Introduction to differential expression analysis

What is differential expression? (Video)

Describe a simple model of gene regulation (transcription factors, DNA, RNA, protein) and the functional genomics technologies used to measure the various features. Focus on the type of data they return (continuous vs. discrete).

Functional genomics technologies (MultipleChoiceExercise)

Choose which is true. The options will be various combinations of genomics technologies, what they measure, and the type of data they return. For example, “Microarrays measure RNA levels and return continuous data. ChIP-seq measures transcription factor binding to DNA and returns discrete data.” is true.

Visualize one differentially expressed gene

Visualize one differentially expressed gene with boxplots (geom_boxplot).

library("ggplot2")

print(single_gene)

  group      gene
1   con  8.555641
2   con  7.977472
3   con 10.388985
4 treat 13.629643
5 treat 15.940644
6 treat 13.341958

# Create boxplots of gene expression for both groups
ggplot(single_gene, aes(x = group, y = gene)) +
  geom_boxplot()

Visualize many differentially expressed genes

Visualize many differentially expressed genes with a heatmap (stats::heatmap).

# Check the dimensions
dim(gexp)

[1] 100   6

# View the first few rows
head(gexp)

            con1      con2      con3    treat1    treat2   treat3
gene01 11.034305  7.693824 15.220120  5.539417  7.662184 13.37325
gene02  6.681872  8.144218 14.673822 14.551493 12.759165 10.32855
gene03 11.550606 15.551538 10.857151 13.525733 11.723725 12.09515
gene04 14.316391  8.371045 14.139368  7.759454 13.570094 16.37769
gene05  8.845249  7.663989  9.342168 10.534286 13.306553  9.02550
gene06  2.684912 10.244040  4.705581  8.920392  9.902485 10.62007

# Plot a heatmap
heatmap(gexp)

What is a linear model? (Video)

Review the basics of multiple linear regression (dependent vs independent variables, continuous vs discrete covariates, intercept term, error term). Demonstrate how to specify equations in R (e.g. lm and model.matrix).

library("dplyr")

Attaching package: 'dplyr'

The following objects are masked from 'package:stats':

    filter, lag

The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union

library("ggplot2")
df_vis <- data.frame(status = rep(c("con", "treat"), each = 50)) %>%
  mutate(gene_de = c(rpois(n() / 2, lambda = 12), rpois(n() / 2, lambda = 30)),
         gene_var =c(rpois(n() / 2, lambda = 15) + rnorm(n() / 2, sd = 10),
                     rpois(n() / 2, lambda = 25) + rnorm(n() / 2, sd = 10)))
ggplot(df_vis, aes(x = status, y = gene_de)) +
  geom_boxplot() +
  theme_classic(base_size = 16) +
  ylim(0, 40) +
  labs(x = "Treatment status", y = "Gene expression level")

Warning: Removed 1 rows containing non-finite values (stat_boxplot).

ggsave("../figure/ch01/gene-de.png", width = 5, height = 6)

Warning: Removed 1 rows containing non-finite values (stat_boxplot).

ggplot(df_vis, aes(x = status, y = gene_var)) +
  geom_boxplot() +
  theme_classic(base_size = 16) +
  ylim(0, 40) +
  labs(x = "Treatment status", y = "Gene expression level")

Warning: Removed 8 rows containing non-finite values (stat_boxplot).

ggsave("../figure/ch01/gene-ambiguous.png", width = 5, height = 6)

Warning: Removed 8 rows containing non-finite values (stat_boxplot).

Transcript: As you just visualized, differential expression describes the situation in which a gene has a different mean expression level between conditions. While some gene expression patterns are easily diagnosed as differential expression or not from a quick visualization, you also saw some examples that were more ambiguous. Furthermore, you need a method that is more robust than a quick visual inspection and also scales to thousands of genes. For this you will use the tools of statistical inference to determine if the difference in mean expression level is larger than that expected by chance. Specifically, you will use linear models to perform the hypothesis tests. Linear models are an ideal choice for genomics experiments because their flexibility and robustness to assumptions allow you to conveniently model data from various study designs and data sources.

You should have already been introduced to linear models, for example in a DataCamp course such as Correlation and Regression, or in an introductory statistics course. Here I’ll review the terminology we will use in the remainder of the course, how to specify a linear model in R, and the intuition behind linear models.

Temperature experiment

\[ Y = \beta_0 + \beta_1 X_1 + \epsilon \]

In this equation of a linear model, Y is the response variable. It must be a continuous variable. In the context of differential expression, it is a relative measure of either RNA or protein expression level for one gene. \(X_1\) is an explanatory variable, which can be continuous or discrete, for example, control group versus treatment, or mutant versus wild type. \(\beta_1\) quantifies the effect of the explanatory variable on the response variable. Furthermore, we can add additional explanatory variables to the equation for more complicated experimental designs. Lastly, models the random noise in the measurements.

In R, you specify a linear model with the function lm. This uses R’s formula syntax. On the left is the object that contains the response variable, and to the right of the tilde are the objects that contain the explanatory variables.

lm(y ~ x1)

A second explanatory variable can be added with a plus sign.

\[ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon \]

lm(y ~ x1 + x2)

Building some intuition.

library("broom")
library("cowplot")
library("dplyr")
library("ggplot2")
theme_set(theme_classic(base_size = 16))

# Simulate a linear regression.
#
# n = sample size
# effect = slope
# error = standard deviation of distribution of residuals
# seed = seed for random number generator
#
# Returns a data.frame with the following columns:
#
# x          Explanatory variable
# y          Response variable
# y_bar      Mean of response variable
# intercept  Intercept of least squares regression line
# slope      Slope of least squares regression line
# y_hat      Fitted values
# fstat      F-statistic
# ss_exp     Explained sum of squares
# ss_res     Residual sum of squares (noise)
sim_lm <- function(n, effect, error, seed = 1) {
  stopifnot(is.numeric(n), n > 0, length(n) == 1)
  stopifnot(is.numeric(effect), length(effect) == 1)
  stopifnot(is.numeric(error), error > 0, length(error) == 1)
  stopifnot(is.numeric(seed), length(seed) == 1)
  
  set.seed(seed)
  x = runif(n, min = -25, max = 25)
  y = x * effect + rnorm(n, sd = error)
  y_bar = mean(y)
  mod <- lm(y ~ x)
  coefs <- coef(mod)
  intercept <- coefs[1]
  slope <- coefs[2]
  y_hat = fitted(mod)
  anova_tidy <- tidy(anova(mod))
  fstat <- anova_tidy$statistic[1]
  ss <- anova_tidy$sumsq
  ss_exp <- ss[1]
  ss_res <- ss[2]
  stopifnot(ss_exp - sum((y_hat - y_bar)^2) < 0.01)
  stopifnot(ss_res - sum(residuals(mod)^2) < 0.01)
  return(data.frame(x, y, y_bar, intercept, slope, y_hat, fstat, ss_exp, ss_res,
                    row.names = 1:n))
}

# Visualize the residual sum of squares
plot_ss_res <- function(d) {
  ggplot(d, aes(x = x, y = y)) +
    geom_point() +
    geom_abline(aes(intercept = intercept, slope = slope)) +
    geom_linerange(aes(ymin = y, ymax = y_hat), color = "red",
                   linetype = "dashed") +
    geom_text(aes(x = quantile(x, 0.25), y = quantile(y, 0.75),
                  label = round(ss_res, 2)), color = "red") +
    labs(title = "Residual sum of squares (noise)") +
    ylim(-60, 60)
}

# Visualize the explained sum of squares
plot_ss_exp <- function(d) {
  ggplot(d, aes(x = x, y = y)) +
    geom_abline(aes(intercept = intercept, slope = slope)) +
    geom_hline(aes(yintercept = y_bar)) +
    geom_linerange(aes(ymin = y_hat, ymax = y_bar), color = "blue",
                   linetype = "dashed") +
    geom_text(aes(x = quantile(x, 0.25), y = quantile(y, 0.75),
                  label = round(ss_exp, 2)), color = "blue") +
    labs(title = "Explained sum of squares") +
    ylim(-60, 60)
}

# baseline
baseline <- sim_lm(n = 10, effect = 2, error = 5)
baseline_ss_res <- plot_ss_res(baseline)
dir.create("../figure/ch01/", recursive = TRUE, showWarnings = FALSE)
ggsave("../figure/ch01/baseline_ss_res.png", baseline_ss_res, width = 5, height = 6)
baseline_ss_exp <- plot_ss_exp(baseline)
ggsave("../figure/ch01/baseline_ss_exp.png", baseline_ss_exp, width = 5, height = 6)
plot_grid(baseline_ss_res, baseline_ss_exp)

baseline$fstat[1]

[1] 280.3557

# Increased error
more_error <- sim_lm(n = 10, effect = 2, error = 10)
more_error_ss_res <- plot_ss_res(more_error)
dir.create("../figure/ch01/", recursive = TRUE, showWarnings = FALSE)
ggsave("../figure/ch01/more_error_ss_res.png", more_error_ss_res, width = 5, height = 6)
more_error_ss_exp <- plot_ss_exp(more_error)
ggsave("../figure/ch01/more_error_ss_exp.png", more_error_ss_exp, width = 5, height = 6)
plot_grid(more_error_ss_res, more_error_ss_exp)

more_error$fstat[1]

[1] 72.89125

# Decreased signal
less_signal <- sim_lm(n = 10, effect = 1, error = 5)
less_signal_ss_res <- plot_ss_res(less_signal)
dir.create("../figure/ch01/", recursive = TRUE, showWarnings = FALSE)
ggsave("../figure/ch01/less_signal_ss_res.png", less_signal_ss_res, width = 5, height = 6)
less_signal_ss_exp <- plot_ss_exp(less_signal)
ggsave("../figure/ch01/less_signal_ss_exp.png", less_signal_ss_exp, width = 5, height = 6)
plot_grid(less_signal_ss_res, less_signal_ss_exp)

less_signal$fstat[1]

[1] 72.89125

Standard linear model for differential expression of one gene

Run lm on one gene.

head(single_gene)

  group      gene
1   con  8.555641
2   con  7.977472
3   con 10.388985
4 treat 13.629643
5 treat 15.940644
6 treat 13.341958

# Fit the linear model
mod <- lm(gene ~ group, data = single_gene)

# Summarize the results
result <- summary(mod)
result$coefficients

            Estimate Std. Error   t value     Pr(>|t|)
(Intercept) 8.974033  0.7761637 11.562035 0.0003196394
grouptreat  5.330049  1.0976613  4.855823 0.0083043428

Standard linear model for differential expression of many genes

Use for loop to run lm on many genes (exact number of genes will be determined by 5 sec time limit for DataCamp exercises).

group <- rep(c("con", "treat"), each = ncol(gexp) / 2)

dim(gexp)

[1] 100   6

print(group)

[1] "con"   "con"   "con"   "treat" "treat" "treat"

# Initialize a vector to save the p-value from testing each gene
p <- numeric(length = nrow(gexp))

# Use a for loop to test each gene
for (i in 1:length(p)) {
  # Fit the model
  mod <- lm(gexp[i, ] ~ group)
  # Extract the p-value
  result <- summary(mod)
  p[i] <- result$coefficients[2, 4]
}

# Visualize p-value distribution with a histogram
hist(p)

Visualize results of standard differential expression analysis

Use geom_point to plot effect size versus variance and color by statistical significance.

head(de)

  effect_size  std_dev    lm limma
1  -2.4577980 3.750555 FALSE FALSE
2   2.7130980 3.353835 FALSE FALSE
3  -0.2048963 1.715542 FALSE FALSE
4   0.2934781 3.511502 FALSE FALSE
5   2.3383110 1.955378 FALSE FALSE
6   3.9361382 3.326517 FALSE FALSE

# Plot effect size (y-axis) vs. standard deviation (x-axis)
ggplot(de, aes(x = std_dev, y = effect_size, color = lm)) +
  geom_point()

limma: linear models for functional genomics (Video)

Explain the advantages of using limma for functional genomics experiments over traditional linear models: 1) share information across genes to reduce uncertainty of variance, 2) flexible system for specifying contrasts of interest, 3) integrated into standard differential expression workflow. Introduce a limma workflow diagram to orient the individual steps in the following chapters within the larger analysis.

limma for differential expression of many genes

Run pre-written limma code to easily test thousands of genes.

library("limma")
print(group)

[1] "con"   "con"   "con"   "treat" "treat" "treat"

# Setup the study design matrix
design <- model.matrix(~group)
colnames(design) <- c("Intercept", "group")

# Fit the model
fit <- lmFit(gexp, design)
fit <- eBayes(fit)

# Summarize results
results <- decideTests(fit[, "group"])
summary(results)

   group
-1    15
0     71
1     14

Visualize results of limma differential expression analysis

Use geom_point to plot effect size versus variance and color by statistical significance.

head(de)

  effect_size  std_dev    lm limma
1  -2.4577980 3.750555 FALSE FALSE
2   2.7130980 3.353835 FALSE FALSE
3  -0.2048963 1.715542 FALSE FALSE
4   0.2934781 3.511502 FALSE FALSE
5   2.3383110 1.955378 FALSE FALSE
6   3.9361382 3.326517 FALSE FALSE

# View the number of discrepancies
table(de$lm, de$limma)

       
        FALSE TRUE
  FALSE    68    7
  TRUE      3   22

# Plot effect size (y-axis) vs. standard deviation (x-axis)
ggplot(de, aes(x = std_dev, y = effect_size, color = limma)) +
  geom_point()

The benefit of sharing information across genes

Visualize the gene expression pattern of a gene called DE by a traditional linear model but not by limma.

head(compare)

        type group      gene
1    neither   con  6.681872
2       both   con  8.555641
3    lm-only   con  9.959914
4 limma-only   con 11.391149
5    neither   con  8.144218
6       both   con  7.977472

# Plot gene expression (gene; y-axis) vs. group (x-axis)
ggplot(compare, aes(x = group, y = gene)) +
  geom_boxplot() +
  facet_wrap(~type, nrow = 1)

Session information

sessionInfo()

R version 3.3.3 (2017-03-06)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X Yosemite 10.10.5

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] limma_3.30.13 cowplot_0.9.2 broom_0.4.4   bindrcpp_0.2  dplyr_0.7.4  
[6] ggplot2_2.2.1

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.14      pillar_1.2.2      git2r_0.21.0     
 [4] plyr_1.8.4        workflowr_1.0.1   bindr_0.1.1      
 [7] R.methodsS3_1.7.1 R.utils_2.6.0     tools_3.3.3      
[10] digest_0.6.13     evaluate_0.10.1   tibble_1.4.2     
[13] gtable_0.2.0      nlme_3.1-131      lattice_0.20-35  
[16] pkgconfig_2.0.1   rlang_0.2.0       psych_1.8.3.3    
[19] parallel_3.3.3    yaml_2.1.16       stringr_1.3.0    
[22] knitr_1.20        rprojroot_1.3-2   grid_3.3.3       
[25] glue_1.2.0        R6_2.2.2          foreign_0.8-69   
[28] rmarkdown_1.9.12  reshape2_1.4.3    tidyr_0.7.2      
[31] purrr_0.2.4       magrittr_1.5      whisker_0.3-2    
[34] backports_1.1.2   scales_0.5.0      htmltools_0.3.6  
[37] mnormt_1.5-5      assertthat_0.2.0  colorspace_1.3-2 
[40] labeling_0.3      stringi_1.1.7     lazyeval_0.2.1   
[43] munsell_0.4.3     R.oo_1.22.0