0. Introduction

Many bioinformatics applications involving repeatedly fitting linear models to data. Examples include:

RNA-Seq differential expression analyses
GWAS (for continuous traits)
eQTL analyses
Microarray data analyses

Understanding linear modelling in R can help in implementing these types of analyses.

Scope

Basics of linear models
R model syntax
Understanding contrasts
Models with continuous covariates

We will not discuss:

Diagnostic plots
Data-driven model selection
Anything that doesn’t scale well when applied to 1000’s of genes/SNPs/proteins

1. Linear models

A linear model is a model for a continuous outcome Y of the form \[Y = \beta_0 + \beta_{1}X_{1} + \beta_{2}X_{2} + \dots + \beta_{p}X_{p} + \epsilon\] The covariates X can be:

a continuous variable (age, weight, temperature, etc.)
Dummy variables coding a categorical covariate (more later)

The \(\beta\)’s are unknown parameters to be estimated.

The error term \(\epsilon\) is assumed to be normally distributed with a variance that is constant across the range of the data.

Models with all categorical covariates are referred to as ANOVA models and models with continuous covariates are referred to as linear regression models. These are all linear models, and R doesn’t distinguish between them.

2. Linear models in R

R uses the function lm to fit linear models.

Read in ’lm_example_data.csv`:

dat <- read.csv("/share/biocore/workshops/2018_March_Prerequisites/lm_example_data.csv")
head(dat)

##   sample expression  batch treatment  time temperature
## 1      1  1.2139625 Batch1         A time1    11.76575
## 2      2  1.4796581 Batch1         A time2    12.16330
## 3      3  1.0878287 Batch1         A time1    10.54195
## 4      4  1.4438585 Batch1         A time2    10.07642
## 5      5  0.6371621 Batch1         A time1    12.03721
## 6      6  2.1226740 Batch1         B time2    13.49573

str(dat)

## 'data.frame':    25 obs. of  6 variables:
##  $ sample     : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ expression : num  1.214 1.48 1.088 1.444 0.637 ...
##  $ batch      : Factor w/ 2 levels "Batch1","Batch2": 1 1 1 1 1 1 1 1 1 1 ...
##  $ treatment  : Factor w/ 5 levels "A","B","C","D",..: 1 1 1 1 1 2 2 2 2 2 ...
##  $ time       : Factor w/ 2 levels "time1","time2": 1 2 1 2 1 2 1 2 1 2 ...
##  $ temperature: num  11.8 12.2 10.5 10.1 12 ...

Fit a linear model using expression as the outcome and treatment as a categorical covariate:

oneway.model <- lm(expression ~ treatment, data = dat)

In R model syntax, the outcome is on the left side, with covariates (separated by +) following the ~

oneway.model

## 
## Call:
## lm(formula = expression ~ treatment, data = dat)
## 
## Coefficients:
## (Intercept)   treatmentB   treatmentC   treatmentD   treatmentE  
##      1.1725       0.4455       0.9028       2.5537       7.4140

class(oneway.model)

## [1] "lm"

Note that this is a one-way ANOVA model.

summary() applied to an lm object will give p-values and other relevant information:

summary(oneway.model)

## 
## Call:
## lm(formula = expression ~ treatment, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.9310 -0.5353  0.1790  0.7725  3.6114 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   1.1725     0.7783   1.506    0.148    
## treatmentB    0.4455     1.1007   0.405    0.690    
## treatmentC    0.9028     1.1007   0.820    0.422    
## treatmentD    2.5537     1.1007   2.320    0.031 *  
## treatmentE    7.4140     1.1007   6.735 1.49e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.74 on 20 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7033 
## F-statistic: 15.22 on 4 and 20 DF,  p-value: 7.275e-06

In the output:

“Coefficients” refer to the \(\beta\)’s
“Estimate” is the estimate of each coefficient
“Std. Error” is the standard error of the estimate
“t value” is the coefficient divided by its standard error
“Pr(>|t|)” is the p-value for the coefficient
The residual standard error is the estimate of the variance of \(\epsilon\)
Degrees of freedom is the sample size minus # of coefficients estimated
R-squared is (roughly) the proportion of variance in the outcome explained by the model
The F-statistic compares the fit of the model as a whole to the null model (with no covariates)

coef() gives you model coefficients:

coef(oneway.model)

## (Intercept)  treatmentB  treatmentC  treatmentD  treatmentE 
##   1.1724940   0.4455249   0.9027755   2.5536669   7.4139642

What do the model coefficients mean?

By default, R uses reference group coding or “treatment contrasts”. For categorical covariates, the first level alphabetically (or first factor level) is treated as the reference group. The reference group doesn’t get its own coefficient, it is represented by the intercept. Coefficients for other groups are the difference from the reference:

For our simple design:

(Intercept) is the mean of expression for treatment = A
treatmentB is the mean of expression for treatment = B minus the mean for treatment = A
treatmentC is the mean of expression for treatment = C minus the mean for treatment = A
etc.

# Get means in each treatment
treatmentmeans <- tapply(dat$expression, dat$treatment, mean)
treatmentmeans["A"]

##        A 
## 1.172494

# Difference in means gives you the "treatmentB" coefficient from oneway.model
treatmentmeans["B"] - treatmentmeans["A"]

##         B 
## 0.4455249

What if you don’t want reference group coding? Another option is to fit a model without an intercept:

no.intercept.model <- lm(expression ~ 0 + treatment, data = dat) # '0' means 'no intercept' here
summary(no.intercept.model)

## 
## Call:
## lm(formula = expression ~ 0 + treatment, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.9310 -0.5353  0.1790  0.7725  3.6114 
## 
## Coefficients:
##            Estimate Std. Error t value Pr(>|t|)    
## treatmentA   1.1725     0.7783   1.506 0.147594    
## treatmentB   1.6180     0.7783   2.079 0.050717 .  
## treatmentC   2.0753     0.7783   2.666 0.014831 *  
## treatmentD   3.7262     0.7783   4.787 0.000112 ***
## treatmentE   8.5865     0.7783  11.032 5.92e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.74 on 20 degrees of freedom
## Multiple R-squared:  0.8878, Adjusted R-squared:  0.8598 
## F-statistic: 31.66 on 5 and 20 DF,  p-value: 7.605e-09

coef(no.intercept.model)

## treatmentA treatmentB treatmentC treatmentD treatmentE 
##   1.172494   1.618019   2.075270   3.726161   8.586458

Without the intercept, the coefficients here estimate the mean in each level of treatment:

treatmentmeans

##        A        B        C        D        E 
## 1.172494 1.618019 2.075270 3.726161 8.586458

The no-intercept model is the SAME model as the reference group coded model, in the sense that it gives the same estimate for any comparison between groups:

Treatment B - treatment A, reference group coded model:

coefs <- coef(oneway.model)
coefs["treatmentB"]

## treatmentB 
##  0.4455249

Treatment B - treatment A, no-intercept model:

coefs <- coef(no.intercept.model)
coefs["treatmentB"] - coefs["treatmentA"]

## treatmentB 
##  0.4455249

The Design Matrix

For the RNASeq analysis programs limma and edgeR, the model is specified through the design matrix.

The design matrix \(\mathbf{X}\) has one row for each observation and one column for each model coefficient.

Sound complicated? The good news is that the design matrix can be specified through the model.matrix function using the same syntax as for lm, just without a response:

Design matrix for reference group coded model:

X <- model.matrix(~treatment, data = dat)
X

##    (Intercept) treatmentB treatmentC treatmentD treatmentE
## 1            1          0          0          0          0
## 2            1          0          0          0          0
## 3            1          0          0          0          0
## 4            1          0          0          0          0
## 5            1          0          0          0          0
## 6            1          1          0          0          0
## 7            1          1          0          0          0
## 8            1          1          0          0          0
## 9            1          1          0          0          0
## 10           1          1          0          0          0
## 11           1          0          1          0          0
## 12           1          0          1          0          0
## 13           1          0          1          0          0
## 14           1          0          1          0          0
## 15           1          0          1          0          0
## 16           1          0          0          1          0
## 17           1          0          0          1          0
## 18           1          0          0          1          0
## 19           1          0          0          1          0
## 20           1          0          0          1          0
## 21           1          0          0          0          1
## 22           1          0          0          0          1
## 23           1          0          0          0          1
## 24           1          0          0          0          1
## 25           1          0          0          0          1
## attr(,"assign")
## [1] 0 1 1 1 1
## attr(,"contrasts")
## attr(,"contrasts")$treatment
## [1] "contr.treatment"

(Note that “contr.treatment”, or treatment contrasts, is how R refers to reference group coding)

The first column will always be 1 in every row if your model has an intercept
The column treatmentB is 1 if an observation has treatment B and 0 otherwise
The column treatmentC is 1 if an observation has treatment C and 0 otherwise
etc.

The design matrix \(\mathbf{X}\) (plus a bit of linear algebra) is used inside lm to estimate the coefficients (this is a detail you don’t have to worry about in an actual analysis).

The coefficients can be estimated directly from the design matrix and the response as \[\mathbf{\hat{\beta}} = (\mathbf{X}^{T}\mathbf{X})^{-1}\mathbf{X}^{T}\mathbf{Y}\] or in R:

Y <- as.matrix(dat$expression, ncol = 1)
solve(t(X)%*%X)%*%t(X)%*%Y

##                  [,1]
## (Intercept) 1.1724940
## treatmentB  0.4455249
## treatmentC  0.9027755
## treatmentD  2.5536669
## treatmentE  7.4139642

coef(oneway.model)

## (Intercept)  treatmentB  treatmentC  treatmentD  treatmentE 
##   1.1724940   0.4455249   0.9027755   2.5536669   7.4139642

3. Adding More Covariates

Batch Adjustment

Suppose we want to adjust for batch differences in our model. We do this by adding the covariate “batch” to the model formula:

batch.model <- lm(expression ~ treatment + batch, data = dat)
summary(batch.model)

## 
## Call:
## lm(formula = expression ~ treatment + batch, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.9310 -0.8337  0.0415  0.7725  3.6114 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   1.1725     0.7757   1.512 0.147108    
## treatmentB    0.4455     1.0970   0.406 0.689186    
## treatmentC    1.9154     1.4512   1.320 0.202561    
## treatmentD    4.2414     1.9263   2.202 0.040231 *  
## treatmentE    9.1017     1.9263   4.725 0.000147 ***
## batchBatch2  -1.6877     1.5834  -1.066 0.299837    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.735 on 19 degrees of freedom
## Multiple R-squared:  0.7667, Adjusted R-squared:  0.7053 
## F-statistic: 12.49 on 5 and 19 DF,  p-value: 1.835e-05

coef(batch.model)

## (Intercept)  treatmentB  treatmentC  treatmentD  treatmentE batchBatch2 
##   1.1724940   0.4455249   1.9153967   4.2413688   9.1016661  -1.6877019

For a model with more than one coefficient, summary provides estimates and tests for each coefficient adjusted for all the other coefficients in the model.

Two-Way ANOVA Models

Suppose our experiment involves two factors, treatment and time. lm can be used to fit a two-way ANOVA model:

twoway.model <- lm(expression ~ treatment*time, data = dat)
summary(twoway.model)

## 
## Call:
## lm(formula = expression ~ treatment * time, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.0287 -0.4463  0.1082  0.4915  1.7623 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           0.97965    0.69239   1.415  0.17752    
## treatmentB            0.40637    1.09476   0.371  0.71568    
## treatmentC            1.00813    0.97918   1.030  0.31953    
## treatmentD            3.07266    1.09476   2.807  0.01328 *  
## treatmentE            9.86180    0.97918  10.071 4.55e-08 ***
## timetime2             0.48211    1.09476   0.440  0.66594    
## treatmentB:timetime2 -0.09544    1.54822  -0.062  0.95166    
## treatmentC:timetime2 -0.26339    1.54822  -0.170  0.86718    
## treatmentD:timetime2 -1.02568    1.54822  -0.662  0.51771    
## treatmentE:timetime2 -6.11958    1.54822  -3.953  0.00128 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.199 on 15 degrees of freedom
## Multiple R-squared:  0.912,  Adjusted R-squared:  0.8591 
## F-statistic: 17.26 on 9 and 15 DF,  p-value: 2.242e-06

coef(twoway.model)

##          (Intercept)           treatmentB           treatmentC 
##           0.97965110           0.40636785           1.00813264 
##           treatmentD           treatmentE            timetime2 
##           3.07265513           9.86179766           0.48210723 
## treatmentB:timetime2 treatmentC:timetime2 treatmentD:timetime2 
##          -0.09544075          -0.26339279          -1.02568281 
## treatmentE:timetime2 
##          -6.11958364

The notation treatment*time refers to treatment, time, and the interaction effect of treatment by time. (This is different from other statistical software).

Interpretation of coefficients:

Each coefficient for treatment represents the difference between the indicated group and the reference group at the reference level for the other covariates
For example, “treatmentB” is the difference in expression between treatment B and treatment A at time 1
Similarly, “timetime2” is the difference in expression between time2 and time1 for treatment A
The interaction effects (coefficients with “:”) estimate the difference between treatment groups in the effect of time
The interaction effects ALSO estimate the difference between times in the effect of treatment

To estimate the difference between treatment B and treatment A at time 2, we need to include the interaction effects:

# A - B at time 2
coefs <- coef(twoway.model)
coefs["treatmentB"] + coefs["treatmentB:timetime2"]

## treatmentB 
##  0.3109271

We can see from summary that one of the interaction effects is significant. Here’s what that interaction effect looks like graphically:

interaction.plot(x.factor = dat$time, trace.factor = dat$treatment, response = dat$expression)

Another Parameterization

In a multifactor model, estimating contrasts can be fiddly, especially with lots of factors or levels. Here is an equivalent way to estimate the same two-way ANOVA model that gives easier contrasts:

First, define a new variable that combines the information from the treatment and time variables

dat$tx.time <- interaction(dat$treatment, dat$time)
dat$tx.time

##  [1] A.time1 A.time2 A.time1 A.time2 A.time1 B.time2 B.time1 B.time2
##  [9] B.time1 B.time2 C.time1 C.time2 C.time1 C.time2 C.time1 D.time2
## [17] D.time1 D.time2 D.time1 D.time2 E.time1 E.time2 E.time1 E.time2
## [25] E.time1
## 10 Levels: A.time1 B.time1 C.time1 D.time1 E.time1 A.time2 ... E.time2

table(dat$tx.time, dat$treatment)

##          
##           A B C D E
##   A.time1 3 0 0 0 0
##   B.time1 0 2 0 0 0
##   C.time1 0 0 3 0 0
##   D.time1 0 0 0 2 0
##   E.time1 0 0 0 0 3
##   A.time2 2 0 0 0 0
##   B.time2 0 3 0 0 0
##   C.time2 0 0 2 0 0
##   D.time2 0 0 0 3 0
##   E.time2 0 0 0 0 2

table(dat$tx.time, dat$time)

##          
##           time1 time2
##   A.time1     3     0
##   B.time1     2     0
##   C.time1     3     0
##   D.time1     2     0
##   E.time1     3     0
##   A.time2     0     2
##   B.time2     0     3
##   C.time2     0     2
##   D.time2     0     3
##   E.time2     0     2

Next, fit a one-way ANOVA model with the new covariate. Don’t include an intercept in the model.

other.2way.model <- lm(expression ~ 0 + tx.time, data = dat)
summary(other.2way.model)

## 
## Call:
## lm(formula = expression ~ 0 + tx.time, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.0287 -0.4463  0.1082  0.4915  1.7623 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## tx.timeA.time1   0.9797     0.6924   1.415 0.177524    
## tx.timeB.time1   1.3860     0.8480   1.634 0.122968    
## tx.timeC.time1   1.9878     0.6924   2.871 0.011662 *  
## tx.timeD.time1   4.0523     0.8480   4.779 0.000244 ***
## tx.timeE.time1  10.8414     0.6924  15.658 1.06e-10 ***
## tx.timeA.time2   1.4618     0.8480   1.724 0.105290    
## tx.timeB.time2   1.7727     0.6924   2.560 0.021751 *  
## tx.timeC.time2   2.2065     0.8480   2.602 0.020018 *  
## tx.timeD.time2   3.5087     0.6924   5.068 0.000139 ***
## tx.timeE.time2   5.2040     0.8480   6.137 1.90e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.199 on 15 degrees of freedom
## Multiple R-squared:  0.9601, Adjusted R-squared:  0.9334 
## F-statistic: 36.06 on 10 and 15 DF,  p-value: 1.14e-08

coef(other.2way.model)

## tx.timeA.time1 tx.timeB.time1 tx.timeC.time1 tx.timeD.time1 tx.timeE.time1 
##      0.9796511      1.3860189      1.9877837      4.0523062     10.8414488 
## tx.timeA.time2 tx.timeB.time2 tx.timeC.time2 tx.timeD.time2 tx.timeE.time2 
##      1.4617583      1.7726854      2.2064982      3.5087306      5.2039723

We get the same estimates for the effect of treatment B vs. A at time 1:

c1 <- coef(twoway.model)
c1["treatmentB"]

## treatmentB 
##  0.4063679

c2 <- coef(other.2way.model)
c2["tx.timeB.time1"] - c2["tx.timeA.time1"]

## tx.timeB.time1 
##      0.4063679

We get the same estimates for the effect of treatment B vs. A at time 2:

c1 <- coef(twoway.model)
c1["treatmentB"] + c1["treatmentB:timetime2"]

## treatmentB 
##  0.3109271

c2 <- coef(other.2way.model)
c2["tx.timeB.time2"] - c2["tx.timeA.time2"]

## tx.timeB.time2 
##      0.3109271

And we get the same estimates for the interaction effect (remembering that an interaction effect here is a difference of differences):

c1 <- coef(twoway.model)
c1["treatmentB:timetime2"]

## treatmentB:timetime2 
##          -0.09544075

c2 <- coef(other.2way.model)
(c2["tx.timeB.time2"] - c2["tx.timeA.time2"]) - (c2["tx.timeB.time1"] - c2["tx.timeA.time1"])

## tx.timeB.time2 
##    -0.09544075

(See https://www.bioconductor.org/packages/3.7/bioc/vignettes/limma/inst/doc/usersguide.pdf for more details on this parameterization)

4. Continuous Covariates

Linear models with continuous covariates (“regression models”) are fitted in much the same way:

continuous.model <- lm(expression ~ temperature, data = dat)
summary(continuous.model)

## 
## Call:
## lm(formula = expression ~ temperature, data = dat)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.87373 -0.67875 -0.07922  1.00672  1.89564 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -9.40718    0.93724  -10.04 7.13e-10 ***
## temperature  0.97697    0.06947   14.06 8.77e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.054 on 23 degrees of freedom
## Multiple R-squared:  0.8958, Adjusted R-squared:  0.8913 
## F-statistic: 197.8 on 1 and 23 DF,  p-value: 8.768e-13

coef(continuous.model)

## (Intercept) temperature 
##  -9.4071796   0.9769656

For the above model, the intercept is the expression at temperature 0 and the “temperature” coefficient is the slope, or how much expression increases for each unit increase in temperature:

coefs <- coef(continuous.model)
plot(expression ~ temperature, data = dat)
abline(coefs)
text(x = 12, y = 10, paste0("expression = ", round(coefs[1], 2),  "+", round(coefs[2], 2), "*temperature"))

The slope from a linear regression model is related to but not identical to the Pearson correlation coefficient:

cor.test(dat$expression, dat$temperature)

## 
##  Pearson's product-moment correlation
## 
## data:  dat$expression and dat$temperature
## t = 14.063, df = 23, p-value = 8.768e-13
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.8807176 0.9764371
## sample estimates:
##       cor 
## 0.9464761

summary(continuous.model)

## 
## Call:
## lm(formula = expression ~ temperature, data = dat)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.87373 -0.67875 -0.07922  1.00672  1.89564 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -9.40718    0.93724  -10.04 7.13e-10 ***
## temperature  0.97697    0.06947   14.06 8.77e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.054 on 23 degrees of freedom
## Multiple R-squared:  0.8958, Adjusted R-squared:  0.8913 
## F-statistic: 197.8 on 1 and 23 DF,  p-value: 8.768e-13

Notice that the p-values for the correlation and the regression slope are identical.

Scaling and centering both variables yields a regression slope equal to the correlation coefficient:

scaled.mod <- lm(scale(expression) ~ scale(temperature), data = dat)
coef(scaled.mod)[2]

## scale(temperature) 
##          0.9464761

cor(dat$expression, dat$temperature)

## [1] 0.9464761

A Brief Introduction to Linear Models in R