [1] "contr.treatment"

R tries to be clever about factor encoding if the intercept is omitted.

Without an intercept, each coefficient is simply the mean for that level.

Can’t do this with multiple factors at once.* Without the intercept term, R will only do this for the first factor. Further factors are encoded as before, omitting the first level. If R didn’t do this, there would be multiple choices of coefficients able to make the exact same prediction (the predictors would not be linearly independent).

f <- factor(c("a","a","b","b","c","c"))

model.matrix(~ f)

##   (Intercept) fb fc
## 1           1  0  0
## 2           1  0  0
## 3           1  1  0
## 4           1  1  0
## 5           1  0  1
## 6           1  0  1
## attr(,"assign")
## [1] 0 1 1
## attr(,"contrasts")
## attr(,"contrasts")$f
## [1] "contr.treatment"

model.matrix(~ 0 + f)

##   fa fb fc
## 1  1  0  0
## 2  1  0  0
## 3  0  1  0
## 4  0  1  0
## 5  0  0  1
## 6  0  0  1
## attr(,"assign")
## [1] 1 1 1
## attr(,"contrasts")
## attr(,"contrasts")$f
## [1] "contr.treatment"

Do section: Single factor predictor, two levels

df0 <- df.residual(fit0)       # n-p0
RSS0 <- sum(residuals(fit0)^2)

df1 <- df.residual(fit1)       # n-p1
RSS1 <- sum(residuals(fit1)^2)

F <- (RSS0-RSS1)/(df0-df1) / (RSS1/df1) 
F

## [1] 16

pf(F, df0-df1, df1, lower.tail=FALSE)

## [1] 0.1559583

\[ RSS_0 = \hat\varepsilon_0^\top \hat\varepsilon_0 \] \[ RSS_1 = \hat\varepsilon_1^\top \hat\varepsilon_1 \]

\[ F = { {(RSS_0-RSS_1) / (p_1-p_0)} \over {RSS_1 / (n-p_1)} } \]

RSS = Residual Sum of Squares

Is the increase in RSS in H0 merely proportionate
to the extra residual degree(s) of freedom?

Then \(F\) will be close to 1.

“linear hypothesis” “general linear hypothesis” “linear restriction” “contrast”

A linear hypothesis test tests an H0 in which some linear combination of coefficients in H1 is restricted to be zero.

From a linear combination of coefficients we obtain:

• estimated value (calculated with dot product)
• Confidence Interval (CI)
• hypothesis test p-value

Examples:

• a contrast compares specific levels of an experimental factor* Terminology is a little tricky here. The limma package refers to linear combinations in general as contrasts.
• the difference of response between two imagined individuals

Use multcomp package.

Do section: Multiple factors, many levels

To test our H0, it must nest within H1.

• It must be possible for any set of predictions \(X\beta\) from H0 also to be made by H1.

Testing linear restrictions using glht naturally obeys this, as the H0 is defined by restrictions on the coefficients of H1.

With the anova test, you need to ensure nesting yourself.

Generally we ensure this by making the predictors for H0 a subset of the predictors for H1.

• Exactly equivalent to restricting the coefficients for these predictors to be zero, using glht.

anova will not give an error if your models don’t nest, but results will be invalid.

A p-value lets us choose our probability of falsely rejecting a true hypothesis (type I error).

If H1 is true, \(p\) heads to zero as we get more and more data.

If H0 is true, \(p\) is a uniformly random number between 0 and 1.

If we decide to reject H0 when \(p \leq 0.05\), we are guaranteed a probability of 0.05 of rejecting H0 if it is actually true.

Suppose we perform 20 such tests. There is a high chance of falsely rejecting at least one null hypothesis.

Need to adjust p-values for this problem (see p.adjust( )).

Suppose some set of null hypotheses are true. What chance do we run of rejecting one or more of them?

Family-Wise Error Rate (FWER) control

“Any false rejection will ruin my argument.”

• Tukey’s Honestly Significant Differences
• Bonferroni correction (etc)

False Discovery Rate (FDR) control

“I can tolerate a certain rate of false rejections amongst rejected hypotheses.”

• Benjamini and Hochberg correction (etc)

“Wilikins-Rogers notation”

Merely a convenience, if you already have model matrix X can just use y ~ 0 + X.

We’ve seen:

• intercept term
• numerical predictors
• factor predictors
• interactions between factors

Let’s look at other things formula syntax can do.

numeric vector is represented as itself.

factor is represented as \(n_\text{levels}-1\) columns containing 1s and 0s.

logical vector is treated like a factor with levels FALSE and TRUE.

matrix is represented as multiple columns.

Can calculate with function calls, or enclose maths in I( ).

Handy functions such as poly and splines::ns produce matrices for fitting curves.

x <- 1:6

model.matrix(~ x + I(x^2) + log2(x))

##   (Intercept) x I(x^2)  log2(x)
## 1           1 1      1 0.000000
## 2           1 2      4 1.000000
## 3           1 3      9 1.584963
## 4           1 4     16 2.000000
## 5           1 5     25 2.321928
## 6           1 6     36 2.584963
## attr(,"assign")
## [1] 0 1 2 3

model.matrix(~ poly(x,3))

##   (Intercept) poly(x, 3)1 poly(x, 3)2 poly(x, 3)3
## 1           1  -0.5976143   0.5455447  -0.3726780
## 2           1  -0.3585686  -0.1091089   0.5217492
## 3           1  -0.1195229  -0.4364358   0.2981424
## 4           1   0.1195229  -0.4364358  -0.2981424
## 5           1   0.3585686  -0.1091089  -0.5217492
## 6           1   0.5976143   0.5455447   0.3726780
## attr(,"assign")
## [1] 0 1 1 1

a:b specifies an interaction between a and b.

All pairings of predictors from the two terms are multiplied together.

For logical and factor vectors, this produces the logical “and” of each pairing.
 

model.matrix(~ f + x + f:x)

##   (Intercept) fb fc x fb:x fc:x
## 1           1  0  0 1    0    0
## 2           1  0  0 2    0    0
## 3           1  1  0 3    3    0
## 4           1  1  0 4    4    0
## 5           1  0  1 5    0    5
## 6           1  0  1 6    0    6
## attr(,"assign")
## [1] 0 1 1 2 3 3
## attr(,"contrasts")
## attr(,"contrasts")$f
## [1] "contr.treatment"

a*b is shorthand to also include main effects a + b + a:b.

(More obscurely a/b is shorthand for a + a:b, if b only makes sense in the context of each specific a.)

(a+b+c)^2 is shorthand for a + b + c + a:b + a:c + b:c.
a*b*c is shorthand for a + b + c + a:b + a:c + b:c + a:b:c.

model.matrix(~ f*x)

##   (Intercept) fb fc x fb:x fc:x
## 1           1  0  0 1    0    0
## 2           1  0  0 2    0    0
## 3           1  1  0 3    3    0
## 4           1  1  0 4    4    0
## 5           1  0  1 5    0    5
## 6           1  0  1 6    0    6
## attr(,"assign")
## [1] 0 1 1 2 3 3
## attr(,"contrasts")
## attr(,"contrasts")$f
## [1] "contr.treatment"

Do section: Gene expression example

Linear models are a building block or basis for many more complex statistical models.

Of particular importance (in my opinion):

• Generalized Linear Models
• Linear Mixed Models
• Screening many different response variables

(Not covered today.)

We are still trying to predict the mean value, but now the response has a noise distribution that changes based on its mean value.

• The noise distributions are characterized by the mean-variance relationship.

We want to fit the model on a transformed scale even if the data contains values that can’t be transformed.

Examples:

• Fit a model on a log scale to count data that includes zeros using a Poisson or negative-binomial GLM.
• Fit a model of the log odds of a binary outcome using logistic regression.

Fit using the function glm.

(Not covered today.)

The models we have fitted today contain only fixed effects.

A mixed model contains both fixed effects and random effects.

With a random effect, the levels of a factor represent individuals from some large population.

Using random effects:

• Lets us characterize a population. The model is not about particular individuals, but about individuals randomly chosen from a large population. How does an average individual behave, and what is the standard deviation?
• Allows analysis of experiments with nested factors, such as repeated measurements (technical replicates).

Fit using a package such as lme4. See example code in workshop appendix.

“High throughput” biological datasets we can view as containing many sets of responses.

Typical example is RNA-Seq gene expression.

“High throughput” biological datasets we can view as containing many sets of responses.

Typical example is RNA-Seq gene expression.

• Perform RNA-Seq on a set of biological samples.
• Number of RNA molecules seen for each gene in each sample is counted.

• Linear model formula based on experiment design.
• Each gene can be viewed as a separate set of responses.

Key features:

• A test is performed for each gene. Multiple testing adjustment is crucial!
• Residual standard deviation is similar between genes. “Empirical Bayes” shrinking of these standard deviations overcomes problems with small numbers of samples.

Use Bioconductor packages such as limma (fit many linear models) or edgeR (fit many GLMs).

Do section: Testing many genes with limma

Should now be able to

• Fit linear models.

• Identify and correct problems with a model, or recognize the need for a more advanced method.

• Test a hypothesis using two nested linear models.
• Test a hypothesis and calculate a CI on a linear combination of coefficients.

• Test a hypothesis for many sets of responses using limma.
• Understand the need for multiple-testing correction.

Primary reference is:

Chambers and Hastie (1991) “Statistical Models in S”

Also good:

Faraway (2014) “Linear Models with R”

James, Witten, Hastie and Tibshirani (2013) “An Introduction to Statistical Learning”

Harrel (2015) “Regression Modeling Strategies”

See links on the workshop home page.

Insufficient data or \(p > n\)

• Regularization with ridge regression and/or LASSO: glmnet::glmnet( )
• Partial Least Squares (PLS)
  (pick best direction to start moving in and take best leap along it, repeat)
• Principal Component Regression (PCR)
  (direction of largest variation in X first)

Noise not normal (eg binary or count data)

• Generalized Linear Models: glm( )
  (logistic regression, binomial, Poisson, negative binomial, gamma, etc)
• Quasi-Likelihood F-test in case of over-dispersion

Survival time, with censoring

• Cox survival model: survival::coxph( )

Noise at a coarser grain than single measurements

Individuals tracked over time, repeated measurements, blocks/batches/plots, etc.

• Linear Mixed Model (some coefficients are drawn from a random distribution): lme4, lmerTest
• Time series models.

Noise in data is poorly understood, or possibly misspecified

• Quantile regression may be more robust to outliers: quantreg::rq( )
  (eg try to predict the median)
• “Sandwich” estimator gives consistent estimates of standard error, even if variance is not uniform.
• “Bootstrapping” can estimate bias, provide confidence intervals for almost any method of estimation.

Response is non-linear in predictors

• Feature engineering: poly( ), splines::ns( )
• Local regression (LOESS): loess( )
• Generalized Additive Models (GAM): mgcv::gam( )
  (splines with regularization)
• Neural network
  (many linear models connected with non-linear functions)

Also consider tree methods. Gradient boosted tree methods are unreasonably effective: xgboost

“coefficient of determination”

How much of the variation in the data is explained by a model?

• We can think of there being a pool of variance in the data, varying around the overall mean* If the model omits the intercept with ~0+..., summary( ) will instead view this pool of variance as varying around 0..

• Our model explains some proportion of this variance, called \(R^2\).

     \(TSS\) = Total Sum of Squares (vs mean)

     \(RSS\) = Residual Sum of Squares

\[ R^2 = \frac{TSS-RSS}{TSS} \]

\(R^2\) is reported by summary( ) as “Multiple R-squared”.

“coefficient of determination”

With a single predictor, \(R^2\) is exactly the squared Pearson correlation \(r^2\).

See also standardized coefficients.

Overfitting:

Even a random predictor will allow a portion of the pool of variance to be removed.

• Account for this using adjusted \(R^2\), also reported by summary( ).

Coefficients are estimates, adding further error to predictions.

• Estimate model performance by Cross Validation, or rank models by AIC.