# Fit linear model for each gene

### Description

For each gene, `lmPerGene`

fits the same, user-specified linear model.
It returns the estimates of the model parameters and their variances
for each fitted model. The function uses matrix algebra so it is much
faster than repeated calls to `lm`

.

### Usage

1 |

### Arguments

`eSet` |
An |

`formula` |
an object of class |

`na.rm` |
Whether to remove missing observations. |

`pooled` |
Whether to pool the variance calculation across all genes. |

### Details

This function efficiently computes the least squares fit of a linear
regression to a set of gene expression values. We assume that there
are `G`

genes, on `n`

samples, and that there are `p`

variables in
the regression equation. So the result is that `G`

different regressions
are computed, and various summary statistics are returned.

Since the independent variables are the same in each model fitting,
instead of repeatedly fitting linear model for each gene,
`lmPerGene`

accelarates the fitting process by calculating the
hat matrix *X(X'X)^(-1)X'* first. Then matrix multiplication, and
`solve`

are to compute estimates of the model parameters.

Leaving the formula blank (the default) will calculate an intercept-only model, useful for generic pattern and outlier identification.

### Value

A list with components:

`eS` |
The |

`x` |
The design matrix of the coded predictor variables. |

`Hmat` |
The Hat matrix. |

`coefficients` |
A matrix of dimension |

`pooled` |
Whether the variance was pooled (this affects “coef.var” and “tstat”, but not “sigmaSqr”). |

`sigmaSqr` |
A vector of length $G$ containing the mean square error
for that model, the sum of the residuals squared divided by |

`coef.var` |
A matrix of dimension |

`tstat` |
A matrix of the same dimension as |

### Author(s)

Robert Gentleman, Assaf Oron

### See Also

`getResidPerGene`

to extract row-by-row residuals; `gsealmPerm`

for
code that utilizes 'lmPerGene' for gene-set-enrichment analysis (GSEA); and `CooksDPerGene`

for diagnostic functions on
an object produced by 'lmPerGene'. Applying a by-gene regression in
the manner performed here is a special case of a more generic
linear-model framework available in the `GlobalAncova`

package; our assumption here is equivalent to a diagonal covariance structure
between genes, with unequal variances.

### Examples

1 2 3 4 5 6 7 8 | ```
data(sample.ExpressionSet)
layout(1)
lm1 = lmPerGene(sample.ExpressionSet,~sex)
qqnorm(lm1$coefficients[2,]/sqrt(lm1$coef.var[2,]),main="Sample Dataset: Sex Effect by Gene",ylab="Individual Gene t-statistic",xlab="Normal Quantile")
abline(0,1,col=2)
lm2 = lmPerGene(sample.ExpressionSet,~type+sex)
qqnorm(lm2$coefficients[2,]/sqrt(lm2$coef.var[2,]),main="Sample Dataset: Case vs. Control Effect by Gene, Adjusted for Sex",ylab="Individual Gene t-statistic",xlab="Normal Quantile")
abline(0,1,col=2)
``` |

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