Select data

We first select all entries belonging to the desired protein and dataset:

stk4 <- filter(df, uniqueID == "STK4_IPI00011488")

The table stk4 has 40 rows with measurements of four experimental groups. They consist of two treatment groups (vehicle: 0 μM staurosporine, treatment: 20 μM staurosporine) with two replicates each. Let us look at the treatment group of replicate 1 for an example:

stk4 %>% filter(compoundConcentration == 20, replicate == 1) %>% 
  dplyr::select(-dataset) %>% kable(digits = 2)

To obtain a first impression of the measurements in each experimental group, we generate a plot of the measurements:

stk4_plot_orig <- ggplot(stk4, aes(x = temperature, y = relAbundance)) +
  geom_point(aes(shape = factor(replicate), color = factor(compoundConcentration)), size = 2) +
  theme_bw() +
  ggtitle("STK4") +
  scale_color_manual("staurosporine (mu M)", values = c("#808080", "#da7f2d")) +
  scale_shape_manual("replicate", values = c(19, 17))

print(stk4_plot_orig)

We will show how to add the fitted curves to this plot in the following steps.

Define function for model fitting

To assess whether there is a significant difference between both treatment groups, we will fit a null model and an alternative models to the data. The null model fits a sigmoid melting curve through all data points irrespective of experimental condition. The alternative model fits separate melting curves per experimental group .

Fit and plot null models

We use the NPARC package function fitSingleSigmoid to fit the null model:

nullFit <- NPARC:::fitSingleSigmoid(x = stk4$temperature, y = stk4$relAbundance)

The function returns an object of class nls:

summary(nullFit)

## 
## Formula: y ~ (1 - Pl)/(1 + exp((b - a/x))) + Pl
## 
## Parameters:
##    Estimate Std. Error t value Pr(>|t|)   
## Pl   0.0000     0.1795   0.000  1.00000   
## a  692.6739   226.9107   3.053  0.00419 **
## b   12.5048     4.4989   2.780  0.00851 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1814 on 37 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)

The function augment from the broom package provides a convenient way to obtain the predictions and residuals at each temperature in tabular format. By appending the returned predictions and residuals to our measurements, we ensure that relevant data is collected in the same table and can be added to the plot for visualization. The residuals will be needed later for construction of the test statistic:

nullPredictions <- broom::augment(nullFit)

Let us look at the values returned by augment at two consecutive temperatures. Note that, while the predictions will be the same for each experiment at a given temperature, the residuals will differ because they were computed by comparing the predictions to the actual measurements:

nullPredictions %>% filter(x %in% c(46, 49)) %>% kable()

Now we can append these values to our data frame and show the predicted curve in the plot:

stk4$nullPrediction <- nullPredictions$.fitted
stk4$nullResiduals <- nullPredictions$.resid

stk4_plot <- stk4_plot_orig + geom_line(data = stk4, aes(y = nullPrediction))

print(stk4_plot)

Fit and plot alternative models

Next we fit the alternative model. Again, we compute the predicted values and the corresponding residuals by the broom::augment() function. To take the compound concentration as a factor into account, we iterate over both concentrations and fit separate models to each subset. We implement this by first grouping the data using the function dplyr::group_by(), and starting the model fitting by dplyr::do().

alternativePredictions <- stk4 %>%
# Fit separate curves per treatment group:
  group_by(compoundConcentration) %>%
  do({
    fit = NPARC:::fitSingleSigmoid(x = .$temperature, y = .$relAbundance, start=c(Pl = 0, a = 550, b = 10))
    broom::augment(fit)
  }) %>%
  ungroup %>%
  # Rename columns for merge to data frame:
  dplyr::rename(alternativePrediction = .fitted,
                alternativeResiduals = .resid,
                temperature = x,
                relAbundance = y)

Add the predicted values and corresponding residuals to our data frame:

stk4 <- stk4 %>%
  left_join(alternativePredictions, 
            by = c("relAbundance", "temperature", 
                   "compoundConcentration")) %>%
  distinct()

## Warning in left_join(., alternativePredictions, by = c("relAbundance", "temperature", : Detected an unexpected many-to-many relationship between `x` and `y`.
## ℹ Row 1 of `x` matches multiple rows in `y`.
## ℹ Row 21 of `y` matches multiple rows in `x`.
## ℹ If a many-to-many relationship is expected, set `relationship =
##   "many-to-many"` to silence this warning.

Add the curves predicted by the alternative model to the plot. Conceptually, it corresponds to the plot shown in Figures 2 (A)/(B) of the paper.

stk4_plot <- stk4_plot +
  geom_line(data = distinct(stk4, temperature, compoundConcentration, alternativePrediction), 
            aes(y = alternativePrediction, color = factor(compoundConcentration)))

print(stk4_plot)

This plot summarizes Figures 2(A) and 2(B) in the corresponding publication (Childs et al. 2019).

Compute RSS values

In order to quantify the improvement in goodness-of-fit of the alternative model relative to the null model, we compute the sum of squared residuals (RSS):

rssPerModel <- stk4 %>%
  summarise(rssNull = sum(nullResiduals^2),
            rssAlternative = sum(alternativeResiduals^2))

kable(rssPerModel, digits = 4)

These values will be used to construct the F-statistic according to

To compute this statistic and to derive a p-value, we need the degrees of freedom d₁ and d₂. As described in the paper, they cannot be analytically derived due to the correlated nature of the measurements. The paper describes how to estimate these values from the RSS-values of all proteins in the dataset. In the following Section, we illustrate how to repeat the model fitting for all proteins of a dataset and how to perform hypothesis testing on these models.

Start fitting

In order to analyze all datasets as described in the paper, we fit null and alternative models to all proteins using the package function NPARCfit:

BPPARAM <- BiocParallel::SerialParam(progressbar = FALSE)

fits <- NPARCfit(x = df$temperature, 
                 y = df$relAbundance, 
                 id = df$uniqueID, 
                 groupsNull = NULL, 
                 groupsAlt = df$compoundConcentration, 
                 BPPARAM = BPPARAM,
                 returnModels = FALSE)

## Starting model fitting...

## ... complete

## Elapsed time: 16.72 secs

## Flagging successful model fits...

## ... complete.

## Evaluating model fits...

## Evaluating models ...

## ... complete.

## Computing model predictions and residuals ...

## ... complete.

## Starting model fitting...

## ... complete

## Elapsed time: 40.77 secs

## Flagging successful model fits...

## ... complete.

## Evaluating model fits...

## Evaluating models ...

## ... complete.

## Computing model predictions and residuals ...

## ... complete.

str(fits, 1)

## List of 2
##  $ predictions: tibble [360,095 × 7] (S3: tbl_df/tbl/data.frame)
##  $ metrics    : tibble [13,515 × 15] (S3: tbl_df/tbl/data.frame)

The returned object fits contains two tables. The table metrics contains the fitted parameters and goodness-of-fit measures for the null and alternative models per protein and group. The table predictions contains the corresponding predicted values and residuals per model.

fits$metrics %>% 
  mutate(modelType = factor(modelType), nCoeffs = factor(nCoeffs), nFitted = factor(nFitted), group = factor((group))) %>% 
  summary

##        modelType         id                  tm               a          
##  alternative:9010   Length:13515       Min.   : 43.56   Min.   :    0.0  
##  null       :4505   Class :character   1st Qu.: 50.21   1st Qu.:  837.2  
##                     Mode  :character   Median : 52.85   Median : 1103.1  
##                                        Mean   : 54.09   Mean   : 1432.8  
##                                        3rd Qu.: 56.29   3rd Qu.: 1489.3  
##                                        Max.   :298.79   Max.   :15000.0  
##                                        NA's   :788      NA's   :15       
##        b                   pl               aumc          resid_sd       
##  Min.   :  0.00001   Min.   :0.00000   Min.   : 4.87   Min.   :0.007522  
##  1st Qu.: 16.12284   1st Qu.:0.05278   1st Qu.:11.56   1st Qu.:0.035041  
##  Median : 21.44876   Median :0.07971   Median :14.46   Median :0.049796  
##  Mean   : 27.02964   Mean   :0.14072   Mean   :15.03   Mean   :0.064139  
##  3rd Qu.: 28.40480   3rd Qu.:0.14426   3rd Qu.:18.06   3rd Qu.:0.074662  
##  Max.   :250.00000   Max.   :1.50000   Max.   :37.42   Max.   :1.896479  
##  NA's   :15          NA's   :15        NA's   :15      NA's   :15        
##       rss               loglik           tm_sd           nCoeffs     
##  Min.   : 0.00113   Min.   :-70.47   Min.   :0.000e+00   3   :13500  
##  1st Qu.: 0.02921   1st Qu.: 27.32   1st Qu.:0.000e+00   NA's:   15  
##  Median : 0.06361   Median : 37.21   Median :0.000e+00               
##  Mean   : 0.19872   Mean   : 40.00   Mean   :1.373e+12               
##  3rd Qu.: 0.15090   3rd Qu.: 49.53   3rd Qu.:1.000e+00               
##  Max.   :79.39804   Max.   :123.31   Max.   :9.409e+15               
##  NA's   :15         NA's   :15       NA's   :788                     
##  nFitted        conv          group     
##  20  :8996   Mode :logical   0   :4505  
##  40  :4504   FALSE:15        20  :4505  
##  NA's:  15   TRUE :13500     NA's:4505  
##                                         
##                                         
##                                         
## 

fits$predictions %>% 
  mutate(modelType = factor(modelType), group = factor((group))) %>% 
  summary

##        modelType           id                  x              y         
##  alternative:179934   Length:360095      Min.   :40.0   Min.   :0.0000  
##  null       :180161   Class :character   1st Qu.:46.0   1st Qu.:0.1563  
##                       Mode  :character   Median :53.5   Median :0.5783  
##                                          Mean   :53.5   Mean   :0.5644  
##                                          3rd Qu.:61.0   3rd Qu.:0.9570  
##                                          Max.   :67.0   Max.   :8.2379  
##                                          NA's   :15     NA's   :15      
##     .fitted            .resid           group       
##  Min.   :0.01099   Min.   :-1.156741   0   : 89929  
##  1st Qu.:0.15960   1st Qu.:-0.025276   20  : 90005  
##  Median :0.58445   Median : 0.000310   NA's:180161  
##  Mean   :0.56001   Mean   : 0.004339                
##  3rd Qu.:0.96271   3rd Qu.: 0.028110                
##  Max.   :1.50000   Max.   : 7.240277                
##  NA's   :15        NA's   :15

Check example

The results of the STK4 example from earlier can be selected from this object as follows.

First, we check the RSS values of the null and alterantive models:

stk4Metrics <- filter(fits$metrics, id == "STK4_IPI00011488")

rssNull <- filter(stk4Metrics, modelType == "null")$rss
rssAlt <- sum(filter(stk4Metrics, modelType == "alternative")$rss) # Summarize over both experimental groups

rssNull

## [1] 1.218132

rssAlt

## [1] 0.08314745

Next, we plot the predicted curves per model and experimental group:

stk4Predictions <- filter(fits$predictions, modelType == "alternative", id == "STK4_IPI00011488")

stk4_plot_orig +
  geom_line(data = filter(stk4Predictions, modelType == "alternative"), 
            aes(x = x, y = .fitted, color = factor(group))) +
    geom_line(data = filter(stk4Predictions, modelType == "null"), 
            aes(x = x, y = .fitted))

Why we need to estimate the degrees of freedom

In order to compute F-statistics per protein and dataset according to Equation (), we need to know the degrees of freedom of the corresponding null distribution. If we could assume independent and identically distributed (iid) residuals, we could compute them from the number of fitted values and model parameters. In the following, we will show why this simple equation is not appropriate for the curve data we are working with.

First, we compute the test statistics and p-values with theoretical degrees of freedom. These would be true for the case of iid residuals:

modelMetrics <- fits$metrics 
fStats <- NPARCtest(modelMetrics, dfType = "theoretical")

Let us take a look at the computed degrees of freedom:

fStats %>% 
  filter(!is.na(pAdj)) %>%
  distinct(nFittedNull, nFittedAlt, nCoeffsNull, nCoeffsAlt, df1, df2) %>%
  kable()

We plot the F-statistics against the theoretical F-distribution to check how well the null distribution is approximated now:

ggplot(filter(fStats, !is.na(pAdj))) +
  geom_density(aes(x = fStat), fill = "steelblue", alpha = 0.5) +
  geom_line(aes(x = fStat, y = df(fStat, df1 = df1, df2 = df2)), color = "darkred", size = 1.5) +
  theme_bw() +
  # Zoom in to small values to increase resolution for the proteins under H0:
  xlim(c(0, 10))

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

## Warning: Removed 180 rows containing non-finite outside the scale range
## (`stat_density()`).

## Warning: Removed 180 rows containing missing values or values outside the scale range
## (`geom_line()`).

The densities of the theoretical F-distribution (red) do not fit the observed values (blue) very well. While the theoretical distribution tends to overestimate the number of proteins with test statistics smaller than 2.5, it appears to underestimate the amount of proteins with larger values. This would imply that even for highly specific drugs, we observe many more significant differences than we would expect by chance. This hints at an anti-conservative behaviour of our test with the calculated degree of freedom parameters. This is reflected in the p-value distributions. If the distribution assumptions were met, we would expect the null cases to follow a uniform distribution, with a peak on the left for the non-null cases. Instead, we observe a tendency to obtain fewer values than expected in the middle range (around 0.5), but distinct peaks to the left.

ggplot(filter(fStats, !is.na(pAdj))) +
  geom_histogram(aes(x = pVal, y = ..density..), fill = "steelblue", alpha = 0.5, boundary = 0, bins = 30) +
  geom_line(aes(x = pVal, y = dunif(pVal)), color = "darkred", size = 1.5) +
  theme_bw()

## Warning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
## ℹ Please use `after_stat(density)` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

How to estimate the degrees of freedom

In the paper, we describe an alternative way to estimate the degrees of freedom by fitting χ² distributions to the numerator and denominator across all proteins in a dataset. To enable fitting of the distributions, we first need to re-scale the variables by a scaling factor. Because the scaling factors are characteristic for each dataset (it depends on the variances of the residuals in the respective dataset), we estimate them from the data according to:

where V is the variance of the distribution, and M is the mean of the distribution.

The following functin estimates V and M from the empirical distributions of the RSS differences (RSS¹ − RSS⁰). To increase robustness, it estimate M and V by their D-estimates Marazzi (2002) (median and median absolute deviation). It then scales the numerator and denominator of the F-statistic by these scaling factors and estimate the degree of freedom parameters by fitting unscaled χ² distributions. Finally, it fits the degrees of freedom parameters numerically, computes the test statistics according to Equation () and derives p-values.

modelMetrics <- fits$metrics 
fStats <- NPARCtest(modelMetrics, dfType = "empirical")

We plot the F-statistics against the theoretical F-distribution to check how well the null distribution is approximated now:

ggplot(filter(fStats, !is.na(pAdj))) +
  geom_density(aes(x = fStat), fill = "steelblue", alpha = 0.5) +
  geom_line(aes(x = fStat, y = df(fStat, df1 = df1, df2 = df2)), color = "darkred", size = 1.5) +
  theme_bw() +
  # Zoom in to small values to increase resolution for the proteins under H0:
  xlim(c(0, 10))

## Warning: Removed 113 rows containing non-finite outside the scale range
## (`stat_density()`).

## Warning: Removed 113 rows containing missing values or values outside the scale range
## (`geom_line()`).

Also check the p-value histograms. We expect the null cases to follow a uniform distribution, with a peak on the left for the non-null cases:

ggplot(filter(fStats, !is.na(pAdj))) +
  geom_histogram(aes(x = pVal, y = ..density..), fill = "steelblue", alpha = 0.5, boundary = 0, bins = 30) +
  geom_line(aes(x = pVal, y = dunif(pVal)), color = "darkred", size = 1.5) +
  theme_bw()

The F-statistics and p-values approximate the expected distributions substantially closer when based on the estimated degrees of freedom than when based on the theoretical degrees of freedom.