Accuracy – How to Calculate Specificity from Accuracy and Sensitivity

accuracysensitivity-specificity

By having accuracy and sensitivity, can we calculate specificity?
How about calculating sensitivity from accuracy and specificity?

My second question is, if sensitivity and specificity both increase 1% how much does accuracy change (1% or more)?

Best Answer

Regarding your first question: only if you know how many positive and negative samples you have. Look at three special cases:

You have the same amount of positive and negative samples. Then accuracy is the mean of sensitivity and specificity.
All samples are positive. Accuracy and sensitivity are the same.
All samples are negative. Accuracy and specificity are the same.

In general, accuracy is a weighted sum of sensitivity and specificity with the weights being the proportion of positive and negative samples respectively.

Second question: It is always exactly 1%. This is because the weighted sum of two identical numbers does not depend on the weights, as long as they add up to 1.

Related Solutions

Solved – Calculation of accuracy (and Cohen’s kappa) using sensitivity, specificity, positive and negative predictive values

You generally know TP, FN, FP, and TN, so based on this wiki:

Po = (TP + TN) / (TP + TN + FP + FN),

Pe = ((TP + FN) * (TP + FP) + (FP + TN) * (FN + TN)) / (TP + TN + FP + FN)^2

Kappa = (Po - Pe) / (1 - Pe)

Our friend Wolfram can then help to simplify this, leading to:

Kappa = 2 * (TP * TN - FN * FP) / (TP * FN + TP * FP + 2 * TP * TN + FN^2 + FN * TN + FP^2 + FP * TN)

So in R, the function would be:

cohens_kappa <- function(TP, FN, FP, TN) {
  return(2 * (TP * TN - FN * FP) / (TP * FN + TP * FP + 2 * TP * TN + FN^2 + FN * TN + FP^2 + FP * TN))
}

Calculating the Standard Error of Sensitivity and Specificity – Statistical Analysis Guide

Let's define some notation.

$y\in\{-1,+1\}$ is the true state.

$\hat y\in\{-1,+1\}$ is the predicted state.

Then...

$$ \text{Sensitivity} = \mathbb{P}\big(\hat y = +1 \vert y = +1)\\ \text{Specificity} = \mathbb{P}\big(\hat y = -1 \vert y = -1) $$

So both of these are just proportions. Treat them like you would any other proportion parameter of a Bernoulli distribution. That's where you get the standard error formula you gave. The common way to calculate standard error (SD) is the following: The outcome of each experiment is Bernoulli distributed with parameter $p$ $X_1,\dots,X_n \overset{iid}{\sim} Bern(p)$. The number of successes is then binomially distributed (since order does not matter) and we need to deal with the https://en.m.wikipedia.org/wiki/Binomial_proportion_confidence_interval The standard error for the proportion is then $SD(\hat{p})=\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$. Other possible ways to compute a confidence interval are highlighted in Wikipedia.

(I dispute that the $1.96$ should be included. Standard errors should not be tied to particular confidence levels. What if I want a $90\%$ confidence interval or a $99\%$ confidence interval?))

To get from there to your formula, the $\hat{p}$ is the sensitivity, and the $n$ is the number of positive cases, which you are calling $n_p$. Likewise, the standard error for specificity would use $\hat{p} = \text{specificity}$ and $n$ equal to the number of true negative cases.

Best Answer

Related Solutions

Solved – Calculation of accuracy (and Cohen’s kappa) using sensitivity, specificity, positive and negative predictive values

Calculating the Standard Error of Sensitivity and Specificity – Statistical Analysis Guide

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