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Hotelling's T-squared distribution
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<h2 id="motivation">Motivation</h2> <p>The distribution arises in <a href="/facts/Multivariate_statistics/tDuNcWrL">multivariate statistics</a> in undertaking <a href="/facts/Statistical_hypothesis_test/yv9RPF6U">tests</a> of the differences between the (multivariate) means of different populations, where tests for univariate problems would make use of a <a href="/facts/T-test/dMH8uC28"><i>t</i>-test</a>. The distribution is named for <a href="/facts/Harold_Hotelling/gGTx0uY8">Harold Hotelling</a>, who developed it as a generalization of Student's <i>t</i>-distribution.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="definition">Definition</h2> <p>If the vector d {\displaystyle d} is <a href="/facts/Multivariate_normal_distribution/2Xfegqz2">Gaussian multivariate-distributed</a> with zero mean and unit <a href="/facts/Covariance_matrix/kLhgjHC6">covariance matrix</a> N ( 0 p , I p , p ) {\displaystyle N(\mathbf {0} _{p},\mathbf {I} _{p,p})} and M {\displaystyle M} is a p × p {\displaystyle p\times p} random matrix with a <a href="/facts/Wishart_distribution/MEcqTump">Wishart distribution</a> W ( I p , p , m ) {\displaystyle W(\mathbf {I} _{p,p},m)} with unit <a href="/facts/Scale_matrix/pFm6fvpu">scale matrix</a> and <i>m</i> <a href="/facts/Degrees_of_freedom_(statistics)/fvwKCU86">degrees of freedom</a>, and <i>d</i> and <i>M</i> are independent of each other, then the <a href="/facts/Quadratic_form_(statistics)/qH3sK2VG">quadratic form</a> X {\displaystyle X} has a Hotelling distribution (with parameters p {\displaystyle p} and m {\displaystyle m} ):<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> X = m d T M − 1 d ∼ T 2 ( p , m ) . {\displaystyle X=md^{T}M^{-1}d\sim T^{2}(p,m).} <p>It can be shown that if a random variable <i>X</i> has Hotelling's <i>T</i>-squared distribution, X ∼ T p , m 2 {\displaystyle X\sim T_{p,m}^{2}} , then:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> m − p + 1 p m X ∼ F p , m − p + 1 {\displaystyle {\frac {m-p+1}{pm}}X\sim F_{p,m-p+1}} <p>where F p , m − p + 1 {\displaystyle F_{p,m-p+1}} is the <a href="/facts/F-distribution/XNuuJtWY"><i>F</i>-distribution</a> with parameters <i>p</i> and <i>m</i> − <i>p</i> + 1. </p> <h2 id="hotelling-t-squared-statistic">Hotelling <i>t</i>-squared statistic</h2> <p>Let Σ ^ {\displaystyle {\hat {\mathbf {\Sigma } }}} be the <a href="/facts/Sample_covariance/Ah8VVVDT">sample covariance</a>: </p> Σ ^ = 1 n − 1 ∑ i = 1 n ( x i − x ¯ ) ( x i − x ¯ ) ′ {\displaystyle {\hat {\mathbf {\Sigma } }}={\frac {1}{n-1}}\sum _{i=1}^{n}(\mathbf {x} _{i}-{\overline {\mathbf {x} }})(\mathbf {x} _{i}-{\overline {\mathbf {x} }})'} <p>where we denote <a href="/facts/Transpose/8wmsagGS">transpose</a> by an <a href="/facts/Apostrophe/Te0sg9km">apostrophe</a>. It can be shown that Σ ^ {\displaystyle {\hat {\mathbf {\Sigma } }}} is a <a href="/facts/Positive-definite_matrix/Hbr6AuPS">positive (semi) definite</a> matrix and ( n − 1 ) Σ ^ {\displaystyle (n-1){\hat {\mathbf {\Sigma } }}} follows a <i>p</i>-variate <a href="/facts/Wishart_distribution/MEcqTump">Wishart distribution</a> with <i>n</i> − 1 degrees of freedom.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> The sample covariance matrix of the mean reads Σ ^ x ¯ = Σ ^ / n {\displaystyle {\hat {\mathbf {\Sigma } }}_{\overline {\mathbf {x} }}={\hat {\mathbf {\Sigma } }}/n} .<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>The Hotelling's <i>t</i>-squared statistic is then defined as:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> t 2 = ( x ¯ − μ ) ′ Σ ^ x ¯ − 1 ( x ¯ − μ ) = n ( x ¯ − μ ) ′ Σ ^ − 1 ( x ¯ − μ ) , {\displaystyle t^{2}=({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'{\hat {\mathbf {\Sigma } }}_{\overline {\mathbf {x} }}^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})=n({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'{\hat {\mathbf {\Sigma } }}^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }}),} <p>which is proportional to the <a href="/facts/Mahalanobis_distance/UneEsDng">Mahalanobis distance</a> between the sample mean and μ {\displaystyle {\boldsymbol {\mu }}} . Because of this, one should expect the statistic to assume low values if x ¯ ≈ μ {\displaystyle {\overline {\mathbf {x} }}\approx {\boldsymbol {\mu }}} , and high values if they are different. </p><p>From the distribution, </p> t 2 ∼ T p , n − 1 2 = p ( n − 1 ) n − p F p , n − p , {\displaystyle t^{2}\sim T_{p,n-1}^{2}={\frac {p(n-1)}{n-p}}F_{p,n-p},} <p>where F p , n − p {\displaystyle F_{p,n-p}} is the <a href="/facts/F-distribution/XNuuJtWY"><i>F</i>-distribution</a> with parameters <i>p</i> and <i>n</i> − <i>p</i>. </p><p>In order to calculate a <a href="/facts/P-value/InnYd1bz"><i>p</i>-value</a> (unrelated to <i>p</i> variable here), note that the distribution of t 2 {\displaystyle t^{2}} equivalently implies that </p> n − p p ( n − 1 ) t 2 ∼ F p , n − p . {\displaystyle {\frac {n-p}{p(n-1)}}t^{2}\sim F_{p,n-p}.} <p>Then, use the quantity on the left hand side to evaluate the <i>p</i>-value corresponding to the sample, which comes from the <i>F</i>-distribution. A <a href="/facts/Confidence_region/8wJdMBtg">confidence region</a> may also be determined using similar logic. </p> <h3>Motivation</h3> <p class="note">Further information: <a href="/facts/Multivariate_normal_distribution/2Xfegqz2">Multivariate normal distribution § Interval</a></p> <p>Let N p ( μ , Σ ) {\displaystyle {\mathcal {N}}_{p}({\boldsymbol {\mu }},{\mathbf {\Sigma } })} denote a <a href="/facts/Multivariate_normal_distribution/2Xfegqz2"><i>p</i>-variate normal distribution</a> with <a href="/facts/Location_parameter/UQTyS3JU">location</a> μ {\displaystyle {\boldsymbol {\mu }}} and known <a href="/facts/Covariance_matrix/kLhgjHC6">covariance</a> Σ {\displaystyle {\mathbf {\Sigma } }} . Let </p> x 1 , … , x n ∼ N p ( μ , Σ ) {\displaystyle {\mathbf {x} }_{1},\dots ,{\mathbf {x} }_{n}\sim {\mathcal {N}}_{p}({\boldsymbol {\mu }},{\mathbf {\Sigma } })} <p>be <i>n</i> independent identically distributed (iid) <a href="/facts/Random_variable/TwTBXnLT">random variables</a>, which may be represented as p × 1 {\displaystyle p\times 1} column vectors of real numbers. Define </p> x ¯ = x 1 + ⋯ + x n n {\displaystyle {\overline {\mathbf {x} }}={\frac {\mathbf {x} _{1}+\cdots +\mathbf {x} _{n}}{n}}} <p>to be the <a href="/facts/Sample_mean/Ah8VVVDT">sample mean</a> with covariance Σ x ¯ = Σ / n {\displaystyle {\mathbf {\Sigma } }_{\overline {\mathbf {x} }}={\mathbf {\Sigma } }/n} . It can be shown that </p> ( x ¯ − μ ) ′ Σ x ¯ − 1 ( x ¯ − μ ) ∼ χ p 2 , {\displaystyle ({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'{\mathbf {\Sigma } }_{\overline {\mathbf {x} }}^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})\sim \chi _{p}^{2},} <p>where χ p 2 {\displaystyle \chi _{p}^{2}} is the <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared distribution</a> with <i>p</i> degrees of freedom.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> <table><tbody><tr><th>Proof</th></tr><tr><td>Proof<p>Every positive-semidefinite symmetric matrix M {\textstyle {\boldsymbol {M}}} has a positive-semidefinite symmetric square root M 1 / 2 {\textstyle {\boldsymbol {M}}^{1/2}} , and if it is nonsingular, then its inverse has a positive-definite square root M − 1 / 2 {\textstyle {\boldsymbol {M}}^{-1/2}} .</p><p>Since var ( x ¯ ) = Σ x ¯ {\textstyle \operatorname {var} \left({\overline {\boldsymbol {x}}}\right)=\mathbf {\Sigma } _{\overline {\mathbf {x} }}} , we have var ( Σ x ¯ − 1 / 2 x ¯ ) = Σ x ¯ − 1 / 2 ( var ( x ¯ ) ) ( Σ x ¯ − 1 / 2 ) T = Σ x ¯ − 1 / 2 ( var ( x ¯ ) ) Σ x ¯ − 1 / 2 because Σ x ¯ is symmetric = ( Σ x ¯ − 1 / 2 Σ x ¯ 1 / 2 ) ( Σ x ¯ 1 / 2 Σ x ¯ − 1 / 2 ) = I p . {\displaystyle {\begin{aligned}\operatorname {var} \left(\mathbf {\Sigma } _{\overline {\boldsymbol {x}}}^{-1/2}{\overline {\boldsymbol {x}}}\right)&=\mathbf {\Sigma } _{\overline {\boldsymbol {x}}}^{-1/2}{\Big (}\operatorname {var} \left({\overline {\boldsymbol {x}}}\right){\Big )}\left(\mathbf {\Sigma } _{\overline {\boldsymbol {x}}}^{-1/2}\right)^{T}\\[5pt]&=\mathbf {\Sigma } _{\overline {\boldsymbol {x}}}^{-1/2}{\Big (}\operatorname {var} \left({\overline {\boldsymbol {x}}}\right){\Big )}\mathbf {\Sigma } _{\overline {\boldsymbol {x}}}^{-1/2}{\text{ because }}\mathbf {\Sigma } _{\overline {\boldsymbol {x}}}{\text{ is symmetric}}\\[5pt]&=\left(\mathbf {\Sigma } _{\overline {\boldsymbol {x}}}^{-1/2}\mathbf {\Sigma } _{\overline {\boldsymbol {x}}}^{1/2}\right)\left(\mathbf {\Sigma } _{\overline {\boldsymbol {x}}}^{1/2}\mathbf {\Sigma } _{\overline {\boldsymbol {x}}}^{-1/2}\right)\\[5pt]&=\mathbf {I} _{p}.\end{aligned}}} Consequently ( x ¯ − μ ) T Σ x ¯ − 1 ( x ¯ − μ ) = ( Σ x ¯ − 1 / 2 ( x ¯ − μ ) ) T ( Σ x ¯ − 1 / 2 ( x ¯ − μ ) ) {\displaystyle ({\overline {\boldsymbol {x}}}-{\boldsymbol {\mu }})^{T}\mathbf {\Sigma } _{\overline {x}}^{-1}({\overline {\boldsymbol {x}}}-{\boldsymbol {\mu }})=\left(\mathbf {\Sigma } _{\overline {x}}^{-1/2}({\overline {\boldsymbol {x}}}-{\boldsymbol {\mu }})\right)^{T}\left(\mathbf {\Sigma } _{\overline {x}}^{-1/2}({\overline {\boldsymbol {x}}}-{\boldsymbol {\mu }})\right)} and this is simply the sum of squares of p {\textstyle p} independent standard normal random variables. Thus its distribution is χ p 2 . {\textstyle \chi _{p}^{2}.} </p><p>Alternatively, one can argue using density functions and characteristic functions, as follows.</p>Proof<p>To show this use the fact that x ¯ ∼ N p ( μ , Σ / n ) {\displaystyle {\overline {\mathbf {x} }}\sim {\mathcal {N}}_{p}({\boldsymbol {\mu }},{\mathbf {\Sigma } }/n)} and derive the <a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">characteristic function</a> of the random variable y = ( x ¯ − μ ) ′ Σ x ¯ − 1 ( x ¯ − μ ) = ( x ¯ − μ ) ′ ( Σ / n ) − 1 ( x ¯ − μ ) {\displaystyle \mathbf {y} =({\bar {\mathbf {x} }}-{\boldsymbol {\mu }})'{\mathbf {\Sigma } }_{\bar {\mathbf {x} }}^{-1}({\bar {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})=({\bar {\mathbf {x} }}-{\boldsymbol {\mu }})'({\mathbf {\Sigma } }/n)^{-1}({\bar {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})} . As usual, let | ⋅ | {\displaystyle |\cdot |} denote the <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of the argument, as in | Σ | {\displaystyle |{\boldsymbol {\Sigma }}|} .</p><p>By definition of characteristic function, we have:<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a></p> φ y ( θ ) = E e i θ y , = E e i θ ( x ¯ − μ ) ′ ( Σ / n ) − 1 ( x ¯ − μ ) = ∫ e i θ ( x ¯ − μ ) ′ n Σ − 1 ( x ¯ − μ ) ( 2 π ) − p / 2 | Σ / n | − 1 / 2 e − ( 1 / 2 ) ( x ¯ − μ ) ′ n Σ − 1 ( x ¯ − μ ) d x 1 ⋯ d x p {\displaystyle {\begin{aligned}\varphi _{\mathbf {y} }(\theta )&=\operatorname {E} e^{i\theta \mathbf {y} },\\[5pt]&=\operatorname {E} e^{i\theta ({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'({\mathbf {\Sigma } }/n)^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})}\\[5pt]&=\int e^{i\theta ({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'n{\mathbf {\Sigma } }^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})}(2\pi )^{-p/2}|{\boldsymbol {\Sigma }}/n|^{-1/2}\,e^{-(1/2)({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'n{\boldsymbol {\Sigma }}^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})}\,dx_{1}\cdots dx_{p}\end{aligned}}} <p>There are two exponentials inside the integral, so by multiplying the exponentials we add the exponents together, obtaining:</p> = ∫ ( 2 π ) − p / 2 | Σ / n | − 1 / 2 e − ( 1 / 2 ) ( x ¯ − μ ) ′ n ( Σ − 1 − 2 i θ Σ − 1 ) ( x ¯ − μ ) d x 1 ⋯ d x p {\displaystyle {\begin{aligned}&=\int (2\pi )^{-p/2}|{\boldsymbol {\Sigma }}/n|^{-1/2}\,e^{-(1/2)({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'n({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})}\,dx_{1}\cdots dx_{p}\end{aligned}}} <p>Now take the term | Σ / n | − 1 / 2 {\displaystyle |{\boldsymbol {\Sigma }}/n|^{-1/2}} off the integral, and multiply everything by an identity I = | ( Σ − 1 − 2 i θ Σ − 1 ) − 1 / n | 1 / 2 ⋅ | ( Σ − 1 − 2 i θ Σ − 1 ) − 1 / n | − 1 / 2 {\displaystyle I=|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n|^{1/2}\;\cdot \;|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n|^{-1/2}} , bringing one of them inside the integral:</p> = | ( Σ − 1 − 2 i θ Σ − 1 ) − 1 / n | 1 / 2 | Σ / n | − 1 / 2 ∫ ( 2 π ) − p / 2 | ( Σ − 1 − 2 i θ Σ − 1 ) − 1 / n | − 1 / 2 e − ( 1 / 2 ) n ( x ¯ − μ ) ′ ( Σ − 1 − 2 i θ Σ − 1 ) ( x ¯ − μ ) d x 1 ⋯ d x p {\displaystyle {\begin{aligned}&=|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n|^{1/2}|{\boldsymbol {\Sigma }}/n|^{-1/2}\int (2\pi )^{-p/2}|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n|^{-1/2}\,e^{-(1/2)n({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})}\,dx_{1}\cdots dx_{p}\end{aligned}}} <p>But the term inside the integral is precisely the probability density function of a <a href="/facts/Multivariate_normal_distribution/2Xfegqz2">multivariate normal distribution</a> with covariance matrix ( Σ − 1 − 2 i θ Σ − 1 ) − 1 / n = [ n ( Σ − 1 − 2 i θ Σ − 1 ) ] − 1 {\displaystyle ({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n=\left[n({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})\right]^{-1}} and mean μ {\displaystyle \mu } , so when integrating over all x 1 , … , x p {\displaystyle x_{1},\dots ,x_{p}} , it must yield 1 {\displaystyle 1} per the <a href="/facts/Probability_axioms/j6IkXsIT">probability axioms</a>. We thus end up with:</p> = | ( Σ − 1 − 2 i θ Σ − 1 ) − 1 ⋅ 1 n | 1 / 2 | Σ / n | − 1 / 2 = | ( Σ − 1 − 2 i θ Σ − 1 ) − 1 ⋅ 1 n ⋅ n ⋅ Σ − 1 | 1 / 2 = | [ ( Σ − 1 − 2 i θ Σ − 1 ) Σ ] − 1 | 1 / 2 = | I p − 2 i θ I p | − 1 / 2 {\displaystyle {\begin{aligned}&=\left|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}\cdot {\frac {1}{n}}\right|^{1/2}|{\boldsymbol {\Sigma }}/n|^{-1/2}\\&=\left|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}\cdot {\frac {1}{\cancel {n}}}\cdot {\cancel {n}}\cdot {\boldsymbol {\Sigma }}^{-1}\right|^{1/2}\\&=\left|\left[({\cancel {{\boldsymbol {\Sigma }}^{-1}}}-2i\theta {\cancel {{\boldsymbol {\Sigma }}^{-1}}}){\cancel {\boldsymbol {\Sigma }}}\right]^{-1}\right|^{1/2}\\&=|\mathbf {I} _{p}-2i\theta \mathbf {I} _{p}|^{-1/2}\end{aligned}}} <p>where I p {\displaystyle I_{p}} is an identity matrix of dimension p {\displaystyle p} . Finally, calculating the determinant, we obtain:</p> = ( 1 − 2 i θ ) − p / 2 {\displaystyle {\begin{aligned}&=(1-2i\theta )^{-p/2}\end{aligned}}} <p>which is the characteristic function for a <a href="/facts/Chi-square_distribution/wYnWWgUe">chi-square distribution</a> with p {\displaystyle p} degrees of freedom. ◼ {\displaystyle \;\;\;\blacksquare } </p></td></tr></tbody></table> <h2 id="two-sample-statistic">Two-sample statistic</h2> <p>If x 1 , … , x n x ∼ N p ( μ , Σ ) {\displaystyle {\mathbf {x} }_{1},\dots ,{\mathbf {x} }_{n_{x}}\sim N_{p}({\boldsymbol {\mu }},{\mathbf {\Sigma } })} and y 1 , … , y n y ∼ N p ( μ , Σ ) {\displaystyle {\mathbf {y} }_{1},\dots ,{\mathbf {y} }_{n_{y}}\sim N_{p}({\boldsymbol {\mu }},{\mathbf {\Sigma } })} , with the samples <a href="/facts/Statistical_independence/NUzQtnUL">independently</a> drawn from two <a href="/facts/Statistical_independence/NUzQtnUL">independent</a> <a href="/facts/Multivariate_normal_distribution/2Xfegqz2">multivariate normal distributions</a> with the same mean and covariance, and we define </p> x ¯ = 1 n x ∑ i = 1 n x x i y ¯ = 1 n y ∑ i = 1 n y y i {\displaystyle {\overline {\mathbf {x} }}={\frac {1}{n_{x}}}\sum _{i=1}^{n_{x}}\mathbf {x} _{i}\qquad {\overline {\mathbf {y} }}={\frac {1}{n_{y}}}\sum _{i=1}^{n_{y}}\mathbf {y} _{i}} <p>as the sample means, and </p> Σ ^ x = 1 n x − 1 ∑ i = 1 n x ( x i − x ¯ ) ( x i − x ¯ ) ′ {\displaystyle {\hat {\mathbf {\Sigma } }}_{\mathbf {x} }={\frac {1}{n_{x}-1}}\sum _{i=1}^{n_{x}}(\mathbf {x} _{i}-{\overline {\mathbf {x} }})(\mathbf {x} _{i}-{\overline {\mathbf {x} }})'} Σ ^ y = 1 n y − 1 ∑ i = 1 n y ( y i − y ¯ ) ( y i − y ¯ ) ′ {\displaystyle {\hat {\mathbf {\Sigma } }}_{\mathbf {y} }={\frac {1}{n_{y}-1}}\sum _{i=1}^{n_{y}}(\mathbf {y} _{i}-{\overline {\mathbf {y} }})(\mathbf {y} _{i}-{\overline {\mathbf {y} }})'} <p>as the respective sample covariance matrices. Then </p> Σ ^ = ( n x − 1 ) Σ ^ x + ( n y − 1 ) Σ ^ y n x + n y − 2 {\displaystyle {\hat {\mathbf {\Sigma } }}={\frac {(n_{x}-1){\hat {\mathbf {\Sigma } }}_{\mathbf {x} }+(n_{y}-1){\hat {\mathbf {\Sigma } }}_{\mathbf {y} }}{n_{x}+n_{y}-2}}} <p>is the unbiased pooled covariance matrix estimate (an extension of <a href="/facts/Pooled_variance/gzBnIsxE">pooled variance</a>). </p><p>Finally, the Hotelling's two-sample <i>t</i>-squared statistic is </p> t 2 = n x n y n x + n y ( x ¯ − y ¯ ) ′ Σ ^ − 1 ( x ¯ − y ¯ ) ∼ T 2 ( p , n x + n y − 2 ) {\displaystyle t^{2}={\frac {n_{x}n_{y}}{n_{x}+n_{y}}}({\overline {\mathbf {x} }}-{\overline {\mathbf {y} }})'{\hat {\mathbf {\Sigma } }}^{-1}({\overline {\mathbf {x} }}-{\overline {\mathbf {y} }})\sim T^{2}(p,n_{x}+n_{y}-2)} <h3>Related concepts</h3> <p>It can be related to the F-distribution by<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> n x + n y − p − 1 ( n x + n y − 2 ) p t 2 ∼ F ( p , n x + n y − 1 − p ) . {\displaystyle {\frac {n_{x}+n_{y}-p-1}{(n_{x}+n_{y}-2)p}}t^{2}\sim F(p,n_{x}+n_{y}-1-p).} <p>The non-null distribution of this statistic is the <a href="/facts/Noncentral_F-distribution/YCyc6Bus">noncentral F-distribution</a> (the ratio of a <a href="/facts/Noncentral_chi-squared_distribution/saAcjyRh">non-central Chi-squared</a> random variable and an independent central <a href="/facts/Chi-squared_distribution/wYnWWgUe">Chi-squared</a> random variable) </p> n x + n y − p − 1 ( n x + n y − 2 ) p t 2 ∼ F ( p , n x + n y − 1 − p ; δ ) , {\displaystyle {\frac {n_{x}+n_{y}-p-1}{(n_{x}+n_{y}-2)p}}t^{2}\sim F(p,n_{x}+n_{y}-1-p;\delta ),} <p>with </p> δ = n x n y n x + n y d ′ Σ − 1 d , {\displaystyle \delta ={\frac {n_{x}n_{y}}{n_{x}+n_{y}}}{\boldsymbol {d}}'\mathbf {\Sigma } ^{-1}{\boldsymbol {d}},} <p>where d = x ¯ − y ¯ {\displaystyle {\boldsymbol {d}}=\mathbf {{\overline {x}}-{\overline {y}}} } is the difference vector between the population means. </p><p>In the two-variable case, the formula simplifies nicely allowing appreciation of how the correlation, ρ {\displaystyle \rho } , between the variables affects t 2 {\displaystyle t^{2}} . If we define </p> d 1 = x ¯ 1 − y ¯ 1 , d 2 = x ¯ 2 − y ¯ 2 {\displaystyle d_{1}={\overline {x}}_{1}-{\overline {y}}_{1},\qquad d_{2}={\overline {x}}_{2}-{\overline {y}}_{2}} <p>and </p> s 1 = Σ 11 s 2 = Σ 22 ρ = Σ 12 / ( s 1 s 2 ) = Σ 21 / ( s 1 s 2 ) {\displaystyle s_{1}={\sqrt {\Sigma _{11}}}\qquad s_{2}={\sqrt {\Sigma _{22}}}\qquad \rho =\Sigma _{12}/(s_{1}s_{2})=\Sigma _{21}/(s_{1}s_{2})} <p>then </p> t 2 = n x n y ( n x + n y ) ( 1 − ρ 2 ) [ ( d 1 s 1 ) 2 + ( d 2 s 2 ) 2 − 2 ρ ( d 1 s 1 ) ( d 2 s 2 ) ] {\displaystyle t^{2}={\frac {n_{x}n_{y}}{(n_{x}+n_{y})(1-\rho ^{2})}}\left[\left({\frac {d_{1}}{s_{1}}}\right)^{2}+\left({\frac {d_{2}}{s_{2}}}\right)^{2}-2\rho \left({\frac {d_{1}}{s_{1}}}\right)\left({\frac {d_{2}}{s_{2}}}\right)\right]} <p>Thus, if the differences in the two rows of the vector d = x ¯ − y ¯ {\displaystyle \mathbf {d} ={\overline {\mathbf {x} }}-{\overline {\mathbf {y} }}} are of the same sign, in general, t 2 {\displaystyle t^{2}} becomes smaller as ρ {\displaystyle \rho } becomes more positive. If the differences are of opposite sign t 2 {\displaystyle t^{2}} becomes larger as ρ {\displaystyle \rho } becomes more positive. </p><p>A univariate special case can be found in <a href="/facts/Welch%27s_t-test/jbqtSaV8">Welch's t-test</a>. </p><p>More robust and powerful tests than Hotelling's two-sample test have been proposed in the literature, see for example the interpoint distance based tests which can be applied also when the number of variables is comparable with, or even larger than, the number of subjects.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Student%27s_t-test/dMH8uC28">Student's <i>t</i>-test</a> in univariate statistics</li> <li><a href="/facts/Student%27s_t-distribution/DeT1SDqH">Student's <i>t</i>-distribution</a> in univariate probability theory</li> <li><a href="/facts/Multivariate_Student_distribution/WspgrIac">Multivariate Student distribution</a></li> <li><a href="/facts/F-distribution/XNuuJtWY"><i>F</i>-distribution</a> (commonly tabulated or available in software libraries, and hence used for testing the <i>T</i>-squared statistic using the relationship given above)</li> <li><a href="/facts/Wilks%27s_lambda_distribution/zzSYXWbF">Wilks's lambda distribution</a> (in <a href="/facts/Multivariate_statistics/tDuNcWrL">multivariate statistics</a>, <a href="/facts/Samuel_S._Wilks/SIH0MW7a">Wilks's</a> <i>Λ</i> is to Hotelling's <i>T</i>2 as <a href="/facts/George_W._Snedecor/8JGVkuUm">Snedecor's</a> <i>F</i> is to <a href="/facts/William_Sealy_Gosset/5v21pjTh">Student's</a> <i>t</i> in univariate statistics)</li></ul> <h2 id="external-links">External links</h2> <ul><li>Prokhorov, A.V. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Hotelling_"><i>T</i>2-distribution "Hotelling <i>T</i>2-distribution"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Hotelling, H. (1931). "The generalization of Student's ratio". Annals of Mathematical Statistics. 2 (3): 360–378. doi:10.1214/aoms/1177732979. <a href="/wiki/Harold_Hotelling" target="_blank">/wiki/Harold_Hotelling</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Johnson, R.A.; Wichern, D.W. (2002). Applied multivariate statistical analysis. Vol. 5. Prentice hall. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Hotelling, H. (1931). "The generalization of Student's ratio". Annals of Mathematical Statistics. 2 (3): 360–378. doi:10.1214/aoms/1177732979. <a href="/wiki/Harold_Hotelling" target="_blank">/wiki/Harold_Hotelling</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Eric W. Weisstein, MathWorld <a href="http://mathworld.wolfram.com/HotellingT-SquaredDistribution.html" target="_blank">http://mathworld.wolfram.com/HotellingT-SquaredDistribution.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Hotelling, H. (1931). "The generalization of Student's ratio". Annals of Mathematical Statistics. 2 (3): 360–378. doi:10.1214/aoms/1177732979. <a href="/wiki/Harold_Hotelling" target="_blank">/wiki/Harold_Hotelling</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Mardia, K. V.; Kent, J. T.; Bibby, J. M. (1979). Multivariate Analysis. Academic Press. ISBN 978-0-12-471250-8. <a href="978-0-12-471250-8" target="_blank">978-0-12-471250-8</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Fogelmark, Karl; Lomholt, Michael; Irbäck, Anders; Ambjörnsson, Tobias (3 May 2018). "Fitting a function to time-dependent ensemble averaged data". Scientific Reports. 8 (1): 6984. doi:10.1038/s41598-018-24983-y. PMC 5934400. Retrieved 19 August 2024. <a href="https://www.nature.com/articles/s41598-018-24983-y" target="_blank">https://www.nature.com/articles/s41598-018-24983-y</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>"6.5.4.3. Hotelling's T squared". <a href="http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc543.htm" target="_blank">http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc543.htm</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>End of chapter 4.2 of Johnson, R.A. & Wichern, D.W. (2002) - Johnson, R.A.; Wichern, D.W. (2002). Applied multivariate statistical analysis. Vol. 5. Prentice hall. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Billingsley, P. (1995). "26. Characteristic Functions". Probability and measure (3rd ed.). Wiley. ISBN 978-0-471-00710-4. <a href="978-0-471-00710-4" target="_blank">978-0-471-00710-4</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Mardia, K. V.; Kent, J. T.; Bibby, J. M. (1979). Multivariate Analysis. Academic Press. ISBN 978-0-12-471250-8. <a href="978-0-12-471250-8" target="_blank">978-0-12-471250-8</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Marozzi, M. (2016). "Multivariate tests based on interpoint distances with application to magnetic resonance imaging". Statistical Methods in Medical Research. 25 (6): 2593–2610. doi:10.1177/0962280214529104. PMID 24740998. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Marozzi, M. (2015). "Multivariate multidistance tests for high-dimensional low sample size case-control studies". Statistics in Medicine. 34 (9): 1511–1526. doi:10.1002/sim.6418. PMID 25630579. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> </ol>