Sensitivity Analysis | Malone Math | Malone Global University

The Central Idea

A model takes inputs and produces outputs.

\[ \boxed{\text{Inputs}\longrightarrow\text{Model}\longrightarrow\text{Outputs}} \]

Sensitivity analysis changes one or more inputs and observes how the output responds.

The central question is: Would a reasonable change in our assumptions change the decision?

Why Sensitivity Analysis Matters

Most practical inputs are not known perfectly.

We may only have estimates for:

probabilities;
future prices;
costs;
demand;
failure rates;
growth rates;
time requirements;
discount rates;
utilities or losses.

Sensitivity analysis reveals whether uncertainty in these inputs materially affects the recommendation.

Baseline or Base Case

Sensitivity analysis usually begins with a baseline or base case.

The base case contains the model's current best estimates.

Other cases are then compared with this reference point.

Example: Base-Case Decision

Suppose Decision A has expected payoff

\[ E[A]=42 \]

and Decision B has expected payoff

\[ E[B]=38. \]

The base-case recommendation is

\[ \boxed{A}. \]

Sensitivity analysis asks what assumptions would have to change before B becomes preferable.

One-Way Sensitivity Analysis

One-way sensitivity analysis changes one input while holding the other inputs fixed.

This isolates the effect of that one assumption.

Common one-way inputs include:

a probability;
a cost;
a payoff;
a growth rate;
a failure rate.

Example: Varying a Probability

Suppose Decision A pays $30 regardless of the state.

Decision B pays:

$50 if demand is high;
$10 if demand is low.

Let

\[ p=P(\text{high demand}). \]

Then

\[ P(\text{low demand})=1-p. \]

Expected Value as a Function of the Input

For Decision A,

\[ E[A]=30. \]

For Decision B,

\[ E[B]=50p+10(1-p). \]

Simplify:

\[ E[B]=50p+10-10p \] \[ \boxed{E[B]=10+40p}. \]

Threshold Probability

The threshold is the input value at which the preferred decision changes.

Set the two expected values equal:

\[ 30=10+40p. \]

Then

\[ 20=40p \] \[ \boxed{p=0.5}. \]

Interpreting the Threshold

\[ p<0.5, \]

then

\[ E[B]<30, \]

so A is preferred.

\[ p>0.5, \]

then B is preferred.

At $p=0.5$, the decision maker is indifferent under expected monetary value.

Break-Even Analysis

A break-even value is an input value at which two alternatives produce the same objective value.

Threshold probability is one type of break-even analysis.

Break-even values can also be found for:

price;
cost;
demand;
volume;
failure probability;
time;
utility parameters.

Example: Break-Even Cost

Suppose Project A has expected gross benefit

\[ 80 \]

and fixed cost $c$.

Its expected net value is

\[ 80-c. \]

Project B has expected net value

\[ 55. \]

Set them equal:

\[ 80-c=55. \]

Therefore,

\[ \boxed{c=25}. \]

Interpret the Break-Even Cost

\[ c<25, \]

Project A has larger expected net value.

\[ c>25, \]

Project B has larger expected net value.

The break-even value divides the input space into decision regions.

Decision Regions

A decision region is a range of input values for which the same decision remains optimal under the chosen criterion.

For the probability example:

\[ 0\le p<0.5 \]

favors A, while

\[ 0.5<p\le1 \]

favors B.

Sensitivity Range

A sensitivity range describes how far an input can move before the recommendation changes.

If the current estimate is $p=0.7$ and the threshold is $0.5$, then the recommendation can tolerate a decrease of

\[ 0.7-0.5=0.2 \]

before reaching the decision boundary.

Robust and Fragile Decisions

A decision is relatively robust when large changes in plausible inputs do not change the recommendation.

A decision is relatively fragile when small changes in plausible inputs reverse the recommendation.

Fragile decisions deserve more scrutiny.

Local Sensitivity

Local sensitivity examines how the output responds to small changes near the base-case input.

If a model output is

\[ y=f(x), \]

then the derivative

\[ \boxed{\frac{dy}{dx}=f'(x)} \]

measures local sensitivity when the derivative exists.

Example: Derivative-Based Sensitivity

Suppose

\[ y=5x^2. \]

Then

\[ \frac{dy}{dx}=10x. \]

\[ x=3, \]

local sensitivity is

\[ \boxed{30}. \]

Near $x=3$, a small change in $x$ changes $y$ by roughly $30$ times that amount.

Finite-Change Sensitivity

When derivatives are unavailable or unnecessary, sensitivity can be approximated with finite changes:

\[ \boxed{ \frac{\Delta y}{\Delta x} = \frac{f(x_2)-f(x_1)}{x_2-x_1} }. \]

This measures average response over a chosen interval.

Absolute Sensitivity

Absolute sensitivity records output change per unit of input change.

For example,

\[ \frac{\Delta y}{\Delta x}=4 \]

means a one-unit increase in $x$ is associated with about a four-unit increase in $y$ over the interval being studied.

Relative Sensitivity

When inputs use different units, percentage changes can be more informative.

A simple relative sensitivity measure is

\[ \boxed{ \frac{\Delta y/y}{\Delta x/x} }. \]

This compares the percentage change in the output with the percentage change in the input.

Elasticity

In continuous settings, relative sensitivity is closely related to elasticity:

\[ \boxed{ \frac{dy}{dx}\frac{x}{y} }. \]

An elasticity of $2$ means a small $1\%$ change in the input is associated with approximately a $2\%$ change in the output.

Two-Way Sensitivity Analysis

Two-way sensitivity analysis changes two inputs together.

This is useful when the decision depends on interactions between uncertain assumptions.

For example, a project might depend simultaneously on:

market demand probability;
production cost.

Example: Two Inputs

Suppose

\[ V_A=60p-c \]

and

\[ V_B=30. \]

The indifference boundary satisfies

\[ 60p-c=30. \]

Solving for $c$,

\[ \boxed{c=60p-30}. \]

Different combinations of $p$ and $c$ fall on different sides of this decision boundary.

Multi-Way Sensitivity Analysis

Multi-way sensitivity analysis varies several inputs at once.

This can reveal interactions that one-way analysis misses.

However, the number of combinations grows rapidly as the number of uncertain inputs increases.

Scenario Analysis as Structured Multi-Way Sensitivity

Scenario analysis changes several related assumptions together.

For example:

Scenario	Demand	Price	Cost
Weak	Low	Low	High
Base	Moderate	Moderate	Moderate
Strong	High	High	Low

The inputs move together in economically coherent bundles.

Best-Case and Worst-Case Analysis

A simple form of sensitivity analysis evaluates favorable and unfavorable combinations of inputs.

This can identify the possible range of outcomes, but extreme combinations should not automatically be treated as equally plausible.

Tornado Diagrams

A tornado diagram ranks uncertain inputs according to how much they change a chosen output when varied over specified ranges.

Inputs producing the widest output ranges appear near the top.

The diagram helps prioritize which assumptions deserve further analysis or better information.

What a Tornado Diagram Does Not Show

A tornado diagram usually varies one input at a time.

It therefore may not capture:

interactions among inputs;
correlations;
joint extreme cases;
nonlinear combinations of assumptions.

Sensitivity of Expected Value

Suppose

\[ E[X]=px_1+(1-p)x_2. \]

Rewrite:

\[ E[X]=x_2+p(x_1-x_2). \]

Therefore,

\[ \boxed{ \frac{dE[X]}{dp}=x_1-x_2 }. \]

The expected value is more sensitive to $p$ when the two outcomes are far apart.

Example: Probability Sensitivity

Suppose

\[ x_1=100,\qquad x_2=20. \]

Then

\[ E[X]=20+80p, \]

\[ \frac{dE[X]}{dp}=80. \]

A change of $0.01$ in $p$ changes expected value by approximately

\[ 80(0.01)=0.8. \]

Sensitivity of Expected Loss

\[ E[L]=pL_1+(1-p)L_2, \]

then

\[ \boxed{ \frac{dE[L]}{dp}=L_1-L_2 }. \]

When the losses differ greatly, uncertainty about the probability has a larger effect on expected loss.

Threshold Analysis in Classification

Suppose predicting positive creates false-positive loss $L_{FP}$, while predicting negative creates false-negative loss $L_{FN}$.

A probability threshold can be derived by comparing expected losses.

This shows how classification decisions depend directly on the relative costs of different errors.

Sensitivity to Loss Assumptions

A decision may look stable under one loss function and unstable under another.

Sensitivity analysis should therefore examine not only probabilities and payoffs, but also whether the chosen loss or utility structure is itself uncertain.

Sensitivity to Model Structure

Not every assumption is a numerical parameter.

Structural assumptions may include:

which variables are included;
whether effects are linear;
whether events are independent;
which distribution is used;
whether a trend continues.

Testing alternate structures is a broader form of sensitivity analysis.

Sensitivity and Uncertainty Are Different

Uncertainty describes how little we know about an input.

Sensitivity describes how strongly the output responds to that input.

An input can be highly uncertain but have little effect on the decision.

Another input can be estimated precisely but have enormous leverage over the result.

High Uncertainty and High Sensitivity

Inputs that are both highly uncertain and highly influential deserve special attention.

These are often the best candidates for:

additional research;
better measurement;
experimentation;
monitoring;
contingency planning.

Sensitivity Analysis and Value of Information

Sensitivity analysis identifies which uncertainties matter most.

The next question is whether reducing one of those uncertainties is worth the cost.

This leads naturally to value of information.

Common Error: Changing Too Many Inputs Without Structure

If many inputs change at once without a clear scenario design, it can become difficult to identify which assumption caused the result to change.

Begin with one-way analysis, then add multi-way or scenario analysis where interactions matter.

Common Error: Using Implausible Input Ranges

Sensitivity analysis is only useful when the tested ranges are meaningful.

Extremely wide or arbitrary ranges can exaggerate sensitivity and distract from realistic uncertainty.

Common Error: Assuming a Stable Recommendation Means a Correct Model

A decision can be insensitive to the parameters being tested while still being based on a poor model structure.

Robustness within a model does not prove that the model itself is correct.

Common Error: Focusing Only on the Output Value

A model output may change substantially while the preferred decision remains the same.

Conversely, a small numerical change near a threshold may reverse the decision.

Decision sensitivity is often more important than raw output sensitivity.

Common Error: Ignoring Correlated Inputs

Some inputs move together.

Varying them independently may produce combinations that are unrealistic.

Scenario analysis or joint modeling is often preferable when dependencies are strong.

Common Error: Confusing Sensitivity with Probability

A highly sensitive parameter is not necessarily highly uncertain or highly likely to change.

Sensitivity measures impact, not likelihood.

A Reliable Procedure for Sensitivity Analysis

Define the decision criterion and base case.
Identify uncertain or important inputs.
Choose realistic ranges for those inputs.
Vary one input at a time first.
Record how the output and recommendation change.
Find threshold or break-even values where useful.
Rank the most influential assumptions.
Test important interactions with two-way or multi-way analysis.
Use coherent scenarios for correlated inputs.
Identify which uncertainties are worth reducing.

Practice 1: One-Way Sensitivity

A model is

\[ y=4x+10. \]

Find $y$ when $x=5$ and when $x=7$.

Practice 2: Finite Sensitivity

Using the values from Practice 1, find

\[ \frac{\Delta y}{\Delta x}. \]

Practice 3: Threshold Probability

Decision A has value

\[ 25. \]

Decision B has expected value

\[ 5+40p. \]

Find the threshold probability at which the two decisions are equal.

Practice 4: Break-Even Cost

Project A has expected gross value $90$ and cost $c$. Project B has net value $60$.

Find the break-even value of $c$.

Practice 5: Decision Region

Suppose A is preferred when $p<0.4$, while B is preferred when $p>0.4$.

Which decision is preferred when $p=0.65$?

Practice 6: Local Sensitivity

Let

\[ y=3x^2. \]

Find

\[ \frac{dy}{dx} \]

at $x=4$.

Practice 7: Probability Sensitivity

Suppose

\[ E[X]=15+50p. \]

By how much does expected value change when $p$ rises from $0.40$ to $0.46$?

Practice 8: Relative Sensitivity

An input rises from $100$ to $110$, while an output rises from $200$ to $230$.

Find the approximate relative sensitivity

\[ \frac{\Delta y/y}{\Delta x/x}, \]

using the initial values as the reference.

Practice 9: Robust or Fragile?

A recommendation changes when an uncertain probability moves from $0.51$ to $0.50$.

Is the decision relatively robust or fragile with respect to that input?

Practice 10: Uncertainty versus Sensitivity

Input A is very uncertain but barely changes the output. Input B is estimated fairly precisely, but small changes in B sharply change the recommendation.

Which input is more sensitive? Which is more uncertain?

Practice Answers

Practice 1

At $x=5$:

\[ y=4(5)+10=30. \]

At $x=7$:

\[ y=4(7)+10=38. \]

Practice 2

\[ \frac{\Delta y}{\Delta x} = \frac{38-30}{7-5} \] \[ =\frac{8}{2} \] \[ \boxed{4}. \]

Practice 3

Set the values equal:

\[ 25=5+40p. \] \[ 20=40p. \] \[ \boxed{p=0.5}. \]

Practice 4

\[ 90-c=60. \] \[ c=30. \] \[ \boxed{30}. \]

Practice 5

Since

\[ 0.65>0.4, \]

the preferred decision is

\[ \boxed{B}. \]

Practice 6

\[ y=3x^2 \]

gives

\[ \frac{dy}{dx}=6x. \]

At $x=4$,

\[ \boxed{24}. \]

Practice 7

The change in probability is

\[ 0.46-0.40=0.06. \]

Therefore expected value changes by

\[ 50(0.06)=\boxed{3}. \]

Practice 8

The input changes by

\[ \frac{110-100}{100}=0.10. \]

The output changes by

\[ \frac{230-200}{200}=0.15. \]

Therefore,

\[ \frac{0.15}{0.10}=\boxed{1.5}. \]

Practice 9

The recommendation changes after a very small change in the probability, so it is relatively

\[ \boxed{\text{fragile}}. \]

Practice 10

Input B is more sensitive because small changes in B strongly affect the recommendation.

Input A is more uncertain because its value is less well known.

What to Remember

Sensitivity analysis studies how conclusions respond when assumptions change.

Begin with a clear base case.
One-way analysis changes one input at a time.
Thresholds mark where the preferred decision changes.
Break-even values divide the input space into decision regions.
Local sensitivity can be measured with derivatives.
Relative sensitivity compares percentage changes.
Two-way and multi-way analysis reveal interactions.
Scenario analysis changes coherent groups of assumptions together.
Robust decisions remain preferred across broad plausible ranges.
Sensitivity and uncertainty are different: one measures impact, the other measures lack of knowledge.

The main operational lesson is: focus attention on assumptions that are both uncertain and capable of changing the decision.

Where We Go Next

Sensitivity analysis tells us which uncertainties matter most. Once those uncertainties have been identified, we can ask whether learning more about them would improve the decision enough to justify the cost of obtaining that information.

The next lesson, Value of Information, develops a formal framework for measuring the benefit of reducing uncertainty before making a decision.