Statistics & Gene Expression Data Analysis Note 1: Basic Statistics

♠ Affymetrix expression data

• See Affymetrix tutorial, literature, websites, etc

• Expression summaries: summary expression estimates and other information

• Array normalisation; Hybridisation problems - low levels of intensity

♠ One gene: Sample statistics, summaries and exploration

• Sample x_{1}, …, x_{n} of expression on one gene

• Sample mean x̄ = n^{−1} ∑_{j=1}^{n} x_{j}

• Sample variances s = (n − 1)^{−1} ∑_{j=1}^{n}(x_{j} − x̄)^{2}

• Sample standard deviation (sd or std) \sqrt{s}

• Median, quartiles

• Histograms

♠ Multiple genes: Sample statistics, summaries and exploration

• Gene i, i = 1, …, p

• Expression level x_{i,j} on array j

• Column vector x_{j} of dimension p × 1 – all genes on array j

• p × n (tall, skinny) matrix X = [x_{1}, …, x_{n}]

• Gene i sample summaries x̄_{i}, s_{i}

• Sample covariance between genes i, k is s_{i,k} = (n − 1)^{−1} ∑_{j=1}^{n}(x_{i,j} − x̄_{i})(x_{k,j} − x̄_{k})

note: s_{i,i} = s_{i} • Sample correlation between genes i, k is r_{i,k} = s_{i,k}/\sqrt{s_{i}s_{k}}

note: r_{i,i} = 1 • Matrix notation: Sample variance (or covariance) matrix S = (s_{i,k})

S = ( \begin{matrix}s_{1} & s_{1,2} & ⋯ & s_{1,p} \\ s_{1,2} & s_{2} & ⋯ & s_{2,p} \\ ⋮ & ⋮ & ... & ⋮ \\ s_{1,p} & s_{2,p} & ⋯ & s_{p} \end{matrix} )

• S is p × p symmetric, non-negative definite matrix

• Matrix notation: Sample correlation matrix R = (r_{i,k})

R = ( \begin{matrix}r_{1,1} & r_{1,2} & ⋯ & r_{1,p} \\ r_{1,2} & r_{2,2} & ⋯ & r_{2,p} \\ ⋮ & ⋮ & ... & ⋮ \\ r_{1,p} & r_{2,p} & ⋯ & r_{p,p} \end{matrix} ) = ( \begin{matrix}1 & r_{1,2} & ⋯ & r_{1,p} \\ r_{1,2} & 1 & ⋯ & r_{2,p} \\ ⋮ & ⋮ & ... & ⋮ \\ r_{1,p} & r_{2,p} & ⋯ & 1 \end{matrix} )

• R is p × p symmetric, non-negative definite matrix

♠ Matrix notation

• Suppose X has variables all standardised to zero mean; that is, x̄_{i} = 0 for all genes i = 1, …, p (just subtract a non-zero mean to begin to assure this). Then the variances and covariances simplify, s_{i,k} = (n − 1)^{−1} ∑_{j=1}^{n} x_{i,j}x_{k,j}. Then S = (n − 1)^{−1} ∑_{j=1}^{n} x_{j}x′_{j} or S = (n − 1)^{−1}XX′. So correlation patterns among genes are derived from XX′.

Mike West  June 2, 2004  Page 0