Hypothesis Testing

Introduction

Organization

15h of lectures, 18h of TD
$12^{th}$ may: Exam (2h)
Lecture notes and slides on the website
About english and programming languages
Wooclap sessions [Test]

Objective

Given a general decision problem
- Introduce precise notations to describe the pb
- formulate mathematically hypotheses $H_0$ (a priori) and $H_1$ (alternative)
Choose a statistic adapted to the problem
Compute this statistic and its pvalue (or an approx.)
Conclude and make a decision

General Principles

Fix an objective: test whether Bob has diabetes
Design an experiment: measure of glucose level
Define hypotheses
- Null hypothesis: $H_0$: a priori, Bob has no diabetes
- Alternative hypothesis: $H_1$: Bob has diabete
Define a decision dule: function of the glucose level
Collect Data: do the measure of glucose level
Apply the decision rule: reject $H_0$ or not
Draw a conclusion: should Bob follow a treatment or make other tests ?

Good and Bad Decisions

Decision	$H_0$ True	$H_1$ True
$T=0$	True Negative (TN)	False Negative (FN) Second Type Error Missed Detection
$T=1$	False Positive (FP) First Type Error False Alarm	True Positive (TP)

Dice Biased Toward $6$

Objective: test if Bob is cheating with a dice.
Experiment: Bob rolls the dice $10$ times.
Hypotheses:
- $H_0$: the probability of getting $6$ is $1/6$
- $H_1$: the probability of getting $6$ is larger than $1/6$
Decision rule: the probability of getting a number of $6$ at least equal to the number of observed $6$ is $< 0.05$
Data: the dice falls $10$ times on $6$
Decision: the probability is $1/6^{10} < 0.05$
Conclusion?

Fairness of Dice

We observe $(X_1, \dots, X_n)$ iid where $X_i \in \{1, \dots, 6\}$ where each $\mathbb P(X_i = k) = p_k$

Same data, two conclusions

$H_0: p_6 = 1/6$ VS $H_1: p_6 > 1/6$
Do not reject $H_0$ ($H_0$ is “likely”)

$H_0: (p_1=\dots=p_6 = 1/6)$ (dice is fair) VS $H_1: \exists k: p_k > 1/6$
Reject $H_1$ ($H_1$ is “unlikely”)

Medical Test

Objective: test if a fetus has Down syndrome
Experiment: measure the Nucal translucency
Hypotheses:
- $H_0$: nucal translucency is normal $\sim \mathcal N(1.5,0.8)$
- $H_1$: nucal translucency is large

Decision rule: reject if $P_0(X \geq x_{obs}) \leq 0.05$
Collect data: $x_{obs}=3.02$
Make a decision: calculate the value of $P_0(X \geq 3.02)=0.029$
Conclusion ?

Probability Basics

Recall of Proba

Consider a probability measure $P$ on $\mathbb R$.

CDF (Cumulative Distribution Function): \[x \to P(~(-\infty,x]~) = \mathbb P(X \leq x) ~~~~\text{(if $X \sim P$ under $\mathbb P$)}\]

Continous Measures

density wrp to Lebesgue: $\mathbb P(X\in[x,x+dx])=dP(x) = p(x)dx$
PDF (Proba Density Function): $x \to p(x)$
CDF: $x \to \int_{\infty}^x p(x')dx'$
$\alpha$-quantile $q_{\alpha}$: $\int_{\infty}^{q_{\alpha}} p(x)dx = \alpha$
or $\mathbb P(X \leq q_{\alpha}) = \alpha$

Discrete Measures

density wrp to counting measure: $\mathbb P(X=x) = P(\{x\})=p(x)$
CDF: $x \to \sum_{x' \leq x} p(x')dx'$
$\alpha$-quantile $q_{\alpha}$: \[\inf_{q \in \mathbb R}\{q:~\sum_{x_i \leq q}p(x_i) > \alpha\}\]

Examples

Gaussian $\mathcal N(\mu,\sigma)$: \[p(x) = \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]

Approximation of sum of iid RV (TCL)

Binomial $\mathrm{Bin}(n,q)$: \[p(x)= \binom{n}{x}q^x (1-q)^{n-x}\]

Number of success among $n$ Bernoulli $q$

Examples

Exponential $\mathcal E(\lambda)$: \[p(x) = \lambda e^{-\lambda x}\]

Waiting time for an atomic clock of rate $\lambda$

Geometric $\mathcal{G}(q)$: \[p(x)= q(1-q)^{x-1}\]

Index of first success for iid Bernoulli $q$

Examples Gamma/Poisson

Gamma $\Gamma(k, \lambda)$: \[p(x) = \frac{\lambda^k x^{k-1}e^{-\lambda x}}{(k-1)!}\]

Waiting time for $k$ atomic clocks of rate $\lambda$

Poisson $\mathcal{P}(\lambda)$: \[p(x)=\frac{\lambda^x}{x!}e^{-\lambda}\]

Numb. of tics before time $1$ of clock $\lambda$

Basics of Hypothesis Testing

Estimation VS Test

We observe some data $X$ in a measurable space $(\mathcal X, \mathcal A)$.
Example $\mathcal X = \mathbb R^n$: $X= (X_1, \dots, X_n)$.

Estimation

One set of distributions $\mathcal P$
Parameterized by $\Theta$ \[ \mathcal P = \{P_{\theta},~ \theta \in \Theta\} \; . \]
$\exists \theta \in \Theta$ such that $X \sim P_{\theta}$

Goal: estimate a given function of $P_{\theta}$, e.g.:

$F(P_{\theta}) = \int x dP_{\theta}$
$F(P_{\theta}) = \int x^2 dP_{\theta}$

Test

Two sets of distributions $\mathcal P_0$, $\mathcal P_1$
Parameterized by disjoints $\Theta_0$, $\Theta_1$ \[ \begin{aligned} \mathcal P_0 = \{P_{\theta} : \theta \in \Theta_0\}, ~~~~ \mathcal P_1 = \{P_{\theta} : \theta \in \Theta_1\}\; . \end{aligned} \]
$\exists \theta \in \Theta_0 \cup \Theta_1$ such that $X \sim P_{\theta}$

Goal: decide between \[H_0: \theta \in \Theta_0 \text{ or } H_1: \theta \in \Theta_1\]

Question Tests

Estimation

Test

Problems

Simple VS Simple:\[\Theta_0 = \{\theta_0\} \text{ and } \Theta_1 = \{\theta_1\}\]
Simple VS Multiple:\[\Theta_0 = \{\theta_0\}\]
Else: Multiple VS Multiple

Test Model

Two sets of distributions $\mathcal P_0$, $\mathcal P_1$
Parameterized by disjoints $\Theta_0$, $\Theta_1$ \[ \begin{aligned} \mathcal P_0 = \{P_{\theta} : \theta \in \Theta_0\}, ~~~~ \mathcal P_1 = \{P_{\theta} : \theta \in \Theta_1\}\; . \end{aligned} \]
$\exists \theta \in \Theta_0 \cup \Theta_1$ such that $X \sim P_{\theta}$

Goal: decide between \[H_0: \theta \in \Theta_0 \text{ or } H_1: \theta \in \Theta_1\]

Simple VS Simple

Multiple VS Multiple

Parametric VS Non-Parametric

parametric: $\Theta_0$ and $\Theta_1$ included in subspaces of finite dimension.
non-parametric: Otherwise

Example of Multiple VS Multiple Parametric Problem:

$H_0: X \sim \mathcal N(\theta,\sigma)$, unknown $\theta < 0$ and unknown $\sigma > 0$: $\Theta_0 \subset \mathbb R^2$
$H_1: X \sim \mathcal N(\theta,\sigma)$, unknown $\theta > 0$ and unknown $\sigma > 0$: $\Theta_1 \subset \mathbb R^2$

Simple VS Multiple Non-Parametric Problems

Decision Rule and Test Statistic

Decision Rule

A Decision Rule or Test $T$ is a measurable function from $\mathcal X$ to $\{0,1\}$: \[ T : \mathcal X \to \{0,1\}\; .\]
It can depend on the sets $\mathcal P_0$ and $\mathcal P_1$
but not on any unknown parameter.
$T(x) = 0$ (or $1$) for all $x$ is the trivial decision rule. Question: Decision Rule

Test Statistic

a Test Statistic $\psi$ is a measurable function from $\mathcal X$ to $\mathbb R$: \[ \psi : \mathcal X \to \mathbb R\; .\]
It can depend on the sets $\mathcal P_0$ and $\mathcal P_1$
but not on any unknown parameter. Question: Test Statistic

Rejection Region

For a given test $T$, the rejection region $\mathcal R \subset \mathbb R$ is the set \[\{\psi(x) \in \mathbb R:~ T(x)=1\} \; .\]
Example of $\mathcal R$, for a given threshold $t>0$, \[ \begin{aligned} T(x) &= \mathbf{1}\{\psi(x) > t\}:~~~~~~\mathcal R = (t,+\infty)\\ T(x) &= \mathbf{1}\{\psi(x) < t\}:~~~~~~\mathcal R = (-\infty,t)\\ T(x) &= \mathbf{1}\{|\psi(x)| > t\}:~~~~~~\mathcal R = (-\infty,t)\cup (t, +\infty)\\ T(x) &= \mathbf{1}\{\psi(x) \not \in [t_1, t_2]\}:~~~~~~\mathcal R = (-\infty,t_1)\cup (t_2, +\infty)\; \end{aligned} \]

Simple VS Simple

Simple VS Simple Problem

We observe $X \in \mathcal X=\mathbb R^n$.
$H_0: X \sim P$ or $H_1: X \sim Q$.
We know $P$ and $Q$ but we do not know whether $X \sim P$ or $X \sim Q$

For a given test $T$ we define:

level of $T$: $\alpha = P(T(X)=1)$ (also: type-1 error)
power of $T$: $\beta = Q(T(X)=1) = 1-Q(T(X)=0)$ (1-$\beta$ is the type-2 error)

Decision	$H_0: X \sim P$	$H_1: X \sim Q$
$T=0$	$1-\alpha$	$1-\beta$
$T=1$	$\alpha$	$\beta$

unbiased: $\beta \geq \alpha$
$\alpha = 0$ for trivial test $T(x)=0$ ! But $\beta =0$ too…

Likelihood Ratio Test

Test $T$ that maximizes $\beta$ at fixed $\alpha$ ?
Idea: Consider the likelihood ratio test statistic \[\psi(x)=\frac{dQ}{dP}(x) = \frac{q(x)}{p(x)}\]
We consider the likelihood ratio test \[ T^*(x)=\mathbf 1\left\{\frac{q(x)}{p(x)} > t_{\alpha}\right\} \;\]
$t_{\alpha}$ is the $\alpha$-quantile of the distrib $\frac{q(X)}{p(X)}$ if $X\sim P$ \[ \mathbb P_{X \sim P}\left(\frac{q(X)}{p(X)} > t_{\alpha}\right) = \alpha\]

We observe $X \in \mathcal X=\mathbb R^n$.
$H_0: X \sim P$ or $H_1: X \sim Q$.
We know $P$ and $Q$ but we do not know whether $X \sim P$ or $X \sim Q$

Decision	$H_0: X \sim P$	$H_1: X \sim Q$
$T=0$	$1-\alpha$	$1-\beta$
$T=1$	$\alpha$	$\beta$

unbiased: $\beta > \alpha$
$\alpha = 0$ for trivial test $T(x)=0$ !
But $\beta =0$ too…

Neyman Pearson’s Theorem

Neyman Pearson’s Theorem

The likelihood Ratio Test of level $\alpha$ maximizes the power among all tests of level $\alpha$.

Example: $X \sim \mathcal N(\theta, 1)$.
$H_0: \theta=\theta_0$, $H_1: \theta=\theta_1$.
Check that the log-likelihood ratio is \[ (\theta_1 - \theta_0)x + \frac{\theta_0^2 -\theta_1^2}{2} \; .\]

If $\theta_1 > \theta_0$, an optimal test if of the form \[ T(x) = \mathbf 1\{ x > t \} \; .\]

Decision	$H_0: X \sim P$	$H_1: X \sim Q$
$T=0$	$1-\alpha$	$1-\beta$
$T=1$	$\alpha$	$\beta$

\[ T^*(x)=\mathbf 1\left\{\frac{q(x)}{p(x)} > t_{\alpha}\right\} \;\]

Where, if $X\sim P$, \[ P(T^*(X)=1)=\mathbb P_{X \sim P}\left(\frac{q(X)}{p(X)} > t_{\alpha}\right) = \alpha\]

Equivalent to Log-Likelihood Ratio Test: \[T^*(x)=\mathbf 1\left\{\log\left(\frac{q(x)}{p(x)}\right) > \log(t_{\alpha})\right\}\]

Example with Gaussians

Let $P_{\theta}$ be the distribution $\mathcal N(\theta,1)$.
Observe $n$ iid data $X = (X_1, \dots, X_n)$
$H_0: X \sim P^{\otimes n}_{\theta_0}$ or $H_1: X \sim P^{\otimes n}_{\theta_1}$
Remark: $P^{\otimes n}_{\theta}= \mathcal N((\theta,\dots, \theta), I_n)$
Density of $P^{\otimes n}_{\theta}$:

\[ \begin{aligned} \frac{d P^{\otimes n}_{\theta}}{dx} &= \frac{d P_{\theta}}{dx_1}\dots\frac{d P_{\theta}}{dx_n} \\ &= \frac{1}{\sqrt{2\pi}^n}\exp\left({-\sum_{i=1}^n\frac{(x_i - \theta)^2}{2}}\right) \\ &= \frac{1}{\sqrt{2\pi}^n}\exp\left(-\frac{\|x\|^2}{2} + n\theta \overline x - \frac{\theta^2}{2}\right)\; . \end{aligned} \]

Log-Likelihood Ratio Test:
$T(x) = \mathbf 1\{\overline x > t_{\alpha}\}$ if $\theta_1 > \theta_0$
$T(x) = \mathbf 1\{\overline x < t_{\alpha}\}$ otherwise
Distrib of $\overline X$ ?
$t_{\alpha}$ ?

Generalization: Exponential Families

A set of distributions $\{P_{\theta}\}$ is an exponential family if each density $p_{\theta}(x)$ is of the form \[ p_{\theta}(x) = a(\theta)b(x) \exp(c(\theta)d(x)) \; , \]
We observe $X = (X_1, \dots, X_n)$. Consider the following testing problem: \[H_0: X \sim P_{\theta_0}^{\otimes n}~~~~ \text{or}~~~~ H_1:X \sim P_{\theta_1}^{\otimes n} \; .\]
Likelihood Ratio: \[ \frac{dP^{\otimes n}_{\theta_1}}{dP^{\otimes n}_{\theta_0}} = \left(\frac{a(\theta_1)}{a(\theta_0)}\right)^n\exp\left((c(\theta_1)-c(\theta_0))\sum_{i=1}^n d(x_i)\right) \; . \]
Likelihood Ratio Test: (Q: Select Exp. Families) \[ T(X) = \mathbf 1\left\{\frac{1}{n}\sum_{i=1}^n d(X_i) > t\right\} \;. ~~~~\text{(calibrate $t$)}\]

Example: Radioactive Source

The number of particle emitted in $1$ unit of time is follows distribution $P \sim \mathcal P(\lambda)$.
We observe $20$ time units, that is $N \sim \mathcal P(20\lambda)$.
Type A sources emit an average of $\lambda_0 = 0.6$ particles/time unit
Type B sources emit an average of $\lambda_1 = 0.8$ particles/time unit
$H_0$: $N \sim \mathcal P(20\lambda_0)$ or $H_1$: $N\sim \mathcal P(20\lambda_1)$
Likelihood Ratio Test: \[T(X)=\mathbf 1\left\{\sum_{i=1}^{20}X_i > t_{\alpha}\right\} \; .\]
$t_{0.95}$: quantile(Poisson(20*0.6), 0.95) gives $18$
$\mathbb P(\mathcal P(20*0.6) > 17)$: 1-cdf(Poisson(20*0.6), 17) gives $0.063$
$\mathbb P(\mathcal P(20*0.6) > 18)$: 1-cdf(Poisson(20*0.6), 18) gives $0.038$: Reject if $N \geq 19$

Multiple-Multiple Tests

Definitions

$H_0 = \mathcal P_0=\{P_{\theta}, \theta \in \Theta_0 \}$ is not a singleton
No meaning of $\mathbb P_{H_0}(X \in A)$
level of $T$: \[ \alpha = \sup_{\theta \in \Theta_0}P_{\theta}(T(X)=1) \; . \]
power function $\beta: \Theta_1 \to [0,1]$ \[ \beta(\theta) = P_{\theta}(T(X)=1) \]
$T$ is unbiased if $\beta (\theta) \geq \alpha$ for all $\theta \in \Theta_1$.

If $T_1$, $T_2$ are two tests of level $\alpha_1$, $\alpha_2$
$T_2$ is uniformly more powerfull ($UMP$) than $T_1$ if
1. $\alpha_2 \leq \alpha_1$
2. $\beta_2(\theta) \geq \beta_1(\theta)$ for all $\theta \in \Theta_1$
$T^*$ is $UMP_{\alpha}$ if it is $UMP$ than any other test $T$ of level $\alpha$.

Multiple-Multiple Tests in $\mathbb R$

Assumption: $\Theta_0 \cup \Theta_1 \subset \mathbb R$.

One-tailed tests: \[ \begin{aligned} H_0: \theta \leq \theta_0 ~~~~ &\text{ or } ~~~ H_1: \theta > \theta_0 ~~~ \text{(right-tailed: unilatéral droit)}\\ H_0: \theta \geq \theta_0 ~~~ &\text{ or } ~~~ H_1: \theta < \theta_0 ~~~ \text{(left-tailed: unilatéral gauche)} \end{aligned} \]
Two-tailed tests: \[ \begin{aligned} H_0: \theta = \theta_0 ~~~ &\text{ or } ~~~ H_1: \theta \neq \theta_0 ~~~ \text{(simple/multiple)}\\ H_0: \theta \in [\theta_1, \theta_2] ~~~ &\text{ or } ~~~ H_1: \theta \not \in [\theta_1, \theta_2] ~~~ \text{( multiple/multiple)} \end{aligned} \]

Theorem

Assume that $p_{\theta}(x) = a(\theta)b(x)\exp(c(\theta)d(x))$ and that $c$ is a non-decreasing (croissante) function, and consider a one-tailed test problem.
There exists a UMP$\alpha$ test. It is $\mathbf 1\{\sum d(X_i) > t \}$ if $H_1: \theta > \theta_0$ (right-tailed test).

Pivotal Test Statistic and P-value

Here, $\Theta_0$ is not necessarily a singleton. $\mathbb P_{H_0}(X \in A)$ has no meaning.

Pivotal Test Statistic

$\psi: \mathcal X \to \mathbb R$ is pivotal if the distribution of $\psi(X)$ under $H_0$ does not depend on $\theta \in \Theta_0$:

for any $\theta, \theta' \in \Theta_0$, and any event $A$, \[ \mathbb P_{\theta}(\psi(X) \in A) = \mathbb P_{\theta'}(\psi(X) \in A) \; .\]

Example: If $X=(X_1, \dots, X_n)$ are iid $\mathcal N(0, \sigma)$, the distrib of \[ \psi(X) = \frac{\sum_{i=1}^n \overline X}{\sqrt{\sum_{i=1}^n X_i^2}}\] does not depend on $\sigma$.

Pivotal Test Statistic and P-value

P-value: definition

We define $p_{value}(x_{\mathrm{obs}}) =\mathbb P(\psi(X) \geq x_{\mathrm{obs}})$ for a right-tailed test.

For a two-tailed test, $p_{value}(x_{\mathrm{obs}}) =2\min(\mathbb P(\psi(X) \geq x_{\mathrm{obs}}),\mathbb P(\psi(X) \leq x_{\mathrm{obs}}))$

P-value under $H_0$

Under $H_0$, for a pivotal test statistic $\psi$, $p_{value}(X)$ has a uniform distribution $\mathcal U([0,1])$.

In practice: reject if $p_{value}(x_{\mathrm{obs}}) \leq \alpha = 0.05$
$\alpha$ is the level or type-1-error of the test

Illustration if $\psi(X) \sim \mathcal N(0,1)$:

Decision	\(H_0: X \sim P\)	\(H_1: X \sim Q\)
\(T=0\)	\(1-\alpha\)	\(1-\beta\)
\(T=1\)	\(\alpha\)	\(\beta\)

Decision	\(H_0: X \sim P\)	\(H_1: X \sim Q\)
\(T=0\)	\(1-\alpha\)	\(1-\beta\)
\(T=1\)	\(\alpha\)	\(\beta\)

Decision	\(H_0: X \sim P\)	\(H_1: X \sim Q\)
\(T=0\)	\(1-\alpha\)	\(1-\beta\)
\(T=1\)	\(\alpha\)	\(\beta\)

Hypothesis Testing

Introduction

Organization

Objective

General Principles

Good and Bad Decisions

Dice Biased Toward \(6\)

Fairness of Dice

Medical Test

Probability Basics

Recall of Proba

Continous Measures

Discrete Measures

Examples

Examples

Examples Gamma/Poisson

Basics of Hypothesis Testing

Estimation VS Test

Estimation

Test

Estimation

Test

Problems

Test Model

Simple VS Simple

Multiple VS Multiple

Parametric VS Non-Parametric

Decision Rule and Test Statistic

Rejection Region

Simple VS Simple

Simple VS Simple Problem

Likelihood Ratio Test

Neyman Pearson’s Theorem

Example with Gaussians

Generalization: Exponential Families

Example: Radioactive Source

Multiple-Multiple Tests

Definitions

Multiple-Multiple Tests in \(\mathbb R\)

Pivotal Test Statistic and P-value

Pivotal Test Statistic and P-value