20.1 – Area under the curve

Introduction.

The classical physiological area under the curve, AUC, represents the total change in given change in . For example, if is time, and is oxygen consumption — eg, respirometry: recorded as an analog voltage output (millivolts or volts) from a gas analyzer to a data acquisition (DAQ) board — an AUC would be appropriate to quantify the total oxygen consumption following strenuous exercise (Excess post-exercise oxygen consumption, EPOC) or following a large meal (Specific Dynamic Action, SDA).  The area represents the total, cumulative volume of oxygen consumed over that specific timeframe, eg, calculating an individual’s total EPOC.

In machince learning, the axes are probabilities and error rates. The area tells you how accurate a decision-making model is. We plot True Positive Rate (Sensitivity) against False Positive Rate (1 – Specificity) across every possible mathematical decision threshold (from 0% to 100% confidence). The curve now represents the shifting balance between catching true positives and committing false positives and the area summarizes how cleanly the model separates two distinct classes, eg, Sick vs. healthy. An area of 1.0 means perfect separation with zero overlap.

Area under the relative (receiver) operating carrier, AUROC, shows characteristics of a diagnostic model, a graphic used to show trade off between sensitivity and specificity. Classifier performance. Used to find the appropriate cut-off. Plot true positive rates against false positive rates as cumulative functions, shows the relationship between sensitivity and specificity for every possible cut off value. Can then calculate AUC to get a measure of the intervention’s ability to discriminate between true and false positive rates.

Related, area under precision-recall curve, AUPRC, is a single scalar metric that measures a binary classification model’s ability to balance precision (how many predicted positives are actually positive) and recall (how many actual positives are caught) across all possible decision thresholds. Here, precision and recall refer to both the accuracy of the model’s positive predictions and its ability to find all the positive cases.

From Chapter 7 we defined

where is true positives and is false positives.

We now introduce recall, which is also refered to as sensitivity

Precision measures the quality of your model’s positive guesses; Recall measures how many actual positive cases in the dataset were identified by the model.

AUC evaluates a model’s ability to distinguish between both and equally, while AUPRC focuses strictly on the model’s performance regarding the positive (rare) class.

AUC would be preferred when your dataset is balanced (equal positives and negatives) OR when you care equally about performance on both classes. For example, dataset of patients with flu or Covid from a clinic during winter months where there is roughly balanced 50/50 split of patients presenting with either Flu or Covid. You want a model that accurately classifies both which favors AUC approach.

AUPRC would be the choice when your dataset is highly imbalanced, ie, the positive class is rare. In contrast, AUPRC would be favored for scenarios where a researcher is screening 100,000 asymptomatic patients for a rare disease that only 10 people actually have.

Physiology Area under the curve

Download and install R package MESS; requires geepack, geeM, and Matrix packages

R code

x <- seq(1:10) 
y <- c(1,4,5,2,11,22,9,7,5,1) 
#length(x)==length(y)
#smooth the data 
loxy <- loess(y~x)
#Make a plot (Fig. 1)
plot(x,y, pch=19, cex=2, col="blue") 
lines(predict(loxy), type="l", col="red")

where == is an R comparison operator.

And R output

Figure 1. Area under the curve example.

library(MESS) 
auc(x,y,from=0,rule=2) 
auc(x,loxy$fitted,from=0,rule=2)

And R output

#area under curve for raw data
[1] 67
#area under curve for smoothed data
[1] 66.77616

Machine Learning area under the curve.

# Install and load the ROCR package if you haven't already
# install.packages("ROCR")
# modified from https://rviews.rstudio.com/2019/03/01/some-r-packages-for-roc-curves/

library(ROCR)
df <- data.frame(ROCR.simple)
pred <- prediction(df$predictions, df$labels)
perf <- performance(pred,"tpr","fpr")

# 1. Calculate the actual numeric AUC value
auc_perf <- performance(pred, "auc")
auc_value <- round(unlist(auc_perf@y.values), 4)

# 2. Plot the ROC curve with colorized thresholds and custom labels
plot(perf, 
colorize = TRUE, 
main = "ROC Curve",
xlab = "False Positive Rate (1 - Specificity)",
ylab = "True Positive Rate (Sensitivity)")

# 3. Add the AUC value as a subtitle
mtext(paste("AUC =", auc_value), side = 3, line = 0.5)

# 4. Add a diagonal 50/50 reference line
abline(a = 0, b = 1, lty = 2, col = "darkgray")

End R code.

R output.

The calculated AUC was . The right-hand axes are color codes by AUC values (Fig 2): good tests AUC between 0.8 and 0.9, very good tests greater than 0.9.

Figure 2. Example ROC curve.

Area under the precision recall curve

AUPRC measures the overall performance of a binary classification model by summarizing how well it balances precision and recall across all possible decision thresholds. A score closer to 1.0 means the model is very good at finding real positive cases without making many mistakes, which is especially useful when your dataset has very few positive examples. A score closer to 1.0 means the model is very good at finding real positive cases without making many mistakes.

R code.

library(ROCR)

# 1. Prepare data and prediction object
df <- data.frame(ROCR.simple)
pred <- prediction(df$predictions, df$labels)

# 2. Extract Precision and Recall for the plot
perf_pr <- performance(pred, measure = "prec", x.measure = "rec")

# 3. Calculate the numeric AUPRC value
# In ROCR, AUPRC is calculated by finding the Area Under the Precision/Recall curve ("aucpr")
perf_auprc <- performance(pred, measure = "aucpr")
auprc_value <- round(unlist(perf_auprc@y.values), 4)

# Print the final AUPRC score to the console
cat("The calculated AUPRC is:", auprc_value, "\n")

# 4. Plot the Precision-Recall Curve using base R graphics
plot(perf_pr, 
colorize = TRUE, 
main = paste("Precision-Recall Curve (AUPRC =", auprc_value, ")"),
xlab = "Recall (Sensitivity)", 
ylab = "Precision (Positive Predictive Value)")

End R code.

R output.

The calculated AUPRC was .

Figure 3. Precision recall curve.

The differences: AUC evaluates how well the model separates both classes equally, while AUPRC evaluates how reliably the model handles the positive class alone.

The Interpretive baselines:

• Figure 2: AUC Baseline is always 0.5. A completely random model yields a diagonal line from (0,0) to (1,1), resulting in an AUC of 0.5. This baseline never changes, regardless of how rare your disease is.
• Figure 3: AUPRC Baseline depends on prevalence. A random model yields a flat horizontal line across the plot. The height of this line equals the proportion of positive cases in your dataset (). If only 1% of the patients have a rare condition, a random model’s AUPRC baseline is 0.01.

What a shifting threshold means:

• On an AUC Curve: As you lower your decision threshold to catch more sick people, you track how fast your True Positive Rate rises compared to your False Positive Rate. You are looking at the overall trade-off between the entire healthy population and the entire sick population.
• On an AUPRC Curve: As you lower your threshold to catch more sick people, you track Recall against Precision. This evaluates the quality of the “Positive” alerts. It directly measures how quickly the pool of patients you flagged as “high risk” becomes diluted with false alarms.

References and suggested readings

Çorbacıoğlu, Ş. K., & Aksel, G. (2023). Receiver operating characteristic curve analysis in diagnostic accuracy studies: A guide to interpreting the area under the curve value. Turkish Journal of Emergency Medicine, 23(4), 195–198.

Lobo, J. M., Jiménez-Valverde, A., & Real, R. (2008). AUC: A misleading measure of the performance of predictive distribution models. Global Ecology and Biogeography, 17(2), 145–151.

Martin, B., Bennett, T. D., DeWitt, P. E., Russell, S., & Sanchez-Pinto, L. N. (2025). Use of the Area Under the Precision-Recall Curve to Evaluate Prediction Models of Rare Critical Illness Events. Pediatric Critical Care Medicine, 26(6), e855–e859.

Ozenne, B., Subtil, F., & Maucort-Boulch, D. (2015). The precision–recall curve overcame the optimism of the receiver operating characteristic curve in rare diseases. Journal of Clinical Epidemiology, 68(8), 855–859. https://doi.org/10.1016/j.jclinepi.2015.02.010

Steyerberg, E. W., & Vergouwe, Y. (2014). Towards better clinical prediction models: Seven steps for development and an ABCD for validation. European Heart Journal, 35(29), 1925–1931.

Questions

1. Write up three learning outcomes for this page. Hint: Point your favorite generative AI to this page and ask for help

Chapter 20 contents

• Additional topics
• Area under the curve
• Peak detection
• Baseline correction
• Surveys
• Time series
• Dimensional analysis
• Estimating population size
• Diversity indexes
• Survival analysis
• Growth equations and dose response calculations
• Plot a Newick tree
• Phylogenetically independent contrasts
• How to get the distances from a distance tree
• Binary classification
• Meta-analysis

/MD

12.7 – Many tests one model

Introduction

In our introduction to parametric tests we so far have covered one and two sample t-tests and now the multiple sample or one-way analysis of variance (ANOVA). In subsequent sections we will cover additional tests, each with their own name. It is time to let you in on a little secret. All of these tests, t-tests, ANOVA, and linear and multiple regression we will work on later in the book belong to one family of statistical models. That model is called the general Linear Model (LM), not to be confused with the Generalized Linear Model (GLM) (Burton et al 1998; Guisan et al 2002). This greatly simplifies our approach to learning how to implement statistical tests in R (or other statistical programs) —  you only need to learn one approach: the general Linear Model (LM) function lm().

Brief overview of linear models

With the inventions of correlation, linear regression, t-tests, and analysis of variance in the period between 1890 and 1920, subsequent work led to the realization that these tests (and many others!) were special cases of a general model, the general linear model, or LM. The LM itself is a special case of the generalized linear model, or GLM; among the differences between LM and GLM, in LM, the dependent variable is ratio scale and errors in the response (dependent) variable(s) are assumed to come from a Gaussian (normal) distribution. In contrast, for GLM, the response variable may be categorical or continuous, and error distributions other than normal (Gaussian), may be applied. The GLM user must specify both the error distribution family (eg, Gaussian) and the link function, which specifies the relationship among the response and predictor variables. While we will use the GLM functions when we attempt to model growth functions and calculate EC₅₀ in dose-response problems, we will not cover GLM this semester.

The general Linear Model, LM

In matrix form, the LM can be written as

where is a matrix of response variables predicted by independent variables contained in matrix  and weighted by linear coefficients in the vector b. Basically, all of the predictor variables are combined to produce a single linear predictor . By adding an error component we have the complete linear model.

In the linear model, the error distribution is assumed to be normally distributed, or “Gaussian.”

R code

The bad news is that LM in R (and in any statistical package, actually) is a fairly involved set of commands; the good news is that once you understand how to use this command, and can work with the Options, you will be able to conduct virtually all of the tests we will use this semester, from two-sample t-tests to multiple linear regression. In the end, all you need is the one Rcmdr command to perform all of these tests.

We begin with a data set, ohia.ch12. Scroll down this page or click here to get the R code.

I found a nice report on a common garden experiment with o`hia (Corn and Hiesey 1973). O`hia (Metrosideros polymorpha) is an endemic, but wide-spread tree in the Hawaiian islands (Fig 1). O`hia exhibits pronounced intraspecific variation: individuals differ from each other. O`hia grows over wide range of environments, from low elevations along the ocean right up the sides of the volcanoes, and takes on many different growth forms, from shrubs to trees. Substantial areas of o`hia trees on the Big Island are dying, attributed to two exotic fungal species of the genus Ceratocystis (Asner et al., 2018).

Figure 1. O’hia, Metrosideros polymorpha. Public domain image from Wikipedia. 

The Biology. Individuals from distinct populations may differ because the populations differ genetically, or because the environments differ, or, and this is more realistic, both. Phenotypic plasticity is the ability of one genotype to produce more than one phenotype when exposed to different environments. Environmental differences are inevitable when populations are from different geographic areas. Thus, in population comparisons, genetic and environmental differences are confounded. A common garden experiment is a crucial genetic experiment to separate variation in phenotypes, P, among populations into causal genetic or environmental components.

If you recall from your genetics course,

where G stands for genetic (alleles) differences among individuals and E stands for environmental differences among individuals. In brief, the common garden experiment begins with individuals from the different populations are brought to the same location to control environmental differences. If the individuals sampled from the populations continue to differ despite the common environment, then the original differences between the populations must have a genetic basis, although the actual genetic scenario may be complicated (the short answer is that if “genotype by environment interaction exists, then results from a common garden experiment cannot be generalized back to the natural populations/locations — this will make more sense when we talk about two-way ANOVA). For more about common garden experiments, see de Villemereuil et al (2016). Nuismer and Gandon (2008) discuss statistical aspects of the common garden approach to studying local adaptation of populations and the more powerful “reciprocal translocation” experimental design.

Managing data for linear models

First, your data must be stacked in the worksheet. That means one column is for group labels (independent variable), the other column is for the response (dependent) variable.

If you have not already downloaded the data set, ohia.ch12, do so now. Scroll down this page or click here to get the R code.

Confirm that the worksheet is stacked. If it is not, then you would rearrange your data set using Rcmdr: Data → Active data set → Stack variables in data set…

The data set contains one factor, “Site” with three levels (M-1, 2, 3). M stands for Maui, and collection sites were noted in Figure 2 of Corn and Hiesey (1973). Once the dataset is in Rcmdr, click on View to see the data (Fig 2). There are two response variables, Height (shown in red below) and Width (shown in blue below).

Figure 2. The o`hia dataset as viewed in R Commander.

The data are from Table 5 of Corn and Hiesey 1973.  (I simulated data based on their mean/SD reported in Table 5). This was a very cool experiment: they collected o`hia  seeds from three elevations on Maui, then grew the seeds in a common garden in Honolulu. Thus, the researchers controlled the environment; what varied, then were the genotypes.

As always, you should look at the data. Box plots are good to compare central tendency (Fig 3).

Figure 3. Box plots of growth responses of o`hia seedlings collected from three Maui sites, M-1 (elevation 750 ft), M-2 (elevation 1100 ft), and M-3 (elevation 6600 ft). Data adapted from Table 5 of Corn and Hiersey 1973.

R code to make Figure 4 plots

par(mfrow=c(1,2))
Boxplot(Height ~ Site, data = ohia, id = list(method = "y"))
Boxplot(Width ~ Site, data = ohia, id = list(method = "y"))

This dataset would typically be described as a one-way ANOVA problem. There was one treatment variable (population source) with three levels (M-1, M-2, M-3). From Rcmdr we select the one-way ANOVA: Statistics → Means → One-way ANOVA… and after selecting the Groups (from the Site variable) and the Response variable (eg, Height), we have

AnovaModel.1 <- aov(Height ~ Site, data = ohia.ch12)
summary(AnovaModel.1)
           Df Sum Sq   Mean Sq  F value          Pr(>F) 
Site        2   4070    2034.8    22.63 0.000000131 ***
Residuals  47   4227      89.9 
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Let us proceed to test the null hypothesis (what was it???) using instead the lm() function. Four steps in all.

Step 1. Rcmdr: Statistics → Fit models → Linear model … (Fig 4).

Figure 4. R Commander, select to fit a Linear model.

Step 2. The pop up menu below (Fig 5) follows.

First, What is our response (dependent) variable? What is our predictor (independent) variable? We then input our model. In this case, with only the one predictor variable, Sites, our model formula is simple to enter (Fig 5): Height ~ Site.

Figure 5. Input linear model formula, Height ~ Site.

Step 3. Click OK to carry out the command.

Here is the R output and the statistical results from the application of the linear model.

LinearModel.1 <- lm(Height ~ Site, data=ohia.ch12) 
summary(LinearModel.1)

Call:
lm(formula = Height ~ Site, data = ohia.ch12)

Residuals:
    Min     1Q   Median     3Q     Max 
-18.808 -4.761   -1.755  4.758  29.257

Coefficients:
             Estimate  Std. Error  t value      Pr(>|t|)
(Intercept)    15.314       2.121    7.222 0.00000000377 ***
Site[T.M-2]    19.261       2.999    6.423 0.00000006153 ***
Site[T.M-3]     2.924       3.673    0.796          0.43    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 9.483 on 47 degrees of freedom
Multiple R-squared:  0.4905,	Adjusted R-squared:  0.4688 
F-statistic: 22.63 on 2 and 47 DF,  p-value: 0.0000001311

End R output

The linear model has produced a series of estimates of coefficients for the linear model, statistical tests of the significance of each component of the model, and the coefficient of determination, R², which is a descriptive statistic of how well model fits the data. Instead of our single factor variable for Source Population like in ANOVA we have a series of what are called dummy variables or contrasts between the populations. Thus, there is a coefficient for the difference between M-1 and M-2. “Site[T.M-2]” in the output, between M-1 and M-3, and between M-2 and M-3.

It also produced our first statistic that assesses how well the model fits the data called the coefficient of determination, R². An of of 1.0 would indicate that all variation in the data set can be explained by the predictor variable(s) in the model with no residual error remaining. Our value of 49% indicates that nearly 50% of the variation in height of the seedlings grown under common environments are due to the source population (= genetics).

Step 4. But we are not quite there — we want the traditional ANOVA results (recall the ANOVA table).

To get the ANOVA Table we have to ask Rcmdr (and therefore R) to give us this. Select

Rcmdr: Models → Hypothesis tests → ANOVA table … (Fig 6).

Figure 6. To retrieve an ANOVA table, select Models, Hypothesis tests, then ANOVA table…

Here’s the type of tests for the ANOVA table, select the default (Fig 7).

Figure 7. Options for types of tests.

Now, in the future when we work with more complicated experimental designs, we will also need to tell R how to conduct the test. For now, we will accept the default Type II type of test and ignore sandwich estimators. You should confirm that for a one-way ANOVA, Type I and Type II choices give you the same results.

The reason they do is because there is only one factor — when there are more than one factors, and if one or both of the factors are random effects, then Type I, II, and III will give you different answers. We will discuss this more as needed, but see the note below about default choices.

By default, Rcmdr makes Type II. In most of the situations we will find ourselves this semester, this is the correct choice.

Below is the output from the ANOVA table request. Confirm that the information is identical to the output from the call to aov() function.

Anova(LinearModel.1, type = "II")
Anova Table (Type II tests)
Response: Height
          Sum Sq  Df   F value        Pr(>F) 
Site      4069.7   2    22.626  0.0000001311 ***
Residuals 4226.9  47 
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

And the other stuff we got from the linear model command? Ignore for now but make note that this is a hint that regression and ANOVA are special cases of the same model, the linear model.

We do have some more work to do with ANOVA, but this is a good start.

Why use the linear model approach?

Chief among the reasons to use the lm() approach is to emphasize that a model approach is in use. One purpose of developing a model is to provide a formula to predict new values. Prediction from linear models is more fully developed in Chapter 17 and Chapter 18, but for now, we introduce the predict() function with our O`hia example.

myModel <- predict(LinearModel.1, interval = "confidence")
head(myModel, 3)    #print out first 3 rows
#Add the output to the data set
ohiaPred <- data.frame(ohia,myModel)
with(ohiaPred, tapply(fit, list(Site), mean, na.rm = TRUE))   #print out predicted values for the three sites

Output from R

myModel <- predict(LinearModel.1, interval = "confidence")
head(myModel, 3)
       fit      lwr      upr
1 15.31374 11.04775 19.57974
2 15.31374 11.04775 19.57974
3 15.31374 11.04775 19.57974
with(myModel, tapply(fit, list(Site), mean, na.rm = TRUE))
     M-1      M-2      M-3
15.31374 34.57474 18.23796

Questions

1. Revisit ANOVA problems in homework and questions from early parts of this chapter and apply lm() followed by Hypothesis testing (Rcmdr: Models → Hypothesis tests → ANOVA table) approach instead of one-way ANOVA command. Compare results using lm() to results from One-way ANOVA and other ANOVA problems.

Quiz Chapter 12.7

Many tests one model

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Data set and R code used in this page

Corn and Hiesey (1973)

Chapter 12 contents

• Introduction
• The need for ANOVA
• One way ANOVA
• Fixed effects, random effects, and agreement
• ANOVA from “sufficient statistics”
• Effect size for ANOVA
• ANOVA posthoc tests
• Many tests one model
• References and suggested readings